Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 49

The other way of getting the discrete transfer function results from discrete
convolution. It is known that convolution in frequency domain results in a product
of transforms. Calculating Z transform of both sides of Eq. 4.32 gives:
Y ð z Þ¼ H ð z Þ X ð z Þ;

ð 4 : 38 Þ

where X(z), Y(z) are Z transforms of the input and output signals, and H(z)isa
discrete transfer function of the system being Z transform of its impulse response:
H ð z Þ¼ X

1

h ð k Þ z k : ð 4 : 39 Þ

k ¼ 0

Example 4.14 Determine the discrete transfer functions of the systems described
by the following difference equations:

(a) y ð n Þ¼ ax ð n Þ

(b) y ð n Þ¼ x ð n 1 Þ

(c) y ð n Þ¼ b 0 x ð n Þþ b 1 x ð n 1 Þþ b 2 x ð n 2 Þ

y ð n Þ¼ x ð n Þþ a 1 y ð n 1 Þþ a 2 y ð n 2 Þ

(d)

Solutions Calculating transforms of the difference equations (assuming zero initial
conditions in all cases), after simple rearrangements the following transfer func-
tions are obtained:

(a) H ð z Þ¼ a

H ð z Þ¼ z 1

(b)

H ð z Þ¼ b 0 þ b 1 z 1 þ b 2 z 2 (c)

H ð z Þ¼

1
1 a 1 z 1 a 2 z 2

(d)

Frequency Response
For the input signal of a discrete system being a phasor expressed as x ð n Þ¼
exp ð jnxT S Þ; where exp ð jnxT S Þ¼ cos ð nxT S Þþ j sin ð nxT S Þ; applying the convo-
lution ( 4.32 ) yields:
y ð n Þ¼ X

1

h ð k Þ x ð n k Þ¼ X

1

h ð k Þ exp ½ j ð n k Þ xT S

k ¼ 0 k ¼ 0

¼ exp ð jnxT S Þ X

1

h ð k Þ exp ð jkxT S Þ; ð 4 : 40 Þ

k ¼ 0

where h(k) is a impulse response of the system.

The second part of ( 4.40 ) is just the frequency response, describing the output
signal when the input is a phasor of given frequency. The frequency response can
thus be given by:

H ð jx Þ¼ X

1

h ð k Þ exp ð jkxT S Þ: ð 4 : 41 Þ

k ¼ 0