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DIGITAL COMMUNICATION RECEIVER DESIGN C. RICHARD

1. Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 h setup constants for each of the three operating modes HEADER_SEARCH_MODE 1 TRAINING_MODE 2 DATA_MODE 3 operationMode HEADER_SEARCH_MODE we start in HEADER_SEARCH_MODE allocate memory and initialize variables theta zeros length r 1 stores outputs of PLL x_down zeros length r 1 Stores downconverted signal pre matched filter x_bb zeros length r 1 stores baseband signal post matc hed filter x_sampled zeros ceil length r T_t f_s 1 stores sampled signa 1 post timing recovery Corr zeros ceil length r T_t f_s 1 stores correlation v alues for header search tau zeros ceil length r T_t f_s 1 stores timing recovery only used for plotting eqOut zeros ceil length r T_t f_s 1 stores output of equalizr e_lms zeros ceil length r T_t f_s 1 stores LMS error dec zeros ceil length r T_t f_s 1 stores PAM 4 decisions packet Index 0 packet counter tnow 2 srrcLength 2 T_t f_s starting location for timing BBsampleIdx 0 intialize baseband sample counter start BPFfilterOrder 1 h outer loop starting point for IFsampleIdx start length r h pre process signal for PLL r2 IFsampleIdx h r IFsampleIdx 1 I
2. Johnson Introducing Receiver Design Apr May 06 DAY 2 LAB 1 Laboratory Exercises Day 2 Introduction to Digital Communication Receiver Design Task 1 Understanding the Subsampled IF Receiver Architecture In an IF receiver also called a heterodyne receiver the downconversion from RF is done in 2 steps e An analog circuit downconverts to some intermediate frequency where the signal is sampled e The resulting signal is then digitally downconverted to baseband The advantage of this 2 step method is that the analog downconversion can be performed with minimal precision and hence inexpensively while the sampling can be done at a reasonable rate In a standard sampled IF receiver the sampling frequency is typically chosen to be twice the IF frequency i e the Nyquist rate However another class of IF receivers called subsampled IF receiver uses a sampling frequency lower than the Nyquist rate which results in aliasing However the aliasing is introduced in a way that reconstruction of the signal is still possible Recall that sampling introduces copies of the signal at every multiple of the sampling rate To illustrate the subsampled IF receiver architecture we consider an specific example with the following parameters parameter value carrier frequency frr 1 GHz intermediate frequency frr 2 MHz receiver sampling rate fs 850 kHz signal bandwidth B 100 kHz where the signal bandwidth of the baseband
3. We can implement the convolution sum very efficiently in MATLAB using vector inner products For example the filter output at time n is given by y n h x n 1 n N Johnson Introducing Receiver Design Apr May 06 DAY 1 LAB 2 Your task Using for loops and vector inner products write a few lines of code that are equivalent to the command y filter b 1 x Compare your result with the the previous problem where you used the filter command and calculate the mean squared error between the two Note you may ignore the first N samples in the error calculation where N is the number of taps in the filter Task 3 Detection via Correlation In packet based wireless communication systems the beginning of the transmission usually contains a marker sequence The receiver is constantly looking for such a marker sequence when it detects that a marker sequence has been sent it knows that data is about to be transmitted and it knows the location of the start of the packet The standard technique for identifying a marker sequence is called correlation Correla tion is much like convolution but with a sign change in the indexing If y n is the received signal marker n is the known marker sequence of length N the correlator output z at time n is given by ai 5 marker k yln k k 0 When the correlator output z n exceeds some pre determined threshold the receiver de cides that the marker was identified at that value of n
4. receiver Transmitter pulse firing trigger or baud timing offset is unknown to receiver Pulse shape is truncated SRRC with rolloff factor of 0 3 Johnson Introducing Receiver Design Apr May 06 DAY 4 22 Received Signal Construction cont d Frequency division multiplexing slots exceed double half power bandwidth of pulse shape Transmitter carrier frequency is known precisely at receiver Transmitter carrier phase unknown to receiver and expected to be slowly wandering The channel can possess eye closing ISI Only the maximum delay spread of the potential ISI is known to the receiver in advance Broadband noise is present but modest Narrowband interferers may be present as well Johnson Introducing Receiver Design Apr May 06 DAY 4 23 Received Signal Construction cont d e Downconversion to IF by front end hits specified target frequency exactly e Automatic gain control in front end is presumed converged and static e Sampler is free running at a frequency well over twice the bandwidth of the pulse shape e Sampler is sub Nyquist for IF which means that downconversion will be performed on passband spectrum replica nearest baseband Johnson Introducing Receiver Design Apr May 06 DAY 4 24 Received Signal Construction cont d Received Sampled Signal Specifications Table left column transmitter symbol period offset frame marker training sequenc
5. 3 1 1 3 Transmitted users Noise sequence b Analog signal P0 One Pulse Carrier shaping Antenna Digital down Pulse matched specification Analog Lae conversion filter conversion to baseband to IF Analog T received Input to the signal software receiver Carrier synchronization m O m e 3 1 1 3 T Source and Reconstructed error coding message Timing synchronization frame synchronization Downsampling Johnson Introducing Receiver Design Apr May 06 DAY 2 28 CARRIER RECOVERY Carrier Phase Tracking x Adaptive Algorithm Development x Carrier Extraction x Phase locked Loop Costas Loop Johnson Introducing Receiver Design Apr May 06 DAY 2 29 Carrier Phase Tracking Binary Other FDM message we 3 1 1 3 Transmitted users Noise sequence b Analg signal Pulse Carrier shaping specification t Antenna Analog Digital down Pulse conversion conversion matched to IF T to baseband filter Analog received Input to the Carrier signal software synchronization receiver T m Q m e 3 1 1 3 Downsampling Decision Decoding Timing Source and Reconstructed synchronization error coding message frame synchronization e A fixed phase offset between the transmitter and carrier oscillators results in an attenuation in the downconverted signal by the cosine of this phase difference We seek
6. Paaeee Lattice of T spaced optimal sampling times with ISI Lattice of T spaced optimal sampling times with no ISI SS Sti of received pulses p t c t p t 0 6 p t A 0 6 p t A The digital channel model is given by T spaced samples of c t Johnson Introducing Receiver Design Apr May 06 DAY 4 Trained Linear Equalization e Objective Given prearranged intermittently transmitted training sequence available at receiver choose impulse response f of equalizer so z k s k 6 so e 0 for some 0 Additive interferers Received Equalizer signal r k output y k Impulse response f Training signal Johnson Introducing Receiver Design Apr May 06 DAY 4 7 Trained Adaptive Least Mean Square LMS Equalization We choose to minimize ave e k k ko with e k s k 6 gt gt _ fir k i using a gradient descent scheme file ilk py With differentiation and average approximately commutable Oe k heti S k p agd r r Dropping the outer average produces LMS file 1 file 27 ee SE Lear filk w s k vik r k i with ylk Efo filelr amp j Johnson Introducing Receiver Design Apr May 06 DAY 4 Trained Adaptive Least Mean Square LMS Equalization cont d With the definition of the FIR equalizer output vik gt Filklr k j Sampled received Signal nd y k Decision Equalizer dev e
7. Your task Load the file day1 correl_ex mat by typing load correl_ex This file contains two variables a length 100 marker sequence called marker and a length 2000 received sequence called y Write a few lines of code to perform the correlation and determine the starting location of the marker sequence Also show a plot of z n Task 4 Amplitude Modulation Consult the file day1 AM m This code generates a message w t and modulates it with a carrier at frequency fe The demodulation is done with a cosine of frequency fe y and a phase offset of When y 0 and 0 i e in the ideal conditions the output is identical to the original message except for the inevitable delay caused by the linear filter Your tasks 1 Plot the signals w t v t x t and m t and describe what you see 2 Using the plotspec command plot the spectra of these same signals Describe what you see 3 Change the phase offset Describe the effect for different values 4 Change the frequency offset y Describe the effect for different values Johnson Introducing Receiver Design Apr May 06 DAY 1 LAB 3 Task 5 Sinc Interpolation As you should be aware sampling a signal faster than the Nyquist rate allows for perfect reconstruction since no information is lost However once we have a sampled digital signal how do we reconstruct the data between samples The answer is sinc interpolation We will use sinc interpolation quite often in
8. extract regressor vector of receive d signal 124 eqOut BBsampleIdx f rr equalizer output 125 dec symbol Index quantalph eqOut BBsampleIdx 3 1 1 3 make decisions 126 127 if mod symbolIndex 4 0 if we ve completed a w hole letter i e 4 PAM symbols convert PAM symbols to letters 128 decoded_msg packet Index symbol Index 4 pam2lett ers dec symbolIndex 3 symbolIndex re assemble text message 129 end 130 symbolIndex symbolIndex 1 h increment training index location 131 if symbolIndex gt dataLength are we done with data yet 132 operationMode HEADER_SEARCH_MODE yep switc h back to header search mode 133 end 134 end 135 end Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 13 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 end i plot results Sass4nases4e sass ese a se ase See ee eae ee eees ee seses e figure 1 plot theta title carrier phase estimate ylabel theta xlabel time figure 2 plot tau title timing offset estimates ylabel tau xlabel time figure 3 plot Corr hold on plot 1 length Corr correlThresh correlThresh plot thre shold title correlator output for finding start of training xlabel time ylabel correlation value
9. figure 4 plot eqOut b plot constellation diagram title constellation diagram equalizer output ylabel estimated symbol values xlabel time figure 5 plot e_lms title error at equalizer output during training ylabel e_1ms xlabel time globalParams m oP UNBE global srrcLength marker training f_s T_t f_if rolloff dataLength srrcLength 4 truncated srrc length divided by 2 marker letters2pam 0 marker sequence training letters2pam Oh well whatever Nevermind training se quence f_s 850e3 sampling frequency Hz Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 14 7 T_t 6 4e 6 symbol period seconds 8 f_if 2e6 intermediate frequency Hz 9 rolloff 0 3 srrc rolloff factor 10 dataLength 400 number of PAM symbols per data frame Johnson Introducing Receiver Design Apr May 06 DAY 3 1 DAY 3 e PULSE SHAPING AND RECEIVE FILTERING e BAUD TIMING FOR CLOCK RECOVERY Johnson Introducing Receiver Design Apr May 06 DAY 3 PULSE SHAPING AND RECEIVE FILTERING Pulse and Pulse Amplitude Modulated Message Spectrum Eye Diagram Nyquist Pulses Matched Filtering Matched Nyquist Transmit and Receive Filter Combination Johnson Introducing Receiver Design Apr May 06 DAY 3 Pulse Shaping and Receive Filtering Binary Other FDM message we 3 1 1 3 Transmitted users Noise sequen
10. v ne 5 T g lt 100 Seconds 10 0 Frequency Johnson Introducing Receiver Design Apr May 06 DAY 2 9 A System cont d Receiver post mixer LPF frequency response 50 faa T lt O Nn ie or n o ia oO uo D oD S 0 3 0 4 0 5 0 6 0 7 0 8 0 9 Normalized frequency Nyquist 1 Phase degrees 0 3 0 4 0 5 0 6 0 7 0 8 0 9 Normalized frequency Nyquist 1 Johnson Introducing Receiver Design Apr May 06 DAY 2 10 A System cont d Receiver downconverter LPF output and magnitude spectrum i Mo m Ni i Ht Ki Johnson Introducing Receiver Design Apr May 06 DAY 2 11 A System cont d First 400 samples of pulse correlator filter output Best times to take samples iso S oD Nn O gt Q Q o ial Q KD an a lt 100 150 200 250 300 350 400 T spaced samples T 2T 3T 4T T spaced samples This reveals 125 for first symbol sample or baud time 125 half length of lowpass filter in downconverter and half length of correlator filter and half a symbol period Johnson Introducing Receiver Design Apr May 06 DAY 2 12 A System cont d Overlay of successive 4T wide correlator output segments starting on first baud time Note recurrence of pulse peaks at successive T wide intervals Johnson Introducing Receiver Design Apr May 06 DAY 2 13 A Syst
11. Ordering the basic components cont d e Timing and equalization can occur in the passband before carrier recovery A fractionally spaced equalizer can absorb the matched filter and resampling operations of the baud timing component Sometimes ordering is based on design tradeofts at hand sometimes on designer preference or personal experience and sometime s on factors outside receiver designer s control e g legacy product lines and intellectual property constraints Johnson Introducing Receiver Design Apr May 06 DAY 4 29 Design Methodology cont d Stage Two Selecting components Downconversion like the other operations of basic components can be done through many methods Here the sub Nyquist sampling of the IF signal places replicas closer to baseband The closest is to be downconverted by a mixer with an adapted phase followed by a suitable lowpass filter The presumption is that the components chosen when properly tuned result in acceptable performance The proper operation of the components selected can be confirmed by simulations in an interference free ideal full knowledge setting Johnson Introducing Receiver Design Apr May 06 DAY 4 30 Design Methodology cont d Stage Three Countering anticipated impairments o residual interference from adjacent FDM band signals AGC jitter quantization noise in sampler round off noise in filters
