The lazy precoder with a zero-forcing equalizer - Signal Processing and Optimization for Transceiver Systems - page 306

Example 8.3: Channels with zeros on the unit circle
Consider the channel C ( z )= n =0 z −n , which has all seven zeros on the

unit circle at the points

z k = e −j 2 πk/ 7 , 0 ≤ k ≤ 6 .

We have L = 7, and choosing M =8wehave P = M + L =15 . Calculations
show:

A K 2 /M K

8 1 . 87

9 1 . 75

10 1 . 59

11 1 . 45

12 1 . 31

13 1 . 18

14 1 . 05

15 0 . 89

A K 2 /M gets smaller, though not as
dramatically as the case where the zeros of C ( z ) are outside the unit circle.
For unit circle zeros with higher multiplicity, the choice of large K becomes
very crucial. For example let

Thus as K increases the quantity

C ( z )=1+3 z − 1 +3 z − 2 + z − 3 ,

which has three zeros at z = − 1 . With M =8sothat P = M + L = 11,

calculations show:

A K 2 /M K

8 734 . 25

9 21 . 03

10 5 . 03

11 3 . 81

In this example there is a major improvement as K increases from M to
M + 1. Again, the noise gain is least when K = M + L =11 .

In practice of course, an arbitrary channel can have zeros inside, on or outside
the unit circle. The above examples show that, as long as we choose K = P (i.e.,
use all the P samples in the output block at the receiver), the quantity A K 2

is insensitive to these details, and the equalization can be done without undue
noise amplification.