Unraveling Convolution Achievements in this
Contents
1. The math for the theoretical verification is done by means of the application of the formula for reducing the sum of sinusoids In the next Part we investigate an example of effects that can be obtained by means of suitably chosen values for the tap gains Part 6 mystery application Here we explore a special application of convolution The setup is the same as for Part 5 except for new tap gains in the delay line Two sets will be tested The first is bo 0 3 b 0 424 b2 0 3 The second set is bo 0 3 b 0 424 b 0 3 27 When the tap gains have been adjusted to the first set of values bo 0 3 b 0 424 bz 0 3 display the output and measure the amplitude and phase relative to the input as in Part 5 Confirm that this is in agreement with theory 28 Alter the tap values to the second set of values bo 0 3 b 0 424 bz 0 3 and repeat the measurements Is the outcome predicted by theory 29 Vary the frequency of the sinewave over a suitable range to demonstrate that sinewave inputs with frequencies near 100 Hz are heavily attenuated Question 12 Why is the outcome obtained above described as filtering In other labs you will discover how to choose the tap gains to design different kinds of responses versus frequency Tutorial questions QI Look up your textbook to find out the name given to systems that have the structure in Figure 1 i e without feedback Consider an alternative2. y n h 0 x n h 1 x n 1 h 2 x n 2 i e y n sum over range of k h k x n k This simple formula is known as convolution For the discrete signal case we have implemented it is also known as the convolution sum or the superposition sum It can also be expressed in this more compact form y n 2 aCe af k y n x n h n where represents the convolution operator It provides a neat way of packaging the procedures you carried out in your investigation and it allows the range of k to be extended to any desired value It is the mathematical representation of the interaction between the input sequence and the unit pulse response of the system Note in particular the term x n k Since we are computing in terms of k this is a time reversed version of x k This issue is put into context further ina later experiment introducing correlation an operation which has a similar looking equation This subject is covered in detail in the text by Oppenheimer p90 see Reference section for details Question 8 Explain why this term is reversed and what does this mean For the continuous signal case the summation is achieved using integration and is known as the convolution integral or the superposition integral This topic is covered in detail in many texts including Lathi and Oppenheimer see Reference section for details In brief the convolution integral for continuous signals comes about by increasing the
3. discrete time system that has a single delay element with feedback refer to your textbook if needed Show how to apply the convolution formula in this case Q2 Consider the condition s required to avoid infinite outout values in applications where the unit pulse response is not time limited Show to avoid an unstable output in the feedback system in Q1 Q3 Consider a process that consists of taking a running average of a data sequence such as atmospheric pressure Suppose we do this by taking the sum of 50 of the middle value and 25 of the preceding and following values Can this process be described as convolution If so write down the unit pulse response Q4 Using integration instead of discrete summation write down a continuous time version of the convolution formula in terms of the system impulse response
4. it to fully boot up so that it s ready to connect to external USB devices 5 Turn on the NI ELVIS unit but not the Prototyping Board switch yet You should observe the USB light turn on top right corner of ELVIS unit The PC may make a sound to indicate that the ELVIS unit has been detected if the speakers are activated 6 Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board Check that all three power LEDs are on If not call the instructor for assistance 7 Launch the SIGEx Main VI 8 When you re asked to select a device number enter the number that corresponds with the NI ELVIS that you re using 9 You re now ready to work with the NI ELVIS SIGEx bundle 10 Select the EXPT 5 tab on the SIGEx SFP Note To stop the SIGEx VI when you ve finished the experiment it s preferable to use the STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the top of the window This will allow the program to conduct an orderly shutdown and close the various DAQmx channels it has opened Experiment Part 1 Setting up 11 Patch up the model as shown in Figure 2 Settings are as follows PULSE GENERATOR FREQUENCY 1000Hz DUTY CYCLE 0 5 50 SEQUENCE GENERATOR DIPS UP UP SCOPE Timebase 10ms Rising edge trigger on CHO Trigger level 1V SEQUENCE GENERATOR DIGITAL OUT PULSE GEN D OUT D OUT L S H CLK Figure 2 patching diagram of the system in Figure 1 12
