1967 , Volume , Issue Sept-1967
Contents
1. Fig 7 Model 37 top one clock random sequence Noise Generator produces syne pulse at point each pseudo signals can be used in analog systems in hybrid sys tems c g process control system containing solenoid operated on off valves in digital systems a PCM channel Although binary and Gaussian noise look quite dif ferent it is possible to get a random Gaussian signal by andom binary mal through low pass sending a filter see Fig 5 The new noise generator produces both binary and Gaussian pseudo random and random outputs Using digital techniques it synthesizes the binary waveform then low pass filters the binary signal to get the Gaussian output Fig 6 shows how the instrument works A binary waveform generator timed by controlled clock synthesizes the basic binary signal The changes of state of the binary signal always take place when a clock pulse occurs but a change doesn t occur on every clock pulse The clock period and hence the in terval between possible changes of state of the binary signal is selectable from to 333 seconds Alter natively the instrument may be timed by an external clock of frequency up to MHz Depending upon the setting of a front panel SE QUENCE LENGTH switch the binary waveform gen erator produces either repetitive or non repetitive output patterns The repetitive or pseudo2. equation 2 approximates the integral equation 1 When x t is a binary signal as it is in the new noise generator the delay line can be a shift register This in fact is how the noise generator s digital low pass filter is constructed It uses a 32 stage shift register as a delay line The first 20 stages of the same register do double duty as the binary waveform generator as we have already explained The desired frequency response of the digital filter is the rectangular response of an ideal low pass filter There fore the coefficients a are selected to approximate an impulse response of sin impulse re sponse of an ideal low pass filter 252 J shapes spectrum of Binary Output Fig 14 Frequency response of digital low pass filter is nearly rectangular Small high frequency components are caused by steps in digital filter output they are subsequently removed by analog filtering Copr 1949 1998 HeWiet Packard Co George Anderson After graduating in 1954 from the Heriot Watt University Edinburgh George Anderson completed a two year graduate apprenticeship course in electrical engineering This was followed by varied industrial work and a three year period with the Royal Observatory where he developed data recording systems for the Seismology Unit George who was the 3722A project leader joined HP in 1966 Brian W Finnie Brian received the degree of BS f
3. is multiplied by a delayed version of the other and the product is averaged The result is a function of the de lay In physically realizable systems the result also depends on the averaging time T Ideally T should be infinite but this would mean that it would take an infinite amount of time to get an answer Fortunately the sta tistical variance caused by using a finite T can usually be made acceptably small by making T fairly large Copr 1949 1998 Hewlett Packard Co Tf y t x t the cross correlation function becomes the autocorrelation function of x t defined as 1 2 RG lim x t r x t dt 2 The autocorrelation function of a signal is the Fourier transform of the power density spectrum Hence the autocorrelation function of white noise is just a single delta function at gt 0 this means that any two samples of the same white noise signal are uncorrelated as long as there is a nonzero time interval between them Since the autocorrelation function is the transform of the power density spectrum it gives us no information that isn t contained in the spectrum However it is an extremely useful function and is often simpler to compute than the power density spectrum Pseudo Random Noise Noise makes a good test signal for two reasons it is broadband and it realistically simulates naturally occur ting disturbances However its randomness is not very helpful to the experimenter Theoret
4. 4 or 8 pseudo random sequences can be selected Another control feature is a HOLD button which when pressed stops the pseudo random waveform Sub sequently pressing the RUN button restarts the waveform from the same point in the sequence that had been reached when the HOLD button was pressed There is also a RESET button which sets the waveform gener ator to the 0 state and removes its supply of clock pulses Pressing the RUN button then starts the gener ator by restoring the clock pulses and placing a in the first stage of the waveform generator RUN HOLD and RESET can all be remotely pro grammed Copr 1949 1998 Hewlett Packard Shift Register Waveform Generator Many binary waveforms have the properties of pseudo random sequences One family called maximal length sequences be generated by a shift register with propriate feedback The binary waveform generator in the new noise generator consists of the first 20 stages of a 32 stage shift register These 20 stages and the last 12 stages also form part of the digital low pass filter which will be discussed later For now we will concentrate on the first 20 stages A shift register stage is a special purpose flip flop It is an information store and each stage of a shift register can store one binary bit of information 0 or 17 The length of time that a bit of information remains in the stage is equal to the time interva
5. These discrete steps in the multi level output of the digital filter are removed by low pass analog filtering if the selected clock period is less than one second and the resulting smooth Gaussian signal is another output of the noise generator It is available at a fixed amplitude of 3 16 V rms with low source imped ance or at a selected amplitude with 600 impedance Fig 6 shows a typical Gaussian output waveform from the noise generator along with its spectrum We will have more to say about this signal when we discuss the digital low pass filter Control and Synchronization Since pseudo random signals are periodic it is possible to obtain a stationary display of them on an oscilloscope or to synchronize other equipment with them For such purposes the noise generator produces a sync pulse one clock period wide at a particular point in each pseudo random sequence Fig 7 0280000115501 Fig 8 Fifteen bit pseudo random binary sequence is gen erated by four stages of shift register with feedback lele a Fig 9 Fifteen bit pseudo random binary sequence generated by system of Fig 8 Fig 10 If is number of stages involved in feedback loop length of pseudo random sequence is N 2 1 clock periods This is a 31 bit sequence generator ie n 5 Besides the sync pulse there is also a GATE output which can be used for controlling external equipment e g a computer Gate lengths of 1 2
