Digital communication systems and filter banks - Signal Processing and Optimization for Transceiver Systems - page 108

j ω

X ( e )

(a)

ω

−π −2π/3 2π/3 π

0

j ω

Y ( e )

(b)

ω

−π π

0

s ( n ) r ( n )

(c)

x ( n ) y ( n )

H ( z )

L M

Figure P3.4.

3.5. Nyquist property and power complementarity. Consider a filter F ( z )writ-
ten in polyphase form F ( z )= M− 1
k =0

z −k R k ( z M ) .

1. Show that

M− 1

[ F ( z ) F ( z )] ↓M =

R k ( z ) R k ( z ) ,

k =0

where the tilde notation F ( z )isasdefinedinSec. 1.6.
2. Let G ( e jω )= F ( e jω ) | 2 . Show that G ( z ) has the Nyquist( M ) property,

|

that is,

g ( Mn )= δ ( n ) ,
if and only if the polyphase components of F ( z )arepowercomple-
mentary, that is, they satisfy the property

M− 1

R k ( e jω ) | 2 =1

| (P3 . 5)

k =0

for all ω.

3.6. Sum of allpass filters. Consider two filters H 0 ( z )and H 1 ( z ) defined as
follows:

H 0 ( z )= A 0 ( z )+ A 1 ( z )
2

H 1 ( z )= A 0 ( z )

− A 1 ( z )

, ,

2

A k ( e jω )

where A 0 ( z )and A 1 ( z ) are allpass filters, that is, =1forall ω.

Show that H 0 ( z )and H 1 ( z ) are power complementary, that is,

| |

H 0 ( e jω ) | 2 +

|

| 2 =1forall ω. Note also that H 0 ( z )and H 1 ( z ) are allpass com-
plementary, that is, H 0 ( z )+ H 1 ( z ) = allpass.

H 1 ( e jω ) |