Classical transceiver techniques - Signal Processing and Optimization for Transceiver Systems - page 200

is minimized. Defining

r =[ r (0) r (1) ... r ( K − D − 1) ]

we can therefore say that y est is a minimum distance estimate of y based on
r . When the noise is AWGN, this estimate is therefore an ML estimate (Sec.
5.5.4). Thus

f ( r y est ) f ( r y ) ≥ (5 . 79)

for any other feasible noise-free output vector y . Observe now that the vector y
is related to the transmitted symbol vector s as follows:
⎡ ⎤ ⎡ ⎤ ⎡ ⎤

y (0)
y (1)
.
y ( N )

h (0) 0 ... 0 s (0)

s (1)
.
s ( N )

h (1) h (0) ... 0

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= , (5 . 80)

. . .

. . .

h ( N ) h ( N − 1) ... h (0)

y s A

where N = K 0 . Since s ( n )

belong to a constellation with M possible discrete values, the vector s can take
M N +1 discrete values. So the output vector y can take at most M N +1 discrete
values. In fact it takes exactly M N +1 distinct values assuming that the matrix
above is nonsingular, that is, h (0) =0. 4 Thus, even though each sample y ( n )
caninprinciplehavemanymorethan M possible values (because it is a linear
combination of the samples s ( n

− D − 1 . In fact Eq. (5.80) is true for any N ≥

k )), the vector y comes from a set with precisely
M N +1 discrete values. A number of points should now be noted.

−

1. ML property. Since each s from the discrete set maps to a unique y and
vice versa, we see that (5.79) also implies
f 1 ( r s est ) f 1 ( r s ) ,

≥ (5 . 81)

where f 1 ( . . ) represent the conditional pdf for r given s . Thus, the fact

that y est is an ML estimate of y implies that s est is an ML estimate of s
(based on the received vector r ).

|

2. MAP property. Next, assume that the symbols s ( n ) are independent and
identically distributed, with identical probabilities for all symbols in the
constellation. Then the M N +1 discrete values of s have identical probabil-
ities. Thus the ML property of the estimate s est also implies that it is an
MAP estimate (Sec. 5.5.1).

3. Error-event probability. From the discussion of Sec. 5.5.3 it therefore follows
that this estimated vector s est has the minimum error probability property.
We simply say that the error-event probability has been minimized [Forney,
1972]. That is, the probability of error in the estimation of the vector,
viewed as one entity, is minimized.

4 If s 1 and s 2 are two distinct values of s then y 1 − y 2 = A ( s 1 − s 2 ). If s 1 − s 2 = 0 and A
is nonsingular, it follows that y 1 − y 2 = 0 . So y takes M N +1 distinct values like s .