7.1. DIG ITAL FILTER RE S P O N S E 1277.1 Digital Filter R e sponseA digital filter can be described in several ways. These include, but are notlimited to, difference equations, bloc k diagram, impulse response, and thesystem function. Eac h m odel is useful in the description of systems and theirbeha vior, and they are all related. They pro v ide differen t views of the sameentity.A good place to start is with a difference equation. The difference equa-tion is an important m odel in modeling rando m processes as well as in de-scribing digital systems. It is a completely general model of a time-invar iantdiscrete linear system. We will develop difference equations as a digital sys-tem model and then relate that approach to other commo n models.After w e ha ve developed the description tools, we will show how they canbe used to model random processes b y using to construct random processeswith va rious properties. These w ill enable us to em ply various analyticaltools on processes with known properties so that w e can gain insight in to theusefulness and limitation of eac h tool. This will prepare us for encounterswith random processes with unknown structures.7.1.1 Difference Equation M odelLet the input sequence to a digital filter be denoted b y x(n) and the corre-sponding response b y y(n). The curren t output value can depend upon thecurrent input v a lue, past input values and past output values. This can beexpressed asy(n)+a1y(n − 1) + a2y(n − 2) + ···+ apy(n − p) (7.1)= b0x(n)+b1x(n − 1) + ···+ bqx(n −q)The differ ence equation is completely described b y the coefficien ts {1,a1,...,ap}and {b0,b1,...,bp}. Note that we will alwa ys assume (without loss of gener-alit y) that a0=1. The abo ve equations can be solv ed for the curren t outputvalue.y(n)=b0x(n)+b1x(n − 1) + ···+ bqx(n − q) − a1y(n − 1) (7.2)−a2y(n − 2) −···− apy(n − p)A bloc k diagram of this equation is shown in Figure 7.1 on page 128. Thestorage elemen ts in the top row preserve a history of the past inputs andthose in the bottom ro w preserv e a history of the past outputs.128Figure 7.1: Bloc k diagram of a digital filter realization of the differenceequantion.Example 7.1.1Sin g le Po le Filter This filter is characterize d by a simpledifference e qu ationy(n)+ay(n − 1) = x(n) (7.3)To see its b ehavior, consider the simple input se quence [···0, 1, 0, 0, 0, 0 ···]which has a single nonzero input value. Without loss of generality we canchoose the index origin so that x(0) = 1 and x (n)=0for n 6=0.Thecurrentoutput c an be written in terms of the current input and the immediate pastoutput.y(n)=x(n) − ay(n − 1)This equatio n is illustrated by the block diagram in Figure 7.2 on page 129.We will assum e the initial state y(−1) = 0. Then it is a simple matter towrite out a few of the values of the output sequence beginning with n =0.n0 1 2 3 4y(n) 1−aa2-a3a4By direct substitutio n into the difference equation you can show that the gen-eral solution for this input and initial condition is y(n)=−(a)nfor n ≥ 0.The output is clearly b ounded if and only if |a| ≤ 1. We will see later when7.1. DIG ITAL FILTER RESPONSE 129Figure 7.2: Block diagram of a single pole filter. Th e system stores one value,the immed iately preceeding output. This value establishes the system state.we discuss the system function that this system has a “single pole” whoselocation is determined by a, and which is stab le of the “pole” is inside theunit circle.This exam p le illustrates the use of an impulse input to gain insight in tothe behavior of a system. T he imp ulse response can be used as a general toolfor this purpose.Impulse ResponseThe im pulse response of a digital filter is the sequ ence t hat is gen erated w h enthe input sequence is x(n)=δ(n),whereδ(n)=½0,n6=01,n=0(7.4)The response to this input sequence is traditionally referred to as h(n).Theimpu lse response is sufficien t to provide a complete descrip tion of the be-ha vior of an y linear time-invariant system, and is therefore equivalent to thedifference equ antion description or the block diagram . The response of asystem to a general input sequence x(n) isy(n)=∞Xm=−∞h(m)x(n − m) (7.5)Clearly, y(n)=h(n) when x(n)=δ(n).130Figure 7.3: Block diagram of a finite impulse response digital filter. Theresponse of a single input sample persists for q additional steps.A special class of systems has an impulse response of finite duration.These systems are characterized by the absence of a feedback path. Thatis, a1= a2= ··· = ap=0. The block diagram of a finite impulse response(FIR) system is show n in Figure.7 .3.The impulse response of the FIR filter ish(n)=0,n<0b(n), 0 ≤ n ≤ q0,n>q(7.6)This relationship prov id es one means to realize any impulse response thatone may need. The algorithm m ust store as man y of the past inputs asnecessary to provide a good approximation through use of a finite n umber ofcoefficients. The practical issue is the size of the mem ory, determined by q,that is practical.FIR App licat ion sThe FIR filter is o ften used to smooth a random process to suppress noise andbring out a slower-varying signal. This is sho w n in the first example belo w.Another application is in the detection of a signal in a noisy background withamatchedfilter. We will look briefly at these applications.Example 7.1.2Weighted A verage Suppose that we would like to smootha waveform by averaging over several values of the input. We may want to7.1. DIG ITAL FILTER RESPONSE 131reduce the weight given to input values that ar e farther in the past. Let r beanumberintherange0 ≤ r ≤ 1. Let the impulse response b e defined byh(n)=0,n<0rn, 0 ≤ n ≤ q0,n>q(7.7)Then the output, using (7.5), isy(n)=qXm=0rmx(n − m) (7.8)If r =1we give equal weight to the q past outputs and if r<1 we giveless weight to those farther in the past. This can be effective in reducing theamount of noise on a slowly varying signal, as shown in Figur e??on page??. In the example shown in the figure the signal is a sine wave that has aperiod of 100 samples. The noise can be reduced by averaging over an intervalof about 10 samples without seriously distorting the signal.Ma tched FilterAFIRfilter can also be used for an operation calledmatche d filte r in g. Suppose that one wants to detect the presence of a wave-form s(n) that is of some finite duration, but that it is observed in the pres-ence of