6. course small impulses could be used but if they are small enough to be safe they Copr 1949 1998 Hewlett Packard Co usually produce outputs which are so small that they are obscured by background disturbances One of the really interesting features of statistical tech niques is that we can inject low amplitude noise into a system and by suitably processing the output obtain the system impulse response without subjecting the system to a damaging high level test signal This technique has two other advantages The test may be performed while the system is operat ing on line This is possible because the intensity of the noise test signal be low enough so it doesn t affect normal operation of the system The results are largely unaffected by background dis turbances in the system This is because the results are obtained by correlation and the disturbances aren t correlated with the test noise Fig 2 shows a setup for obtaining impulse responses from noise responses The output of the system is cross correlated with the noise input that is the output is multiplied by a delayed version of the input and the product is averaged The average as a function of the delay 7 is the same as the impulse response of the system as a function of time provided that the autocorrelation function of the noise input is an impulse 1 the noise should be wideband compared with the system s fre quency response If the autocorre
7. random patterns are periodic but they look random there is apparently a 50 probability that the binary waveform will change state on any given clock pulse These waveforms repeat after a fixed number N of clock periods crystal The number N of clock periods in the pseudo random sequences is selectable from 2 1 to 27 1 i e from 15 to 1 048 575 The length of one sequence is the prod uct of N and the clock period so the number of seconds in the pseudo random sequences be as short as ps 15 15 or as long as 333 5 X 1 048 575 more than 11 years When the SEQUENCE LENGTH switch is set to its INFINITE position the binary waveform generator is random noise source In this signal is truly random and never primed by a solid condition the bina repeats As Fig 6 shows the binary signal is one of the outputs from the noise generator It is available at 10 V with very low impedance or at a selected amplitude with 600 impedance A relay contact version of it is also avail able if the selected clock period is greater than 100 ms Spectrum of the Binary Output A pseudo random binary sequence has a line power spectrum the envelope of which is a sin curve as shown in Fig 6 Note that most of the power in the first lob contained nd that the nulls occur at intervals of f the clock frequency The harmonic line spacing is a function of sequence length and cl
8. spectrum doesn t specify the signal uniquely is a consequence of the fact that it contains no phase information Two periodic signals for example have the same power spectrum if they both contain the same fre quency components at the same amplitudes But if the Copr 1949 1998 H wlett Packard b Fig 3 Probability densiry function tells what proportion of time is spent by signal ar various amplitudes Shaded area in is equal to proportion oj time spent by signal between x and x Gaussian probability density function b is common to many natural disturbances phase of just one component of one signal is shifted with respect to the phase of the corresponding component of the other the two signals can have drastically different waveforms A statistic of a signal that gives waveshape information and is independent of the spectrum is the probability density function or pdf see Fig 3 The pdf tells us what proportion of time on the average is spent by the signal at various amplitudes The area under a pdf between any two amplitudes x and is equal to the proportion of time that the signal spends between x and x Equivalently this area is the probability that the signal s amplitude at any arbitrary time will be between x and x The total area under a pdf is always one general the probability density function and power spectrum or power density spectrum are two diffe
9. thermal noise atmospheric noise etc Naturally occurring noise can have a frequency content similar to that of binary noise but it is random in amplitude not confined to just two levels The noise generator provides in addition to the basic binary signal pseudo random or random signals of the more familiar multi level or Gaussian type Gaussian in this context means that the probability density func tion of the output tends to be the classical bell shaped curve see Fig 3 As we have shown Fig 5 a multi level waveform can be derived from a binary signal by conventional ana log low pass filtering However it takes a filter cutoff frequency that is about 1 20 of the clock frequency to give a reasonably Gaussian pdf Since the lowest clock frequency in the new noise generator is about one cycle in five minutes the lowest filter cutoff frequency has to be about one cycle per 100 minutes It simply isn t practical to make analog filters with such low cutoff frequencies To convert the output of the binary waveform gener ator to a multi level signal we use a low pass digital filter which is not subject to the same limitations as a conventional low pass filter The 3 dB bandwidth of the filtered signal defined as to the half power frequency is nominally 1 20 of the clock frequency f The output of the digital filter is not a smooth signal but a series of steps like any waveform that has been generated digitally
10. HEWLETT PACKARD JOURNAL SERLEMBER 4967 Pseudo Random and Random Signals Using digital techniques this precision low frequency noise generator can synthesize repeatable controllable pseudo random noise patterns as well as truly random noise By George C Anderson Brian W Finnie and Gordon T Roberts LMOST EVERY NATURAL AND MAN MADE SYSTEM IS A subject to random disturbances under normal oper ating conditions Consequently it is often appropriate and sometimes essential to test a system with random test signals rather than with the sine waves that are so tamiliar to electrical engineers Many of the areas of application for random test signals lie outside the field of electrical engineering Examples are biomedical phenomena vibration However a dynamics and seismolog growing number of electrical problems fall into this same category For example it is much more appropriate to test a multi annel telephone system with random noise sim ulating each speech signal than to use a number of sine waves The problem of communicating with deep space probes is another subject that can be adequately treated only by means of statistical techniques From the mathematical viewpoint there fore there are good reasons for using noise as a test signal Yet despite the fact that adequate theories have been developed the introduction of test methods based on these theories has bee