5. number of discrete samples whilst reducing their width to a limit of O such that the sum of many products can be represented by the integration of the product Its equation is y t J x z a t 2 dt In the above exercise we obtained the result by means of a superposition of dissected parts of the unit pulse response In Part 3 we also used superposition however it did not involve breaking up the unit pulse response This method was easier to apply for that case because the input sequence had only two pulses Compare the two methods when used with the input in Part 4 do this with the help of sketch graphs as before Consider possible advantages of one over the other as a vehicle for deriving the convolution formula 21 Repeat this process with a new system response Let each tap be equal to 0 333 ie bO b1 b2 0 333 Question 9 What is a common label for this response Part 5D sinewave input Up to this point we have used relatively short input sequences This made it easy to trace the passage of the pulses through the system since output segments remained distinct and could be readily referenced to the corresponding input segment However we need to consider whether the formula that we obtained on this basis is also valid in the more general and usual situation when the input signal is an ongoing stream without breaks For this purpose we will go over the steps in Part 4 using a sinewave as input This means there is no nat
6. process mathematically An obvious way to start is to write a separate formula for the six nonzero output pulse amplitudes Let us name them y 1 y 2 y 6 19 Label them on your sketch above Pay attention to the orientation you use Each consists of a sum of three products i e of a tap gain bo b or bz and an input pulse amplitude The input sequence has eight elements as previously observed four of these are zero Label these x 1 x 2 x 8 you should find it convenient to choose x 1 x 2 x 7 x 8 0 20 Label these on your sketch above Numeric indexing is useful as an aid in looking for the general pattern The next step is to use symbolic indexing so that the set of formulas can be condensed into one Here are some of the formulas as they appear with numeric indexing y 3 bo x 3 b x 2 b2 x 1 y 4 bo x 4 by x 3 bz x 2 y 5 bo x 5 by x 4 b2 x 3 y 6 bo x 6 b x 5 bz x 4 Question 7 Write down the formula for y 2 and y 1 Discuss any unexpected differences The general pattern is readily apparent With symbolic indexing we can replace these with the single formula y n bo x n b x n 1 b2 x n 2 where n represents the position in the sequence as a discrete time index In Part 2 we found that the unit pulse response h k for this system is h 0 bo h 1 b i h 2 b Hence we can express the formula for y n as
7. Experiment 4 Unraveling Convolution Achievements in this experiment Carry out a step by step dissection of the convolution process ina discrete time system Use this to discover the convolution formula Demonstrate that convolution can be visualized as a running average of successive values of the input signal Observe a special property applying to sinewaves Demonstrate the operation of a filter in the time domain Preliminary discussion For many students the first encounter with convolution is an abstract mathematical formula in a textbook This lab offers a more evocative experience By tracing the passage of some basic signals through a simple linear system you will be able to observe the underlying process in action and with a little arithmetic discover a formula as it emerges from the hardware Refresh your basic trigonometry you will need sin w t A cos w t gt expressed as a single sinusoid If any of the modules is unfamiliar spend a little time with the SIGEx User Manual This will give you a headstart in setting up the lab We patch up a delay line with two unit delays and three taps with independently adjustable gains as in Figure 1 In the first exercise you will set these gains to given values and observe the output when the input is a single pulse more exactly a periodic sequence of single pulses This is an important preliminary as it introduces the unit pulse response Figure 1a general block diagram
8. GEN D OUT 1 souto J Figure 4 SIGEx patching model for rectified sinewave input 15 Maintaining the same values for the b gains as in Part 1 i e bo 0 3 b 0 5 and b 0 2 display the input i e SAMPLE HOLD output as shown in Figure 5a and output signals Sketch the original half rectified sinusoid and discrete output from the SAMPLE HOLD in the graph below Graph 2 inputs and sampled outputs Confirm that there are 8 samples of each half wave sine input This is expected as the sampling clock is 800Hz and the input sinewave is 100 Hz Note there are four nonzero pulses in the delay line input sequence and six in the summer output The reason for this will emerge as we proceed Again we will carry out a deconstruction of the output signal in terms of the time offset contributions i e we will trace the output pulse amplitudes in terms of the input values We could use the same method as in Part 3 however there is an interesting alternative We will separately observe and compare the three individual contributions into the output ADDER i e the signals added through gains bo b and bz in turn Examples of these signals are shown in Figure 5b below Ik o piuriya gp EERE 4 wal majad Ang et eb PEAS BF J cianasiasndhansshenntinsssienesicesticnnsisessienssiseenicnseisiashssssisemiecsel Mi00us A Chi s 1 00V il 1 00V 8ich2 Figure 5a example input signal and 4 level sampled si
