1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

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9.1 • THE Z..TRANSFORM 345

Solution Let both Xn =land Yn = 1 be the unit step sequence, and X(z) =

2

Y(z) = (z:i)· Then W(z) = X(z)Y(z) = (•:l)'' so that Wn is given by the

convolution

n n

Wn = Xn * Yn = L XiYn-i = L 1 = n + 1.

i=O i=O

9.1.5 Application to Signal Processing

Digital signal processing often involves the design of finite impulse response (FIR)

filters. A simple 3-point FIR filter can be described as

y [n] = x[n] +a. x{n - l] + b x[n - 2]. (9-4)

Here, we choose real coefficients a and b so that the homogeneous difference

equation

x[n] +a. x[n - l] + b x[n - 2] = 0 (9-5)

has solutions x[n] = cos(w7rn) and x[n] = sin(wirn). That is, if the linear com-

bination x(n] = c 1 cos(w7rn) + c2sin(wirn) is input on the right side of the FIR

filter equation, the output y[n] on the left side of the equation will be zero.

Sequence z-transform

Definition Xn = x{n] X(z) = I:::"=oXnZ- n

Addition Xn+Yn X(z) + Y (z)

Constant multiple CXn cX(z)

Linearity CXn + dy,, cX(z) + dY(z)

Delayed unit step u [n-m]

z t-m

z-1

Time delay 1 tap Xn-1u[n - 1] ~X(z)

Time delayed shift Xn-mu[n-m] z -mx(z)

Forward 1 tap Xn+J z (X(z) - xo)

Forward 2 taps Xn+2 z^2 (X(z) - xo -x1z-^1 )

Time forward Xn+m Z m(x() Z - L:m-1 i=O X;Z -i)

Complex translation e""Xn X (ze- ")

Frequency scale b"xn x (~)

Differentiation nxn - zX '(z)

Integration ;Xn l -f Xiz) dz

Integration shift n+n:;l Xn -z- m J ;.~~ dz

Discrete-time convolution Xn * Yn = L:7=o X;Yn-i X(z)Y(z)

Convolution with Yn = 1 L:~;QXi z:J X(z)

Initial time XO limz-oo X(z)

Final value lilll.n-oo Xn lim,._ 1 (z - l)X(z)

Table 9.2 Some properties of the z-tra.nsform.