12. T 14 2 7 T 1 1 4 2 2 1 2 r T 1 2 r T Ir T e Desired offset of r 0 4nT occurs with maximization of average squared sampler output ave x 1 0 5 3 F2 T T 2 T 2 Timing offset t Johnson Introducing Receiver Design Apr May 06 DAY 3 30 Output Power Maximization e Moving average of square of sampler output kotN 1 Jop tT x k avg x k k ko e To maximize Jop using a gradient ascent rik 1 71K ae lave a KH with small u we interchange the average and the differentiation and drop the outer average yielding Tk 1 7 ee rk 27 a EEL bart where for small da k _ dx 47 AP 47 dt ae a eT Ole tae a D 26 Johnson Introducing Receiver Design Apr May 06 DAY 3 31 Output Power Maximization cont d e Output power maximizing baud timing adaptation algorithm with alk x kT M r k T k 1 7 k pak 57 T k 6 n 7 k 5 e Output power maximizing baud timing adjusted oversampler schematic Sampler ore Resample War gt x kTIM 7 k Johnson Introducing Receiver Design Apr May 06 DAY 3 32 Output Power Maximization cont d Example from clockrecOP Source 2 PAM Baud timing adaptor stepsize u 0 05 Derivative approximation increment 6 0 1 Pulse shape SRRC with 6 0 5 Free running receiver sampler offset 0 3 gt desired baud
13. The W f portion of D f about zero frequency can be extracted by filtering d t through an ideal filter that has a flat magnitude and a linear phase for low frequencies and near zero magnitude for high frequencies i e an ideal lowpass filter Johnson Introducing Receiver Design Apr May 06 DAY 1 17 Message Recovery via Filtering cont d WP 2 Pak 0 f a IS e A ne fo fo F S t cos 27f t filter 2fo 0 2fo c a original spectrum of the message b message modulated by the carrier c demodulated signal has original spectrum after ideal lowpass filtering Johnson Introducing Receiver Design Apr May 06 DAY 1 18 Synchronized Demodulation of Amplitude Modulation with Suppressed Carrier e analog message signal w t e transmitted modulated signal u t A w t cos 27z fet transmitted signal spectrum V f 5 AeW f a T ae AW f Je ideal demodulation with synchronized mixing and LPF m E erie E SAWI P main disadvantage carrier phase and frequency synchronization needed at receiver Johnson Introducing Receiver Design Apr May 06 DAY 1 19 Synchronized Demodulation of Amplitude Modulation with Suppressed Carrier cont d e Example Perfect delayed recovery with perfect synchronization using AM Amplitude a message signal v 5 5 T zi lt b message after modulation Amplitude c demodulated signal Amplitu
14. and A T To Sampled impulse response h t iD t eT 47 A R i T To k it 1 k i 0 otherwise e7 lt 0 Two nonzero points in sampled impulse response A 2T To and A T 70 Sampled impulse response A t iT ar tro Johnson Introducing Receiver Design Apr May 06 DAY 3 24 A Baud Timing Example cont d Any sampled output z k is based only on at most two symbol spaced samples for any choice of T e For example with 7 gt 0 for k 6 a 6 x s i h 6 i T To s 6 h 70 s 5 h T 7 562 55 2 e For example with r lt 0 fork 6 x 6 X s h 6 iT 70 s 5 h T To s 4 A 2T To sg sa Bh e For a binary input there are 4 possible symbol pairs 1 1 1 1 1 1 and 1 1 that are assumed equally likely Johnson Introducing Receiver Design Apr May 06 DAY 3 25 A Baud Timing Example cont d e For example with 7 gt 0 for k 6 s 5 s 6 1 1 2 6 2 1 2 1 s 5 8 6 1 1 gt psl s 5 sl6 1 1 gt B 1 p 14 8 s 5 s 6 1 1 z 6 14 1 Two of the possibilities for x 6 give correct values for s 5 while two are incorrect As long as 27 lt T then the sign x 6 matches s 5 for all four possibilities If To exceeds T 2 the sign of x 6 would be associated with an earlier s than s 5 Johnson Introducing Receiver Desi
15. p a0 Because w p is nonzero only when p kT w T X ws kTs sinc T kTs k o0 Prescription for perfection As long as fs gt 2B where B is the highest frequency present in w t this doubly infinite sinc interpolator is exact Filtering interpretation Creation of w T can be interpreted as a convolution of w with a sinc shaped impulse response Johnson Introducing Receiver Design Apr May 06 DAY 1 34 Interpolation cont d e Ideal LPF Interpolator Convolution in time domain is multiplication in frequency domain Spectrum of sinc is a rectangle i e an ideal LPF Thus an ideal lowpass filter with appropriate cutoff frequency is a perfect interpolator for a Nyquist sampled signal Perfection inhibiting practicalities In practice it is necessary to truncate the doubly infinite convolutional sum Furthermore w t can always be expected to have traces of frequencies above B Therefore in practice we must settle for an approximation Non ideal LPF interpolator Fortunately any suitable LPF with nonzero flat magnitude and linear phase up to frequency B and fully rejecting before reaching next higher frequency chunk in spectrum of ws will provide accurate interpolation Johnson Introducing Receiver Design Apr May 06 DAY 1 35 Interpolation cont d Example Using sininterp which uses interpsinc to reconstruct a sinusoid sampled five times per
16. Apr May 06 DAY 4 15 Example using dae e Source binary 1 e Channel Impulse Response 1 9 81 73 64 55 46 37 28 4 138 Sinusoidal interferer frequency 1 4 radians sample Some broadband noise present Equalizer length 33 Johnson Introducing Receiver Design Apr May 06 DAY 4 16 Example cont d Trained LMS ay So gt Squared prediction error jo 4 tH e tH D tH D Pag D g tH a 10 D tH a ion N eo D g g N 1000 2000 3000 4000 1000 2000 3000 Iterations Iterations Combined magnitude response Adaptive equalizer output 1000 2000 3000 4000 2 3 Iterations Normalized frequency Johnson Introducing Receiver Design Apr May 06 DAY 4 17 Example cont d Decision directed 10 107 H io oO H O Q pa D D z 5 S 5 Q am Ko D D pa a on 3 7A 2 3 gs 5 V T g A N o Manaa aii ilike 1000 2000 3000 4000 4000 Iterations Iterations Combined magnitude response Adaptive equalizer output 1000 2000 3000 4000 3 Iterations Normalized frequency Johnson Introducing Receiver Design Apr May 06 DAY 4 18 Example cont d Dispersion minimization 10 Squared prediction error ior o iti i eer AST 1000 2000 3000 4000 1000 2000 Iterations Iterations 1071 H e H D ial oO oO fa or a ge D io ioe g n uo oO g N Combined magn
17. Jy has double dip egg carton style cross section e For specific data set with N 1000 from aes 0 02 i i 0 0 2 adaptive gain a 48 Johnson Introducing Receiver Design Apr May 06 DAY 1 49 AGC cont d Computation of the gradient requires that a remain constant over the N samples over which avg s is composed Consider squeezing the averaging window to a single sample so N 1 and afi psign a stil a This is the algorithm developed heuristically and tested previously This algorithm also emerges from first reducing the averaging window to N 1 in the cost function and then taking the gradient and forming a gradient descent iteration This technique of shrinking the averaging window so averaging is explicitly removed works because LPF action of adaptation acts similarly to averaging before updating Johnson Introducing Receiver Design Apr May 06 DAY 1 50 Tracking Example Time Varying Fade e To demonstrate desired tracking capability use agcvsfading to test afi psign a i stil a with u 0 01 d 0 5 afl 1 and a large slow oscillating channel gain Input r k e Fade must be changing sufficiently slowly and the input must never die for the AGC with small stepsize to track adequately Johnson Introducing Receiver Design Apr May 06 DAY 1 LAB 1 Laboratory Exercises Day 1 Introduction to Digital Communication Rece
18. McClellan R W Schafer and M A Yoder Signal Processing First Pearson Prentice Hall 2003 DAY 1 lectures present a basic pulse amplitude modulated PAM radio system and discuss how such impairments if uncompensated can deteriorate communication system performance A basic adaptive algorithm creation strategy is described in particular for automatic gain control to track the compensator parameter needed to counteract an encountered impair ment the specifics of which are initially unknown to the user and expected to be time varying DAY 2 lectures feature simulated system performance degradation due to various impairments and the successful automatic gain control compensation of flat fading The adaptive stochastic gradient descent based strategy is applied on DAY 2 to carrier phase tracking by the receiver mixer resulting in the popular phase locked and Costas loop algo rithms on DAY 3 to baud timing for clock recovery based on downsampled signal power optimization and on DAY 4 to equalization of frequency selective channel impairments via both trained and blind schemes DAY 5 consists of a final project assignment that puts it all together The first 4 days are designed for 3 hours of lecture followed by 3 hours of supervised lab instruction There are no lectures on the 5th day just a lab session by the end of which the modified radio developed individually by each student will be tested in comparison to the base radio develo
19. e Generation of 4 PAM sequence lines 30 32 This part encodes the ASCII test message into 4 PAM signals using letters2pam and inserts the header and marker sequences The result is a serial stream of 4 PAM symbols stored in the variable called s e Pulse Shaping lines 33 35 This part performs upsampling of the 4 PAM signal and then filters the signal with the pulse shape obtained from srrc The result is stored in the variable x e Analog Upconversion lines 36 39 This part modulates the signal up to the carrier frequency and includes the effect of phase noise The result is stored in x_rf e Channel lines 40 42 This part convolves the upconverted signal with the chan nel storing the result in the variable x2 e Noise lines 43 47 This part adds the additive white Gaussian noise of specified SNR e Analog Conversion from RF to IF lines 48 52 This part acts as the frontend of the receiver and performs analog conversion of the RF signal down to IF While the RF signal in reality would be analog our computer simulation uses a digital representation throughout thus the sampled IF receiver is obtained from the RF signal by simple downsampling The result is the digital signal which gets passed into Rx m Receiver Details Rx m Similar to the previous section this section briefly describes each of the main components of the receiver and points to their corresponding line numbers in the code Most o
20. kT Si ygC08 47 fokTs 20 vu kT cos 47 fokT 20 A narrow bandpass filter centered at 2fo with phase shift p at 2f9 extracts Sess A 1 2 s7 c0s 4r fokTs 26 p from r while passing a bit of v about 2 fo Digital BPF implementation presumes that 2 fo lies within the Nyquist frequency 1 27 Johnson Introducing Receiver Design Apr May 06 DAY 2 33 Carrier Extraction cont d For 1 second of a 4 PAM signal with Hamming blip symbol width T 0 005 sample period with an oversample factor of 50 T 0 0001 and a carrier with frequency fo 1000 and phase 1 from pllcrt the received signal and its spectrum are o ne 5 zx zou E Gs i i jl 0 5 0 6 0 7 0 8 0 9 seconds magnitude 0 5000 4000 3000 2000 1000 0 1000 2000 3000 4000 5000 frequency Johnson Introducing Receiver Design Apr May 06 DAY 2 34 Carrier Extraction cont d Passing the received signal with f 1000 r kT s kT cos 27fokT through a squarer and a BPF centered at 2000 Hz with approximately 100 Hz passband and mod p 27 0 where mod a b produces the remainder after division of a by b yields x in time and frequency amplitude 5000 4000 F tude 3000 2000 magni 1000 F 0 1 1 L 1 li ll 5000 4000 3000 2000 1000 0 1000 2000 3000 4000 5000 frequency Johnson Introducing Receiver Design Apr May 06 DA
21. maximize the power of the signal v t at a specific time t T ie v T relative to the total power of w t where the power spectral density of n t is a constant 7 over all frequencies Johnson Introducing Receiver Design Apr May 06 DAY 3 14 Matched Filter cont d With spectrally flat channel noise the SNR maximizing receive filter impulse response is the time reversal of that of the pulse shape e Example pitt e PTA e 2 t 2 Minimum 7 for causality of matched filter is pulse width for pulse initiated at t 0 e NOTE Minimum delay matched filter is same as pulse if pulse is causal and even symmetric Johnson Introducing Receiver Design Apr May 06 DAY 3 15 Matched Nyquist Transmit and Receive Filter Combinations A preferred receive filter impulse response in the absence of channel ISI but with broadband channel noise i will match the reversed impulse response of the transmitter pulse shape and ii when convolved with the transmitter pulse shape will form a Nyquist pulse e Want convolution of candidate pulse shape g t and its matched filter g t T to equal even symmetric Nyquist pulse p t e Since convolution of two even symmetric pulse shapes is even symmetric presume g t is even symmetric so with particular 7 g t g r t e Objective becomes p t g xg gt PF G Johnson Introducing Receiver Design Apr May 06 DAY 3 16 Matched C
22. residual interference from doubly upconverted spectrum carrier phase jitter baud timing offset residual MSE from equalizer equalizer parameter jitter noise enhancement by equalizer Legend major o minor Johnson Introducing Receiver Design Apr May 06 DAY 4 31 Design Methodology cont d Stage Three Countering impairments cont d e We anticipate the need for carrier phase adaptation baud timing adaptation equalizer adaptation post decision frame synchronization e Choices so far Carrier phase recovery phase locked loop and Costas loop Baud timing recovery on oversampled matched filter output output power absolute value fourth power and dispersion Equalizer adaptation trained LMS decision directed dispersion minimizing Frame synchronization and training segment location marker correlation Johnson Introducing Receiver Design Apr May 06 DAY 4 32 Design Methodology cont d Stage Four Tuning and Testing In order of appearance e Step One Tuning the Carrier Recovery e Step Two Tuning the Clock Recovery e Step Three Tuning the Equalizer e Step Four Frame synchronization for decoder Sampled received signal Matched Interpolator Downconversion filter downsampler A uye Carrier Timing os recovery recovery Recovered soi source Decision Equalizer Decoder device Training Equalizer Frame segment adaptation synchron
23. signal is defined as having spectral content between B and B Your task Draw the spectrum of the signal at each of the following steps 1 The original baseband signal with bandwidth B 2 The signal after modulation to the RF frequency accomplished by mixing with a sinusoid of frequency fre 3 The signal after downconversion to the IF frequency accomplished by mixing with a sinusoid of frequency frr frr 4 The signal after bandpass filtering which removes the unwanted image Johnson Introducing Receiver Design Apr May 06 DAY 2 LAB 2 5 The signal after sub sampling at rate f Note For this one you only need to draw the spectrum between f 2 and f 2 If the next step were to perform downconversion of the signal to baseband what frequency would you choose for the sinusoid used in the downconversion In spite of the fact that subsampling introduces aliasing is it still possible to recover the original baseband signal Or is the signal distorted A subsampled IF receiver is attractive because it can be implemented even more inex pensively than a standard sampled IF receiver However there is one major drawback to this use of this receiver architecture in the presence of noise AWGN Can you think of what this drawback might be Task 2 Implementing the Costas Loop The receiver you have been given currently uses a PLL for carrier recovery in system_code Rx m Your task is to replace the PLL with a Co
