The Noise Lab
Contents
1. Do the same experiment using the Square and Triangle waves Do the har monics decay like the power spectra shown earlier in the lab 4 3 2 Exercise Some Discrete Effects The goal of this section is to figure out what effect digitization has on our ability to resolve closely spaced frequencies We can get this sort of input by using two function generators wired in series remember to float the second function generator Set the frequency difference to 100Hz and look at the resulting spectrum Turn off averaging but turn on the Hamming window Vary the 4We will look at the effect of windowing a little later For now we just want to concentrate on the harmonics 34 bandwidth and frequency difference and comment on what happens In general what requirements do we have to satisfy if we want to be able to resolve closely spaced frequencies 4 3 3 Exercise Commensurate and Incommensurate Fre quencies Turn off windowing and averaging for the moment As always the vertical axis of the power spectrum should be logarithmic Setup a signal generator on the input of the SA to generate a sine wave at 2kHz Carefully adjust the frequency nob of the signal generator to try and get the broad background to collapse into as small of a peak as possible What happens to the spectrum as you watch it Can you get the spectrum to remain very sharply peaked without adjustment 4 3 4 Exercise Exploring the Transfer Function of the SA Earl2. 2 2 2 Moments of a Random Variable The expectation value of a random variable X with a continuous set of output states is defined as 00 X X E X i x px x dx 2 19 oco There are a number of different conventions used to denote the expectation value These generally vary by field We can define the nt moment of the random variable as 00 X E X x px a da 2 20 Co and the nt central moment of the distribution as FOS X X E X x J x X px x da 2 21 oo Some of the moments have common names The 1 moment is the mean value of a function The second central moment is the variance of a function Many of the moments may vanish for symmetry or other reasons In the case of Gaussian random variables every central moment beyond the second vanishes a property that characterizes gaussian random variables Exercise Prove that the variance X X can be rewritten as X XY using the definition of the expectation value The expectation values can also be defined for multiple random variables The joint moments are defined as 00 xr ym f a y pxv v y dedy 2 22 Q0 and the joint central moment is defined as 00 X XY OS J x X y Y pxy a y dx dy 2 23 O 13 We have particular interest in the second joint moment Rxy XY and second central joint moment Cxy X X Y
3. Example Adding Random Variables Let W be a random variable defined as W X Y 2 15 where X and Y are independent random variables with arbitrary PDF Again we start with the definition of the distribution function w y 00 Fw PX Y lt f f pxv ey drdy 216 00 The instantaneous value of W is w x y Using the fact that X and Y are independent we can rewrite the distribution function as 00 w y Fy to J B px e de dy 2 17 Using Leibniz s rule we find the PDF 00 w f wow a 2 18 is the convolution of the density functions Exercise Find the variance of a random process H K L where K and L are gaussian random variables K L 0 K og and L op 8 Unit step function is the name commonly used in engineering literature In physics it is usually refered to as the heavyside function O a 12 Having the tools to understand multiple random variables is important in understanding among other things the central limit theorem This is an impor tant component in the Johnson Noise derivation and in many other phenomena The central limit theorem states that the sum of N independent random vari ables approaches a Gaussian probability distribution in the limit as N oo Exercise Go through the proof of the central limit theorem and explain why it works and why N oo makes this gaussian Going through the next section will be helpful in following the derivation
4. For now lets shoot for a 1 statistical error in determining kp When possible we will choose the contribution of all other noise sources to be less than this I will spoil the surprise The statistical error will not be our dominant prob lem However it is well worth reducing this contribution as much as possible so it is easier to see other systematic forms of error Statistical Errors If we want to measure ky to 1 how accurately do we have to measure Vpgp 2 How many times do we have to measure average Vpsp 2 What if we set the Marker Size to Wide so we include more points in the average How much time will a single measurement take at 1 accuracy with the different acquisition times corresponding to different bandwidths Range of Resistive Loads The resistors that we have available for measurement are 2M 1M 500k 200k 100k 50k 20k and 10k Of course these resistances are in parallel with the 1M input impedance We can also put combinations of these resistances in parallel to extend fill in the range of resistances Considerations due to the LPF The LPF formed by the input of the SA has the largest effect when Ry is high In the case where the input is left unconnected and we are measuring the thermal noise due to the input impedance alone the cutoff frequency is on the order of 10kHz This corresponds to the noise power dropping off by 3dB or the voltage dropping to 1 2 In principle the behavior of the L
5. gt f lt 2squarepulses gt Do this again graphically This is done explicitly in The Fast Fourier Transform and Its Applications by Brigham in Section 4 This book does an excellent job explaining graphical convolution and I will defer to this text Exercise Find the FT of the sine wave using the shifting property and the FT of the cosine wave Exercise Compute the FT of the square wave using the square pulse the comb function and the FT properties Exercise Note that each spectrum contains both positive and negative frequencies What does it mean to have negative frequen cies The FT of the sine and cosine waves may be useful in answering the question Exercise What is the relationship between cn in the FS and the FT of a square wave As a general note notice that a narrow function in time corresponds to a wide function in frequency and vice versa Also to make a function that is not periodic we have a continuous spectrum not a set of discrete delta functions Finally we define the normalized Power Spectrum PS Self FIGAP 3 5 where T is the time over the total measurement of g t As the name implies relates to the power of the signal S f is always real and the magnitude is normalized with respect to time 3 1 1 Autocorrelation Function in the Frequency Domain The autocorrelation function gives us a nice physical interpretation of the tem poral aspects of a random signal To translate this view into