11. actice in noise theory to consider amplitude as the unit of power For electrical signals this gives the power density spectrum units of V Hz A power density spectrum is shown in Fig 2 The total area under this curve gives the total power con tained in the signal The power contributed by all fre quency components in any band say from f to is equal to the area under the power density curve between f and f shaded area in Fig 2 Power density spectra can be measured experimentally with a narrow band constant bandwidth wave analyzer followed by a true square law meter with a long averaging time This inconsistency in the units of power is unacceptable to some engineers they reconcile the difficulty by assuming one ohm load resistance Cover 180A Oscilloscope bottom displays portion of pseudo random Gaussian noise pattern erated by Mode 3722A Noise Generator center Top instrument is a display unit from new Model 5400 Multi channel Analyzer which will be described future issue of the Hewlett Packard Journal Here the Analyzer displays the probability density function of the noise generator s Gaussian output 2 5 FREQUENCY Hz Fig 2 Typical power density spectrum for random sig nal Total area under curve is mean square value of signal usually spoken of as power in noise theory Shaded area is power in the frequen
12. bances and acceler meter measures structure s response 18 Copr 1949 1998 Hewlett Packard Co x t 15 pseudo random binary output of 3722A Autocorrelation function approximates an impulse See Figure 3 AVERAGING CIRCUIT iel TJ MULTIPLIER CORRELATOR Fig 2 System for obtaining impulse responses with noise and correlation techniques Flat Spectrum at Low Frequencies In most of the applications of noise as broadband t signal the preferred shape of the power density 5 flat at least through the band of interest tes spectrum This is a difficult requirement for conventional natural noise sources to meet especially at low audio and sub audio frequencies where flicker noise noise hum ambient temperature fluctuations vibrations and micro phonics all degrade the spectrum In addition a noise source usually produces a small amplitude signal If low frequencies are important this signal must be amplified by ade coupled amplifier and the random drifts of such an amplifier cannot be distinguished from the low fre quency portion of the original noise signal Low frequency noise however is a necessary product of useful noise source The main use of very low fre quency noise e g in the 0 to 50 Hz range is in testing systems which have long time constants These include such things as massive mechanical arrays nuclear re actors and chemical proc
13. cy band fs to fo It is important to notice that the power density spec trum is not the same as the power spectrum The former has units of V Hz The latter is just the square of the amplitude spectrum and has units of The power spectrum is used to describe signals which have a finite number of discrete frequency components The ampli tude or amplitude of each component can be repre sented by a line of the proper length on the graph But when the signal is a complex random waveform the power spectrum has to have an infinite number of lines all of zero amplitude Thus the power spectrum shrinks to zero for a random signal The power density spectrum however does not disappear Noise which contains equal amounts of all frequencies is called white noise by analogy to white light White noise has a power density spectrum which is simply a horizontal line representing some non zero value of power per unit bandwidth Truly white noise which has infinite bandwidth and therefore infinite power is never found in physical systems which always have finite band widths We usually call noise white if it has a flat power density spectrum over the band of interest Probability Density Functions The power density spectrum tells us how the energy of a signal is distributed in frequency But it doesn t specify the signal uniquely nor does it tell us very much about how the amplitude of the signal varies with time That the
14. dwidth 220 05 Hz This specification is valid only when sequence length gt 1 023 Output Impedance lt 1 0 Load Impedance 600 2 minimum Zero Drift lt 5 mV change zero level in 10 C range from 0 to 85 C Power Density Approximately equal to clock period 200 VI Hz at low frequency end of spectrum Power Spectrum Rectangular pass nominal upper frequency f 98 point equal to of clock frequency Spectrum flat within 0 3 dB Crest Factor Up to 3 75 dependent on Probability Density Function error curves page 16 VARIABLE OUTPUT Binary or Gaussian Amplitude Open Circult BINARY 4 ranges 1 3 23 16 and 10 with each range from 0 1 to X 1 0 GAUSSIAN ranges 1 rms rms and 3 16 rms with ten steps in each range from X 0 1 to X 1 0 Calibration Accuracy Better than 2 5 plus tolerance on binary or Gaussian output Output Impedance 600 0 1 MAIN CONTROLS Sequence Length Switch First 17 positions select different pseudo random sequence lengths tinal position selects random mode ot operation INFINITE se quence length Sequence length is number of clock periods In sequence possible values of N are 15 31 63 127 255 511 1023 2047 4095 8191 16383 32767 65535 131071 262143 524287 1048575 2 1 where is the range 4 to 20 Inclusive CLOCK PERIOD SWITCH Selects 18 frequ
15. e of filter approximate sin x x shape negative level This is the waveform obtained at the BINARY connector of the noise generator with the SEQUENCE L 3TH switch set to 15 The next setting 31 of the SEQUENCE LENGTH switch selects for modulo two addition the outputs from stages 3 and 5 as shown in Fig 10 With five stages the maximum number of I and 0 combinations is 32 but as before the all zero condition cannot occur The result ing sequence is therefore 31 bits long The number of sta included in the feedback loop is increased by one at each setting of the SEQUENCE LENGTH switch Feedback is always taken from the last of the active stages and from one or more of the preceding stages For the 127 bit sequence for example feedback is taken from stage 7 7 is the number graved on the front panel and also from stages 3 4 and 5 Where more than two outputs are modulo two added extra EXCLUSIVE OR gates are used The number of bits N in pseudo random sequences is always one less than the maximum number of 1 and 0 combinations possible with the selected length of register Thus if n is the number of active stages N 2 1 In the new noise generator is variable from 4 to 20 and N ranges between 15 and 1 048 575 Random Operation of the Shift Register With the SEQUENCE LENGTH switch set to IN FINITE the feedback system is disconnected and the fir