24. signal set e probe receiver limits e g assess how much noise causes performance failure e implement debug mode that plots pertinent signals e test an adaptive element in two scenarios i start at right answer with zero stepsize and see if achieved performance is as expected and then ii start near right answer with nonzero stepsize and see if algorithm shrinks into tight orbit about right answer Johnson Introducing Receiver Design Apr May 06 DAY 4 LAB 1 Laboratory Exercises Day 4 Introduction to Digital Communication Receiver Design Implementation of Decision Directed LMS The existing receiver only uses the trained LMS algorithm for equalizer adaptation Your task is to add the decision directed DD LMS algorithm to the receiver Note that you should not remove the trained LMS algorithm which operates during TRAINING_MODE Rather your task is to add the DD LMS algorithm which will operate when in DATA_MODE As before you should start with the most benign channel conditions then gradually increase the impairment After successfully adding the DD algorithm you should tune the stepsize Then complete the following tasks 1 Static channel e Starting with the default simulation parameters set the trained LMS stepsize and DD LMS stepsizes to zero which effectively disables the equalizer adapta tion Change the channel in main m to c 1 0 6 0 3 Run main m and plot the smoothed squared DD equa
25. timing adjustment of T 0 3 Constellation diagram Estimated symbol values 1000 2000 3000 4000 5000 Offset estimates 1000 2000 3000 4000 5000 Iterations Johnson Introducing Receiver Design Apr May 06 DAY 3 33 Output Power Maximization cont d Example from clockrecOPcost Cost functions for desired 7 of zero with SRRC pulse shape with roll off factor G 0 5 e absolute value Jay Sa ei e fourth power Jpp 4 yo ee e output poe aka output energy kotN 1 Jop t brog 127 1K e dispersion aka constant modulus Ip t H Dak 2 A 1 Fourth Power i Ol a Abs Value N j gt oo gt N Dispersion gt P gt N value of performance functions i 0 01 0 2 03 04 0 5 0 6 0 7 0 8 0 9 1 timing offset t Johnson Introducing Receiver Design Apr May 06 DAY 3 34 Output Power Maximization cont d What happens with ISI using clockrecOP e Channel 1 0 7 0 0 0 5 e All else same 2 PAM source u 0 05 0 1 SRRC pulse with 0 5 Free running receiver sampler offset 0 3 Constellation diagram M tbe m ENE ratom vee FIT HENNE NOTA ORAS IE y maina s rih ID SNORE LG IP barir 70 rp ereanatan a ONLO tt st ANWR WIA ORE ees Las Lea Ny AEA EPRI LOTTE PPD Ie Fo Ak at ipa A ene el AR DREE LSE A CE OES Be n Q a Ss gt f Tal E al n ge O iso Ss z aa 2000 3000 4000 5000 Offset estimates 2000
26. zeroing the average correction term zeros the average of d s But indeed that is what we seek Johnson Introducing Receiver Design Apr May 06 DAY 1 46 AGC cont d Gradient Descent Algorithm Development e As a more generalizable approach to adaptor algorithm development consider specifying a cost function and using an iterative optimizer based on gradient descent OJn a Ja a a t e Try Jn a ave a s k 3 d with the definition of avg as ali 1 az u k N 1 avs zikl G N gt afl i k e For small stepsize u differentiation and averaging are approximately interchangeable O wete EO ay s aval ila ED e p Johnson Introducing Receiver Design Apr May 06 DAY 1 47 AGC cont d olal _ o dw _ dw dy e With 5 gt sign a and 2 age a ave a 1 3 2ar2 kT sign a 1 3 a r7 kT sign a d e With sign a a a OJn a a avg sign a a r kT d e With a r s OJn a x ave sign a s k d a So the stationary points of zero gradient are in the right places with avg s d e With O sign a Oa 0 everywhere but a 0 the second derivative is approximately wale sign a a r kT d avg 2a sign a r kT avg 2la r7 kT gt 0 So stationary points at a 0 are minima Johnson Introducing Receiver Design Apr May 06 DAY 1 AGC cont d e With constant avg r7 and d
27. 000 9000 10000 Input r k 5 T T T 0 5 1 f f 1 f f f f f 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Output s k 5 1 f 1 f fi 1 i 1 f 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 iterations Johnson Introducing Receiver Design Apr May 06 DAY 1 44 AGC cont d e Start at x 0 05 with u 0 001 Adaptive gain parameter T T T i i i 1 1000 2000 4000 5000 Input r k 1 1 1 fi 1000 2000 4000 5000 Output s k T f f fi fi 1000 2000 4000 5000 iterations Start at x 2 with u 0 02 Adaptive gain parameter T T T f f fi 4000 5000 6000 Input r k fi fi fi 4000 5000 6000 Output s k 1 1 1 1 4000 5000 6000 9000 10000 iterations Johnson Introducing Receiver Design Apr May 06 DAY 1 45 AGC cont d Observations e Asymptotically this algorithm hovers in a small region about the desired answer The asymptotic hovering region s size can be decreased by reducing the stepsize u which also reduces the algorithm convergence rate When the average value of the hovering parameter has effectively reached a fixed value the average of a i 1 will equal the average of a z such that from our algorithm ali 1 afi psign ali d s i the average of the correction term usign a i d s 7 must be zero With u gt 0 and the asymptotic hovering ali not changing sign
28. 30 40 50 Frequency Johnson Introducing Receiver Design Apr May 06 DAY 2 18 What if cont d Channel noise cont d Received signal eye diagram of 4 symbol wide overlays Johnson Introducing Receiver Design Apr May 06 DAY 2 19 What if cont d Channel noise cont d Pulse correlator filter synchronized output signal Johnson Introducing Receiver Design Apr May 06 DAY 2 20 What if cont d Multipath Mild multipath soft decisions 20 40 60 80 100 120 140 160 180 200 The appearance of 4 distinct stripes indicates no decision errors Johnson Introducing Receiver Design Apr May 06 DAY 2 21 What if cont d Multipath cont d Harsh multipath soft decisions 20 40 60 80 100 120 140 160 180 200 The lack of emergence of 4 distinct stripes indicates the likely presence of decision errors Johnson Introducing Receiver Design Apr May 06 DAY 2 22 What if cont d Carrier phase offset Severe offset 80 100 120 140 160 180 200 The attenuation due to carrier phase offset reduces all soft decisions below magnitude 2 resulting in no 3 as decision device outputs plenty of errors If scaled back up so stripes of largest magnitude values are above magnitude 2 the SNR will suffer relative to case without carrier phase offset Johnson Introducing Receiver Design Apr May 06 DAY 2 23 What if cont d Carrier frequency offset Soft de
29. 3000 4000 5000 Iterations e Initially closed eye is opened within 500 iterations e Asymptotic offset not same as without ISI Johnson Introducing Receiver Design Apr May 06 DAY 3 LAB 1 Laboratory Exercises Day 3 Introduction to Digital Communication Receiver Design Task 1 Algorithm Derivation for Dispersion Minimization Based Baud Timing In the lecture notes you were shown how to implement the Output Power Maximization OP technique for clock recovery Recall that the cost function to maximize in the OP technique is given by Jop t avg 2 A which can be implemented via steepest ascent resulting in the adaptation algorithm k l clk pak a 7 k 5 z a 7 k s Your task Derive the steepest descent baud timing algorithm for the dispersion min imization DM cost given by Jour avg 2k 1 You may find the discussion on the first two pages of the Output Power Maximization section of the DAY 2 lecture notes to be useful Task 2 Implementation of Dispersion Minimization Based Baud Timing The receiver you have been given currently uses the OP technique for clock recovery in system_code Rx m Your task is to replace the OP algorithm with the dispersion minimization algorithm you developed above You will then compare the performance of the two schemes After you have successfully added the DM algorithm to the system you should tune its parameters stepsize etc fo
30. AE frequency F X tt x Xz lt Ej tal Xa l6 i band f tf ie zrk t pess 5 6 z tofl i R cos 2rfg K 4 MG T Xalt Xs kt Xo kT ideali x 7 kh X_ xT 2s MY LPF T Johnson Introducing Receiver Design Apr May 06 DAY 1 30 Sub Nyquist Sampling cont d Another Example cont d e For the following specifications in kHz fi 50 f2 1690 fz 1920 fa 1460 fs 1620 fe 1760 fz 800 fg 90 fo 60 given X f as even symmetric triangular shaped and centered at zero frequency and M 2 we can draw X f for i 1 2 8 to show that Xg f matches up to a scalar gain factor the magnitude spectrum of x t sampled at the symbol rate Johnson Introducing Receiver Design Apr May 06 DAY 1 31 Sub Nyquist Sampling cont d Another Example cont d x F Ise Jo 170 Xale Johnson Introducing Receiver Design Apr May 06 DAY 1 32 Sub Nyquist Sampling cont d Another Example cont d Ixsce lo iSo eo uo qo ADH Ho go iso ete f Ixece Afo o0 b30 400 180 5o S 80 f2 630 X7 a 50 206 4oo Johnson Introducing Receiver Design Apr May 06 DAY 1 33 Interpolation e Objective Use signal samples from times kT to reconstruct the analog signal value at a time instant not among the set of sample times Sinc interpolator CO w t liar w r w p sine r p dp
31. AY 3 20 A Baud Timing Example We will analyze the special case for h t gr t c t g7 c t R t Sampler Transmit Receive x 6 x kT M 7 Channel Filter filter when e the noise w is absent and e the analog pulse shaping filter the channel transfer function and the receive filter combine into an impulse response that is a triangle spanning two symbol intervals 1 0 Johnson Introducing Receiver Design Apr May 06 DAY 3 21 A Baud Timing Example cont d e With perfect baud timing 7 0 baud space sampled M 1 combined analog pulse channel receive filter impulse response shape is a Nyquist pulse l k l 0 k 1 h kT In general without perfect baud timing the sampler output is a weighted combination of several source symbol values zik X s i h t iT a t kT 7 Consider three cases r 0 O T gt O0 OTT lt O0 Johnson Introducing Receiver Design Apr May 06 DAY 3 22 A Baud Timing Example cont d e r 0 Only one nonzero point in sampled impulse response Sampled impulse response h t iT A kT 7 iT h k i T 7 h k 1 T i kers gt i k 1 li kT 7 0 otherwise xfk s k 1 system is pure delay and sampler is synchronized with transmitter pulse Johnson Introducing Receiver Design Apr May 06 DAY 3 23 A Baud Timing Example cont d e7 gt 0 Two nonzero points in sampled impulse response h 7
32. FsampleIdx BPFfilterOrder ae adapt PLL theta IFsampleIdx 1 theta IFsampleIdx mu_PLL r2 IFsampleIdx sin 4 pi f_c IFsampleIdx f_s 2 theta IFsampleIdx phaseBPF perform downconversion x_down IFsampleIdx r IFsampleIdx cos 2 pi f_c IFsamplelIdx f_s theta IFsampleIdx perform matched filtering x_bb IFsampleIdx srrcFlt x_down IFsampleIdx 1 IFsampleIdx sr rcFltLength 1 while tnow lt IFsampleIdx 2 srrcLength T_t f_s 2 do we have a new baseband sample 10 Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 11 87 BBsampleIdx BBsampleIdx 1 ok we re at next baseband sample so increment 88 89 perform timing recovery OP 90 x_sampled BBsampleIdx interpsinc x_bb tnowttau BBsampleIdx srrcLength interpolated value at tnow tau 91 x_deltap interpsinc x_bb tnow tau BBsampleIdx delta srrcLe ngth h get value to the right 92 x_deltam interpsinc x_bb tnow tau BBsampleIdx delta srrcLe ngth h get value to the left 93 dx x_deltap x_deltam calculate numerical derivative 94 tau BBsampleIdx 1 tau BBsampleIdx mu_timing dx x_sampled BBsampleldx alg update OP 95 tnow tnowtT_t f_s update current position 96 97 run correlator matched to marker sequence 98 if BBsampleIdx gt eqDelay markerLength 1 need to skip t he first few sample until we have enough to fill correlator 99 corInputSignal x_sampled BBsampleIdx markerLength 1 eqD elay BBsampleIdx eqDelay extr
33. In this report you should also include the following plots for medium m only Carrier phase Timing offset Equalizer error If your receiver makes errors on any of the test vectors you should make a conjecture about why your receiver is unable to make zero errors Is there a particular component of the receiver that seems to be the source of the errors e The receiver that you design must be your own work Suggestions e You may find it easiest to add the requested modifications incrementally Testing your code after each change will help narrow down the possible sources of an error e You may have to do some adjustment of algorithm stepsizes in your receiver This is a natural part of the design process e Start with the easy mat test vector Once your simulation works with this vector you should progress to the medium mat test vector and then to hard mat e Try to break your receiver See how much noise can be present in the received signal before accurate demodulation seems impossible e g BER gt 107 Try to determine how bad the worst channel can be through which a signal can be transmitted where your receiver correctly decodes the signal e The test vectors and mystery signal may have originated from a time varying chan nel Take note of the ability of your equalizer to track by looking at the equalizer error signal Does the error stay small Or does it increase Again you will want to tune the stepsize
34. Johnson Introducing Receiver Design Apr May 06 1 INTRODUCTION TO DIGITAL COMMUNICATION RECEIVER DESIGN PREPARED BY C RICHARD JOHNSON JR FOR DELIVERY AT UNIVERSITY COLLEGE DUBLIN IRELAND AND TECHNISCHE UNIVERSITEIT DELFT THE NETHERLANDS IN APRIL MAY 2006 UNDER THE SUPPORT OF A FULBRIGHT SCHOLARSHIP AND A WEISS FELLOWSHIP e Lectures drawn from Johnson and Sethares Telecommunication Breakdown Concepts of Communication Transmitted via Software Defined Radio Prentice Hall 2004 e Lab assignments use a Matlab based PAM Radio from Dr Andy Klein e Distribution does not constitute release of copyright All rights reserved Johnson Introducing Receiver Design Apr May 06 2 Five Day Schedule e DAY 1 3 lecture hours 3 lab hours A Naive Digital Radio De Modulation Automatic Gain Control DAY 2 8 lecture hours 3 lab hours An Idealized RF System Simulation Carrier Recovery DAY 3 8 lecture hours 3 lab hours Pulse Shaping and Receive Filtering Baud Timing for Clock Recovery DAY 4 8 lecture hours 3 lab hours Linear Equalization Putting It All Together DAY 5 6 lab hours Design Project and Report Preparation Design Testing and Report Presentation Johnson Introducing Receiver Design Apr May 06 FOREWORD 1 This compacted 5 day introduction to digital communication recevier design was orig inally extracted from C R Johnson Jr and W A Set
