Cha?ter 2 BASICS OF SIGNAL PROCES^STNG 2.1. INTRODUCTION The data arising from an electroPhysiological experiment on the nervous system initially consist of records in continuous ana- log form of stimuLus events and the responses that they give rise to. If these data are to be analyzed in more than a qualitative way, digital computation techniques are usually called for' This means that the analog data have first to be converted to digital ' sampled form. Then the full range of analysis techniques that have been developed to study dynamic processes can be br:ought to bear' These include filtering, averaging, spectral analysis, and covari- ance analysis. In this chapter we discuss first the properties of the analog-to-digital conversion processes with particular re- gard to their effect on the experimental data, and the subsequent of tests the data are subJectecl to. I'lten we move to a discussion filtering operations, analog and digital, with emphasis on the latter and how it fits into computer data analysis procedures' From time to time we consider some of the hardware aspects of filtering since familiarity \"rith them is quite useful for a fuller comprehension of filtering procedures.' ANA],OG-TO-DIGITAL CONVERS ION An analog-to-digital converter (ADC) converts a continuous signal into a sequence of ?- and l'l-dj-screte measurements' The two steps of time sarnpling and amplitude quantizing are usual-ly performed in a combined procedure. The ADC is first given the of command to sample by the computer and then holds the ampl'itude this sample briefly while quantlzing it. We illustrate the ADC in Fiq. 2.I as performing its operations in the sequence PRINCIPLESOF NEUROBIOLOGICALSIGNAL ANALYSIS BASICSOF SIGNALPROCESSING (o) sequence of maintained vol-tage Ievefs lasting the duration between The amplitude of each level- is the SAMPLE sanpling times, Fig. 2.1(b). COMMAND REAO OUT COMMANO signal amplitude at the sampling j-nstant t'A. In \4rhat follows, we assume A to be unity so that toA can be replaced by the integer valued time variable to. Sampling devices are often referred to as sample-and-hold circuits because of their ability to hold the san- pled value without significant decay until quantization has been completed--a time duration that is often considerably shorter than the interval between samples. In a nunber of experimental situations in which a response to a stimulus is being analyzed, the instrumentation is organized so that Lhe stimulator is triggered by the same pulse that initiates A-D conversion of the data. This insures that there will be no jitter (random variation in tirne) or asynchrony between the onset of the stimu.Lus and the data sampling instants. That is, sampling always occurs at fixed delays from stimul-us onset. If' on the other hand, the stimulator is driven independently of the ADc and notj-- fLes that device when to initiate sampling, jitter of the sampling instants can occur and tend to result in some temporal smearing of the digitized data. The jitter effect will be small when the cycle time of the computer is small compared with the sampling interval. Here we j-gnore the effects of iitter in A-D conversion- Fig. 2.7. The sampled signal xa(t) is then quantized to yield an output (a) Block diagram st-ructure of an A_D conyerEer. sampLing is initiated periodicalTg. euantization of the sampJe is x (t") which can take on only a fimited ntunber of, usuallyr urli- foLLowed o' bg coding it into digitar format. when this is compr-ete quan- a read-out conunand causes formly spaced values. The input-output relationship for the deliverg of the converted signaT to the data processor. (b) fne signal s(t) before sampling una it" tizer is shown in rig. 1(c). The quantization step is g volts in sanpl-e.d version su(t). (c) The input-output relation for the the input is greater than O quantJ-zer. ?he step size is q. amplitude. The output is O as long as and no larger than g; it is g as long as the input is greater than indicated there. The organization of the converter is not i.ntended g and no larger than 29 and so on. In equation form, the input- to describe a particular type of ADc, but to ilfustrate the function = output relationship is, at integral values of t = to (with A t) of such a device. In addition, the data analysis problems we are concerned with do not depend upon the detailed circuitry linking the computer x (t") mq< x_ (to) < (m + I)9, (2.r) to the ADC or upon the structuraL features of the con- l'l :, verter itsel_f. The sampled version of the signal i,s x.