16. ed Envelope 5 0 5 Binary Output N 15 31 63 1048575 ore i 2 3 NAT FREQUENCY 1 Clock Frequency Spectrum of Digital Filter Output Fiter Bandwidth Varies with Clock Frequency 2 895 POWER FREQUENCY Hz First Lobe of High Frequency Components in Digital Filter Output Spectrum of Gaussian Output 03 at fo Corner Frequency of 7 Analog Smoothing Filter Characteristic 2 of Analog a 5 Smoothing Gaussian Fiter Output More than 25 dB down at 20 FREQUENCY Hz Fig 6 Model 3722A Noise Generator synthesizes pseudo random or random binary signal in a digital waveform generator which is timed by a crystal controlled clock Clock rate and length of pseudo random sequences are variable Gaussian signal is derived from bi nary output by digital low pass filtering Discrete steps in digital filter output are removed by analog filter Pseudo random binary output of noise generator has line power spectrum having flat envelope from to an upper dB frequency which is selectable from 0 00135 Hz to 450 kHz Spectrum of pseudo random Gaussian output has flat envelope from de to upper 3 dB frequency which is selectable from 0 00015 Hz to 50 kHz Random outputs have continuous power density spectra having same shapes as envelopes of spectra of pseudo random outputs Copr 1949 1998 Hewlett Packard Co
17. encies from internal clock INTERNAL CLOCK Crystal Frequency MHz nominal Frequency Stabitity lt 25 ppm over ambient temperature range 0 to 55 C Output 15 V to 4125 rectangular wave period as selected by CLOCK PERIOD switch Maximum current at 1 5 level 10 EXTERNAL CLOCK Input Frequency 1 MHz maximum for stated specifications Usable BINARY output pseudo random only with external clock frequencies up to 1 5 Mhz Input Level Negative going signal trom 5 V to 3 V initiates clock pulse Maximum input 20 Input Impedance 1 nominal SECONDARY OUTPUTS Syne Negative going pulse 125 to 1 5 occurring once pseudo random sequence duration of pulso equal to selected clock period Maximum current at 1 5 V level 10 mA Gate Gato signal indicates start and completion of selected number of pseudo random sequences 1 2 4 selected by front panel control Two outputs are provided 1 Logie signal output normally 12 5 falls to 1 5 V at start of gate Interval and returns to 12 5 V at end of Interval Maximum current at 1 5 level 10 mA 2 Relay changeover contacts gate relay switching Is syn chronous with logic signal Maximum current controlled by relay 500 mA Maximum voltage across relay contacts 100 V Maximum toad controlled by relay 3 W cont Binary Relay Relay changeover contacts operate in sync with binary output signal available only w
18. ensterry West Lothian Scotland 17 Copr 1949 1998 Hewlett Packard Co Testing with Pseudo Random and Random Noise Pseudo random noise is faster more accurate and more versatile than random noise in most measurement situations NEW NOISE GENERATOR described in the artic beg 2 is different from conventional nning on pa noise sources in that it synthesizes noise by a digital process This not only makes its output statistics more stable and controllable but also allows it to produce pseudo random noise as well as random noise Pseudo random signals are periodic signals that look random they have the same advantages as random noise for test ing but have the disadvantage of randomness Here are some of the ways in which noise is useful as a test signal with emphasis on the uses of pseudo random noise Noise as a Broadband Test Signal Broadband noise makes an excellent test signal for environmental testing For example the vibrations produced by a shake table with a noise input are similar to those a product will meet in service A loud speaker connected to a noise generator makes a useful acoustical noise source for testing microphones mate rials rooms and so on In fatigue testing pseudo random noise is helpful because it has a known num ber of peaks of various amplitudes this means that test time can often be reduced since it is not necessar
19. esses where the effect of chang ing any parameter of the process takes a long time to become evident When testing these systems the lowest frequency content of the test signal must be comparable with the system time constant This also holds true when the system is being simulated on an analog computer The spectrum of the binary output of the new noise generator is virtually flat from de to an upper 3 dB fre quency which can be adjusted from 0 00135 Hz to 450 kHz The Gaussian signal has a spectrum which is flat from de to an upper 3 dB point of 0 00015 Hz to 50 kHz Regardless of selected cutoff frequency the genera total power output is constant in other words when we halve the bandwidth we don t halve the power as occurs when the output from a conventional noise source is low pass filtered Model and Computer Simulation Control systems buildings ships automobiles air craft aerospace guidance systems bridges missiles and a host of other complex objects can often be designed and studied most easily by simulating them in the lab oratory This can be done either by using a scale model of the object or by simulating it on an analog computer In either case the new noise generator can provide realistic simulations of road roughness air turbulence earthquakes storms at sea target evasive action con trolled variable fluctuations and so on Particularly use ful is the pseudo random output o
20. evel and Gaussian multi level outputs are generated plitudes and bandwidths of outputs and lengths of pseudo random patterns are variable 2 Specifying Noise How can noise be specified Simple deterministic signals can be completely speci fied by a small number of parameters For example de is specified by only one parameter A step function is specified by two parameters amplitude and time And sine wave is specified by three parameters amplitude frequency and phase Random signals on the other hand can t be completely specified by a finite number of parameters But we still need some way of describing them so we resort to statis tical descriptions which tell us about the average be havior of the signals The simplest statistic of a noise signal is its mean square value or equivalently its rms value This param eter is quite easy to measure provided that we have an instrument with a true square law response We also have to carry out the averaging process over a long enough time to reduce the statistical variance of the results to an acceptably small value Power Density Spectrum Another statistical description of a random signal that isn t difficult to measure is its power density spectrum This tells us how the noise power contributed by separate frequency components of the signal is distributed over the frequency spectrum It should have units of watts per unit bandwidth but it is common pr
