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Chapter 17 Colored noise SUMMARY: So far we have only considered additive white Gaussian noise. In this chapter we will see that we should use a whitening ﬁlter if the channel noise is nonwhite. We will discuss the corresponding receiver structures. 17.1 Introduction transmitter receiver Figure 17.1: Communication over an additive waveform channel. The noise is nonwhite and Gaussian. Consider ﬁgure 17.1. There the transmitter sends, depending on the message index , one of the signals (waveforms) ), ), ··· over a waveform channel. The channel adds wide-sense-stationary zero-mean non-white Gaussian noise with power spectral density to this waveform . Therefore the output of the channel How should an optimum receiver process this output signal now? We will show next that we can use a so-called whitening ﬁlter to make the colored noise white. Then we proceed as usual. 17.2 Another result on reversibility We could try to construct an optimum receiver in two steps (see ﬁgure 17.2). First an operation is performed on the channel output . This yields a new channel output . Then we construct See appendix D. 150

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CHAPTER 17. COLORED NOISE 151 channel receiver operation Figure 17.2: Insertion of an operation between channel output and receiver. an optimum receiver for the channel with input and output It will be clear that such a two-step procedure cannot result in a smaller average error proba- bility than an optimum one-step procedure would achieve. If an inverse operation exist which permits to be reconstructed from however, the theorem 4.3 of reversibility states that the average error probability need not be increased by a two-step procedure. The second step could in that case consist of the calculation of from , followed by the optimum receiver for . The theorem of reversibility in its most general form states: RESULT 17.1 A reversible operation, transforming into one or more waveforms, may be inserted between the channel output and the receiver without affecting the minimum attainable average error probability. 17.3 Additive nonwhite Gaussian noise Suppose that we use a ﬁlter with impulse response and transfer function to change the power spectral density of the Gaussian noise process . Consider ﬁgure 17.3. The ﬁlter produces the Gaussian noise process at its output. From equation (D.6) in appendix D we know that (17.1) If we use the ﬁlter to whiten the noise, i.e. to produce noise at the output with power spectral density 2, the ﬁlter should be such that 2. In appendix I we have shown how to determine a realizable linear ﬁlter with transfer function , which has a realizable inverse and for which (17.2) The method in the appendix is applicable if whenever can be expressed (or approximated) as a ratio of two polynomials in . We will work out an example next. Figure 17.3: Filtering the noise process

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CHAPTER 17. COLORED NOISE 152 Example 17.1 Consider (see ﬁgure 17.4) noise with power spectral density (17.3) Observe that we can take as transfer function of our whitening ﬁlter −10 −8 −6 −4 −2 10 0.5 1.5 2.5 3.5 Figure 17.4: The power spectrum )/( of the noise and the squared modulus )/( of the transfer function of the corresponding whitening ﬁlter. (17.4) then both and its inverse are realizable (see appendix I). 17.4 Receiver implementation 17.4.1 Correlation type receiver When the whitening ﬁlter has been determined we can construct the optimum receiver. Note that we actually place the whitening ﬁlter at the channel output as in the left part of ﬁgure 17.5 but this is equivalent to the circuit containing two ﬁlters as in the right part of ﬁgure 17.5. Therefore the effect of applying the whitening ﬁlter is that we have transformed the channel into an additive white Gaussian noise waveform channel and that the signals are changed by the whitening ﬁlter into the signals (α) α) for . (17.5)

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CHAPTER 17. COLORED NOISE 153 Figure 17.5: Two equivalent channel/whitening-ﬁlter circuits. This observation leads to the optimum receiver shown in ﬁgure 17.6. This receiver correlates the output of the whitening ﬁlter with the signals for all . The signals are obtained at the output of ﬁlters with impulse response when is input. Note that the constants for are now ln Pr } dt (17.6) since the signal has changed into 17.4.2 Matched-ﬁlter type receiver To determine the matched-ﬁlter type receiver here, we will do some frequency-domain investiga- tions. Suppose that for some the Fourier spectrum of the waveform is denoted by . Assume that 0 except for 0 . The transfer function of the corresponding matched ﬁlter is then given by exp f t dt (α) exp α)) exp f T (α) exp α) exp f T ), (17.7) where is the complex conjugate of Now what is the matched ﬁlter corresponding to the signal . First note that the spectrum of is ). (17.8) Suppose that is only non-zero for 0 . Moreover let all be zero outside the time interval 0 . Then (α) α) can only be non-zero if an

