Channel equalization with transmitter redundancy - Signal Processing and Optimization for Transceiver Systems - page 261

If we introduce additional redundancy in an appropriate way (2 L instead of L
samples per block) it is possible to show that the equalizers are the reciprocals
of

L

c ( n ) e −j 2 πk/ ( M + L ) , C 1 [ k ]= 0 ≤ k ≤ M + L − 1 ,

n =0

which are samples of the channel frequency response taken at a closer spacing
2 π/ ( M + L ). If appropriately designed, it can be shown that only M of these
M + L samples need to be inverted at the receiver. Choosing the M largest
samples we ensure that none of them is zero (because the L th order FIR channel
can have at most L zeros on the unit circle). This clever idea has been used to
design receivers that are resistant to channel nulls [Liang and Tran, 2002]. See
Problem 7.23.

Use of FFT in equalization. The cyclic-prefix transceiver uses DFT and inverse
DFT operations, both of which can be performed eciently using the FFT algo-
rithm. For this it is desirable to choose the block length M to be a power of two,
so that radix-2 FFT can be employed. The number of complex multipliers in an
M -point radix-2 FFT is equal to [Oppenheim and Schafer, 1999]

M
2

3 M
2

log 2 M − +2 .

Since W − 1 = W ∗ /M, the complexity of the inverse DFT operation is the same. 3
Counting the M multipliers 1 /C [ k ] in Fig. 7.13, we have a total of

M log M − 2 M +4 ≈ M log M

2 2

multiplications. From Sec. 7.2 we know that the complexity of the channel equal-
izer based on zero padding is nearly M 2 / 2 per block, which is much larger for
large M (unless we use fast convolution methods in the implementation of the
receiver). Recall from Sec. 7.2 that large M is desirable if we want to keep the
bandwidth expansion ratio γ =( M + L ) /M small.

7.4 The circulant matrix representation

Equation (7.20), which was crucial in the design of the cyclic-prefix receiver, is
a beautiful equation. To appreciate the significance of this in a different light,
recall that cyclic prefixing with M>L ensures that the operation of the channel
is turned into a circular convolution. More specifically, the last M samples of
y ( n ) in each block can be regarded as the circular convolution of c ( n )withthe
M samples of s ( n ) in a block. Written in the form of an equation we have

3 The extra M multiplications due to the scale factor 1 /M are binary shifts if M is a power
of two.