21. f the generator which the same effect on the model as real noise but which can be repeated at will Analog computer users should find the following char cs of the noise generator particularly helpful ccurately defined signals amplitude controls not subject to loading errors ability to change time scale without changing ampli tude or pattern shape remote programming for RUN HOLD RESET gate circuits to control operations in the computer good autocorrelation function see Fig 3 zero moment option see Specifications 17 Fig 1 shows a model simulation of a tall structure mounted on a shake table which is being excited by Gaussian noise from the new noise generator This set up currently in use at Edinburgh University provides ex perimental data on the behavior of tall buildings sub jected to ground disturbances The lower trace shows the acceleration of the first floor of the structure as measured by the accelerometer mounted on the model acte Impulse Responses Without Impulses All the information necessary to characterize a linear system completely is contained in its impulse response Given any unknown system then it would be desirable to be able to find its impulse response One way to do this would be to excite the system with an impulse or a train of impulses and observe the output with an oscil loscope However impulses are dangerous they are likely to cause overload and saturation Of
22. graded to follow the sin x x curve as shown in Fig 12 Notice in Fig 12 that the contribution made by the first and last groups of seven resistors is required to be of the opposite polarity to that made by resistors in the central group This can be arranged by supplying all of the weighting resistors in the central group with direct outputs from the shift register and supplying those in the outer groups with inverse outputs direct and inverse are used here to describe the two outputs from opposite sides of a flip flop A 17 starting at one end of the register and being conveyed to the other by a series of shift pulses will generate the time waveform shown in Fig 13 Fig 15 Bandwidth of ideal low pass filter inversely proportional to time of first null in impulse response In noise generator first null in digital filter pulse response occurs at nine clock periods so cutoff frequency is theo retically 1 18 of clock frequency Actual response is not ideal and has 3 dB frequency equal to 1 20 of clock fre quency Thus bandwidth can be varied simply by changing clock frequency Copr 1949 1998 Hewlett Packard Co The digital filter has an effective frequency response which approximates a rectangular spectrum Fig 14 Owing to the limitation on the size of the shift register which results in truncation of the sin x x curve the corner of the spectrum is not perfectly square There are also
23. hen clock period 2100 ms Relay spool fication as for gate relay above REMOTE CONTROL Control Inputs Remote control Inputs for RUN HOLD RESET and GATE RESET functions are connected to 36 way receptacle on rear panel Command signal each input de voltage between 1 5 and zero volts No command condition open circuit input de voltage between 55 Vand 125 V Input impedance 5 nominal RUN HOLD RESET 1 5 k2 nominal GATE RESET Sequence Length Indication 18 pins plus one common pin on the 36 way receptacle are used for remote signaling of selected sequence length contact closure between common pin and any one of the 18 pins GENERAL Construction Standard 19 in rack width module with stand Ambient Temperature Range 0 to 55 C Power Requirement 115 or 230 10 50 to 1000 Hz 70 W Weight Net 10 5 kg 23 Ib shipping 13 5 kg 30 10 Accessories Furnished Detachable power cord rack mounting kit circuit extender board 36 way male cable plug operating and service manual Price 2 650 00 OPTION 01 Zero Moment Option Shifts relative position of sync pulse and pseudo random binary sequence such that first time moment of sequence taken with respect to sync pulse is zero sequence shift mechanism is oper ative only when selected sequence length is lt 1023 option 01 also provides facility for inverting binary output signal ADD 50 00 MANUFACTURING DIVISION HEWLETT PACKARD LTD South Que
24. high frequency components in the digital filter out put spectrum These components caused by the abrupt changes in output level as pulses pass down the shift register are removed by analog filtering as described later Changes in clock frequency do not affect the rectan gular shape of the spectrum they simply alter the upper frequency limit So here is a low pass filter whose cut off frequency automatically keeps in step with clock fre quency see Fig 15 Probability Density Function The amplitude pdf of the multi level signal is not significantly affected by the values of weighting resistor assigned to the various stages The Gaussian nature of the pdf arises mainly from the apparent randomness of the changing pattern of ones and zeros in the register the pdf becomes more nearly Gaussian as the sequence length and hence the randomness is increased This is a consequence of the Central Limit Theorem of proba bility theory which states that the sum of a large number of independent random variables tends to have a Gaus sian pdf regardless of what the pdf s of the individual variables look like For sequence lengths of 8191 or more the pdf of the multi level signal closely approximates the Gaussian curve and the waveform closely resembles naturally occurring noise Fig 16 Fig 17 shows the measured deviations of the noise generator s output pdf from the true Gaussian curve for sequence lengths of 8191 or greater Wo
25. hout the 32 stages of the shift register There is a delay of one clock period between the pattern from one stage and the pattern from the next The digit sequence from any of the stages is 100110101111000 100110101111000 Time Fig 9 shows this sequence translated into a two level or binary waveform is represented by the relatively 1 ANALOG FILTER 2 DELAY LINE FILTER ya tT 3 DIGITAL FILTER with clock period Same as Delay Line Filter except delay line is shift register and x t is a binary signal y fi Fig 11 To get good Gaussian signals from binary signals lowest cutoff frequency required of low pass filter in Model 3722A Noise Generator is about one cycle per 100 minutes This makes analog filter impractical so generator uses digital approximation to ideal low pass filter Delay line in noise generator is 32 stage shift register and weighting networks a are resistors Copr 1949 1998 Hewlett Packard Co RECIPROCAL RESIS ANCE 1234567 8 9101112 1314 15 161718192021222324 25 26 27 2829 B0 3132 inverse J of Rip Flog Fig 12 For digital filter in Model 3722A Noise Generator outputs of 32 stage flip flop shift register are weighted by resistors and added Values of resistors are graded as shown to make pulse respons
