Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 45

Transmitter ( Tx ) AWGN Channel

Receiver ( Rx )

Sampler
[ at t = t o ]

n ( t )

MF

y ( t ) = s o ( t ) + n o ( t )

r ( t ) = s ( t ) + n ( t )

s ( t ) H ( f )

SNR i ( t ) = | s ( t )| 2 / σ

2

SNR o ( t ) = | s o ( t )| 2 / σ o

2

G n ( f )

s o ( t ) = o/p message
n o ( t ) = o/p noise

s ( t )

η /2

t f

0 T 0

Fig. 1.27 Matched filter and associated waveforms in a communication system

it produces the maximum signal-to-noise-ratio (SNR) enhancement possible for
any linear filter. To determine the form the matched filter must take to produce this
optimal enhancement, the problem of interest must first be properly defined.
Consider that the problem is to detect a particular symbol s(t) of finite duration
T. The following derivation will seek to determine the shape of the impulse
response for the required matched filter. It will actually be found that the optimal
impulse response h(t) is dependent on the symbol shape, and hence the name
matched filter.
Figure 1.27 shows a communication system with a matched filter. It should be
noted that the matched filter is applicable only when the receiving station knows
the symbol library at the transmitter.
The PSD of the output noise n o (t) in Fig. 1.27 is:

G n o ð f Þ¼ G n ð f Þj H ð f Þj 2 : ð 1 : 30 Þ

The output noise power is therefore:
r 2 ¼ Z

1

G n o ð f Þ df : ð 1 : 31 Þ

1

The output signal component is:
s o ð t Þ¼F 1 f H ð f Þ S ð f Þg¼ Z

1

H ð f Þ S ð f Þ e þ j2pft df :

ð 1 : 32 Þ

1

The output instantaneous SNR is:

SNR o ð t Þ¼j s o ð t Þj 2 = r 2 (which is time-dependent) : ð 1 : 33 Þ

Now it is necessary to find the H(f) that maximizes SNR o (t) when t = t o , where
t o is a specific time instant chosen by the sampler. This time instant is either 0 or
T. From ( 1.33 ) it follows that:

SNR o ð t o Þ¼ j s o ð t o Þj 2
r 2

: ð 1 : 34 Þ