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

9.6 Application: Optimum Filtering (Wiener-Hopf Filter) 485
The optimum filter is known as the Wiener-Hopf filter in the literature. Equation (9.46a)
shows that H0 p 1 (J) I (no attenuation) when S m if) » Sn if). But when S m if) « Sn if),
the filter has high attenuation. In other words, the optimum filter attenuates heavily the band
where noise is relatively stronger. This causes some signal distortion, but at the same time it
attenuates the noise more heavily so that the overall SNR is improved.
Comments on the Optimum Filter
If the SNR at the filter input is reasonably large-for example, Sm (J) > l OOSn (J) (SNR
of 20 dB)-the optimum filter [Eq. (9.46a)] in this case is practically an ideal filter, and N a
[Eq. (9.46b)] is given by
N a '.::::'. L: Sn (J) df
Hence for a large input SNR, optimization yields insignificant improvement. The Wiener-Hopf
filter is therefore practical only when the input SNR is small (large-noise case).
Another issue is the realizability of the optimum filter in Eq. (9.46a). Because Sm (J)
and Sn (J) are both even functions of f, the optimum filter H o pt (J) is an even function
of f . Hence, the unit impulse response h 0p 1 (t) is an even function of t (see Prob. 3.1-1).
This makes h 0p 1 (t) noncausal and the filter unrealizable. As noted earlier, such a filter can
be realized approximately if we are willing to tolerate some delay in the output. If delay
cannot be tolerated, the derivation of H 0p 1 (J) must be repeated with a realizability constraint.
Note that the realizable optimum filter can never be superior to the unrealizable
optimum filter [Eq. (9.46a)]. Thus, the filter in Eq. (9.46a) gives the upper bound on performance
(output SNR). Discussion of realizable optimum filters can be readily found in the
literature 1 • 2 .
Example 9 .1 1 A random process m(t) (the signal) is mixed with a white channel noise n(t). Given
2a
Sm (2f ) = a 2 + (2nj) 2
and
find the Wiener-Hopf filter to maximize the SNR. Find the resulting output noise power N a .
From Eq. (9.46a),
4a
H op
i (J) = 4a + N[a 2 + (2rrf) 2 ]
4a
= ------
N[/3 2 + (2nf) 2 ]
(9.47a)
Hence,
h opt (t) = 2a e -fi ltl
N/3
(9.47b)
Figure 9.16a shows h 0p 1 (t). It is evident that this is an unrealizable filter. However, a
delayed version (Fig. 9.16b) of this filter, that is, hopt (t - to), is closely realizable if we
make to 2:: 3/{3 and eliminate the tail for t < 0 (Fig. 9.16c).