26. ically experiments involving random noise should be carried out over an infinite time interval so that only the average characteristics of the noise will affect the result But every real measurement can only be made over a finite time say This means that if random noise is used as a test signal the result of an experiment will in general be different from its expected value Or if an experiment involving random noise is repeated over and over each repetition will yield a different result In other words the randomness of the noise introduces statistical variance into the results Variance can be reduced by extending the measure ment time But it can never be made zero when truly random test signals are used What we need obviously is a test signal which has the good properties of random noise i c broad flat spectrum and resemblance to natural disturbances in waveform and pdf but doesn t have the bad property i e randomness This signal should be one that intro duces no statistical variance into the results even though the measurement is made over a finite time T Such a signal exists Pseudo random noise is a signal which looks and acts like random noise but is in fact periodic This kind of noise is one of the principal prod ucts of the new noise generator Pseudo random waveforms consist of completely de fined patterns of selectable lengths repeated over and They have spectra and pdf s tha
27. l between two successive clock or shift pulses Individual shift register stages are connected in cas cade so that on receipt of shift pulses the information they contain is stepped progressively along the chain as if on a conveyor belt In this case information means the pattern of ones and zeros in the register Pseudo Random Sequence Generation When generating pseudo random binary sequences the shift register operates in a closed loop condition and the input to the first stage is supplied via a feedback path from later stages of the shift register Fig 8 shows a simple form of pseudo random sequence generator In this example only the first four of the shift register stages are actually involved in generation of the sequence Feedback to the first stage is taken from stages 3 and 4 the outputs from which are processed in an EXCLU SIVE OR gate otherwise known as modulo two adder half adder non equivalence or anti coincidence gate This gate gives output only when its two inputs are dissimilar according to the following truth table Truth Table for EXCLUSIVE OR Gate 0 1 1 1 1 1 1 0 L The sequence generated by the four stage arrange ment of Fig 8 can easily be derived For the purpose of illustration the initial contents of the first four stages are taken arbitrarily to be as follows Before Ist shif
28. lation function of the noise isn t true impulse the result will be less than perfectly accurate The accuracy of the correlator output is also affected by the correlator s averaging time Mathematically the setup of Fig 2 works as follows If the noise is x t the unknown impulse response is h t and the response of the system to the noise is y t then y t h u x t udu The cross correlation function of y t with x t is defined lim 1 Ma x t 7 Substituting for y t gives RG h u Ry 5 where R is the autocorrelation function of the noise x t If Ra 7 is a true impulse then Rye a where is the rms value of the noise x t In other words HEWLETT PACKARD JOURNAL 1967 Volume 19 Number 1 TECHNICAL INFORMATION FROW THE LABORATORIES OF THE HEWLETT PACKARD COMPANY PUBLISHED 1 Simt F 2 DOLAN L SHERGALIS SNYOEA A ERICKSON Fig 3 Autocorrelation function of pseudo random binary sequence approximates an impulse the unkown impulse response is proportional to the cross correlation function of the input noise x t with the output y t The binary pseudo random noise synthesized by the new noise generator has an autocorrelation function which while not precisely an impulse is very close to one as shown in Fig 3 What s the averaging
29. n delayed by a lack of suitable convenient test equipment Chief among the many factors responsible for this state of af fairs is that conventional noise generators employ natu tubes and s discha ral noise sources such as ga temperature limited diodes The statistics of the noise gnals produced by these sources are not very stable well defined or controllable The problem is most severe 1949 1998 Hewlett Packard Co at low audio and sub audio frequencies where much of the current interest in noise testing is focused To circumvent these deficiencies the development of a new low frequency noise generator was undertaken The result of this development program is the instrument shown in Fig 1 It is not a natural noise source it is a nerator which synthesizes noise and precision noise noise like pseudo random signals by a controllable dig ital process As a result the characteristics of its output can be specified accurately and varied to fit the measure ment situation This new measurement tool will realize its full potential only after people understand it and begin to see how they can use it to solve their problems We hope to ac celerate this process by describing how the new noise generator works and some of the things it can do se Generator synthesizes repeated pseudo random noise like random noise Binary erns l
30. o random or random binary signals by low pass filtering To give good results filter cutoff frequency must be about 1 20 of clock frequency of binary signal reduce the variance introduced by random noise Pseudo pseudo random signals even though they are periodic random noise therefore can save a great deal of time Measurements using random noise must be made in a The repeatability that pseudo random noise gives an finite time anyway so it makes no difference whether the experiment is especially valuable when parameters of signal repeats not after the measurement time is over the system being tested are varied as on an analog com puter In such tests it is important to know that changes Binary and Gaussian Noise Generated in test results are caused by parameter manipulation and The most useful and most widely used pseudo random not by statistical variance or random test signals are of two types pseudo random Because measurements using pseudo random noise or random binary two level signals and pseudo random normally made over one pattern length we lose none or random Gaussian multi level signals The Gaussian of the advantages of random signals by substituting signals are used in testing analog systems The binary Copr 1949 1998 NOTE Scales on Spectrum Plots are Logarithmic Spectrum of Binary Output Seance 348 at 0 45 1 2 Shap
