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

9.6 Application: Optimum Filtering (Wiener-Hopf Filter) 483
Hence,
and
2 a
S y2 lf) = IH 2 (J)I Sx2 lf) = 2[9(2 nf) 2 + l][a 2 + (21rf) 2 ]
Because the input processes x1 (t) and x 2 (t) are independent, the outputs y1 (t) and y 2 (t)
generated by them will also be independent. Also, the PSDs of Yl (t) and y 2 (t) have no
impulses at f = 0, implying that they have no de components [i.e., Yl (t) = y 2 (t) = 0].
Hence, Yl (t) and y 2 (t) are incoherent, and
S y (J) = Sy l (j) + Sy2 (j)
2K[a 2 + (2nf) 2 ] + 9a
18[9(2nf) 2 + l][a 2 + (2nf) 2 ]
The power P y
(or the mean square value y 2 ) can be determined in two ways. We can find
R y
( r) by taking the inverse transforms of Sy 1 (j) and S Y 2 (j) as
K
- 1 , 1 /3 3a - e-al, I
Ry (r) = 54
e + 4(9a
2
- 1)
.___,_ "-.--'
Ryl (r) R y2 (r)
and
- K 3a - 1
P - y2 - R (0) - - + ---
y
- - Y
- 54 4(9a
2
- 1)
Alternatively, we can determine y 2 by integrating S y (f) with respect to f (or f) [see
Eq. (9.19)].
9.6 APPLICATION: OPTIMUM FILTERING
(WIENER-HOPF FILTER)
When a desired signal is mixed with noise, the SNR can be improved by passing it through a
filter that suppresses frequency components where the signal is weak but the noise is strong.
The SNR improvement in this case can be explained qualitatively by considering a case of
white noise mixed with a signal m(t) whose PSD decreases at high frequencies. If the filter
attenuates higher frequencies more, the signal will be reduced-in fact, distorted. The distortion
component m E (t) may be considered as bad as added noise. Thus, attenuation of higher
frequencies will cause additional noise ( from signal distortion), but, in compensation, it will
reduce the channel noise, which is strong at high frequencies. Because at higher frequencies
the signal has a small power content, the distortion component will be small in comparison to
the reduction in channel noise, and the total distortion may be smaller than before.
Let Hoptlf) be the optimum filter (Fig. 9.15a). This filter, not being ideal, will cause signal
distortion. The distortion signal m E (t) can be found from Fig. 9.15b. The distortion signal
power Nv appearing at the output is given by