CHAPTER Properties of Convolution - Analog Devices

1 123 CHAPTER7 EQUATION 7-1 The delta function is the identity forconvolution. Any signal convolved witha delta function is left [n](*[n]'x[n] Properties of ConvolutionA linear system's characteristics are completely specified by the system's impulse response, asgoverned by the mathematics of Convolution . This is the basis of many signal processingtechniques. For example: Digital filters are created by designing an appropriate impulseresponse. Enemy aircraft are detected with radar by analyzing a measured impulse suppression in long distance telephone calls is accomplished by creating an impulseresponse that counteracts the impulse response of the reverberation. The list goes on and CHAPTER expands on the Properties and usage of Convolution in several areas. First, severalcommon impulse responses are discussed. Second, methods are presented for dealing withcascade and parallel combinations of linear systems.)

2 Third, the technique of correlation isintroduced. Fourth, a nasty problem with Convolution is examined, the computation time can beunacceptably long using conventional algorithms and computers. Common Impulse ResponsesDelta Function The simplest impulse response is nothing more that a delta function, as shownin Fig. 7-1a. That is, an impulse on the input produces an identical impulse onthe output. This means that all signals are passed through the system withoutchange. Convolving any signal with a delta function results in exactly thesame signal. Mathematically, this is written:This property makes the delta function the identity for Convolution . This isanalogous to zero being the identity for addition , and one being the(a%0'a)identity for multiplication . At first glance, this type of system(a 1'a)The Scientist and Engineer's Guide to Digital Signal Processing124 EQUATION 7-2A system that amplifies or attenuates hasa scaled delta function for an impulseresponse.

3 In this equation, k determinesthe amplification or [n](k*[n]'kx[n]EQUATION 7-3A relative shift between the input andoutput signals corresponds to an impulseresponse that is a shifted delta variable, s, determines the amount ofshift in this equation. x[n](*[n%s]'x[n%s]may seem trivial and uninteresting. Not so! Such systems are the ideal fordata storage, communication and measurement. Much of DSP is concernedwith passing information through systems without change or 7-1b shows a slight modification to the delta function impulseresponse. If the delta function is made larger or smaller in amplitude, theresulting system is an amplifier or attenuator, respectively. In equationform, amplification results if k is greater than one, and attenuation resultsif k is less than one:The impulse response in Fig. 7-1c is a delta function with a shift. This resultsin a system that introduces an identical shift between the input and outputsignals.))

4 This could be described as a signal delay, or a signal advance,depending on the direction of the shift. Letting the shift be represented by theparameter, s, this can be written as the equation:Science and engineering are filled with cases where one signal is a shiftedversion of another. For example, consider a radio signal transmitted froma remote space probe, and the corresponding signal received on the time it takes the radio wave to propagate over the distance causes adelay between the transmitted and received signals. In biology, theelectrical signals in adjacent nerve cells are shifted versions of each other,as determined by the time it takes an action potential to cross the synapticjunction that connects the 7-1d shows an impulse response composed of a delta function plus ashifted and scaled delta function. By superposition, the output of this systemis the input signal plus a delayed version of the input signal, , an are important in many DSP applications.

5 The addition of echoes is akey part in making audio recordings sound natural and pleasant. Radar andsonar analyze echoes to detect aircraft and submarines. Geophysicists useechoes to find oil. Echoes are also very important in telephone networks,because you want to avoid 7- Properties of Convolution125 Sample number-2-10123456-2-1012 Sample number-2-10123456-2-1012 Sample number-2-10123456-2-1012 Sample number-2-10123456-2-1012 FIGURE 7-1 Simple impulse responses using shifted and scaled delta IdentityThe delta function is the identity forconvolution. Convolving a signal withthe delta function leaves the signalunchanged. This is the goal of systemsthat transmit or store signals. b. Amplification & AttenuationIncreasing or decreasing the amplitudeof the delta function forms an impulseresponse that amplifies or attenuates,respectively. This impulse response willamplify the signal by ShiftShifting the delta function produces acorresponding shift between the inputand output signals.