6. h t glat gt H f mets Time Shifting h t g t to A f e7 G f Convolution h t f t g t gt A f FPG Multiplication h t f t g t H f F f G f Differentiation PRY o Qnif F f Integration Exercise Prove these properties At some point in your physics career you have undoubtedly computed the FT of several basic functions Here are the results from FT ing several common functions Note that c gt 0 and is real Time Domain Frequency Domain elt 1 R mr FD o Ff Cosine Wave 5 f 6 f Square Pulse Ta galek Ta sinc 2 Pic of pulse cae Asine 2 Ta fal Triangular Pulse Pic of pulse e cltl Pic of exponential Pic of spectrum 1 c i2nf Pic of Lorentzian 2 e7ct Pa Se c Generally we are interested in systems responding to periodic inputs not a single pulse It turns out that periodic functions while they violate the square integrability condition of FTs do have transforms Their transforms are distributions as illustrated by the cosine wave and exist as long as the f h t dt over one period is finite This special case of the Fourier Transform is the familiar Fourier Series FS Our definition is 21 00 gt XO er 3 3 n oo 1 7 2 p Cn g t eo i2ant T 3 4 T 7 2 Exercise Use the properties of the FT to compute the FT of a tri angle pulse starting with lt trianglepulse
7. During this stage 1 f type noise is added to the signal to aproximate the additions from the Mixer and A D converter The digital electronics take over and compute the Power Spectrum The consequences of the mixer and the options in computing the PS will be discussed later The maximum bandwidth of the SR760 is set by the sampling rate Sampling is done at 256kHz which gives a Nyquist frequency of 128kHz However we never see any frequencies above 100kHz The reason A digital filter cuts everything above this frequency Another interesting thing about this SA is the way it reduces the bandwidth The number of samples used to compute the power spectrum is constant 1024 21 samples per FT and the sampling rate is always fixed at 256kHz 1 T The way it reduces the bandwidth is by throwing away some of the samples to effectively decrease the sampling rate Exercise The spectrum analyzer implements the Fast Fourier Trans form FFT algorithm instead of a brute force version of the Discrete Fourier Transform algorithm Given that the SA takes 1024 samples and then computes the spectrum roughly what factor of speedup do achieve by using the FFT algorithm lInside the Spectrum Analyzer 2Basically they choose not to display these frequencies I suspect it has to do with the anti aliasing filter 31 a k M N Attenuator Aree Adjustable And Filter m iia pe Mier j ha p Fi 5 Sa e a 1
8. SA is turned on Do not directly attach to the input of the SA as this quickly fatigues the connector It is non trivial and expensive to fix 3Sometimes manufacture s datasheets just distort the tests to make their device seem much better i e Lie 32 4 2 2 Frequency Menu Here we set the span the bandwidth This sets the linewidth and the acquisi tion time Play with this a little to see how the system reacts Try pressing the Auto Range button This detects how much attenuation is needed so that the preamplifier is not saturated Also try the Auto Scale button 4 2 3 Input Menu We will have Input Source set to A and Grounding set to Ground This prevents the input from floating since we have no other external ground in our system which seems to add excess noise to the signal Also set the Coupling to DC For now let Input Range be set automatically When we want to measure the noise we will explore the effect further Finally set the Auto Offset to On What does this do 4 2 4 Measure Menu The Display menu allows us to change between a variety of plotting styles Log Mag allows us to view signals that span a wide range of intensities You should be using this option The measure option has a sub menu The Spectrum option measures a raw unnormalized spectrum The PSD measures the power spectral density by taking the raw
9. Y Rxy is commonly referred to as the correlation of X and Y Cxy is referred to as the covariance Exercise Prove that X and Y are not necessarily independent ran dom variables if Rxy 0 Also prove that Cxy 0 if X and Y are independent random variables At this point we have developed all of the theory need to work with random variables We now construct a definition of random processes using these ideas 2 3 Random Processes Lets define our random process to be X t s where t is the time we observe the system and the variable s is a reference to the particular configuration in the ensemble of possible systems Figure 2 6 shows several possible configurations in this ensemble If we instead examine the process for fixed t as shown in figure 2 7 we simply have a random variable X s We can think of a random process as a set of different random variables indexed by the time of the system Since we understand quantitative relation between random variables we have the tools to fully explain our qualitative ideas of correlations When discussing correlations in random processes we are usually talking about the autocorrelation function The autocorrelation function is defined as Rx x ti t2 X t1 X te 2 24 Exercise Why am I able to omit s from the definition of the autocorrelation function The physics of the problems can further simplify this equation First we demand that X t is ergodic the average valu
10. a Noise Source This is a really fun idea Try building a noise source and characterizing it with the SA It can be an analog or digital design Try to see how white you can make the source over some bandwidth For example with a digital source you will probably want to make the SA trigger on new values from the source If you elect to create a digital noise source remember that D A converters have inherent non linearities that eventually show up in the PS Try making a calibration routine that accounts for these non linearities to create a perfect source Have fun 7Or feel free to make that an option you can change with some external circuitry or minor program modifications 8Look at the SA manual to figure this out 39 4 7 Acknowledgments I would like to thank all of the people who have helped make this lab possi ble Bob Pizzi Daniel Bridges David Stuart Phil Lubin Beth Gwinn Mark Sherwin and David Cannell The list goes on with friends and guinnea pigs Jason Seifter Daniel Sank Amanda Fournier Chris Kohler Ian Meyer Nile Fairfield John Billings Ben jamin White and Kyle Bebak 4 8 References The Fast Fourier Transform and Its Application by E Oran Brigham This is an excellent and probably necessary reference for things related to FTs It is written with graphical techniques in mind and explains everything in an unusually clear elegant manner A copy should be in the lab room The Art of Electronic
11. at the receiving system The envelope again is m t However there is significant high frequency signal as well at the receiver end modulated by the carrier frequency again with the result shown in Figure 3 4 We succeed at getting some of the signal back around zero frequency but we have to add a LPF as to kill the copy at at 2fe and we recover the original signal m t at the receiver Exercise I have shown the steps in the time domain Sketch the corresponding graphs in the frequency domain Exercise What happens if the receiver end uses a carrier fre quency that is slightly off Show this graphically Exercise With a cosine wave as the input of the mixer we have assumed a perfect oscillator Oscillators generally have phase noise associated with them Thus the spectrum is not a delta function but some Lorentzian like function centered around fe Assuming the message signal consists of two perfect cosine waves what will the output look like using the imperfect oscillator to drive the second input of the mixer Hint What happens when the Lorentzian s ove