35. Y 2 35 Phase locked Loop PLL To introduce a phase locked loop the most widely known carrier recovery scheme we present a candidate cost function producing the PLL e Reconsider the output of the squarer and narrow BPF which is a scaled version of the carrier x kT gcos 4afokT 26 where g is 84 2 times the square of the product of the channel and BPF gains at 2 9 and w is the BPF phase mod 27 at 2fo Consider downconverting kT with our unsynchronized receiver oscillator s output and form kT cos 4r fokT 20 Y g cos 4r fokT 26 cos 4r fokT 20 y cos 26 26 cos 87 fokTs 26 20 2y Johnson Introducing Receiver Design Apr May 06 DAY 2 36 PLL cont d e Lowpass filtering this product with a LPF with cutoff below 4f9 produces LPF a kT cos 4r fokTs 20 w N 5 cos 2 20 which is maximized when 2 20 2nt gt o d nrt e Value of positive finite g does not effect locations of maxima and minima e We will choose to maximize ko P JPLL D x kT cos 4r fokTs 20 4 k ko ave a kT cos 4r fokTs 20 LPF x kT cos 4r fokTs 20 Johnson Introducing Receiver Design Apr May 06 DAY 2 37 PLL cont d As a numerical test for extrema the PLL cost JprL LPF x kT cos 4a fokT 20 w can be formed for various fixed 0 producing via plicrt A maximum near 0 5 with g 1 in this case
36. able f stores the equalizer coefficients and eqOut stores the output of the equalizer Decision Device Frame Sync and Message Decoding lines 125 129 This part quantizes the equalizer output using quantalph resulting in a stream of 4 PAM symbol estimates stored in the variable dec With knowledge of the start of the header sequence from the previous stage frame synchronization is performed after which the decisions pass into the decoder i e pam2letters the output of which is stored in the variable decoded_msg There are some other details of the receiver which are worth noting The receiver consists of two main loops and their corresponding counters 1 IFsampleIdx Each time this loop counter is incremented the receiver has received a new IF sample at the receiver frontend 2 BBsampleIdx This loop counter is incremented every time a new baseband sample is output from the baud timing device Also the receiver operates in 3 distinct modes 1 HEADER_SEARCH_MODE In this mode the receiver is running its correlator to search for the header sequence 2 TRAINING_MODE In this mode the receiver thinks that it is receiving training data and so it is training the equalizer using the LMS algorithm 3 DATA_MODE In this mode the receiver has completed training and believes that it is receiving data The receiver starts in HEADER_SEARCH_MODE Once the header is found it switches to TRAINING_MODE
37. act portion of signal used f or correlation 100 Corr BBsampleIdx marker corInputSignal 2 calculate correlation 101 end 102 103 switch operationMode 104 case HEADER_SEARCH_MODE if we haven t already fou nd marker look for it 105 if Corr BBsampleIdx gt correlThresh has cor relation exceeded threshold 106 operationMode TRAINING_MODE yep so switch to training mode 107 trainingIndex 1 h reset to trainingIndex to first sample of training data 108 packet Index packetIndext1 increment packet counter 109 end 110 111 case TRAINING_MODE if we re in equalizer training mod e train the LMS equalizer 112 rr x_sampled BBsampleIdx 1 BBsampleIdx eqLength 1 extract regressor vector of receive d signal Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 12 113 eqOut BBsampleIdx f rr equalizer output 114 e_lms BBsampleIdx training trainingIndex eqOut BB sampleIdx calculate LMS error term 115 f f mu_eq_lms e_lms BBsampleIdx rr update equalizer coefficients 116 trainingIndex trainingIndex t1 h increment training index location 117 if trainingIndex gt trainingLength h are we done training 118 operationMode DATA_MODE yep switch to data mode 119 symbol Index 1 and re init symbol counter to 1 120 end 121 122 case DATA_MODE we re into data portion of the pack et equalizer and save data 123 rr x_sampled BBsampleIdx 1 BBsampleIdx eqLength 1
38. algorithms for adjusting the receiver mixer s phase that can track slow time variations in the transmitter s phase We treat carrier phase tracking as a single parameter adaptation problem Johnson Introducing Receiver Design Apr May 06 DAY 2 30 Adaptive Algorithm Development Our single parameter adaptive algorithm development strategy e Propose a cost function assessing behavior over measured data set Check location of minima and maxima in terms of adjusted parameter to see if in desired location Pursue small stepsize gradient descent strategy with its commutability of averaging and differentiation The correction term must be calculable from available signals e Test performance Johnson Introducing Receiver Design Apr May 06 DAY 2 31 Carrier Extraction e For AM with suppressed carrier we will process the received upconverted signal r kT s kT cos 27 fokTs which does not include an additive carrier in order to extract a signal related to the carrier Consider squaring the received signal and using cos x 1 2 1 cos 2zx to produce P kl y 1 2 s kT 1 cos 4r fokT 20 Johnson Introducing Receiver Design Apr May 06 DAY 2 32 Carrier extraction cont d e Rewrite s t as the sum of its positive average value and the variation about this average s kTs 84 u kTs so r KD 452 K 1 cos 4n fokT 20 2 car u
39. alized cost t S LPF 2r kTs cos 27 fokT 0 B D 1 LPF s kT where r is the received signal for our continuing Jye example for various fixed 0 producing via ccert This normalized cost function matches 1 cos 2 0 2 as anticipated Johnson Introducing Receiver Design Apr May 06 DAY 2 44 Costas Loop cont d Our next step in our algorithm creation strategy is to interchange the averaging and differentiation in the gradient ascent update Olk 1 0 k p lt lavel LPF 2r kT cos 2m fokTs 9 Hlo o x With LPF 2r kT cos 27 fokT 0 v kT cos 0 the update can be written as O k 1 O k a avet EET cos d 8 o or Ofh p avelo2 kT cos o O 4 d Y and from 7 cos y sin y we wish to form Ojk 1 k u avg fu kT coslo 6 k sin d O K Johnson Introducing Receiver Design Apr May 06 DAY 2 45 Costas Loop cont d Given LPF 2r kT cos 27 fokT 0 u kT cos 0 to compose the update from measurable signals we need to find a realizable expression for u kT sin 0 For a LPF with cutoff under 2f 9 defining v LPF s and using sin x cos y 1 2 sin a y sin x y and sin x sin x produces LPF 2r kT sin 27 fokT 0 LPF s kT cos 27 fokT sin 27 fokT 0 LPF s kT sin d sin 4rfokT 6 6 kT sin
40. an f 2 top plot and slightly larger than f2 2 bottom Iwel Johnson Introducing Receiver Design Apr May 06 DAY 1 27 Sub Nyquist Sampling cont d e Nyquist Sampling Theorem If the signal w t is bandlimited to B W f 0 for all f gt B and if the sampling rate is faster than f 2B then w t can be reconstructed exactly for all t from its samples w kT Sub Nyquist Sampling What if the signal to be sampled is a passband signal but the signal to be reconstructed is this passband signal downconverted to a baseband signal with a much lower maximum frequency Can sub Nyquist sampling of the passband signal be employed without aliasing of the baseband signal The following examples provide a positive answer Johnson Introducing Receiver Design Apr May 06 DAY 1 28 Sub Nyquist Sampling cont d e Example Consider f fc 2 WO 4 1 f B f f B f IYI JUAN 3f 2 fe fl2 0 f2 3f 2 f Eftf Works for fs fe n What if f not exactly fe n Johnson Introducing Receiver Design Apr May 06 DAY 1 29 Sub Nyquist Sampling cont d e Another Example For a PAM system the sampler downconverter and downsampler to symbol period T should produce an output xg with a spectrum matching that of a sampled version with sample period matching symbol period of the baseband source 21 higheet cos 2rf t cos 2
41. and when the training is complete it switches to DATA_MODE Once the data transmission is complete based on the length of the data sequence specified in the system parameters the receiver then returns to HEADER_SEARCH_MODE and repeats Listings main m 1 2 3 Example script that demonstrates how to call transmitter and rece iver code Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO O O CO N 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 written by A Klein 26 Oct 2005 call script to get global simulation parameters i e carrier fre q baud rate training data etc globalParams initialize random seed for repeatability helps for debugging randn state 0 rand state 0 set channel SNR phase noise and message to be sent c 1 0 4 0 2 channel T spaced SNR 18 signal to noise ratio dB of und er sampled signal phase_noise_variance 1e 6 variance of underlying phase noise process uncomment the following lines for the most benign conditions c 1 0 0 no ISI SNR Inf no noise phase_noise_variance 0 perfect oscillators m This is the first frame which you probably shouldn t able to d ecode perfectly unless you cheat and gt give your receiver the initial points Now we re into the second frame You might be able to decod e thi
42. appears at the desired location of 0 1 with w 0 and at locations an integer multiple of m away as predicted in the preceding analysis Johnson Introducing Receiver Design Apr May 06 DAY 2 38 PLL cont d Following a gradient ascent strategy for maximization compose Ok 1 ofk o z lave a kT cos 4r fokT 20 w o 0 k With a small stepsize assuring approximate commutability of differentiation and average O k 1 Ofk yr avel S x kT cos 4r fokTs 20 Y lo 01x where 0 zg ZT cos 4r fokTs 20 Y o 01k 2x kT sin 4r fokTs 20 k 4 This produces 6 k 1 6 k uLPF z kT sin 4r fokT 20 k Johnson Introducing Receiver Design Apr May 06 DAY 2 39 PLL cont d PLL carrier recovery system where input rp is the processed received signal of r t e r t eer r t cos 4r fot 20 ys Squaring Center frequency nonlinearity at 2fo 2 avg has been and normalizing gain 2 s implicitly included in BPF though any substantial gain is acceptable which has phase shift Y at frequency 2 fo When w is nonzero it should be added in carrier recovery system schematic after 20 k term in the oscillator Johnson Introducing Receiver Design Apr May 06 DAY 2 40 PLL cont d For the PLL algorithm with explicit LPF preceding integrator summer removed Olk 1 O k wx kT sin 4r fokT 20 k w a typ
43. ce b signal l Coding 4 P t Analog Channel upconversion Pulse Carrier shaping specification Antenna Analog Digital down Pulse gt conversion conversion matched Andor to IF T to baseband filter S received Input to the Carrier signal software synchronization receiver m Q m e 3 1 1 3 b Downsampling Equalizer Decision Decoding Timing Source and Reconstructed synchronization error coding message frame synchronization We will focus on the situation where up and downconversion have been flawlessly performed and the effect of transmission from baseband PAM message waveform to received signal is presumed described by a linear transfer function and the addition of interferers in particular spectrally flat broadband noise Noise Reconstructed Message Interferers n t message w kT e 3 1 1 3 x m kT 8 e 3 1 1 3 g t y i Pulse Channel gt O Receive Decision shaping filter P t h t halt P f Hf AR f Johnson Introducing Receiver Design Apr May 06 DAY 3 Pulse and pulse amplitude modulated PAM message spectrum Noise Messag Interferers n t leo l 0 e w kT e 3 1 1 3 x Decision The spectral footprint of a baseband PAM signal is no wider than that of the pulse sha
44. cisions for 0 01 frequency offset 3 40 60 80 100 120 140 160 180 The carrier frequency offset appears as a low frequency amplitude modulation of the desired outputs Johnson Introducing Receiver Design Apr May 06 DAY 2 24 What if cont d Downsampler timing offset Eye diagram with debilitating offset Assumed best times to take samples 50 100 150 200 250 300 350 400 With samples for symbol values taken every 100 samples after sample 125 numerous errors occur Johnson Introducing Receiver Design Apr May 06 DAY 2 25 What if cont d Downsampler period offset Eye diagram top and soft decisions bottom with 1 downsampler period offset 120 140 160 180 200 All is lost Johnson Introducing Receiver Design Apr May 06 DAY 2 26 Well then e Coding and matched receive filtering are intended to counter effects of broadband channel noise Equalization compensates for multipath interference and can reject narrowband interferers as well Carrier recovery schemes including phase locked loops and Costas loops adjust receiver oscillator phase to counteract phase offset and mild frequency offset Timing recovery using interpolation is intended for reduction of downsampler timing offset and mild period offset 27 Johnson Introducing Receiver Design Apr May 06 DAY 2 Our Project System Binary Other FDM message we
45. d 0 Johnson Introducing Receiver Design Apr May 06 DAY 2 46 Costas Loop cont d Thus a small stepsize gradient ascent algorithm for maximization of Jc is Ok 1 O k j avg LPF 2r kT cos 27 fokT T O k LPF 2r kT sin 27 fokT O k e The use of lowpass filtering in the update is predicated on a presumption that the LPF output is characterized by its asymptotic response e This effectively presumes 0 k remains fixed for a sufficiently long time for this asymptotic behavior to be achieved e We rely on a small stepsize u to keep Ofk variations modest in the relatively short time frame anticipated for LPF achievement of asymptotic behavior Johnson Introducing Receiver Design Apr May 06 DAY 2 47 Costas Loop cont d Schematic for Costas loop carrier phase recovery with the outer averaging removed which presumes that the integrator summer of the update will provide sufficient averaging 2cos 2afokT O K 2sin 2mrfokT O k Johnson Introducing Receiver Design Apr May 06 DAY 2 48 Costas Loop cont d A typical learning curve for this Costas loop carrier phase recovery scheme as shown in the preceding schematic without explicit averaging in the update on our continuing example with an objective of 1 is from ccrt with a stepsize of u 0 001 Phase Tracking via the Phase Locked Loop l 2 D phase offset