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CHAPTER 17. COLORED NOISE 154 largest select Figure 17.6: Correlation receiver for the additive nonwhite Gaussian noise channel (ﬁltered- reference receiver). exists such that 0 and 0 hence if 0 . The transfer function of the ﬁlter matched to should therefore be exp )) exp )) exp f T exp f T ). (17.9) This corresponds to a cascade of the ﬁlter followed by the ﬁlter , one for each . Thus the ﬁlter matched to consists of a part matched to the channel (thus to the whitening ﬁlter) and a part that is matched to the signal . This leads to the receiver structure in ﬁgure 17.7. It is interesting to see what the front-end of this receiver does. This consists of the cascade of the whitening ﬁlter and the part of the matched ﬁlter matched to the channel (the whitening ﬁlter). The transfer function of this cascade is exp f T = | exp f T exp f T ). (17.10) This cascade may be implemented as a single ﬁlter. This effect of this ﬁlter is to pass energy over frequency bands where the the noise power is small and to suppress energy over frequency bands where it is strong.

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CHAPTER 17. COLORED NOISE 155 sample at largest select Figure 17.7: Matched ﬁlter receiver for the additive nonwhite Gaussian noise channel (ﬁltered- signal receiver). 17.5 Exercises 1. A communication system uses the shown in ﬁgure 17.8 to transmit one of two equally likely messages by means of the signals The channel is also illustrated in ﬁgure 17.8. (a) What is the probability of error for an optimum receiver when 5? (b) Give a detailed block diagram, including waveshapes in the absence of noise, of the optimum correlation receiver (ﬁltered-signal receiver). (Exercise 7.1 from Wozencraft and Jacobs [25].)

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CHAPTER 17. COLORED NOISE 156 9/2 1/2 Figure 17.8: Building blocks waveforms, communication system, and the channel impulse re- sponse

In this chapter we will see that we should use a whitening 64257lter if the channel noise is nonwhite We will discuss the corresponding receiver structures 171 Introduction transmitter receiver Figure 171 Communication over an additive waveform chan ID: 22973

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Chapter 17 Colored noise SUMMARY: So far we have only considered additive white Gaussian noise. In this chapter we will see that we should use a whitening ﬁlter if the channel noise is nonwhite. We will discuss the corresponding receiver structures. 17.1 Introduction transmitter receiver Figure 17.1: Communication over an additive waveform channel. The noise is nonwhite and Gaussian. Consider ﬁgure 17.1. There the transmitter sends, depending on the message index , one of the signals (waveforms) ), ), ··· over a waveform channel. The channel adds wide-sense-stationary zero-mean non-white Gaussian noise with power spectral density to this waveform . Therefore the output of the channel How should an optimum receiver process this output signal now? We will show next that we can use a so-called whitening ﬁlter to make the colored noise white. Then we proceed as usual. 17.2 Another result on reversibility We could try to construct an optimum receiver in two steps (see ﬁgure 17.2). First an operation is performed on the channel output . This yields a new channel output . Then we construct See appendix D. 150

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CHAPTER 17. COLORED NOISE 151 channel receiver operation Figure 17.2: Insertion of an operation between channel output and receiver. an optimum receiver for the channel with input and output It will be clear that such a two-step procedure cannot result in a smaller average error proba- bility than an optimum one-step procedure would achieve. If an inverse operation exist which permits to be reconstructed from however, the theorem 4.3 of reversibility states that the average error probability need not be increased by a two-step procedure. The second step could in that case consist of the calculation of from , followed by the optimum receiver for . The theorem of reversibility in its most general form states: RESULT 17.1 A reversible operation, transforming into one or more waveforms, may be inserted between the channel output and the receiver without affecting the minimum attainable average error probability. 17.3 Additive nonwhite Gaussian noise Suppose that we use a ﬁlter with impulse response and transfer function to change the power spectral density of the Gaussian noise process . Consider ﬁgure 17.3. The ﬁlter produces the Gaussian noise process at its output. From equation (D.6) in appendix D we know that (17.1) If we use the ﬁlter to whiten the noise, i.e. to produce noise at the output with power spectral density 2, the ﬁlter should be such that 2. In appendix I we have shown how to determine a realizable linear ﬁlter with transfer function , which has a realizable inverse and for which (17.2) The method in the appendix is applicable if whenever can be expressed (or approximated) as a ratio of two polynomials in . We will work out an example next. Figure 17.3: Filtering the noise process