6 Depending on thedirection, this can be called a delay oran advance. This impulse responsedelays the signal by four EchoA delta function plus a shifted andscaled delta function results in an echobeing added to the original signal. Inthis example, the echo is delayed by foursamples and has an amplitude of 60% ofthe original OperationsConvolution can change discrete signals in ways that resemble integration anddifferentiation. Since the terms "derivative" and "integral" specifically referto operations on continuous signals, other names are given to their discretecounterparts. The discrete operation that mimics the first derivative is calledthe first difference. Likewise, the discrete form of the integral is called theThe Scientist and Engineer's Guide to Digital Signal Processing126 EQUATION 7-4 Calculation of the first difference. Inthis relation, is the original signal,x[n]and is the first difference.

7 Y[n]y[n]'x[n]&x[n&1]Sample number-2-10123456-2-1012 Sample number-2-10123456-2-1012 FIGURE 7-2 Impulse responses that mimic calculus First DifferenceThis is the discrete version of the firstderivative. Each sample in the outputsignal is equal to the difference betweenadjacent samples in the input signal. Inother words, the output signal is theslope of the input Running SumThe running sum is the discrete versionof the integral. Each sample in theoutput signal is equal to the sum of allsamples in the input signal to the that the impulse response extendsto infinity, a rather nasty sum. It is also common to hear these operations called the discretederivative and the discrete integral, although mathematicians frown whenthey hear these informal terms used. Figure 7-2 shows the impulse responses that implement the first difference andthe running sum. Figure 7-3 shows an example using these operations. In 7-3a, the original signal is composed of several sections with varying this signal with the first difference impulse response produces thesignal in Fig.

8 7-3b. Just as with the first derivative, the amplitude of eachpoint in the first difference signal is equal to the slope at the correspondinglocation in the original signal. The running sum is the inverse operation of thefirst difference. That is, convolving the signal in (b), with the running sum'simpulse response, produces the signal in (a).These impulse responses are simple enough that a full Convolution program isusually not needed to implement them. Rather, think of them in the alternativemode: each sample in the output signal is a sum of weighted samples from theinput. For instance, the first difference can be calculated: That is, each sample in the output signal is equal to the difference between twoadjacent samples in the input signal. For instance, . Ity[40]'x[40]&x[39]should be mentioned that this is not the only way to define a discretederivative.

9 Another common method is to define the slope symmetricallyaround the point being examined, such as: . y[n]'(x[n%1]&x[n&1])/2 CHAPTER 7- Properties of Convolution127 FIGURE 7-3 Example of calculus-like operations. Thesignal in (b) is the first difference of thesignal in (a). Correspondingly, the signal is(a) is the running sum of the signal in (b).These processing methods are used withdiscrete signals the same as differentiationand integration are used with Original signalb. First differenceAmplitudeAmplitudeEQUATION 7-5 Calculation of the running sum. In thisrelation, is the original signal, and x[n]y[n]is the running [n]'x[n]%y[n&1]Using this same approach, each sample in the running sum can be calculatedby summing all points in the original signal to the left of the sample's instance, if is the running sum of , then sample is found byy[n]x[n]y[40]adding samples through . Likewise, sample is found by addingx[0]x[40]y[41]samples through.

10 Of course, it would be very inefficient to calculatex[0]x[41]the running sum in this manner. For example, if has already beeny[40]calculated, can be calculated with only a single addition:y[41]. In equation form:y[41]'x[41]%y[40]Relations of this type are called recursion equations or differenceequations. We will revisit them in CHAPTER 19. For now, the important ideato understand is that these relations are identical to Convolution using theimpulse responses of Fig. 7-2. Table 7-1 provides computer programs thatimplement these calculus-like Scientist and Engineer's Guide to Digital Signal Processing128100 'Calculation of the running sum110 Y[0] = X[0]120 FOR I% = 1 TO N%-1120 Y[I%] = Y[I%-1] + X[I%]130 NEXT I%100 'Calculation of the First Difference110 Y[0] = 0110 FOR I% = 1 TO N%-1120 Y[I%] = X[I%] - Y[I%-1]130 NEXT I%Table 7-1 Programs for calculating the first difference and running sum.