(t) t a xu(t") <-Mq=-Q 56 57 BASICSOF SIGNALPROCESSING PRINCIPLESOF NEUROBIOLOGICALSIGNAL ANALYSIS produce peak value limiting at The ing step but small enough not to maximum and minimurn voltage levels that can be handled hrithout these reasonable assumptions the saturation are -e any time in the converter. Under e and and the totaf nrnnber of levels 2!t that the well: (l) the quanLizing error output signal following statements hold reasonably can take on is usua.j.l-y some integer power .L of 2: neighbors; of a sample is uncorrelated \{ith that of its sequential 2u=2L (2.2) ffi a sample (2) the probability density function for the err^orr zq of The degree of precisi.on of an A-D conversion is referred to in g' That is' it is #' is uniformly distributed over the interval 0 to terms of the nunber of bits in the output word of the converter. in this gF#, equally 1ike1y that the magnitude of the error be anywhere A fO-bit converter will quantize vortages between -r and +1 vort &I (2) quantization rule of Eq' (2'l) ' i6I range. From assumption and the into one of IO24 1evels each of whose magnitude *;.r. is 1.952 mV_ tu1 is a bias term' fiK the nean value of the quantizing noise is q/2' This The final step in the conversion is to code {g xn(t") (on1y the +tr' 2 rq/2 2 values of x- at the sampling times are important) *8. = (2.4) q into a form varrzn)= "i a" f; acceptabfe for use by J _-nr, the digital computer. Most often this means ffi implies that EhaE xq(t"), whether J* The lack of correlation between sample errors positive or negative, is represented in binary {4}, given by form, .L binary the autocovariance function for the noise is digits being adequate for this. Tlpically, one coded :8*: = output line q21L2, for ro o is assigned to each binary digit and the value of the ffi (2.5) c (to) = voltage on this line at the read-out time indicates 2Z other\^tise whether that H 9Iq o, binary digit is a 1 or a O. The time for excluding the dc bias term' is both sampling and read_ Sr'b The power spectnrm of the noise, out are determined by a clock contained within the computer. s, flattoF=I/2.Toseethis,supposethedataconsistoflv "rnterrupt" features of signal of the computer assure that the incoming data samples of the signal and that we assume the combination are accepted ffi = = after each quantization has been performed. 49.' be periodic with period ? IVA N' The substitution &i and noise to ffi' of Eq. (2.5) into Eq. (I.23) results in spectral terms Crn"n(ilt 2.3. QUANTTZATTONNOrSE #; which are all equal and independent of n' This is because = (to) is different from 0 only when to 0' Thus the quantiz- Each conversion has associated with it a discrepancy between # c-'q'q - the N/2 frequency components the quantized and the true val-ue ing noj.se is equally divided among all of the signal. It is useful to :€*W O and N/2: consider this error as a form of noise, catled quantizing noise, i$i between zn(t"L). we can ,'{..d:' (2.6' then write cz (n, = q2 1r2w, O<n<N/2 z = xn(t"A) x(toA) + zn(t"L) (2.3) It YY incoming b{ The ADC converter thus adds noise of its own to the zn is linited in absolute va.Lue to r/2 the size quantizing of the f-c-:. deter- Tr signal, a noise whose covariance and spectral properties are step g. (The properties of quantizing noise in ':!i . the uppermost and quantization' ml-ned solely by the sampling rate and the fineness of Lowermost quantizing level-s are different but'do f:e'' riot substantiarry it is Although quantizing noise has the appearance of being randorn' alter this anarysis.) I^le assume the incoming signal to be a random :!i:. this best to remember that this is not entirely so' To iLIustrate one that is band l-inited to F = t,/2 such that A = 1. This means $i point, suPpose the incoming signal were a repetitive wave synchron- that sampling is done at the Nyquist rate. We also assime that period' Samples ized exactly to some multiple of the sampling the signalrs amplitude is large compared to the size of a quantiz- rlt . '.i: PRINCIPLESOF NEUROBIOLOGICALSIGNAL ANALYSIS BASICSOF SIGNALPROCESSING this can be taken of the waveform during each period at the same time relative response detection being secondary. To find how well the peaks in to the beginning of a period wilJ- always produce the same quantiz- done, it is necessary to kno$/ how fine, relative to must be' when ing error and this would not be removable by the process of averag- the amplitude distribution, the quantization steps signal are avai]- ing over successive waveform repetitions.