12. ideal resistor in parallel with a random current source Transforming between Thevenin and Norton equivalents is performed in the same way as with normal voltage and current sources Adding noise sources is identical to adding random variables For thermal noise each source acts as an independent Gaussian random variable with zero mean and V 4k TRB The result of adding two resistors R and R gives R R R2 with voltage fluctuations 4k TRB Exercise Find the equivalent transformation of two real resistors R and R in parallel 3 2 3 Mixer Mixers are key components in almost all systems working with frequencies greater than a few hundred kilohertz It s an electronic device that multiplies two analog signals This corresponds to a convolution of the two signals in the frequency domain A common use of mixers is in communication systems The simplest example is that of AM radio Let m t be the signal that is to be broadcast and is somewhere in the range of 0 to 10kHz A priori we only know that the signal is in some range not its amplitude or shape To prepare the signal for transmission the signal must be centered around the carrier frequency fe This can be done by multiplying the signal with a cosine or sine wave at the carrier frequency In frequency space this corre sponds to the convolution of the message signal and 4 f fe 6 f fe For the purposes of this example consider m t being a single frequenc
13. integral component of our model for the input of the SA There is a mapping between circuit elements in the time domain and frequency domain For resistors nothing changes For capacitors the impedance resistance in the frequency domain is 445 And for inductors the impedance is tw Circuits can now be formally written down as a network of resistor like components being driven by sinusoidal voltage sources Again any input signal can be constructed as a superposition of these waves The LPF circuit in the frequency domain is shown in figure 3 1 We define the transfer function H f as Vout f H f f 3 7 N 7 C 3 7 From nodal analysis this works out to be H 3 8 1 i2rfRC l 5This is mostly a mathematical ambiguity but it is worth knowing about 6We work under the assumption that the oscillator is on for all time so the source is a perfect delta function Complex values of resistance are like phase changes of a signal The real component corresponds to dissipation of energy 23 van iwc Vout Figure 3 1 This is the diagram of a LPF with a sinusoidal input Vin f driving the circuit Rewriting as a magnitude and phase H f HP 0 3 9 EOI PROT 3 10 o f tan 2r f RC 3 11 The PS is NP R 3 12 Since all of the elements are linear their is no difference between the output and input frequency The only difference then is an amplitude and phase shift This i
14. spectrum and normalizing it with respect to bandwidth As far as units Volts RMS is the most useful for measuring white noise The Window section has several options There are several windows avail able uniform no windowing BMH Blackmann Harris Hamming Window etc We will explore the effect of windowing later when performing measure ments Exercise Using the Spectrum setting how does the graph change when the bandwidth is changed What about when using the PSD setting Exercise Why should we use Volts RMS instead of the peak to peak voltage 4 2 5 Display Menu Set the Format to Single Marker to On Grid div screen to 8 or 10 and Graph Style to line Try playing with the Marker Width and Marker Seeks values along with the jog dial Note If you have recently selected a menu that can be changed with the jog dial you cannot use the jog dial to move the marker around Hit the Marker button on the keypad to use the jog dial to control the marker 33 Exercise Later we will want to use the widest setting for the Marker Width to improve our statistical accuracy of measure ments In this mode everything between the two dashed vertical lines is averaged together In order to do this figure out how many values are used in this average Hint Remember that the the SA digitally filters the PS 4 2 6 Average Menu When
15. the frequency domain we apply the Wiener Khintchine theorem defined as Sxx f Rxx r eT dr 3 6 oCo 4Valid for Wide Sense Stationary processes only 22 This defines Sx x f as the Power Spectral Density PSD From the definition we see that the PSD is nothing but the FT of the autocorrelation function It is important to note that this is a definition The FT of X t in any state of the ensemble is in general neither square integrable nor periodic The PSD is an idealized quantity that doesn t exist in a way that lets us directly measure it We estimate it by measuring the Power Spectrum By measuring longer and longer times we can come to a better estimation of the PSD but their will always be some uncertainty a matter we will deal with later For Johnson Noise the FT of the autocorrelation function is a Lorentzian with a very high cutoff frequency well beyond our spectrum analyzer Thus to us the PS appears uniform in our range of measurement All frequencies are present with equal magnitudes but unknown phase relationships This is called white noise 3 2 Practical Applications in the Frequency Do main 3 2 1 Low Pass Filter LPF One of the great advantages of the frequency domain is that differential equa tions are transformed and can become far simpler Consider the low pass filter circuit driven by an oscillator This is the same LPF circuit used extensively in analog and it will become an
16. will cause our noisy amplifier model to break sooner than it should From this point you are responsible for picking the correct attenuation level Be very careful Exercise 1 f Noise Try grounding the input of the SA and observing the 1 f noise You will want the span to be small and averaging to be on so you can see the decay Eventually we are going to want to measure the noise We should establish a minimum frequency where we say the 1 f noise becomes negligible Assume that the real noise floor is uniform and that the 1 f noise is additive Make a plot of this and save it for later You can also try fitting this to 1 f 8 where a and are constants 4 4 Measurement Routine So we have discussed a variety of problems that fundamentally effect the preci sion of our measurement 5Make sure the preamplifier is not being overloaded You should be able to get to zero attenuation or very close 6Work at a frequency where the LPF is not attenuating the signal too much 36 e Statistics of our measurement e Noise sources from the SA 1 f noise amplifier noise model quantization noise CRT scanning freq etc e Measurement of the Transfer function of the LPF SA This is by no means the complete list We can always get deeper and deeper into the details a danger in any measurement Defining the precision of our experiment To design a measurement routine we need to decided how precise the experiment is going to be
17. 2000 1000 0 1000 2000 Figure 2 8 Left Output of random process Right Autocorrelation function quickly decaying exponential centered around 7 0 1 0 8 0 6 0 4 0 2 0 2000 1000 0 1000 2000 Figure 2 9 Left Output of random process on a timescale short enough to see correlations Right Autocorrelation function decaying exponential centered around 7 0 2 3 1 Statistics of Random Processes 2 4 Physical Interpretation of the Autocorrela tion Function With the quantitative description of the random process lets go back to the example at the beginning of the section Since X t 0 the autocorrelation function is identical to the covariance The earlier figures and their correspond ing autocorrelation functions are sketched in figures 2 8 and 2 9 The autocorrelation function pictured in has an exponential decay with a very short time constant related to the response time of the system Exercise The value of Rx x rT never really goes to zero How ever in practice it becomes uncorrelated enough Assuming Rx x rT e 7 7 calculate by what factor the autocorrelation factor falls of for T nt where n 1 2 3 4 5 Exercise Explain what a non zero value at 7 0 means 17 Figure 2 10 Electric field of a perfect laser a perfect sine wave 2 4 1 Laser Lasers produce coherent light This means that the wavefronts of the ele
