Signals and Systems - Electrical Engineering

548 C H A P T E R 9: The Z-Transform

To find the poles and the zeros ofX(z), given the coefficients{b[k]}and{a[k]}of the numerator and

the denominator, we use the MATLAB functionroots. To get a plot of the poles and the zeros ofX(z),

the MATLAB functionzplane, with inputs the coefficients of the numerator and the denominator of

X(z), is used (conventionally, an’x’is used to denote poles and an’o’for zeros).

Two possible approaches can now be used to compute the inverse Z-transformx[n]. We can com-

pute the inverse (below we call itx 1 [n] to differentiate it from the other possible solution, which

we callx[n]) by using the information on the partial fraction expansion (the residuesr[k]) and

the corresponding poles. An alternative is to use the MATLAB functionfilter, which considersX(z)

as a transfer function, with the numerator and the denominator defined by thebandavectors

of coefficients. If we assume the input is a delta function of Z-transform unity, the functionfilter

computes as output the inverse Z-transformx[n] (i.e., we have trickedfilterto give us the desired

result).

The following script is used to implement the generation of the terms in the numerator and the

denominator to obtain the corresponding coefficients, plot them, and find the inverse in the two

different ways indicated above. For additional help on the functions used here usehelp.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Two methods for inverse Z-transform

%%%%%%%%%%%%%%%%%%%%%%%%%%%%

p1 = poly(0.5); p2 = poly(-0.5); % generation of terms in denominator

a = conv(p1,p2) % denominator coefficients

z1 = poly(0); z2 = poly(-1); % generation of terms in numerator

b = conv(z1,z2) % numerator coefficients

z = roots(b) % zeros of X(z)

[r,p,k] = residuez(b,a) % partial fraction expansion, poles and gain

zplane(b,a) % plot of poles and zeros

d = [1 zeros(1,99)]; % impulse delta[n]

x = filter(b,a,d); % x[n] computation from filter

n = 0:99;

x1 = r(1)∗p(1). ˆ n + r(2)∗p(2). ˆ n; % x[n] computation from residues

a = 1.0000 0 -0.2500

b = 1 1 0

z = 0

-1

r = 1.5000

-0.5000

p = 0.5000

-0.5000

Figure 9.9 displays the plot of the zeros and the poles and the comparison between the inversesx 1 [n]

andx[n] for 0≤n≤99, which coincide sample by sample.