9. The required signal appears at the SEQUENCE GENERATOR SYNC output as a 5V signal and needs to be reduced in amplitude using the available aO GAIN function Using the scope check that you have a periodic sequence of a single 1V pulse ina frame of 31 pulse periods Confirm the pulse width is ims Adjust the aO gain value to have a pulse amplitude of 1V precisely Adjust the SCOPE trig level to suit Note that as the incoming pulse is being clocked by the same clock as the unit delay blocks the pulse is already aligned with the unit delay clock and is thus already a discrete signal For this reason it can be input directly into the unit delay without use of the S H block Part 2 unit pulse response Before proceeding with the examination of the system response the delay line tap gains must be set For the first case we shall use b 0 3 b 0 5 and bz 0 2 see Figure 1 These settings have been chosen arbitrarily as interesting and varied values for this exercise Adjust each gain in turn on the SFP and use the scope to confirm your settings Questionl Describe a procedure for confirming the GAIN at each tap Question 2 Display the delay line input signal i e at the first z block input and the ADDER output signal Measure and record the amplitude of each pulse in the output sequence Note that the system output is a sequence of three contiguous pulses with amplitudes in the same ratio as the adder input gains Could th
10. gnal Figure 5b example individual ba b amp bz inputs to adding junction 16 We begin with the contribution through bz Temporarily disconnect the leads corresponding to inputs bo and b Observe and record the input and output signals in the graph above Confirm for yourself that this result is as expected View the sampled input on one scope channel and the individual output on the other scope channel 17 Now repeat for the b and bo contributions Only the outputs need be recorded since the same input is used Again verify that the results are as expected When completed you will have three scaled replicas of the input with time offsets 18 For each time slot sum the contributions of the three output records and plot the result Verify that this agrees with the output signal produced when the three leads are reconnected to the adder inputs Question 6 How does this process relate to the principle of superposition In summary we have just passed a sequence of discrete values our sampled input through a system with a particular response a series of unit delays with multiplying taps to produce an accumulation the adder of discrete product terms which are then output Each one of these individual contributions is the scaled and shifted version of the unit pulse response of this system Another way of thinking about this system as as it being a series of weighting coefficients Now we need to think about representing this
11. is have been predicted from Figure 1 Indeed since the single pulse at the input is generating delayed and scaled replicas as it travels down the delay line These are then summed in the adder Thus we have the system s response to an isolated pulse From this we can define the unit pulse response h n as the response when the amplitude of the input pulse is unity From your measurements show that the unit pulse h O bo h 1 bi h 2 be in this example The presence of delayed energy is normally expected in real life systems whether electrical or mechanical For example due to inertia in mechanical systems Similarly in electric circuits we have energy storage effects in reactive components such as capacitors and inductors Part 3 The superposition sum 13 Adjust the SEQUENCE GENERATOR DIP switches to position UP DOWN to select the sequence of two contiguous pulses Using the same gain settings as in Part 2 observe the output signal Note that it consists of four nonzero pulses per frame Measure and record the amplitude of each pulse 14 Verify that the output sequence is simply the sum of two offset unit pulse responses Use the graph below to show your computation Graph 1 unit pulse pair summation Question 3 What is meant by superposition Discuss how this exercise above relates to superposition and the additivity principle Question 4 What do you expect to see if this exercise were expanded to tw