18. BNV shows this behavior This mapping starts when sampling waveforms with frequencies higher than the Nyquist frequency fn an Figure BNV Continuous waves with samples overlaid a Slow sine wave sampled at rate Ts b Still resolvable wave at fy fa where fa is a constant c Sine wave at fn d Sine wave at fy a e Sine wave at 2fn Exercise Where do input frequencies higher than 2 fy get mapped Hint Try doing this graphically Windowing When measuring a random process we can only take data for a finite time However the ideal random process exists at for all time Effectively our mea surement is the real random process multiplied by a window lasting for time T This is illustrated in figure CVC Figure CVC Random process being measured with a square window Also show resulting spectrum This convolves the true spectrum we want to measure with the PS of the window a sinc function The width of this function is proportional to 1 T 29 Thus we can reduce the spreading in the frequency domain by measuring longer However this is not always feasible A common technique for reducing these effects is using a different type of window The SA that we will use has several common windows including Hamming Blackmann Harris and Triangular Exercise This question is very subtle Try to explain as much as you can and ask the TA s for help if you get stuck The term Dis crete Fourier Transform is a bit of a
19. M N j Input F i e Terminals S _ Is t Fi Ideal Se 1 f Noise Spectrum A NS Analyzer Figure 4 1 Block Diagram of the Spectrum Analyzer Note nothing is hooked up at the input terminals Exercise The A D converter is always triggered at Ts Throw ing away samples gives us a new effective sampling time T nT where n is the number of samples thrown away Would you expect any differences in the output spectrum noise if the A D converter is instead triggered at T Are their any advantages experimentally with either method Hint A D converters are deceptively compli cated often non linear circuits with poorly understood noise characteristics Without any more information although you are more than welcome to find some you should be able to answer the question Lets go through the relevant menus of the SR760 Spectrum Analyzer to see the functions available to us This is a sort of quick start guide There is more detailed information in the SR760 User Manual which can be found beside the equipment or online The introduction in the manual is quite good and is suggested reading Most of the menus and navigation are accessed via the soft keys to the right of the screen Changing values is usually done with the nob or keypad 4 2 Using the SR760 4 2 1 Starting the SR760 Turn on the SA with NOTHING plugged in Make sure that the coaxial extender is on the input before the
20. PF is known and we can correct for it and perform our measurements anywhere However lets try and work in a regime where the LPF attenuates less than 1 This will give us the highest frequency where we can measure We will want to perform the final measurements at the same frequency for all load resistances 37 Exercise What is the frequency where the LPF attenuates the Johnson noise by 1 when measuring only the input impedance Exercise I suggested working at a frequency where the effect of the LPF is negligible Prove using measurements from the SA if this is necessary This does not need to be an exhaustive study but it should be convincing If you find that we can correct for the LPF behavior at what point does this model break down and what are the real values of R and C Here is one suggestion for proving disproving the LPF model Try measuring kp at several values where the LPF is attenuating the signal Correct for this known behavior Considerations due to the 1 f noise The previous section gives us an upper limit that we can use for the measure ment Now we have to find the lower limit where the 1 f noise will not disturb the measurement Using the results of the 1 f Noise exercise find a minimum frequency where the noise seems to be negligible and express this as a fraction of the largest input signal Exercise We are cheating a bit with this minimum frequency for 1 f noise The relative error introduced by thi
21. The Noise Lab Chris Takacs Chapter 1 Introduction 1 1 Goals of this lab Our goal is to do a precision measurement of ky Boltzmann s constant To find Boltzmann s constant we measure voltage fluctuations known as Johnson Noise over a resistor using a Spectrum Analyzer SA The SA is a complicated piece of equipment and the measurements we want to perform will push their limits Thus we need to understand how these instruments work at their most basic level The first section is about understanding the properties of Johnson Noise as well as other Random Processes Dealing with random processes is very different than deterministic systems Developing tools to work with such systems is useful in a wide variety of fields We will quote the properties of Johnson Noise and leave the physics of this phenomena to the references The reason for this is simple there are already excellent derivations of this phenomena and I want to focus myattention to developing tools to understand these derivations The tools are more similar to those in quantum and statistical mechanics than those of classical mechanics The second section will be focused on creating a set of tools in the frequency domain We start this section off with a review of Fourier Transforms and their properties The emphasis is not on carrying out tedious calculations but using the properties of Fourier Transforms and graphical techniques to quickly find the answers This is to h
22. ad Resistance Temperature The first thing to try is putting the load resistance into Liquid Nitrogen You will have to work out a new noise model and modify the fitting script accordingly but all of the pieces are here It will become a little more complicated as the internal impedance of the SA will be at a temperature different than that of the load From the experimental point of view how far should you immerse the resistor in LNO is up to you Try arguing and experimentally proving that you have negligible thermal leakage into the resistor Remember that the resistors have some temperature dependence 4 6 2 Measuring Very Small Signals We have seen that our noise models break down when the load resistances become very small or low in temperature One way of fighting this is to actually add a known noise source to our signal For example adding a large resistance in series with our desired probe resistance Try using a very small resistor i e 1k or smaller You can work out a new noise model for the system and try putting the probe into liquid nitrogen This is a new measurement technique It has several experimental prob lems to overcome that require us to further refine our understanding of the SA Specifically we must prove that the measurements are repeatable and depend able Creating measurement procedures that calibrate out temperature and background level drifts will be the key to making this technique succeed 4 6 3 Build