31. ock frequency and is equal to f N or 1 NAT where N is the number of bits in the sequence and AT is the clock period The upper 3 dB half power frequency of the binary output is 0 45 Hence by adjusting the clock period the operator can adjust the upper 3 dB frequency of the binary signal from 0 00135 Hz to 450 kHz Regardless of what clock frequency or sequence length N is selected the binary waveform always switches between the same two amplitude levels This means that its rms value and therefore its total power is not changed by a change of bandwidth Halving the bandwidth of the noise from a natural noise source on the other hand also halves the power this is a disad vantage when very low bandwidths are needed since the power available becomes very small The power density spectrum of the purely random binary output sequence length INFINITE is continu ous i e contains no discrete harmonics it has the same shape as the envelope of the pseudo random power spectrum Gaussian Output The basic noise produced by the noise generator is a binary waveform having a nominal bandwidth to the half power point of 0 45 clock frequency While this is noise in the sense that is contains a multiplicity of fre Copr 1949 1998 Hewlett Padkard Co quency components is a two level waveform bearing little resemblance in the time domain to naturally occurring disturbances
32. rator is 3 75 except for the shortest sequences This gives an excellent fit to the Gaussian curve The crest factor of a truly Gaussian signal is of course infinite and some natural noise sources have higher crest factors than 3 75 However it is often necessary to wait a long time to be sure that one of their largest peaks has occurred With the pseudo random output of the noise generator on the other hand a definite number of the highest peaks occur in every sequence Acknowledgments Major contributions to the development of the noise generator were made by Duncan Reid Alistair MacPar land Glyn Harris Michael Perry and Richard Rex Copr 1949 1998 HaWlett Packard Copr 1949 1998 Hewlett Padkard SPECIFICATIONS HP Model 3722A Noise Generator BINARY OUTPUT Fixed Amplitude Amplitude 10V 19 when clock period 333 3 when 1 lt clock period lt 333 5 when clock period 1 us Output Impedance lt 5 1 clock period gt 333 lt 10 0 if clock period 100 Load Impedance 160 minimum Rise Time lt 100 ns ower Density Approxmately equal to clock period 200 Vi Hz at low frequency end of spectrum Power Spectrum sin form first null occurs at clock frequency and 3 4B point occurs at 0 45 x clock frequency GAUSSIAN OUTPUT Fixed Amplitude Amplitude 3 16 rms 2 when bandwidth 0 15 6 2 ban
33. rent unrelated properties of a signal Probably the most familiar pdf is the bell shaped Gaussian curve Fig 3 b which is characteristic of many naturally occurring random disturbances Gaussian means that a curve has the shape y Probability density functions must all have areas equal to one so a Gaussian pdf must be normalized i e where is the rms value of the signal It is important not to confuse the Gaussian pdf with the output of a Gaussian filter A Gaussian filter has an impulse response shaped like e and a frequency response shaped like e The output of a Gaussian filter may indeed have a Gaussian pdf But an arbitrary signal having a Gaussian pdf may have a power density spec trum which bears no resemblance to the frequency re sponse curve of the Gaussian filter It is also important to recognize that n noise does not have to be white noise and vice versa The pdf and the power density spectrum are independent Correlation Functions A statistic which is useful because it tells something about the time or phase relationship between two signals random or not is the cross correlation between them The cross correlation function for two signals x t and y t is defined as 1 2 lim T J T 2 T 2 2 A block diagram of a system which performs this cal culation approximately is shown in Fig 4 One signal
34. rom Manchester University in 1962 He spent the next three years at Edinburgh University where he worked the research team headed by Gordon Roberts He was concerned with an advanced system for real time correlation and was awarded the degree of PhD for his work in this field Brian joined HP in 1965 and was responsible for initial design work on the 3722A He 15 currently Investigating a new range of instrumentation and is working up routines for computer aided design using the HP 2116A Gordon T Roberts 1954 Gordon graduated from the University of Bangor North Wales with the degree of BS in electrical engineering This was followed by a three year period at Manchester University where he investigated problems of noise in non linear systems for this work Gordon was awarded the degree of PhD After five years of industrial work a return to more academic surroundings this time at Edinburgh University where he lectured in control theory and headed a research team investigating the uses of noise signals in systems evaluation Gordon has continued to work in these fields since he joined HP in 1965 He is now technical manager of Hewlett Packard Limited in South Queensferry Scotland The weighting networks used in the noise generator are simply resistors The resistor values are chosen such that the contributions of the outputs of successive shift register stages to the current at the summing point are
35. rst case devia tions are less than 0 020 which corresponds to about 10 Analog Filtering In analog computing applications time derivatives differentiated versions of signals occur frequently and whenever a signal has sharp edges there is the dan ger that derivatives could cause overload In the case of a boxcar waveform with its very fast transit times even the first time derivative would be a series of very large amplitude spikes which could overload the system For this reason a second order analog filter is used to remove sharp edges from the digital filter output waveform As a result neither the first nor the second Fig 16 Part of 8191 bit pseudo random Gaussian pattern Clock period is 1 ys bandwidth is 50 kHz time derivatives of the waveform yield sharp spikes The pdf for both derivatives is reasonably Gaussian see Fig 17 The analog filter cut off frequency is selected by the CLOCK PERIOD switch and is nominally 1 5th of the clock frequency that is four times the half power fre quency of the digital filter This feature is included for all clock periods commonly of interest to analog com puter users i e noise bandwidth from 50 kHz to 0 15 Hz At frequencies of 0 05 Hz and below the analog filter cut off remains at the same frequency as for the 0 15 Hz position Crest Factor of Gaussian Output The crest factor ratio of peak to rms values of the Gaussian output of the noise gene
