When the noise is colored, the common-sense best strategy for optimal detection is to bias the spectrum of the correlation pulse in favor of where the signal power is the strongest, and the noise power is the weakest. If the noise PSD increases with the square of frequency, then using a correlation pulse, or matched filter, that provides good high-frequency attenuation, is desirable. The resulting receiver can be derived from the optimal correlation receiver in AWGN, by using windowing functions to change the correlation pulse in a manner that provides better high-frequency attenua- tion. Fig. 3.21(a) shows a colored noise spectrum processed by a filter matched to a rectangular pulse, while Fig. 3.21(b) shows the same noise spectrum filtered by a half- cosine impulse response filter. It can be seen that windowing the rectangular correlation pulse with the half-cosine pulse provides desirable high-frequency attenuation.
3.8.1 Condition for Maximizing SNR of the Test Statistic
We saw earlier in (3.61) the optimal correlation receiver in AWGN can be written as
Z
T
0 h
s()r(T;;)d (3.161)
so that if we have a matched filter output of the form
p
n(t;) =Z (n+1)Ths()r(t;;)d (3.162)
Optimal Decision Theory 151
then the samples of the signalpn(t;)at values of (n+1)T are equivalent to the optimal test statistics for a correlation receiver in AWGN. If we now are operating in non-white noise, we wish to find the shape of the windowing function that will maximize the signal-to-noise ratio of the test statistic. It can be shown [1, Ch.6, p.
173] that the windowing functionh0()that maximizes the SNR for a noise process with an autocorrelation functionRn()satisfies the condition
Z T
0
h0()Rn()d =s(T;t) for 0tT; (3.163)
3.8.2 Matched Filter Approximation to Optimal Receiver in Colored Noise
Since the integral in (3.163) is only over[0;T]instead of[;1;1], thenh0(t)can
not be considered to be an impulse response of a matched filter. Notice ifh0(t)
extends beyond a bit period, then the filtering operation will overlap adjacent bits and cause intersymbol interference (ISI), unless additional care is taken to insure that
h0(t)is orthogonal to shifted data bits at specified sampling points. We can however, gain additional insight into the the shape ofh0(t)if we make the approximation that
h0(t)Rn(t)is negligible outside the interval[0;T]. In this case we can replace the integral in (3.163) with a convolution;
Z T
0
h0()Rn()d 'Z 1
;1
h0()Rn()d: (3.164)
Therefore,
h0(t)Rn(t)=s(T ;t) for 0tT: (3.165)
Under this approximation,h0(t)can now be considered as the impulse response of a matched filter. Taking the Fourier transform of both sides of (3.165) gives,
H0(j2f)Pn(f)=Fs(j2f)e;j2fT; (3.166)
wherePn(f) is the power spectral density of the noise. Therefore the magnitude response of the filter is given by
jH0(j2f)j= jFs(j2f)j
Pn(f) (3.167)
This result corresponds to the common-sense approach of making the frequency re- sponse of the matched filter large where the SNR is high, and weak where the SNR
152 Chapter 3
-50 -40 -30 -20 -10 0 10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Normalized Frequency (f/BT)
Magnitude (dB)
Frequency Dependent Noise Frequency Spectrum
of Rectangular Data
-50 -40 -30 -20 -10 0 10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Normalized Frequency (f/BT)
Magnitude (dB)
Frequency Spectrum of Matched Filter
(a) (b)
Figure 3.22 Illustration of optimal matched filter frequency response in colored noise: (a) magnitude of rectangular NRZ pulses and colored noise PSD, (b) magnitude response of matched filter.
0 0.2 0.4 0.6 0.8 1
-4 -3 -2 -1 0 1 2 3 4
Normalized Time (t/T)
0 1 2
-1 -2
Normalized Amplitude
Matched Filter Impulse Response Rectangular
Data Pulse Half-Cosine Correlation
Windowing Function
Figure 3.23 Impulse response of a matched filter in colored noise that increase as a function of frequency.
is low. The warping of the frequency spectrum of the matched filter is illustrated in Fig. 3.22 for rectangular NRZ data. The signal spectrum is a sinc function. The PSD of the noise is shown with a break frequency, where the noise begins to rise in proportion of the square of the frequency. The resulting spectrum of the matched filter that maximizes the SNR at sample intervals ofT is then shown in Fig. 3.22b. After taking the inverse FFT of the optimal spectrum, we obtain the impulse responseh0(t)
as is shown in Fig. 3.23. By windowing this impulse response so that it goes to zero outside the interval[0;T], we can obtain a correlation pulse that improves the SNR of the test statistic, and does not introduce any ISI.
Comparison With Optimal Correlator in White Noise In the previous sections we were dealing with white noise with a constant PSD ofN0=2. In this casejH0(j2f)j/
Optimal Decision Theory 153
w(t) m(t)
Whitening Filter
Matched Filter for White Noise
s(t) + n(t, ) w(t) s(t) *
+ White Noise y(t, )
Figure 3.24 Block diagram of a matched filter in colored noise represented as a whitening filter, and a matched filter in white noise.
jFs(j2f)j, and we can show that the optimal filter impulse response for white noise
is h0(t)/s(T;t) for white noise: (3.168)
This is equivalent to the optimal matched filter given in (3.61). Since s(t)is zero
outside the interval[0;T], our assumption that the integral in (3.163) could be replaced by a convolution is valid. The fact thath0(t)is confined to the interval[0;T]for white
noise results fromRn(t)being an impulse function so that the spread in time of the convolution integral is no greater than the integration limits. Conversely, the higher the correlation in the noise, or the larger the spread ofRn(t)compared tos(t), the less
valid is our assumption made in (3.165).
3.8.3 Whitening Filter
The optimal matched filter in colored noise can be understood more intuitively by splitting the filter into two parts as shown in Fig. 3.24. The first filter whitens the noise producing a constant spectral density at the output. Therefore, the PSD at the output is given by
jW(j2f)j2Pn(f)=1; (3.169)
and the magnitude of the whitening filters frequency response must satisfy
jW(j2f)j= Pn(f1)1=2: (3.170)
Now the second filter is just the matched filter in white-noise for a signalw(t)s(t),
which is the original signal warped by the prewhitening filter. We know that the impulse response of the optimal matched filter in white noise is given by
m(T;t)=w(t)s(t): (3.171)
The magnitude response of the second filter is easily found by taking the Fourier transform;
jM(j2f)j=jW(j2f)jjFs(j2f)j (3.172)
154 Chapter 3
Since we know the magnitude response of the whitening filter, then substituting gives
jM(j2f)j=jFs(j2f)j
Pn(f)1=2 (3.173)
The overall transfer function of the two filters is then given by the product of the individual transfer functions, so that
jH0(j2f)j=jW(j2f)jjM(j2f)j= jFs(j2f)j
Pn(f) ; (3.174)
which is the same as that obtained in (3.167).
It is still important to keep in mind that it has been assumed thath(t)is only non-zero fort 2[0;T], When this is not the case, (ISI) will be introduced, and the conditions under which this receiver was assumed optimum will be violated. Nevertheless, this discussion illustrates how the correlation pulse windowing functions can be altered to improve the performance in colored noise. In the following section the performance of a correlation receiver will be evaluated for various windowing functions in one particular type of colored noise. The results will be compared to see the improvement gained over using a correlation receiver that was optimized under the assumption that the noise was white.