23. als we measure Equipment problems like this are an all to common experience in experimental physics 35 Exercise Preamplifier Effects The preamplifier of the SA has a fixed range To accommodate larger signal amplitudes a variable attenuator is placed before the preamplifier to condition the signal Up to this point we have been autoranging the input and letting the SA deciding this level For our noise measurements we will want to manually set this level and fix it for all measurements You can do this in the input menu Exercise Disconnect any resistive loads and ground the input Draw the equivalent circuit of the SA including the model of a noisy amplifier Now start changing the input level Explain why the background spectrum is changing Exercise Disconnect any resistive loads and turn the attenuation down as low as possible Draw the equivalent circuit of the SA in cluding the model of the noisy amplifier Try measuring ky from the input impedance alone ignoring all of the other factors contributing to the measurement It will be within a few percent of the correct value Increase the attenuation in increments 5dB Comment on the changes to the spectrum We will explore these consequences further in the fitting section but it is important to introduce these concepts now as they will influence the way we take data This setting becomes extremely important when the signals become small Attenuating an already small signal
24. becomes correlated with the voltages at later times This should not be confused with the system being deterministic If we wait long enough we will have no idea where the system will be We only have a way of predicting likely values for the system at short timescales By describing the random process through the correlations it posses at different timescales we can characterize a random process 4The derivation of Johnson noise goes deeper into this idea The mobility of electrons in the material will set this bound 2 2 Random Processes A More Formal Defi nition This section is going to give a more quantitative introduction to random pro cesses based on the ideas of probability distributions moments and correlation functions This is an introduction to the subject and not an exhaustive treat ment The math introduced is designed to elucidate the physics and interpreta tion of results later in the text References are given in the end and I urge the reader to consult them if needed or interested In general a random processes has an output that is dependent on a vari ety of inputs which may also be random The formalism that we are going to develop will explain the temporal characteristics of random process as a collec tion of random variables For the most part we are going to be interested in gaussian random processes as they represent many physical systems However the formalism for all of them is the same For the systems of
25. ctro magnetic wave have well defined phase relationships in space and or time For this example lets look at the electric field with time at a fixed location Fig ure 2 10 shows the wave crests for a perfect laser as a function of time The autocorrelation function is plotted to the right We see that only a single frequency is present If we model a real laser one where we are lazing on multiple frequencies we get something like figure 2 11 The periodicity is not quite perfect and within a few periods we cannot predict exactly the electric field The autocorrelation function reflects this behavior It appears to be the correlation function of a perfect laser but with an exponen tially decaying envelope Looking at this envelope we can define a correlation time that describes how quickly the phase information is lost We call this the coherence time of the laser Multiplying this by the speed of light we get the coherence length 18 Figure 2 11 Realistic electric field of a laser that is modulating on more than one frequency The dashed line is the perfect wave 19 Chapter 3 The Frequency Domain Many ideas and problems are more readily thought of in the frequency domain than in the time domain This section is designed to develop graphical tech niques and simple arguments that allow us to quickly rea
26. e of the process must be a constant independent of time Second the autocorrelation function should not depend on the absolute time of the system Let t t and T t2 t Thus the autocorrelation function can be written in the form of Rxx ti t2 Rxx r X t X t 7 2 25 In the jargon of random processes we have restricted our study to Wide Sense Stationary processes Their are a multitude of other types of processes but this one fits most physical systems I will not rigorously prove these claims but physically they seem reasonable for the system we wish to study If you wish to see proof see the references 9The ensemble is the set of all distinct possible configurations of a system over all times Again the system will be in one of these configurations although we may not know which one However all of the configurations have the same properties i e same moments 10 Johnson Noise certainly falls into this category If it was non ergodic it would be possible to extract power from these fluctuations violating a plethora of physical laws 14 Figure 2 6 Time series of a random process for different members of the en semble i e fixed s Units are arbitrary in both directions For illustration purposes only 15 Figure 2 7 Sample output of a random process for several fixed values of t 16 1 0 8 0 6 0 4 0 2 0
27. elp build physical intuition when dealing with the frequency domain The third section is about taking data with the SA The goal of the previous two sections was to develop the conceptual basis to understand the workings of the SA amplifier We start with developing a model based on the first two sections access the limitations and capabilities of each instrument and create a testing regime The final section is a brief overview about extracting ky Most of the fitting is done with a simple Mathematica script provided with the lab From the data you will determine the useful range of the instrument Also additional directions for the lab are suggested 1 2 What I expect of you This is an ambitious lab As such I feel obligated to list the things expected of you before the lab starts e Prior knowledge of analog electronics Phys 127A e Willing to put the effort in to learn these skills e Read the lab and work the exercise before the lab starts Here is a disclaimer many of the exercises in this lab have open ended answers This is done so you can show your depth of understanding which is what determines your grade With that said not every exercise is required nor are all exercises of equal difficulty Exercises that are mandatory will be marked with a There are a fair number of exercises If you are having difficulty answering a question move on to the next if possible and come back to it later The main point is to und