12. o or more contiguous pulses Explain Part 4 rectified sinewave at input In this exercise the input is a little more interesting than in Part 3 a sequence of three or four pulses of different amplitude We obtain the source of this signal from the ANALOG OUT DAC O We then pass this analog signal to the SAMPLE HOLD block to be sampled and this becomes our discrete sequence of pulses Note that the PULSE GENERATOR and DAC signal generator share the same internal clock and hence no slippage occurs in the scope displays Note that although the signal is sampled and becomes discrete it has not become a digital signal This is an important distinction Rather it now exists as sequential discrete samples of the original signal More about sampling and its implications in several later experiments Settings are as follows PULSE GENERATOR FREQUENCY 800Hz DUTY CYCLE 0 5 50 SCOPE Timebase 10ms Rising edge trigger on CHO Trigger level 1V Confirm that the sinewave from the DAC O is 100Hz 2V peak before entering the RECTIFIER We will treat the sinewave as a continuous signal and ignore the very small steps present as these have no consequence to our procedure Question 5 Note the amplitude of the half wave rectified sine and explain why its amplitude is reduced relative to the input OUTPUT Figure 3 Block diagram of system with series of pulses as input ANALOG OUT DAC 1 RECTIFIER DAC 0 DIGITAL OUT PULSE
13. of the system Next you will observe the output when a pair of adjacent pulses is used as input This near trivial example provides us with a springboard to the general case It will demonstrate how convolution operates as an overlapping and superposition of unit pulse responses A second more general input sequence is then used to reveal a deeper insight and to provide a vehicle for setting up the key formulas In the remainder of the lab we use sinewaves as inputs This takes us to the rediscovery of the special role of sinusoids in linear time invariant systems In the final exercise we carry out and unravel a disappearing act It should take you about 40 minutes to complete this experiment Pre requisites Familiarization with the SIGEx conventions and general module usage A brief review of the operation of the S H amp UNIT DELAY module from SIGEx User Manual No theory required Equipment PC with LabVIEW Runtime Engine software appropriate for the version being used NI ELVIS 2 or 2 and USB cable to suit EMONA SIGEx Signal amp Systems add on board Assorted patch leads Two BNC 2mm leads Procedure Part A Setting up the NI ELVIS SIGEx bundle if Turn off the NI ELVIS unit and its Prototyping Board switch 2 Plug the SIGEx board into the NI ELVIS unit Note This may already have been done for you 3 Connect the NI ELVIS to the PC using the USB cable 4 Turn on the PC if not on already and wait for
14. ser look at the output signal obtained in Part 5 and show that it is a sampled sinewave This is a discrete time example of the investigation in Lab 4 where we discovered that when the input of a linear system is sinusoidal the output will also be sinusoidal 25 Confirm that the eight pulses making up one period of the output sequence represent samples of a sinewave A straightforward method is to exploit the sum of squares identity Since there are eight samples per period you can match pairs of samples that are 90 degrees apart how many pairs can you find Note that knowledge of the peak amplitude is not essential for this all that is needed is to show that the sum of the squares is the same for each pair Question 11 Show your working for the sum of squares analysis 26 Use the FUNCTION GENERATOR tuned to 100 Hz to provide the alternative unsynchronized sinewave input Set it to an amplitude of 2Vpeak 4V pp The resulting slippage effectively produces an interpolation of the samples a useful exploitation of something that is usually unwanted Now that we have a simple way of displaying the input and output sinewaves an interesting additional item to examine is the theoretical verification of the measured output input amplitude ratio and phase shift The peak values are clearly apparent so the amplitude ratio measurement is straightforward Similarly the now discernible zero crossings can be used for the phase shift measurement
15. ural reference marker hence you will need to designate one Nevertheless keeping track of time points will be straightforward since the signal is periodic and the number of samples per period is relatively small 22 We use the setup in Part 4 and re use full sinewave output at DAC O by bypassing the RECTIFIER As before since the sinewave frequency is a submultiple of the PULSE GENERATOR clock no slippage occurs in the scope displays 23 Proceed as in Part 4 connecting only one of the three adder inputs As in Part 4 observe and record each of the separate output sequences for each input in turn As you would expect these are scaled and delayed replicas of the input i e they are sampled sinewaves Carry out spot checks to verify that amplitude and phase relative to the input are correct Reminder since there are eight samples per period the phase shift corresponding to a unit delay is 45 degrees 24 For each time slot add up the contributions of the three output records obtained above and plot the result Verify that this agrees with the output signal produced when all the input leads are reconnected to the adder Graph 3 sinewave input signals Now we are ready to revisit the formula obtained in Part 4 Proceed as before and show that the formula remains valid for this case Question 10 Show that the formula remains valid This completes the basics of the convolution formula In the next task we take a clo
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