Signals and Systems - Electrical Engineering

11.2 Frequency-Selective Discrete Filters 645

|H

( 2

e

jω

)|

<H

( 2

e

jω

)

0 1 2 3

0.5

1

1.5

2

ω

|H

( 1

je

ω)

|

<H

( 1

je

ω)

0 1 2 3

−0.6

−0.4

−0.2

0

0.2

ω

0 1 2 3

0

0.2

0.4

0.6

0.8

1

ω

0 1 2 3

− 2

0

2

4

ω

(a)

(b)

FIGURE 11.4
Magnitude and phase responses of an IIR filter with a transfer function of (a)H 1 (z)= 1 /( 1 −0.5z−^1 ), and of an
FIR filter with a transfer function of (b)H 2 (z)=(z− 1 ej2.09)(z− 1 e−j2.09)/ 3 z. Notice the phase responses are
nonlinear.

Thus, the first is an IIR filter and the second is an FIR filter (notice that this filter is noncausal as it

requires future values of the input to compute the present output). The phase responses of these

filters are clearly nonlinear. The transfer functionH 2 (z)has zeros on the unit circlez= 1 e±j2.09,

making the phase of this filter not continuous and so it cannot be unwrapped. Figure 11.4 shows

the magnitude and the phase responses of the filtersH 1 (z)andH 2 (z). n

nExample 11.2

A simple model for the multipath effect in the channel of a wireless system is

y[n]=x[n]−αx[n−N 0 ] α=0.8,N 0 = 11