28. erstand the concepts of each section I expect that you will work through the lab in a mostly linear manner There is a substantial amount of theory in the beginning but I believe this is unavoidable Approximately half the lab Given more time the first day of the lab you would sit down with the SA and start trying to make these measurements for yourself Then motivated by the strangeness of the results learn the tools outlined the theory section Unfortunately we don t have enough time for this We pretty much have to do the experiment correctly the first time through My hope is that the theory and background allow you to quickly understand the SA so the equipments don t seem like a mysterious black box The final lab report is to be focused on the extraction of ky The first goal is to be clear and concise The second goal is to prove your results This is done by explaining the subtleties of extracting the data detailing in what range your results are valid and discussing the assumptions both good and bad that you made about the instrument This lab is not about getting the right answer it is about approaching the problems in an organized methodical way 1 3 What you can get out of the lab This is the exciting part This lab is an excellent example of the way experimen tal science is conducted most of the time is spent understanding the equipment Taking the real data require only a small fraction of the total lab time Als
29. esults Could I have specified m to be negative and had a sensible answer Exercise Construct a Gaussian random variable from the random variable A defined previously Exercise Most programming languages and spreadsheets have a built in function for generating a numbered from 0 to 1 i e The random variable A Using the results from the previous exercise write a short program that takes a random variable and generates a Gaussian distribution Verify the output approaches a gaussian by making a histogram of the output and fitting Dealing with multiple random variables is crucial to understanding Random Pocesses The formalism is simliar to that of a single random variable with a few additions First we define the joint probability density function of two 6Use anything Excell Mathematica MATLAB etc TOf course there can be any number of random variables We deal with two for simplicity 11 random variables Q and R written as pgr q r The joint PDF can in general be dependent on both variables An examples of a dependent joint PDF is par qr u x u y z e7709 2 12 where u x is the unit step function We can identify a joint PDF as be ing formed from independent random variables if we can factor pgr q r as polq pr r The joint PDF must also satisfy the normalization condition 00 00 fronton agar 1 2 13 The joint probability distribution function is defined as T q p Foran f ponr agar 2 14
30. ier when discussing the LPF in the frequency domain we used the idea of sweeping a sine wave to measure the response of the circuit to a single frequency We can try this by inserting a resistor in between the Function Generator and the SA This is shown in figure TRT Figure TRT Circuit schematic for measuring the Transfer Function BUILD some sort of adapter to do this Find the 3dB point for several values of Rz and check if this matches the theoretical cutoff frequency Does the signal decay as expected when the input is well above the cutoff frequency Is this a good way to measure the Transfer Function of our device What could be done to make this better Try writing a more realistic circuit schematic that includes some of the parasitics not present in the real measurements How do these effect our results Use this to argue one way or the other 4 3 5 Exercise Other Noise Sources Bad Design While the SR760 is an impressive instrument it s not without faults Set the span to 100kHz and connect the coaxial to alligator clips cable to the input of the spectrum analyzer Connect the alligator clips together This forms an antenna Try moving the antenna around the SA and around the surrounding equipment You will see a nice sharp spike at 49kHz when moving around the CRT of the SA This frequency corresponds to the scanning rate that is used to draw the image on the screen The CRT should be shielded so it doesn t contaminate the sign
31. interest to us we can describe these outputs as random variables Before discussing random processes lets go through random variables 2 2 1 Probability Basics For this section of the discussion we start with random variable X This variable has no time dependence The Probability Density Function PDF describes the probability of X taking on the specific value x and is given by px a This function will be real positive and must obey the normalization condition AUX Y ex z 1 2 2 If the system has a continuous set of output states running from oo to 00 we can rewrite the sum as the integral 00 J px x dx 1 2 3 co When dealing with a system with a continuous set of output states we always measure the system in some range The probability of the system being in the range Lmin tO Lmax iS Emax Pini lt 2S Emak J px x dz 2 4 Lmin Note that the function px x gives us a great deal of information about the most probable state it contains no information about future or past states We will use the PDF in evaluating the expectation value of random variables 5This is roughly the same idea as you have seen in quantum mechanics 0 67 0 27 Figure 2 5 Unform PDF of A There is another function that is useful when working with random variables called the Probability Distribution Function and is defined as T Fx x Px lt zx af px d 2 5 Note tha
32. lectrons throughout the material The variance of the voltage signal is V 4k TRB 2 1 where the brackets denote an average B is the bandwidth of the system a property we will be discussing later For now it is only important to know that bandwidth is inversely proportional to the sampling time Also there is a filter that prevents any frequency higher than TR On average these voltage fluctuations will average to zero Exercise 1 How much power can you extract from these fluctu ations Why 10f course this is not true in Quantum systems 2This is a perfect filter at the Nyquist sampling frequency 0 1 0 08 F e ak J Figure 2 1 Instantaneous voltage measured over a resistor sampled at a rate T Arbitrary units on the vertical axis The formal derivation of Johnson noise is more detailed and requires some of the tools that will be developed throughout the lab Thus much of the information in this section will be given and not derived so as to not obscure the qualitative features of random processes Figure 2 1 shows the sampled voltage fluctuations of a resistor The sampling time is T and the scale is in arbitrary units For the purposes of this qualitative analysis we are only concerned with scaling of the vertical axis with sampling rate Most of the sampled points are confined within the first few standard de viations The histogram of this data is in figure 2 1 We see that the distri b
