B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

10.1 Optimum Linear Detector for Binary Polar Signaling 511
t m = T 0 gives the minimum delay for decision making using a realizable filter. In our future
discussion, we shall assume t m = T 0 , unless otherwise specified.
Observe that both p(t) and h(t) have a width of T 0 seconds. Hence, p 0 (t), which is a
convolution ofp(t) and h(t), has a width of 2T 0 seconds, with its peak occurring at t = T 0 where
the decision sample is taken. Also, because P 0 (f) = P(f)H (f) = k'IP(f) l 2 e-i 2 11:fT 0 , p 0 (t) is
symmetrical about t = T 0 .*
Since the gain k' does not affect the SNR p, we choose k' = 1. This gives the matched
filter under white noise
or equivalently
h(t) = p (T 0 - t) (10.12a)
H (f) = P(-f)e - } 2 11:fTo
(10.12b)
for which the signal to noise ratio is maximum at the decision-making instant t = T 0 •
The matched filter is optimum in the sense that it maximizes the signal-to-noise ratio
at the decision-making instant. Although it is reasonable to assume that maximization of this
particular signal-to-noise ratio will minimize the detection error probability, we have not proven
that the original structure of linear receiver with threshold detection (sample and decide) is the
optimum structure. The optimality of the matched filter receiver under white Gaussian noise
will be shown later (Section 10.6).
Given the matched filter under white Gaussian noise, the matched filter receiver leads to
Pmax of Eq. (10.lla) as well as the minimum BER of
Pe = Q(Pmax) = Q ( ffi-) (10.13)
Equation (10. 13) is quite remarkable. It shows that, as far as the system performance is concerned,
when the matched filter receiver is used, various waveforms used for p(t) are equivalent
as long as they have the same energy
The matched filter may also be implemented by the alternative arrangement shown in
Fig. 10.3. If the input to the matched filter is y(t), then the output r(t) is given by
r(t) = L: y(x)h(t - x) dx
(10.14)
where h(t) = p (T 0 - t) and
h(t - x) = p [T 0 - (t - x)] = p (x + T 0 - t)
(10.15)
* This follows from the fact that because IP(f)l 2 is an even function off, its inverse transform is symmetrical about
t = 0 (see Prob. 3.1-1). The output from the previous input pulse terminates and has a zero value at t = T 0 .
Similarly, the output from the following pulse starts and has a zero value at t = T 0 • Hence, at the decision-making
instant TO, no intersymbol interference occurs.