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CHAPTER 17. COLORED NOISE 152 Example 17.1 Consider (see ﬁgure 17.4) noise with power spectral density (17.3) Observe that we can take as transfer function of our whitening ﬁlter −10 −8 −6 −4 −2 10 0.5 1.5 2.5 3.5 Figure 17.4: The power spectrum )/( of the noise and the squared modulus )/( of the transfer function of the corresponding whitening ﬁlter. (17.4) then both and its inverse are realizable (see appendix I). 17.4 Receiver implementation 17.4.1 Correlation type receiver When the whitening ﬁlter has been determined we can construct the optimum receiver. Note that we actually place the whitening ﬁlter at the channel output as in the left part of ﬁgure 17.5 but this is equivalent to the circuit containing two ﬁlters as in the right part of ﬁgure 17.5. Therefore the effect of applying the whitening ﬁlter is that we have transformed the channel into an additive white Gaussian noise waveform channel and that the signals are changed by the whitening ﬁlter into the signals (α) α) for . (17.5)

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CHAPTER 17. COLORED NOISE 153 Figure 17.5: Two equivalent channel/whitening-ﬁlter circuits. This observation leads to the optimum receiver shown in ﬁgure 17.6. This receiver correlates the output of the whitening ﬁlter with the signals for all . The signals are obtained at the output of ﬁlters with impulse response when is input. Note that the constants for are now ln Pr } dt (17.6) since the signal has changed into 17.4.2 Matched-ﬁlter type receiver To determine the matched-ﬁlter type receiver here, we will do some frequency-domain investiga- tions. Suppose that for some the Fourier spectrum of the waveform is denoted by . Assume that 0 except for 0 . The transfer function of the corresponding matched ﬁlter is then given by exp f t dt (α) exp α)) exp f T (α) exp α) exp f T ), (17.7) where is the complex conjugate of Now what is the matched ﬁlter corresponding to the signal . First note that the spectrum of is ). (17.8) Suppose that is only non-zero for 0 . Moreover let all be zero outside the time interval 0 . Then (α) α) can only be non-zero if an

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CHAPTER 17. COLORED NOISE 154 largest select Figure 17.6: Correlation receiver for the additive nonwhite Gaussian noise channel (ﬁltered- reference receiver). exists such that 0 and 0 hence if 0 . The transfer function of the ﬁlter matched to should therefore be exp )) exp )) exp f T exp f T ). (17.9) This corresponds to a cascade of the ﬁlter followed by the ﬁlter , one for each . Thus the ﬁlter matched to consists of a part matched to the channel (thus to the whitening ﬁlter) and a part that is matched to the signal . This leads to the receiver structure in ﬁgure 17.7. It is interesting to see what the front-end of this receiver does. This consists of the cascade of the whitening ﬁlter and the part of the matched ﬁlter matched to the channel (the whitening ﬁlter). The transfer function of this cascade is exp f T = | exp f T exp f T ). (17.10) This cascade may be implemented as a single ﬁlter. This effect of this ﬁlter is to pass energy over frequency bands where the the noise power is small and to suppress energy over frequency bands where it is strong.

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CHAPTER 17. COLORED NOISE 155 sample at largest select Figure 17.7: Matched ﬁlter receiver for the additive nonwhite Gaussian noise channel (ﬁltered- signal receiver). 17.5 Exercises 1. A communication system uses the shown in ﬁgure 17.8 to transmit one of two equally likely messages by means of the signals The channel is also illustrated in ﬁgure 17.8. (a) What is the probability of error for an optimum receiver when 5? (b) Give a detailed block diagram, including waveshapes in the absence of noise, of the optimum correlation receiver (ﬁltered-signal receiver). (Exercise 7.1 from Wozencraft and Jacobs [25].)

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CHAPTER 17. COLORED NOISE 156 9/2 1/2 Figure 17.8: Building blocks waveforms, communication system, and the channel impulse re- sponse

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