33. measuring noise we are going to want to do averaging to improve our statistics From this menu we can toggle the averaging setting and set the num ber of averages The Overlap should be set to zero This setting periodically becomes non zero so you must check it when you enable averaging It makes the sampling procedure go faster by using previous measurements when com puting the FFT however this does not give us new statistically independent measurements The situation becomes particularly bad it is set to 98 The mode of averaging should be set to Linear so that each sample is given the same statistical weight The Exponential mode weights newer measurements more than older ones which is not very helpful for the measurments we want to take 4 3 Testing the Spectrum Analyzer 4 3 1 Exercise Exploring the Function Generator Lets start by using the Spectrum Analyzer with a sine wave From the Analog lab we know that a function generator does not generate a perfect sine wave far from it but to our eye it looks pretty close Setup the function generator going directly into the SA through the coaxial extender Make sure the input signal doesn t saturate the preamplifier by Auto Ranging Start with the function generator around 1kHz and using the BHM or Hamming window Set the span such that we can see a few of the harmonics as well Measure the height of the harmonics compared to the fundamental for a few harmonics
34. misnomer since this is more closely related to FS Why Brigham s book will be very help ful You can also see the answer by taking an arbitrary dataset tn DFT ing it shifting the input by N and performing the inverse DFT What is the relationship between this function and the origi nal data By extending this argument what assumptions does the DFT really make about the data Exercise What happens when the data is shifted by a non integer amount What does this tell you about the way the DFT interprets the data In what situations will the DFT correctly interpolate between the points You may need information later in this section to answer this question Exercise What effect does windowing have in discrete transforms What happens to the spectrum when the input frequency is com mensurate and incommensurate with the sampling frequency 30 Chapter 4 Modeling and Measurement 4 1 Overview of the Spectrum Analyzer The SR760 Spectrum Analyzer is a research grade instrument A block diagram model for the SA is shown in figure 4 1 Moving from left to right we see that the signal goes through several stages before the Power Spectrum is computed and displayed At the input of the SA we see a 1M input impedance and 10pf input capacitance The signal is filtered to prevent aliasing and attenuated as to not saturate the preamplifier The low noise preamplifier amplifies the filtered attenuated signal where it is sent into the mixer
35. o the theory and techniques introduced in this lab are used in almost all branches of physics and engineering 1This lab assumes a complete knowledge of analog presented at the level of Phys127A My hope is that no one particular step is too difficult and that each part of the lab builds on itself This is the way it happens in research and industry Great advances are the result of many small successes built upon themselves Chapter 2 Random Processes 2 1 Qualitative Description of a Random Pro cess The term Random Process also referred to as a Stochastic Process is a generic term for the output of a system that is not completely deterministic in nature This output can be anything measurable position voltage current etc In classical systems we understand the physics of most systems well enough to write down a set of equations that will correctly give the exact behavior of the system given knowledge of the previous states of the system However we generally do not have this information nor could we solve the equations even if we did Thus our goal is to find ways of describing the general properties of systems without requiring such detailed knowledge Let s start with describing Johnson noise Given a resistor with resistance R and at a fixed temperature T a randomly fluctuating voltage is present across the resistor even though no voltage is applied These voltage fluctuations result from random movements of e
36. rlap 3 2 4 Amplifier Noise Models Measurement and amplification of small signals usually requires us to take into account the noise from the amplifier The origins of the noise are unimpor tant for our purposes The situation becomes more complicated with cascaded amplifiers where the noise from each amplifier is amplified by the next Fortu nately most low noise systems are designed such that the first stage dominates the noise characteristics of the system This is accomplished by making the first stage have a low intrinsic noise level and a high gain Thus successive stages 27 Vin Figure 3 5 Input referred noise model Note that the input voltage source and source resistance are not pictured add negligible amounts of noise and can be ignored hopefully The input re ferred noise model is shown in Figure Using nodal analysis to find Vout and Vout Vout Vs 3 13 Vout en Vs in Rs 3 14 Figure HHH Input referred noise model Note that the input load is an ideal resistor and the voltage sources have associated spectra with them Exercise Verify this formula First find the explicit expression for Vout and then compute the expectation values i and en are uncorrelated and have zero mean 3 2 5 The Ubiquitous 1 f Noise 1 f noise occurs in a wide variety of physical and electronic phenomena In electronic systems practically every amplifier has this type of noise The reasons have
37. s by Horowitz and Hill The best and most complete electronics book ever written It has a very nice section on noise along with all of the basics As you probably know it can be terse 40
38. s illustrated for a series of driving frequencies below near and above the fsap point of the filter in figure MM Figure MM Series of plots showing what happens to a single frequency incident on the input of the LPF From the principle of superposition we know that any periodic input will be a superposition of sines and cosines Each output frequency will pick up a phase amplitude factor specific to that particular frequency Once we work this out we can inverse transform and examine the output The results for a square wave and triangle wave input are shown in figure JJ Figure JJ Series of plots of a filtered square triangle wave where the period is far below near and much larger than the cutoff frequency Figure IT Plots of the resulting spectra Exercise Find ee using nodal analysis for the LPF circuit in Exercise Look at the time domain output for a square wave input much greater than fsag It appears to roughly integrate the 24 signal Note that in analog the LPF was also an integrator This can be seen by solving the differential equation in the time domain and making some assumptions about the product RC as discussed in Horowitz and Hill Compare the shape of H f with that of an ideal integrator in the frequency space In what regime are they similar Justify the results graphically 3 2 2 Noisy Components A noisy resistor can be represented as an ideal resistor in series with a random voltage source or as an
39. s source will increase as the signal level decreases However all of our measurements are being conducted at a single frequency Thus the 1 f noise should be a constant for all the measurements Explain how this will be taken into account when we add in our model for a noisy preamplifier developed earlier in the lab What would we have to do if we took measurements at different frequencies Putting it all together The measurement routine is now complete The procedure is finished and now it is time to take data Our goal is to see how far we can extend our measurements of k Attach load resistances ranging from 1k and up You do not have to do every possible combination but do enough such that the equivalent load resistance spans several orders of magnitude Don t go to crazy taking data just yet It will be best to take data for a few resistances and fit the data first Once you are confident with your measurement technique and fitting take some more data to fill in between some of the points 4 5 Fitting the Data The supplied Mathematica script is setup and documented for fitting the data to our noisy amplifier model You should examine the output and comment on where the model breaks down 38 4 6 Further Studies Congratulations for getting to this point Ideally you should try the Liquid Nitrogen experiment as it is pretty quick and fun The other experiments are posed here for those who feel motivated 4 6 1 Change the Lo
