Signals and Systems - Electrical Engineering

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

Or the pair

nx[n]u[n] ⇔ −z

dX(z)

dz

(9.18)

For instance, ifX(z)= 1 /( 1 −αz−^1 )=z/(z−α), we find that

dX(z)

dz

=−

α

(z−α)^2

That is, the pair

nαnu[n] ⇔

αz

(z−α)^2

indicates that double poles correspond to multiplication ofx[n]byn.

The above shows that the location of the poles ofX(z)provides basic information about the signal

x[n]. This is illustrated in Figure 9.3, where we display the signal and its corresponding poles.

9.4.3 Convolution Sum and Transfer Function

The most important property of the Z-transform, as it was for the Laplace transform, is the

convolution property.

The outputy[n]of a causal LTI system is computed using the convolution sum

y[n]=[x∗h][n]=

∑n

k= 0

x[k]h[n−k]=

∑n

k= 0

h[k]x[n−k] (9.19)

wherex[n]is a causal input andh[n]is the impulse response of the system. The Z-transform ofy[n]is the

product

Y(z)=Z{[x∗h][n]}=Z{x[n]}Z{h[n]}=X(z)H(z) (9.20)

and the transfer function of the system is thus defined as

H(z)=

Y(z)

X(z)

=

Z[output y[n]]

Z[input x[n]]

(9.21)

That is,H(z)transfers the inputX(z)into the outputY(z).

Remarks

n The convolution sum property can be seen as a way to obtain the coefficients of the product of two polyno-

mials. Whenever we multiply two polynomials X 1 (z)and X 2 (z), of finite or infinite order, the coefficients

of the resulting polynomial can be obtained by means of the convolution sum. For instance, consider

X 1 (z)= 1 +a 1 z−^1 +a 2 z−^2