mohatta2015.pdf

66 ZERO-FORCING DECISION FEEDBACK EQUALIZATION
where r = Q H x. Because Q is orthonormal, e is complex Gaussian with zero mean
and covariance iV () I.
Since R is upper triangular, we can detect SK with no ISI. We then subtract the
influence of SK on the r and detect SK-\- The process continues until all symbols
have been detected. Mathematically,
K-\
Vk„ = i-fe,, - Σ
ñ ( fc n,fc)Sfc (3.46)
fc=fc„ + l
Sk,, = detect(j/fc„,\/^rñ(fc(),fco)), (3.47)
where R(r, c) denotes the element in row r and column c of R.
Now suppose N r > N t . To reduce the number of equations, we initially perform
matched filtering, giving
H"x, (3.48)
which can be modeled as
y \= A/ËT(H H H)S + H H n. (3.49)
Notice that the noise on y is no longer uncorrelated, but now has covariance
7V()H H H. So, before performing QR factorization, we need to whiten the noise.
This can be done by multiplying by square matrix F H , which can be obtained by
Cholesky decomposition of (Η Η Η)"\ i.e., FF H = (H^H)" 1 .
3.5 AN EXAMPLE
In practice, the ZF DFE is usually not used, as it tends to enhance the noise. Thus,
we will wait until Chapter 5 to discuss an example.
3.6 THE LITERATURE
As DFE is revisited in Chapter 5, we will only mention references related specifically
to this chapter.
The idea of connecting the FBF prior to the FF can be found in [Ari92]. In
synchronous CDMA, triangularization and ZF DFE is developed in [Due93].
For block equalization, Cholesky factorization is applied to triangularize the
channel after matched filtering in [Cro92]. In [Kal95], Cholesky factorization is
motivated by the desire to whiten the noise after matched filtering.
PROBLEMS
The idea
3.1 Consider the Alice and Bob example. Suppose instead that r\ = —1 and
Γ2 = 4. Also, so = +1.
a) What is the value of ζ λ with ZF DFE?