46. de 0 01 0 02 0 03 0 04 0 05 0 06 0 07 d recovered message is a LPF applied to c Johnson Introducing Receiver Design Apr May 06 DAY 1 20 Unsynchronized Demodulation w t v t A cos 2r f t cos 2r f y t e b a transmitter modulator b unsynchronized receiver demodulator Johnson Introducing Receiver Design Apr May 06 DAY 1 21 Unsynchronized Demodulation cont d e Using F g t cos 27at 0 eP G f a e OT a x t v t cos 2m fe yt and V f AWCE fo 5AeW UF fe Ae el WUE fe fe 1 W f E fe Fe y PEW ET W f fe e le Wa re WE 2 Ac PEOWER EIFE WF Johnson Introducing Receiver Design Apr May 06 DAY 1 22 Unsynchronized Demodulation cont d e If no frequency offset y 0 then with exponential description of cosine Ac X f E eW el W f 2 fe e WF 2fe Ww fyecos 52 PWG 2h HW 2f Thus with LPF cutoff between W f bandwidth B and 2f B mb PEG w t cos d Recovered signal is attenuated relative to perfectly synchronized demodulation As approaches 7 2 recovered signal vanishes Johnson Introducing Receiver Design Apr May 06 DAY 1 23 Unsynchronized Demodulation cont d e If no carrier offset 0 WEF A EWEA a W fete E EW ET Thus with m t LPF t Ac 4 and using frequency shifting property of WE DE E
47. e channel delay spread maximum sampler frequency Johnson Introducing Receiver Design Apr May 06 DAY 4 25 Received Signal Construction cont d Received Sampled Signal Specifications Table right column lowpass filtered white noise Johnson Introducing Receiver Design Apr May 06 DAY 4 26 Receiver Design Methodology e Stage One Ordering the basic operations e Stage Two Selecting components e Stage Three Countering anticipated impairments e Stage Four Tuning and testing Sampled received signal Matched Interpolator Downconversion filter downsampler ae tve Carrier Timing ayer recovery recovery Recovered ot source Equalizer Decision Decoder device Trainin Equalizer Frame Adaptive 8 PE content layer segment adaptation synchronization locator Johnson Introducing Receiver Design Apr May 06 DAY 4 27 Design Methodology cont d Stage One Ordering the basic components e The basic receiver components are downconversion with carrier recovery baud timing recovery with matched filter and interpolator downsampler trained equalizer with training segment locator decision device and decoder with frame synchronization Our ordering downconversion timed downsampling equalization and decoding is classical and popular but not the only possibility Johnson Introducing Receiver Design Apr May 06 DAY 4 28 Design Methodology cont d Stage One
48. e interpsinc function and consists of several sub steps Parameter Initialization lines 35 38 Get current interpolated value line 90 Calculate approximate derivative lines 91 93 Algorithm Adaptation line 94 Recall the equation for the output power maximizing baud timing adaptation algo rithm has the form rk 1 r k walk fe 5 r k 5 x 7 k s which is seen in lines 91 94 The MATLAB variable tau stores the timing offset tnow stores the current position and x_sampled stores the downsampled signal after timing recovery e Correlation lines 97 102 Always running this part calculates the correlation of the downsampled signal with the known header sequence e Header Search lines 39 41 104 110 This part searches for the header by comparing the correlation value with a threshold e Equalization with Adaptation via LMS lines 111 124 This part performs equalization of the signal which includes adaptation of the equalizer coefficient during training periods see paragraph below about different operating modes of receiver Equalization consists of two sub steps Adapt equalizer using LMS lines 111 121 Generate equalizer output lines 121 124 Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 5 Recall the equation for the LMS algorithm which has the form filk 1 failk u s k 6 ylk r k il and is seen in lines 114 115 The MATLAB vari
49. eferring to the plot of 0 3 What effect to you observe as you decrease the stepsize What happens as you increase the stepsize Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 1 Transmitter Receiver Code Description Introduction to Digital Communication Receiver Design Introduction A transmitter and sampled IF receiver have been implemented in MATLAB and this docu ment describes the corresponding code This operation of the receiver including its chosen parameters are described in the latter half of the lecture notes for DAY 4 under the head ing of Putting It All Together Receiver Design This receiver system will be used in the lab assignments for Days 2 4 by focusing only on the specific segment described in the associated lectures while allowing us to judge the impact on overall system performance The block diagram in Fig 1 shows the steps of generation of the transmitted signal its propagation through the channel and the operations performed by the receiver While Binary Other FDM message we 3 1 1 3 Transmitted users Noise sequence b Analo signal Coding P f 8 Channel gt upconversion L Pulse Carrier shaping specification Antenna Analog Digital down Pulse gt conversion conversion matched gt to IF to baseband filter Analog T received Input to the Carrier si
50. em cont d Soft Decisions Constellation Diagram History 40 60 80 100 120 140 160 180 200 Because the soft decisions are so close to the alphabet levels there are no decision errors and no symbol errors Johnson Introducing Receiver Design Apr May 06 DAY 2 14 Flat Fading Impairment At time representing 20 of duration of simulation window the channel gain changes abruptly from 1 to 0 5 as in idsystagc Effect Soft decisions in ideal system receiver 3 2 20 40 60 80 100 120 140 160 180 The soft decisions have all moved inside 2 in magnitude meaning that decision device will never produce 3 gt lots of errors Johnson Introducing Receiver Design Apr May 06 DAY 2 15 Flat Fading cont d Fixed Soft decisions with inclusion of AGC 20 40 60 80 100 120 140 160 180 200 Decisions correct once top and bottom stripes in constellation diagram history have magnitude TD Johnson Introducing Receiver Design Apr May 06 DAY 2 16 Flat Fading cont d Adapted gain time history Starts at 1 ends near 2 2 6 2 4 2 2 2 1 8 1 6 1 4 1 2 1 0 02 04 06 08 1 12 14 16 1 8 2 x104 Johnson Introducing Receiver Design Apr May 06 DAY 2 17 What if Channel noise Noisy received signal and spectrum from impsys 5 o ao 5 T lt 60 80 100 120 140 160 180 200 Seconds Magnitude 0 di ih 50 40 30 20 10 0 10 20
51. f the components can also be found in the block diagram in Fig 1 e Calculate Intermediate Variables lines 18 25 52 72 This part calculates in termediate variables which are used in the receiver This includes calculation of the effective carrier frequency and image frequency memory allocation variable alloca tion and size determination e Digital Downconversion via PLL lines 26 34 74 82 This part implements the downconversion which is accomplished in the current receiver with a PLL The procedure consists of several sub steps Parameter Initialization and Bandpass Filter Design lines 26 34 PLL Pre processing lines 74 76 PLL Adaptation lines 77 79 Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 4 Mixing lines 80 82 Recall that the equation for PLL adaptation has the form Olk 1 O k wx kT sin 47 fokT O k Y which appears in line 78 The carrier phase estimate is stored in the variable theta while the downconverted signal is stored in the variable x_down e Pulse Matched Filter lines 48 51 83 85 This part performs filtering of the signal with the square root raised cosine filter The filtered signal is stored in the variable x_bb e Downsampling Timing Sync via Output Power OP Method lines 35 38 89 96 This part performs the downsampling and timing synchronization using the method of output power maximization The procedure makes several calls to th
52. gn Apr May 06 DAY 3 26 A Baud Timing Example cont d e Similarly with T lt 0 for k 6 the four equally likely source symbol pairs creating x 6 are SELS U a With the addition of the absolute value on To which does not effect a positive To the formulas for the four choices are the same as for positive T Johnson Introducing Receiver Design Apr May 06 DAY 3 27 A Baud Timing Example cont d e For T 2 lt 7 lt T 2 Q a k s k 1 e So source recovery error equals decision error s k 1 z k Q z k z k when eye is open But if eye is closed cluster variance does not equal average squared recovery error We are now in a position to consider some candidate cost functions for this baud timing example Johnson Introducing Receiver Design Apr May 06 DAY 3 28 A Baud Timing Example cont d e Cluster variance ave Q 2 k k ave Q z 6 2 6 2 fa 1 1 1 y 2 Tol 1 1 yes 1 0 B 1 Ar n Are z 25 gt A4 T2 T2 T2 e The same result occurs for other k e Desired offset of r 0 nT occurs with minimization of average squared decision error in the sampler output avg Q x x 0 5 gt 32 Ee f T 2 T Timing offset t Johnson Introducing Receiver Design Apr May 06 DAY 3 29 A Baud Timing Example cont d e Average squared sampler output or output power avg x k 1 4 1 2 7
53. gnal software synchronization receiver A T m O m e 3 1 1 3 b gt Downsampling Equalizer Decision Decoding Timing Source and Reconstructed synchronization error coding message frame synchronization Figure 1 System Block Diagram the blocks in this figure are quite general design choices were made in the development of this particular transmitter receiver implementation These design choices i e which algorithms have been selected and the code to implement them will be described in the following sections MATLAB Files A brief description of each of the functions used in the complete system is provided here You can find all of the files in the system_code directory For a more detailed description Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 2 of the inputs and outputs for each of these functions you can using the help command at the MATLAB prompt e g by typing help Tx Main Files Listings at the end of this code description document e main m This script is merely an example which shows how to set up the system parameters run the transmitter run the receiver and calculate the bit error rate e Tx m This function contains the transmitter and introduces the impairments e g the channel imperfect receiver frontend etc e Rx m This function decodes the received signal and outputs the message e globalParams m This function contains the parameters f
54. gth globalParams calculate new effective carrier frequency and the image which wi ll appear at 2f_c and may get aliased f_c f_if fix f_if f_s f_s f_image abs mod 2 f_c f_s 2 f_s f_s 2 Calculate sizes markerLength length marker trainingLength length training PLL parameters amp BPF filter design bpf_ctr f_image f_s 2 set center frequency of BPF to 2f BPFfilterOrder 500 should be an even number ff 0 bpf_ctr 0 006 0 003 0 003 0 006 1 BPF trans band fa 0 0110 0 values at transition regions h remez BPFfilterOrder ff fa design filter phaseBPF angle exp 1j 0 length h 1 pixbpf_ctr h calculate phase introduced by BPF mu_PLL 0 001 stepsize for first PLL loop baud timing OP parameters mu_timing 0 1 algorithm stepsize delta 0 1 time for derivative correlation i e header search parameters correlThresh 6500 threshold for determining whether we ve received the header sequence equalizer parameters eqLength 8 equalizer length mu_eq_lms 0 005 trained LMS stepsize eqDelay 3 desired delay for LMS f zeros eqLength 1 f eqDelay 1 1 equalizer initialization m ust have correct length h design SRRC matched filter srrcFlt srrc srrcLength rolloff f_s T_t 0 srrcFltLength length srrcFlt
55. hares Telecommunication Break down Concepts of Communication Transmitted via Software Defined Radio Prentice Hall 2004 under the support of Prof Rick Johnson by a Fulbright Scholarship to France in the latter half of 2005 The accompanying labs were developed in collaboration with Dr A G Klein currently a post doctoral researcher in the Laboratoire de Signaux et Systemes Sup lec Gif sur Yvette France The first version of this compacted course was offered in the ATHENS Programme at Ecole Nationale Sup rieure des T l communications Paris France in November 2005 The current version was prepared for presentation in April and May 2006 at University College Dublin Ireland and Technische Universiteit Delft the Netherlands This spring 2006 teaching activity is supported in part by a Stephen H Weiss Preisdential Fellowship from Cornell University In keeping with the philosophy of Telecommunication Breakdown this compacted ver sion is built around a Matlab based software radio developed by Dr Andy Klein that implements the major digital signal processing operations of a common radio receiver demodulation carrier recovery matched receive filtering baud timing equalization and decoding This radio can compensate for the transmission impairments of carrier phase jit ter channel noise time varying channel intersymbol interference and baud timing offset Relying on a background in signals and systems comparable to that of J H
56. he new mode sequence 1 2 3 2 3 e In the day5 directory you will find 3 test vectors easy mat medium mat and hard mat Each of these MATLAB data files contains an example received signal and each originates from an increasingly hostile communication environment Using the file tester m you can test the performance of your receiver You should place all files in the same directory Evaluation e At the end of class you will be presented with a mystery signal You will be assigned a grade based on how many errors your receiver makes You are not required to decode the message contents for the first 5 frames This is to allow your algorithms time to converge All symbols after the first 5 frame will be used to calculate your grade The difficulty of the mystery signal will be between that of the medium m and hard m e Your grade will be based on the output of the tester m program and it is your responsibility to make sure that your receiver is compatible with this script If your Johnson Introducing Receiver Design Apr May 06 FINAL PROJECT 3 receiver does not operate with any of the provided test vectors then it certainly will not work with the mystery signal You will be required to explain to the instructor the operation of your MATLAB code in Rx m only e Your are also required to include a very brief report 1 2 pages detailing the per formance of your receiver on the 3 test vectors easy m medium m and hard m