40. son through difficult problems without resorting to brute force calculations The result of thinking in this fashion is that solving real problems ones without clear answers becomes more transparent We start with the background material Fourier Transforms Fourier Series and Power Spectra We then introduce the ideas of the Power Spectral Density and link this to the autocorrelation function From there we start a series of practical applications of these ideas Each application is an integral part of the full model we eventually construct for the SA 3 1 Background Lets start with the basics Most of this information is out of The Fast Fourier Transform and Its Applications by Brigham This should be in the lab room and is an incredible reference We define our Fourier Transform FT pair as H f es h t e 27F dt 3 1 ny f H fyer af 3 2 A function is guaranteed to have a FT if f h t dt over all t is finite Here time t and frequency f are the conjugate variables With position the conjugate variable is q or k the wavevector We will stick with time and frequency for our variables as this is what we will eventually measure using the SA Much of the usefulness of working in the frequency domain comes from its properties Let a and b be arbitrary complex numbers 1By working with f instead of w we avoid the factors of 27 floating around 20 Linearity h t af t bg t A f aF f 6G f Scaling
41. t this function will be a monotonically increasing function of x starting at 0 that tends toward 1 as x oo Example Functions of Random Variables Lets do a practical example We often need to construct new random variables from other random variables Explicitly if the random variable A gives us the value a we want the random variable B to return the value b i e a mapping from one to the other Starting with the random variable A with the PDF shown in figure 2 2 1 lets find the transformation to the new random variable B where B T A mA z 2 6 where T A is the transformation function relating random variables A and B m m gt 0 is the slope of a line and z is the offset The transformation function 10 maps the value a from the random variable A to the value b giving us b T a 2 7 Inverting the equation a T 1 b 2 8 The probability distribution functions for A and B yield the constraint Fg b F4 a 2 9 which can be written as b AS a T b ih pa i ab f pala aa f pala d 2 10 o 00 oo This expression can be rewritten to give a function for the PDF of B using the Leibniz Rule pet pa pata 2 11 Exercise Explain why the above constraint on the probability distribution function is correct Exercise Go through the steps of this example explicitly by using the Leibniz rule Exercise What properties should the transformation function have in order to give sensible r
42. to do with device physics and are well beyond the scope of this lab Basically all we can do is avoid it 11 am assumed a unity gain amplifier to simplify the formula Although the gain will be much greater than 1 in a real circuit for our SA our measurements of voltages are automat ically divided by this gain so we can omit it for our calculations 12Strictly speaking the noise falls off as 1 f where a is a number not necessarily an integer depending on the device 28 3 2 6 Going to a Discrete World In reality the functions that we measure are never continuous The act of sampling the signal has several interesting effects on the spectrum These effects are explained incredibly well in Brigham s book so we will not dwell on it too much here However these concepts will show up later when using the spectrum analyzer Discrete Transform The Discrete Fourier Transform DFT is defined as follows N 1 Xk 5 Ine aa wherekgoesfrom0toN 1 3 15 n 0 1 r 2rikn tn F Xk eN wherengoesfrom0toN 1 3 16 k 0 This definition converges to that of the regular FT when we consider the continuous function as a set of periodic discrete 6 functions Aliasing Perhaps the most dramatic feature of sampling a function waveform is aliasing This occurs when high frequency components are present in the signal and the sampling rate is too slow to correctly resolve the frequency Thus it is mapped to a lower frequency Figure
43. ution approaches a Gaussian the reasons for this are explained in the formal derivation of Johnson noise Here is the key statement while the probability distribution of the signal seems to be well defined there is no relation between successive samples If we decrease the sampling time increase the bandwidth as in figure 2 1 we see the variance increase However there is no real difference in the structure of the signal The value of each sampled point is independent of the value of 3I use the word structure to denote a visual pattern 140 120 100 80 60 40 20 0 0 1 0 05 0 0 05 0 1 Figure 2 2 Histogram of the time series Figure 2 3 Time series with decreased sampling time Figure 2 4 T low enough to see the finite response time of the system all previous samples This coupled with the Gaussian probability distribution leads to the term white noise which will be explained later In the limit T 0 V implying that the system is fluctuating ever more wildly This is an odd result It means that the voltage over this resistor is always fluctuating from oco to oo constantly Of course this is an unphysical In practice there is some upper bound on how fast the resistors voltage can fluctuate We see in figure 2 1 what happens when the sampling time is on the order of the response time of the system The voltage at one point in time
44. y as shown in Figure 3 2 The process of graphical convolution is best shown graphically in Brigham s FFT book At the receiver end we have to recover the signal shown in Figure 3 3 It is difficult to do this directly while the signal is still near f 1 The trick is to move the signal back down and center it around zero We use another mixer 8This is roughly the bandwidth of human speech 9I am also assuming that the amplitude of m t is small compared to the amplitude of carrier signal and that fe gt gt 10kHz Also I have added an offset to m t so it is always greater than zero 10This is called a homodyne system when we work with the signal directly at the trans mission frequency It is much less common than the hetrodyne system architecture we are using 25 AS V Figure 3 2 This is our sample function for m t It is a single frequency with a constant offset such that m t is always greater than zero hh Ml mi in mil il y j I i Ii i n il Figure 3 3 This is the time domain signal after being multiplied by the carrier frequency The envelope of the wave is shown and is equivalent to m t 26 i h i i T y Figure 3 4 This is the time domain signal in Figure 3 3 multiplied by a cosine wave at fe
Download Pdf Manuals
Related Search