Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 106

| H ( e j ω )|

| H a ( ω )| = | ω |

π f s

Ω = ω T s

ω

− π π

0

0

(a)

(c)

| H a ( ω )|

∠

| H ( e j ω )|

∠

π /2 π /2

ω Ω

0 0

− π /2 − π /2

(b) (d)

Fig. 2.29 Frequency response for an analog and digital differentiator-II

2.6.6.4 The Digital Matched Filter

Recall from Sect. 1.4.2 that in the analog domain the impulse response h(t) of the
optimum filter for receiving a transmitted symbol b(t)(0B t B T) is given by the
T-shifted (delayed) mirror image of the symbol, i.e.,

h ð t Þ¼ b ð T t Þ P T ð t T = 2 Þ:

For orthogonal binary transmission, two matched filters are needed, one mat-
ched to the ''0'' and the other matched to the ''1''. Figure ( 2.30 ) shows a digital
matched filter for orthogonal binary signalling with the sampling frequency at the
receiver equal to five times the bit rate. If bipolar signalling is used (i.e., +Vcc
represents ''1'' and -Vcc represents ''0''), then one sample per bit is possible, but
in practical signalling schemes there are normally at least 4 samples per bit.
Increasing the sample rate increases the accuracy of detection, especially in noise,
and it does not affect the transmission bandwidth (which depends on the data rate
1/T only). If the sample rate is N samples per bit, then an N-stage shift register is
needed to store the samples before each symbol is detected for each filter (see Fig.
( 2.30 )).
In digital matched filters it is necessary to have good synchronization between
the transmitter and the receiver to decide the start time (t = 0). The coefficients of
the ith matched filter f h i ð n Þj n ¼ 0 ; 1 ; ... ; N 1 g; which is matched to the ith
symbol (bit), are given by:

h i ð n Þ¼ b i ð N 1 n Þ;

n ¼ 0 ; 1 ; ... ; N 1