57. ical learning curve from pllcrt for a stepsize of 0 001 for our continuing example with 0 and an objective of 0 1 is Phase Tracking via the Phase Locked Loop T T T T T Johnson Introducing Receiver Design Apr May 06 DAY 2 Al Costas Loop Now we seek an algorithm not based on a presumption of carrier extraction from the received signal e Reconsider the received signal r kT s kT cos 27 fokT and form 2r kT cos 27 fokT 0 s kT cos 0 cos 4r fokT 6 0 e With a LPF cutoff below 2 fo LPF 2r kT cos 27 fokT 6 u kT cos 0 where u kT LPF s kT If the cutoff frequency of the LPF is above the bandwidth of the baseband waveform s then v is s Johnson Introducing Receiver Design Apr May 06 DAY 2 42 Costas Loop cont d e As a cost function consider ko 1 X LPF 2r kT cos 2m fokTs 9 k ko P 1 ave u kT cos o 0 e Because the squared cosine term is fixed ave u kT cos 0 avg v kT H eE and assuming that the average of v is fixed this cost function will be maximized with a value equal to the average of v which is average value of LPF s at 0 mn or 0 mn for all positive and negative integers n Johnson Introducing Receiver Design Apr May 06 DAY 2 43 Costas Loop cont d We can numerically check the extrema of a norm
58. ication System Transmitter and Transmitted Pulse Sequence Received Signal and Receiver Synchronization Issues Spectrum Sharing RF Communication System Practical Obstacles Analog Digital Signal Processing Split Johnson Introducing Receiver Design Apr May 06 DAY 1 3 An Illustrative Digital Communication System e Objective Send text converted to a stream of bits from place 1 to place 2 through the analog medium in between Coding Use standard ASCII code to convert text to bits using 8 bits per character Transmitter Use sequence of scaled rectangular pulses to convey bits singly e g 1 1 and 0 1 or in clusters e g 10 1 01 1 00 3 and 11 3 We choose pairs so groups of 8 bits become clumps of 4 symbols Receiver Sample received pulse and convert symbols to bits e g 1 3 1 1 3 1000011011 and then back to text Johnson Introducing Receiver Design Apr May 06 DAY 1 4 Transmitter and Transmitted Pulse Sequence e An idealized baseband transmitter Symbols Scaling s k factor T wide Baseband analog signal y t pulse gt Initiation shape p t X trigger generator and transmitted baseband signal y e The transmitted signal consists of a sequence of pulses one corresponding to each symbol e Each pulse has the same rectangular shape though offset in time and scaled in magnitude John
59. ition W f J w t e i dt Ffw t OO Johnson Introducing Receiver Design Apr May 06 DAY 1 14 Up Conversion via Mixing cont d e So S f Fistt y Fiw t cos 27 fot j F w t E mhe 4 e jon fot Frog lt ef fot sit e J fot ec i2T ftdt ties j j2rn f fo t age j2m F Fo t de dt ce J2n f fo t dt ec J2n f fo t dt w t ie Wye a Wf fo Joe Johnson Introducing Receiver Design Apr May 06 DAY 1 15 Downconversion via Mixing e Assume transmitted signal arrives unimpaired e For downconversion use mixer with frequency and phase matching transmitter s d t s t cos 2r fot w t cos 27 fot cos x cos 2z t 5 4 cos 4z fot w t Sw t cos 27 2fo t e Using linearity of Fourier transform and previously extracted result on Fourier transform of mixer output D f Fyalt F Sw t w t cos 27 2fo t SF w t F w t cos 27 2fo t 5W f 4W f 2fo iW f 2fo Johnson Introducing Receiver Design Apr May 06 DAY 1 16 Message Recovery via Filtering e Passing a signal s t through a linear system with transfer function h t results in an output that is the convolution of s t and h t The Fourier transform of a convolution is the product of the Fourier transforms We often distinguish among linear systems based on the range of frequencies they pass or reject e g lowpass highpass bandpass notch
60. itude response Adaptive equalizer output 1000 2000 3000 4000 2 3 Iterations Normalized frequency Johnson Introducing Receiver Design Apr May 06 DAY 4 19 PUTTING IT ALL TOGETHER RECEIVER DESIGN x Received Signal Construction x Receiver Design Methodolgy in 4 Stages Johnson Introducing Receiver Design Apr May 06 DAY 4 20 Received Signal Construction Receiver design responds to the received signal composition Transmitter and channel Symbols Text Tr Si Scaling message Characters Coding including tobinary gt periodic marker and conversion training insertion Trigger le iT e Adjacent Broadband users noise Transmitted Analog Baseband passband received signal Modulation signal signal ee Channel with phase noise Receiver front end Analog received signal Sampled f received Automatic signal r k Bandpass Downconversion filter to IF pan control Johnson Introducing Receiver Design Apr May 06 DAY 4 21 Received Signal Construction cont d Original character string message is coded into 7 bit ASCII format and mapped to 4 PAM Symbol sequence is composed as a 124 symbol marker training segment followed by 400 4 PAM message symbols followed by the same 124 symbol marker training segment followed by another 400 message symbols etc Transmitter pulse period T precisely matches the symbol period specification adopted by
61. iver Design Task 1 Filter Design With remez The MATLAB command remez is useful for generating so called equiripple FIR filters We will rely on it frequently for designing lowpass and bandpass filters The remez command takes three parameters Type help remez to familiarize yourself with the parameters you only need to pay attention to the first paragraph in the help called with 3 parameters N F and A The following code generates 3 seconds worth of a random white signal sampled at 10 kHz and plots the magnitude spectrum time 3 Ts 1 10000 x randn time Ts 1 plotspec x Ts The following lines design a 100 th order low pass filter with a cutoff at 1 kHz and plots the filtered signal h remez 100 0 0 2 0 21 1 1 10 0 y filter h 1 x plotspec y Ts Your task Provide the corresponding lines of code to design a bandpass filter BPF which passes frequencies between 1 5 kHz and 2 5 kHz Plot the result of filtering x with the BPF Plot the result of filtering y with the BPF Task 2 Filtering with Tapped Delay Lines The filter and conv commands are quite useful for filtering signals but they assume you have all of the data available In a real time communication system we may want to put each sample into a filter as we receive it In this case the filter and conv commands are not so useful For a signal x n passing through a filter h n of length N the output at time n is given by the convolution sum
62. ization locator Adaptive layer Johnson Introducing Receiver Design Apr May 06 DAY 4 33 Design Methodology cont d Stage Four Tuning and Testing cont d Plan of action e One at a time e In order of appearance e With preceding steps countering their impairments as intended e Each with its own share of total allowable error Johnson Introducing Receiver Design Apr May 06 DAY 4 34 Design Methodology cont d Stage Four Tuning and Testing cont d Tuning tradeofts e All adaptive components will select stepsize in tradeoff between rapid tracking and dampened jitter e Carrier recovery LPF cutoff frequency and range between in band and stopband gain e Clock recovery in derivative time support of interpolation filter Johnson Introducing Receiver Design Apr May 06 DAY 4 35 Design Methodology cont d Stage Four Tuning and Testing cont d e Equalizer number of taps channel inverse delay spread 2 to 5 times channel maximum delay spread training signal delay half of equalizer length initialization center spike e Frame or training synchronization marker chosen for peaky autocorrelation preferred marker unlikely to occur in message Johnson Introducing Receiver Design Apr May 06 DAY 4 36 Development Tips e simulate transmitter to allow controlled tests on broader set of circumstances than provided by test
63. lizer output error on the same plot as the smoothed squared LMS error but in a different color Is the eye open by the end of the simulation If so at approximately which iteration is the eye open What is the BER e Keep the DD LMS stepsize at zero but set the trained LMS stepsize to 0 001 Re run main m Is the eye open by the end of the simulation If so at approxi mately which iteration is the eye open What is the BER e Now set both the DD LMS stepsize and trained LMS stepsizes to 0 001 Re run main m Is the eye open by the end of the simulation If so at approximately which iteration is the eye open What is the BER e Finally set the DD LMS stepsize to 0 001 but the trained LMS stepsize to zero Re run main m Is the eye open by the end of the simulation If so at approximately which iteration is the eye open What is the BER 2 Time varying Channel e Load the file day4 time_var mat which contains a signal from a time varying channel stored in the variable r Set the trained LMS stepsize and DD LMS stepsizes to zero Test your receiver on the signal Is your receiver able to track the time varying channel Show a plot of the equalizer output error during both trained and DD modes Is the eye open by the end of the simulation If so at approximately which iteration is the eye open What is the BER e Keep the DD LMS stepsize at zero but set the trained LMS stepsize to 0 001 Test your receiver on the signal Is the eye
64. lse shape T wide Hamming blip p carrier frequency fe 20 carrier phase 0 RECEIVER e sampler period T T M e oversample rate M 100 Johnson Introducing Receiver Design Apr May 06 DAY 2 5 A System cont d e free running sampler output N 1 r t lt er gt mli p kT iT cos 2r f kTs mixer frequency fe 20 mixer phase 0 demodulator LPF remez f 1 fbe damps with f1 50 fbe 0 0 5 0 6 1 and damps 1100 pulse correlator filter T wide Hamming blip downsampler baud timing l 125 determined experimentally quantizer to nearest element in 1 3 decoder 4 PAM to 8 bits via reverse ASCII to text with frame synchronization assured by indexing from first symbol set by baud timing Johnson Introducing Receiver Design Apr May 06 DAY 2 A System cont d Transmitter baseband signal and magnitude spectrum 80 100 Seconds Q l eD S on S S Q S o 10 0 10 Frequency Note that spectrum is limited to minus to plus Nyquist frequency i e half of oversample frequency Johnson Introducing Receiver Design Apr May 06 DAY 2 7 A System cont d Transmitter passband signal and magnitude spectrum v 5 5 T lt Seconds Magnitude 230 20 10 0 Frequency Johnson Introducing Receiver Design Apr May 06 DAY 2 A System cont d Receiver mixer output and magnitude spectrum
65. multiplication by a cosine m t w t cos 2r7t Recovered signal is low frequency amplitude modulated relative to perfectly synchronized demodulation periodically every 1 y sec it vanishes e Ergo The need for carrier recovery Johnson Introducing Receiver Design Apr May 06 DAY 1 24 Sub Nyquist Sampling of RF Signal e In a digital radio the sampler can be after analog demodulation to baseband or after partial analog demodulation to an intermediate frequency With sampling after analog demodulation to baseband we can use the Nyquist sampling theorem to select a sample rate that allows perfect reconstruction of analog signal at any point in time just from sampled values If we sample before demodulation to baseband must we sample at the much higher Nyquist rate for the RF signal to achieve successful demodulation Johnson Introducing Receiver Design Apr May 06 DAY 1 25 Sub Nyquist Sampling cont d With w t the input to an impulse sampler the output w t is w t w t XO lt kT k co Analog w t is multiplied point by point by a pulse train Pulse train gt S t kT Impulse sampling w t Point sampling wik w kT wl kT Johnson Introducing Receiver Design Apr May 06 DAY 1 26 Sub Nyquist Sampling cont d e With fs 1 T Relative to W f W f has been scaled by fs and contains replicas at every fs e Largest frequency in W f less th
66. o 1 T Sinc is Nyquist pulse because ps 0 1 and ps kT sin rk O Sinc envelope decays at 1 t Raised cosine pulse sin 27 fot cos 2r fat 2m fot Fecvod Pro t 2fo with roll off factor 6 fa fo Raised cosine is Nyquist pulse for T 1 2fo because pro has a sinc factor sin rk rk which is zero for all nonzero integers k Raised cosine envelope decays at 1 t As 8 0 raised cosine sinc Johnson Introducing Receiver Design Apr May 06 DAY 3 11 Nyquist Pulses cont d e Raised cosine pulse cont d Fourier transform 1 EE Pro f Lreosta PSB 0 fl gt B where B is the absolute bandwidth fo is the 6db bandwidth fa B fo ti fo fa and ml Fi 2fa Johnson Introducing Receiver Design Apr May 06 DAY 3 12 Nyquist Pulses cont d e Raised cosine pulse cont d Time and Frequency Plots 1 fo 2 fo Johnson Introducing Receiver Design Apr May 06 DAY 3 13 Matched Filter Suppose the channel simply adds broadband noise n t The symbol to reconstructed downsample system is described by n t mkT t y y kT m t Pulse Receive Downsample shaping filter n t TRO w t Heen Receive Downsample filter m kT kT kT Tap nan H a Ay y Pulse Receive Downsample shaping filter so y t v t w t Ar t g t hr t n t e Our objective is to choose hr t to
67. o so the new standard has shortened the length of the training sequence This comes at a cost however as the equalizer may not have enough data to open the eye Johnson Introducing Receiver Design Apr May 06 FINAL PROJECT 2 Fortunately your manager has another wise suggestion and has drawn a diagram which shows the frame structure of the transmitted signal consisting of the marker M training T and data D Recall from the documentation of the MATLAB code that the receiver operates in 3 modes 1 header search mode 2 training mode and 3 data mode These modes are also shown in the figure Frame Structure IM Tt o Mi tT MTT D o Old Mode Sequence Your manager has pointed out the following fact after the first marker sequence has been found there is no reason to continue looking for subsequent marker sequences since you know the length of each frame and therefore you know the location of the next training sequence Thus your manager requests that you make the following changes After you find the first marker sequence turn off the correlator This way the receiver will burn less power since the correlator only needs to operate at the start of the reception Since you know the marker sequence you can use it to help train the equalizer thereby compensating for the reduced amount of training data Instead of operating in the old mode sequence 1 2 3 1 2 3 modify the receiver so that it uses t
68. oducing Receiver Design Apr May 06 DAY 4 12 Blind Adaptive Dispersion Minimizing Equalization We choose to minimize avg 1 firik i j 0 ko N 1 ni ye ae 2 hrie k ko using a gradient descent scheme filk 1 filk 8 avef 1 Shao Air 51 2 EE ee ae H af f f lk Commuting average and differentiation and dropping outer average produces file 1 filk 22 1 firlk J 0 Oj 0 Firik i aaa f k Johnson Introducing Receiver Design Apr May 06 DAY 4 13 Blind Equalization cont d Evaluating derivative produces file 1 filk u0 CD f lklrlk 3 Do filer amp irie gt siklik J otk filk 1 file oO y A y A r k i In comparison to LMS the prediction error s k y k has been effectively replaced by 1 y k y k Johnson Introducing Receiver Design Apr May 06 DAY 4 14 Blind Equalization cont d With the definition of vik gt Filklr k j Sampled received signal r k Equalizer Adaptive algorithm Performance evaluation the dispersion minimizing approximate gradient descent adaptation algorithm for the linear FIR equalizer is filk 1 filk wo y k ylk r k i e The adaptive scheme is labelled as blind rather than trained due to the creation of the correction term without a training signal Johnson Introducing Receiver Design
