Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 47

s ( t ) h ( t ) = s ( − t )

(if t o = 0)

t t

0 T −T 0

s ( − t ) h ( t ) = s ( T − t )

(if t o = T)

t t

−T 0 0 T

s ( t o − t )

h ( t ) = s ( t o − t )

t t

t o t o

−T + t o 0 −T + t o 0

Fig. 1.28 Matched filter impulse response with various modifications of the original symbol

waveform

That is, the impulse response of a filter matched for a symbol is a scaled time-
reversed version of the symbol itself. Frequently k is chosen to be 1 in practical
applications (see Fig. 1.28 ).

1.4.2.1 Conclusion

If one wants a filter H(f) to give maximum SNR enhancement for a specific symbol
s(t)att = t o , one should match the filter to the symbol, such thath(t) = ks(t o - t).

1.4.2.2 The Output of the Matched Filter at the Time of Optimal SNR

Assuming the system of interest is LTI, the output y(t) of the matched filter in
Fig. 1.27 is given by:

y ð t Þ¼ r ð t Þ h ð t Þ¼ Z r ð k Þ h ð t k Þ dk ¼ Z

t þ T t o t þ T t o

r ð k Þ ks ½ t ð t k Þ dk ; ð 1 : 42 Þ

t t o t t o

At the sampling instant t = t o one obtains:

Z

1

r ð k Þ s ð k Þ dk ¼ k Z

T

y ð t o Þ¼ k r ð k Þ s ð k Þ dk : ð 1 : 43 Þ

1

0

That is, the output of the matched filter at the time of optimal SNR, t = t o ,is
simply the integration overTof the scaled product of the symbols(t) with the
received signalr(t).