36. st stage of the shift register is controlled by a semicon ductor noise source giving a truly random output signal Just before each shift pulse the random signal is sampled by a level detector which decides on arrival of the shift Fig 13 Single pulse response of digital filter is a discrete step approxi mation to sin x x shaped impulse response of ideal low pass filter Copr 1949 1998 Hewlett Padkard Co pulse whether a or a 0 is to be placed in first stage of the register Since the random signal is non periodic there is no repeated pattern in the resulting series of ones and zeros from the register The power density spectrum of the random signal is continuous and has the same shape as the envelope of the power spectrum of the pseudo random signal Digital Low pass Filter A linear filter having an impulse response h t and input x t has an output gt x t u 1 ao A finite sum approximation to this integral can be synthesized using a delay line Fig 11 shows a filter composed of a delay line a number of multipliers or weighting networks and a summing amplifier The out put of port of the delay line is x t jAT where x t RELATIVE RESPONSE 48 LOG FREQUENCY is the input and AT is the delay between ports The sum ming amplifier output is then n y t a x t jAT 2 i If a and if n is sufficiently large the sum
37. t are similar to those of random noise but because they are synthesized their statistics are much easier to control Most important is the fact that if the measurement time T is made exactly equal to the length of one pseudo random pattern the results of an experiment will be identical on every repetition as long as nothing else has changed There is no statistical variance This means that it isn t necessary to use a long measurement time be cause the reason for the long measurement time was to good reference on pseudo random signals is Korn Random Process Simu lation and Measurements New York McGraw Hill Book Company 1966 x t Autocorrelation y t Cross correlation Approximate Correlation xt Function LA Fig 4 Correlation functions show time relationships between signals They can be com puted by multiplying one signal by a delayed version of the other and averaging the product Copr 1949 1998 Co ms 10 100pF Time Constant las 10 0 001 H Time Constant 108 Part of 2047 Bit Pseudo Random Binary Sequence Clock Period 3 33 Sweep Rate 10 10 Time Constant 200 Fig 5 Pseudo random or random Gaussian signals can be derived from pseud
38. t pulse 0 waiting to Into stage 1 receipt of shift pulse The modulo two sum of the outputs from the last two stages is 0 this can be written 0 0 At the first shift pulse the in the first stage is transferred to the second and is replaced by the 0 in the feedback line This gives the pattern After Ist pulse Again the modulo two summation yields 0 The next pattern is therefore After 2nd pulse With this pattern the outputs from the third and fourth stages are dissimilar so the modulo two sum is 1 The thus placed in the feedback line will enter the first stage on arrival of the next shift pulse The remainder of the sequence be worked out in a similar manner After the 14th pulse the register pattern 15 After 14th pulse a Copr 1949 1998 Hewlett Packard The fifteenth pulse restores the register to the initial state 1000 and thereafter the sequence repeats With the exception of 0000 the register generates the maximum number of 1 and 0 combinations possible with four stages The all zero condition cannot arise if it were to occur all stages of the shift register would remain in the 0 state and the output would thereafter be an infinite sequence of zeros The pattern appearing at the output from the first stage is exactly the same as that from the second the third and the fourth and so on throug
39. time T for the correlation system only needs to be as long as one period of the pseudo random waveform i e as long as one complete pseudo random pattern Unlike random noise pseudo random noise introduces sta tistical variance into the results as long as the averaging time T is exactly one pattern length Calibration Research Training Other uses of the noise generator include research in communication biomedical engineering seismology underwater sound PCM calibration of true rms voltmeters spectrum analyzers and other low frequency test equipment e g the pseudo random signal generates a comb of frequencies useful for checking wave analyzers student familiarization with random signal theory and the behavior of systems with noise inputs It will be interesting to see how this list grows as the potential of controllable repeatable noise becomes more widely realized PAGE MILL ROAD PALO ALTO CALIFORNIA Copr 1949 1998 Hewlett Packard Co
40. y to wait a long time to be sure a certain number of peaks have occurred process control system evaluation Process control sys tems can be tested for their responses to random fluctuations in the controlled variables e g tempera ture pressure flow concentration Pseudo ran dom signals are helpful here because they do not introduce statistical variance into the results Measure ments are completed in the time required for only one pseudo random pattern This is especially important in low speed systems which might have to be tied up for hours if truly random noise were used as a test signal Pseudo random noise is also especially useful in testing large systems As a system ts bigger it harder to test on a lab bench Eventually it must be tested under working conditions A good example is an airplane which in the end must be tested in flight Pseudo random noise can speed these tests for the same reasons given above under process control sys tem evaluation limited time situations Pseudo random noise is better than random noise when the situation to be measured exists only for a short time e g a missile during blastoff Again this is because measurements that use pseudo random noise are made over only one pattern length and no statistical variance is introduced into the results by the noise Fig 1 Model simu of tall structure Noise driven shake table simulates ground distur
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