69. of the low noise amplifier driven by the antenna signal Received me Recovered signal Analog Digital source signal signal processing processing e Sample period lt symbol period Johnson Introducing Receiver Design Apr May 06 DAY 1 11 An ASP DSP Division of Labor ASP e frequency translation to intermediate frequency e out of band signal attenuation e automatic gain control DSP downconversion to baseband via mixer carrier tracking via mixer phase setting symbol timing via interpolation channel compensation via linear filtering symbol decision via quantization frame synchronization via marker correlation decode symbols to message text via table Johnson Introducing Receiver Design Apr May 06 DAY 1 12 DE MODULATION Up Conversion via Mixing Downconversion via Mixing Message Recovery via Filtering Synchronized Demodulation of Amplitude Modulation with Suppressed Carrier Unsynchronized Demodulation Sub Nyquist Sampling of RF Interpolation Johnson Introducing Receiver Design Apr May 06 DAY 1 13 Up Conversion via Mixing e For upconversion mixer multiplies input waveform with a sinusoid s t w t cos 27 fot w t message waveform s t transmitted waveform mixer output e We want to compute the Fourier transform of the transmitted waveform s t using o Exponential definition of a cosine Ls 2 cos z ale e 4 o Fourier transform defin
70. om data available on the DSP side of the sampler Ans Yes s and not r is needed e Will this algorithm converge to the desired a of d 4 Y refi Ans It depends what you mean by converge Johnson Introducing Receiver Design Apr May 06 DAY 1 42 AGC cont d e The candidate algorithm ali 1 afi m sign afi d 5 i cannot be expected to converge to a fixed value Because r ranges widely only on average does a r or s actually equal d The resulting typically nonzero instantaneous error in d s and a nonvanishing stepsize u will result in a change in a even if it is already at the right value for the average behavior of s A sufficiently small should keep this asymptotic rattling within a tolerable level Johnson Introducing Receiver Design Apr May 06 DAY 1 43 AGC cont d Testing e Using agcgrad with ave r 1 and d 0 15 the desired a V0 15 0 38 e Start at x 2 with u 0 001 Adaptive gain parameter 2 T T T T T 15 1 0 5 0 l l l l O 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Input r k 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Output s k 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Iterations e Start of x 2 with u 0 001 Adaptive gain parameter 0 T T T 0 5 A 1 5F 2 1 1 1 f 1 1 1 L 0 1000 2000 3000 4000 5000 6000 7000 8
71. ombinations cont d e So choose g t F VP f e For example consider the square root raised cosine SRRC _i_sin nl a t T 4 Uat T cosmid tatr VT xt T 1 4at T for t 0 UF ee val a 4a r fort 0 Jez sin ga 1 c08 a for t which has a magnitude spectrum the square of which equals the magnitude spectrum of a raised cosine e The square root raised cosine is the most commonly used pulse in bandwidth constrained communication systems Johnson Introducing Receiver Design Apr May 06 DAY 3 17 BAUD TIMING FOR CLOCK RECOVERY x A Baud Timing Example x Output Power Maximization Johnson Introducing Receiver Design Apr May 06 DAY 3 18 Baud Timing e Consider the situation where the up and down conversion is done perfectly so we need only consider a baseband model of the communication system A t gr t c t 87 8r c t BRC Sampler Sli il Transmit Receive x t x kT M 7 Channel Filter filter w t e We are to select T in ek x a T OO SO hE iT w grlt hoigy Johnson Introducing Receiver Design Apr May 06 DAY 3 19 Baud Timing cont d Three possible implementation configurations Sampler a Sampler b Sampler HH c We favor the last with its free running sampler and fine tuning of the baud timing done in the receiver DSP Johnson Introducing Receiver Design Apr May 06 D
72. on to t he medium test vector Good luck call transmitter r s Tx m c SNR phase_noise_variance m m m m h call receiver HSS ee ass BSH a en See eae decoded_msg y Rx r call code to calculate BER BERcalc m function r s Tx m c SNR phase_noise_variance function r s Tx m c SNR phase_noise_variance Inputs m text message to be sent 7 c channel T spaced hh SNR signal to noise ratio phase_noise_variance variance of phase noise added to signal Outputs r received signal at IF 4 s transmitted symbols for calculating SER written by A Klein 26 Oct 2005 global srrcLength marker training f_s T_t f_if rolloff dataLength h determine suitable oversampling downsampling factor M N rat f_s T_t M_scale ceil 2 T_t f_ift i rolloff M M M M_scale N N M_scale h get dimensions lines_of_text size m 1 frame_length dataLength length marker training Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Rx O oNnNoAORUOUNBEe jr e char_str_length frame_length lines_of_text insert training amp header and generate 4 PAM source vector s reshape repmat marker training lines_of_text 1 reshape let
73. open by the end of the simulation If so at approximately which iteration is the eye open What is the BER Johnson Introducing Receiver Design Apr May 06 DAY 4 LAB 2 e Keep the trained LMS stepsize set to 0 001 and pick your own stepsize for the DD LMS algorithm Test your receiver on the signal and tune the stepsize to your liking Is the eye open by the end of the simulation If so at approximately which iteration is the eye open Is the equalizer able to track What is the BER e Finally set the stepsize of the trained LMS algorithm to zero Test your receiver on the time varying signal again Is it able to track now Is the eye open by the end of the simulation If so at approximately which iteration is the eye open What is the BER Johnson Introducing Receiver Design Apr May 06 FINAL PROJECT 1 Final Project Day 5 Introduction to Digital Communication Receiver Design Description of the project e The transmitter receiver code we have been using in the system_code directory was developed by a former employee at the company where you work As such development of the radio has not progressed for some time Meanwhile your manager has just informed you that the chief competitor to your company has released a radio with superior performance Your manager has made several suggestions in hopes of improving upon the existing sampled IF receiver You will use code and knowledge that you have developed in the lab to modify the e
74. or the system e g sam pling period IF frequency marker sequence etc Subroutine Files These files are from Telecommunication Breakdown e letters2pam m This function converts an ASCII text sequence into 4 PAM sym bols Used by Tx m e pam2letters m This function converts a sequence of 4 PAM symbols into an ASCII text string Used by Rx m e quantalph m This function is effectively a minimum Euclidean distance detector or decision device It accepts soft PAM symbols and quantizes the input to the nearest PAM symbol Used by Rx m e srrc m This function generates the impulse response for the square root raised cosing pulse shape Used by both Tx m and Rx m e interpsinc m The function performs sinc interpolation and is used for the baud timing and downsampling in the receiver Used by Rx m Transmitter Details Tx m This section briefly describes each of the main components of the transmitter and points to their corresponding line numbers in the code The components can also be found in the block diagram in Fig 1 Note that in addition to the transmitter Tx m also includes the channel and receiver frontend blocks Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 3 e Calculate Intermediate Variables lines 19 29 This part calculates intermedi ate variables which are used in the transmitter including the upsampling downsampling ratios and the phase noise random process
75. our digital receiver particular during baud timing The function day1 interpsinc m performs sinc interpolation and we will use this frequently Open this file and familiarize yourself with its operation To see an example of using sinc interpolation consider interpolating the points of a sam pled sinusoid The file day1 interp example m generates a sine wave w t of frequency 20 Hz with a sampling rate of 100 Hz The code then shows how to use interpsinc m to interpolate between the samples Your task Generate a new wave w t which is the sum of 2 sinusoids one with frequency 17 Hz and one with frequency 20 Hz Consider t between 10 and 10 Let w kT represent samples of w t with T 0 01 Use interpsinc m to interpolate the values w 0 011 w 0 013 and w 0 015 using 10x oversampling Compare the interpolated values to the actual values Task 6 Automatic Gain Control via Gradient Descent The function day1 agcgrad m implements the AGC gradient descent algorithm which minimizes the cost 2 2 Jn a avg ll is by choice of a The gain parameter a adjusts automatically to make the overall power of the output s roughly equal to the specified parameter ds Run agcgrad m and you will see that a converges to about 0 38 since 0 38 0 15 ds Your task Using agcgrad m answer the following questions 1 What range of stepsize mu works What happens if it is too small too large 2 How does choice of mu effect conve
76. pe e Compose the analog pulse train entering the pulse shaping filter as walt X gt w kT 6 t kT k which is w kT for t kT and 0 for t 4 kT e Pulse shaping filter output s t walt xpt gt X f Walf P P e X f cannot be nonzero at frequencies where P f is zero Johnson Introducing Receiver Design Apr May 06 DAY 3 Pulse message spectrum cont d One symbol wide Hamming blip pulse shape with 10 samples per symbol and frequency response using freqz in pulsespec Q an iso E n Q 4 A Spectrum of the pulse shape 0 2 0 3 0 4 0 5 0 6 0 7 Sample periods a 0 2 0 3 0 4 0 5 0 6 0 7 0 8 Normalized frequency b Johnson Introducing Receiver Design Apr May 06 DAY 3 6 Pulse message spectrum cont d Spectrally flat 4 PAM symbol sequence triggering baud spaced 10 times oversampled Hamming blip pulse shape as baseband output of pulse shaping filter 3 0 2 39 5 10 15 20 25 Output of pulse shaping filter Symbols Spectrum of the output Normalized frequency Message signal spectrum has scalloped contours of Hamming blip pulse frequency response Johnson Introducing Receiver Design Apr May 06 DAY 3 Eye Diagram Eye diagram is a popular robustness evaluation tool For 4 PAM single baud wide Hamming blip with additive broadband channel noise retriggering oscilloscope after every 2 baud intervals produces O
77. ped by Dr Klein This course packet provides the overheads used in the lectures of days 1 4 the associated lab assignments for days 1 4 the description of the final project and a user s manual for Dr Klein s software radio An accompanying CD includes the pertinent Matlab files designed for compatibility with version 6 for demonstrations cited in lecture Dr Klein s software radio the labs and final project The CD also includes a pdf version of the printed course packet As a bonus a movie is also included of a working receiver built in the late 1980s by Applied Signal Technology for 16 QAM where carrier phase offset results in rotation of the recovered 4 by 4 constellation and carrier frequency offset results in recovered constellation Johnson Introducing Receiver Design Apr May 06 FOREWORD 2 rotation A document which is drawn from the CD accompanying Telecommunication Breakdown and also includes a Matlab simulated radio on the extension of the PAM radio of Telecommunication Breakdown and this compacted course to the more pragmatic QAM radio is also included on the CD for this compacted course along with all of the Matlab based software for its simulation Johnson Introducing Receiver Design Apr May 06 DAY 1 DAY 1 e A NAIVE DIGITAL RADIO e DE MODULATION e AUTOMATIC GAIN CONTROL Johnson Introducing Receiver Design Apr May 06 DAY 1 2 A NAIVE DIGITAL RADIO x An Illustrative Digital Commun
78. period as indicated by the choppy staircase zero order hold reconstruction of the Johnson Introducing Receiver Design Apr May 06 DAY 1 36 AUTOMATIC GAIN CONTROL x Automatic Gain Control Algorithm Construction x Tracking Example Time Varying Fade Johnson Introducing Receiver Design Apr May 06 DAY 1 37 Sampling with AGC We now focus on the sampler and its surrounding automatic gain control AGC in a receiver front end Antenna Analog BPF conversion Analog to IF received signal Quality Assessment Our purpose here is more to introduce a strategy for parameter adaptation that will be repeated for carrier and clock recovery and equalization rather than to promote a particular AGC algorithm Johnson Introducing Receiver Design Apr May 06 DAY 1 38 Automatic Gain Control AGC e An AGC maintains the dynamic range of a zero average signal by attenuating when it is too large as in a ne by amplifying when too small as in iil e AGC adjusts gain parameter a so average energy at output remains roughly fixed despite fluctuations in average received energy Sampler S KT s k gt Quality Assessment Johnson Introducing Receiver Design Apr May 06 DAY 1 39 AGC cont d Gain Tuning e We are to choose a for a received waveform r t segment that produces sampler outputs s k with the intent of having the a
79. ptimum sampling times Sensitivity to timing error Distortion at zero crossings j k 1 T Observe illustrative vertical amplitude and horizontal timing margins for correct decision at sample times Johnson Introducing Receiver Design Apr May 06 DAY 3 8 Eye Diagram cont d Consider 20 symbol wide 10 times oversampled truncated sinc pulse sin at T at T with zero crossings at kT for k 1 2 10 for 4 PAM symbol sequence from spsex Using a sinc pulse shape 2 0 2 4 pulse shaped data sequence T A aa i li 5 10 15 symbol number 5 25 3 baud and 30 sample wide eye diagram symbol times indices 10 20 and 30 A multi baud wide pulse shape but no ISI Johnson Introducing Receiver Design Apr May 06 DAY 3 Nyquist Pulses The impulse response of a Nyquist pulse creating no ISI at other sample times is zero at those instants and nonzero only at the one particular sample time e The impulse response p t is a Nyquist pulse for a T spaced symbol sequence if there exists a T such that p t Fare eee e Rectangular pulse 1 O0 lt t lt T PR 0 otherwise Rectangle is Nyquist pulse for almost any sampler timing Johnson Introducing Receiver Design Apr May 06 DAY 3 10 Nyquist Pulses cont d e Sinc pulse sin 7 fot t ps t fod where f
80. r best performance Lastly you should run system_code main m to see how the receiver performs in comparison to the old OP based receiver You should examine a plot of the timing offset estimate r when answering the following questions 1 Consider a case with no noise and no phase noise but with a channel having taps c 0 7 1 0 4 0 2 To what value of tau does the DM algorithm converge to To what value of tau does the OP algorithm converge to Do both algorithms converge to the same value Include a plot for each 2 Decrease the SNR in 3 dB steps and determine which algorithm first begins to make bit errors Johnson Introducing Receiver Design Apr May 06 DAY 3 LAB 2 3 What effect do you observe as you decrease the stepsize What happens as you increase the stepsize 4 Change the SRRC rolloff factor in system_code globalParams m What is the effect on algorithm performance when you increase or decrease the rolloff factor Johnson Introducing Receiver Design Apr May 06 DAY 4 DAY 4 e LINEAR EQUALIZATION e PUTTING IT ALL TOGETHER RECEIVER DESIGN Johnson Introducing Receiver Design Apr May 06 DAY 4 2 LINEAR EQUALIZATION Multipath and Other Interference Trained Linear Equalization Trained Adaptive Least Mean Square Equalization Blind Adaptive Decision Directed Equalization Blind Adaptive Dispersion Minimizing Equalization Johnson Introducing Receiver Design Apr May 06 DAY 4 3 Multipa
81. rgence rate 3 How does the variance of the input effect the convergent value of a 4 Try initializing the estimate a 1 2 Which minimum does the algorithm find What happens to the data record Johnson Introducing Receiver Design Apr May 06 DAY 2 DAY 2 e RF SYSTEM SIMULATION WITH IMPAIRMENTS e CARRIER RECOVERY Johnson Introducing Receiver Design Apr May 06 DAY 2 AN IDEALIZED RF SYSTEM SIMULATION x A Naive Ideal Communication System x Flat Fading x What if Johnson Introducing Receiver Design Apr May 06 DAY 2 3 A Naive Ideal Communication System With a perfect i e gain with delay channel and satisfactory carrier baud timing and frame synchronization we simulate this PAM system using idsys Message T spaced Baseband Passband character symbol signal signal string sequence l a Transmitter T spaced passband baseband signal signal Lowpass filter Demodulator MT spaced MT spaced soft hard T spaced decisions decisions Recovered baseband 1 character signal Pulse string correlator Quantizer Decoder filter n MT IT n 0 1 2 Downsampler b Receiver Johnson Introducing Receiver Design Apr May 06 DAY 2 4 A System cont d TRANSMITTER e text message 01234 I wish I were an Oscar Meyer wiener 56789 coding text characters via 8 bit ASCII to 4 PAM mili baud interval T 1 time unit pu
82. s one and now the third error free But if you didn t then don t worry yet The only fr ames you are required to decode during the actual testing are thos e past the fifth frame So you re still okay We re getting close to the end of the 5th frame so your receiver better start working Congratulations If you can see this then you re receiver has su ccessfully decoded the fifth frame gt You might want to re test your receiver by using different initial parameters and different stepsiz es to see what the effect is It s probably help to plot the tim e history of the adaptive parameter elements too so you can see if they re taking too long to conve rge if they seem unstable etc A nd now some more Nirvana lyrics With the lights out it s less da ngerous Here we are now Entertain u s I feel stupid and contagious Here we are now Entertain us A mula tto An albino A mosquito My Libido And I forget Just what it takes And yet I guess it makes me smile Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 35 36 37 38 39 40 41 42 43 44 45 46 Tx O ONDA FBRWNE NNNNNNNNBPRPRPBRBP BRB NOOR WNFOWCDANODOOKRWNHO I found it hard Its hard to find Well if your receiver has made it this far with no errors and pe rforms error free even when you ch ange the initial parameter values then it s time to move
83. sion achieved by narrower pulses increases exclusionary baseband spectrum requirement If all frequencies in bandlimited baseband spectrum can be translated by same amount several users could be multiplexed to different center frequencies without overlap Johnson Introducing Receiver Design Apr May 06 DAY 1 Radio Frequency RF Communication System e RF transmitter Baseband Passband Symbols Pulse signal signal shape filter Frequency translator e RF receiver Received Baseband Reconstructed signal signal ges Frequency Sampler Quantizer Decoder translator Johnson Introducing Receiver Design Apr May 06 DAY 1 9 Practical Obstacles e precise frequency translation required in receiver precise timing required in receiver multi user interference occurs in received signal e g since each user is not strictly bandlimited in frequency noise contamination of transmitted signal in band out of band narrowband or broadband channel distortion fading or multipath possibly time varying Interference from other sources Transmitted Received signal signal Self interference Multipath Johnson Introducing Receiver Design Apr May 06 DAY 1 10 Analog Digital Signal Processing Split e Due to cost and flexibility benefits modern radio design is pushing the sampler and subsequent digital signal processing closer to the received signal i e the output
84. son Introducing Receiver Design Apr May 06 DAY 1 Received Signal and Receiver e In the ideal case the received signal is the same as the transmitted signal though attenuated in magnitude and delayed in time r t 74 64 27T 7 6 4 3T 7 6 4T Reconstructed Reconstructed symbols text Quantizer Decoder n kT Johnson Introducing Receiver Design Apr May 06 DAY 1 6 Synchronization Issues e Baud symbol timing 7 selection for fixed T top dead center g T 04 T7 2 Peaked rather than rectangular pulse shapes will reduce the spectral footprint of the sequence of pulses but increase the sensitivity to top dead center baud timing e Frame start determination o grouping symbols to decoder o example 1 1 1 3 1 first 4 symbols decode to X and last four decode to a o special marker sequence inserted in source sequence at start of a frame with subsequent frame starts determined by knowledge of the the period of their recurrence Johnson Introducing Receiver Design Apr May 06 DAY 1 Spectrum Sharing Several user pairs should be able to communicate through same medium simultaneously in same geographical region Interference avoidance achieved by disallowing use of same frequencies by different users in same geographical area Bandwidth occupied by pulse shape sequence is inversely related to rectangle width More frequent symbol transmis
85. stas loop and compare the performance of the two schemes Recall from the lecture notes the update equation for the Costas loop has the form 6 k 1 O k u LPF 2r kT cos 2m fokT O k LPF 2r kT sin 2n fokT O k Since the existing receiver code has a PLL it is useful to compare and contrast the two algorithms in terms of their implementation While the PLL requires a pre processing step the Costas loop does not require pre processing The Costas loop makes use of a low pass filter which is not present in the PLL and you will need to use remez to design this filter The schematic for the Costas loop on the next to last page of the lecture notes for DAY 2 may be helpful as well In testing your Costas loop implementation you should start by using the most benign conditions no noise no channel no phase noise Once your implementation is working you should gradually add more realistic channel impairments Additionally you will need to tune the Costas loop parameters stepsize filter parameters etc for best performance In the following exercises you should check how the receiver performs in comparison to the old PLL based receiver You should always include a plot of the phase estimate 0 1 Modify the channel decrease the SNR or increase the phase noise variance In general do you find one receiver to be more robust 2 Is one receiver better at tracking the variation due to phase noise Explain this by r
86. t ers2pam reshape m lines_of_text dataLength 4 1 dataLength lines_ of _text lines_of_text dataLength length marker training 1 generate pulse shaped signal x conv srrc srrcLength rolloff M 0 upsample s M mix signal to RF analog upconversion p_noise cumsum randn size x sqrt phase_noise_variance N gene rate phase noise process x_rf x cos 2 pi f_if 1 length x T_t M p_noise pass through BP channel x2 conv x_rf upsample c M add channel noise of appropriate SNR x2_size size x2 x2_nrm sqrt x2 srrcLength M 1 x2_size 1 srrcLength M x2 srrcLeng th M 1 x2_size 1 srrcLength M x2_size 1 X_r x2t randn size x2 10 SNR 20 x2_nrm perform analog conversion to IF and do AGC r x_r N N end r_nrm r r length r r r sqrt r_nrm add some zeros to front and back r zeros floor rand 10 M 1 r zeros 10 M 1 m function decoded_msg eqOut Rx r function decoded_msg eqOut Rx r Inputs r received signal at IF Outputs decoded_msg received signal at IF eq0ut output of equalizer Johnson Introducing Receiver Design Apr May 05 KLEIN S RADIO 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 written by A Klein 26 Oct 2005 global srrcLength marker training f_s T_t f_if rolloff dataLen
87. th and Other Interference e Assume up and down conversion and carrier and clock recovery including matched filtering and downsampling all executed transparently Impairment of interest is multipath interference linear filtering by analog channel and receiver front end preceding equalizer and other additive interference broadband noise and narrowband interferers Noise and interferers Received Digital analog source Pulse Analog signal shaping channel Received Sampled analog received signal signal Linear _ lt digital equalizer Decision device Johnson Introducing Receiver Design Apr May 06 DAY 4 4 Multipath Interference cont d e FIR channel model y kT ayu kT agu k 1 T tanu k n T kT where n kT is sample of other interference Order n of discrete time FIR channel model dependent on physical delay spread of channel For 4 usec delay spread by physical channel T 0 04 psec 25 Msymbols sec n 100 T 0 4 psec 2 5 Msymbols sec gt n 10 T 4 psec 0 25 Msymbols sec gt n 1 Johnson Introducing Receiver Design Apr May 06 DAY 4 Multipath Interference cont d e Multipath FIR model coefficients depend on actual baud timing choice of clock recovery algorithm which need not match timing in non ISI situation e Example Two ray analog channel c t p t p t A with A 0 7T
88. verage s value over that dataset match a preselected constant d Because s k ar kT we can choose d d N SA elk il ave r k preferring a gt 0 to make as desired lt i k i a a Unfortunately we need the samples of r which are not available on the DSP side of the receiver to solve this formula for a e Our search for a gain tuner continues Johnson Introducing Receiver Design Apr May 06 DAY 1 40 AGC cont d Heuristic Algorithm Development As an alternative consider the following strategy e select an initial positive a e As a sample s arrives compare its square to d If s at that particular sample instant is greater than d we will reduce a positive a to a smaller positive value If a is negative we would decrease its magnitude i e increase it toward zero Plus the correction term should be larger the further d is from s Similarly if s lt d we will increase a positive a by an amount proportional to d s If a is negative a should be decreased i e made more negative so its magnitude increases Johnson Introducing Receiver Design Apr May 06 DAY 1 Al AGC cont d An algorithm that performs this strategy is ali 1 ali sign a i d s li where u is a suitably small positive stepsize The sign a z term can be removed if afi starts and stays positive e Can this algorithm be implemented fr
89. vice ia Performance s k algorithm training evaluation signal the trained approximate gradient descent adaptation algorithm LMS for the linear equalizer filk 1 falk w slk ylk rik il e Should be engaged only during processing of portion of received signal due to training segment e g using marker correlation Johnson Introducing Receiver Design Apr May 06 DAY 4 9 Blind Adaptive Decision Directed Equalization We choose to minimize avg Q 3 fir k ko N 1 N du UD sir k ko using a gradient cued scheme file 1 lk ag aval QC Johnson Introducing Receiver Design Apr May 06 DAY 4 10 Blind Adaptive Decision Directed Equalization cont d Commute average and partial derivative drop outer average and presume QD firk j Afi 0 to produce filk 1 filk 27 Q Stat sO D firik jl a Il Johnson Introducing Receiver Design Apr May 06 DAY 4 11 Blind Equalization cont d With the definition of vik D_ Filklr k j Sampled received signal k l Decision Equalizer deyi evice Adaptive Performance algorithm evaluation the decision directed approximate gradient descent adaptation algorithm for the linear FIR equalizer is filk filk QUA ylk rik il e Relative to trained adaptation via LMS the decision device output just replaces the training signal Johnson Intr
90. xisting MATLAB simulation to build a receiver that performs better than the competition e The original radio was based on a draft ETSI standard that had not yet been ratified A standards battle ensued and when the standard was finally ratified some of the parameters of the radio changed While the radio is still a 4 PAM radio thanks to the strength of your marketing department these are the parameters that have changed parameter value assigned intermediate frequency 2 2 MHz nominal symbol period 5 microseconds NYUCKiitick LarryCurlyMoe 472 0 25 1 MHz frame marker training sequence training sequence recurrence period SRRC pulse shape rolloff factor sampler frequency These parameters can be found in the file day5 globalParams_project m e To improve the performance of your receiver your manager asks you to make the following changes to Rx m which were already completed in the lab 1 Change the carrier recovery scheme It is currently implemented using a PLL and you are asked to change it to a Costas loop refer to lab from Day 2 2 Add a decision directed equalizer Currently the equalizer is only adapted dur ing the training periods Your manager believes your company will have a com petitive advantage if you are able to make the radio work well in environments where the channel is time varying refer to lab from Day 4 e Sending large amounts of training data reduces the effective throughput of the radi

DIGITAL COMMUNICATION RECEIVER DESIGN C. RICHARD

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