statical signal processing.pdf

C how does the detector change when the value of m is

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(c) How does the detector change when the value of m is not known? 5.52 Noise generated by a system having zeros complicates the calculations for the colored noise detection problem. To illustrate these difficulties, assume the observation noise is produced as the output of a filter governed by the difference equation N ( ) = aN ( - 1 )+ w ( )+ bw ( - 1 ) , a 6 = - b , where w ( ) is white, Gaussian noise. Assume an observation duration sufficiently long to capture the colored effects. (a) Find the covariance matrix of this noise process. (b) Calculate the Cholesky factorization of the covariance matrix. (c) Find the unit-sample response of the optimal detector’s whitening filter. If it weren’t for the finite observation interval, would it indeed have an infinite-duration unit-sample response as claimed on Page 181? Describe the edge-effects of your filter, contrasting them with the case when b = 0. 5.53 It is frequently claimed that the relation between noise bandwidth and reciprocal duration of the ob- servation interval play a key role in determining whether DFT values are approximately uncorrelated. While the statements sound plausible, their veracity should be checked. Let the covariance function of the observation noise be K N ( ) = a | | .
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222 Detection Theory Chap. 5 (a) How is the bandwidth (defined by the half-power point) of this noise’s power spectrum related to the parameter a ? How is the duration (defined to be two time constants) of the covariance function related to a ? (b) Find the variance of the length- L DFT of this noise process as a function of the frequency index k . This result should be compared with the power spectrum calculated in part (a); they should resemble each other when the “memory” of the noise—the duration of the covariance function— is much less than L while demonstrating differences as the memory becomes comparable to or exceeds L . (c) Calculate the covariance between adjacent frequency indices. Under what conditions will they be approximately uncorrelated? Relate your answer to the relations of a to L found in the previous part. 5.54 The results derived in Problem 5.53 assumed that a length- L Fourier Transform was computed from a length- L segment of the noise process. What will happen if the transform has length 2 L with the observation interval remaining unchanged? (a) Find the variance of DFT values at index k . (b) Assuming the conditions in Problem 5.53 for uncorrelated adjacent samples, now what is the correlation between adjacent DFT values? 5.55 In a discrete-time detection problem, one of two, equally likely, length- L sinusoids (each having a frequency equal to a known multiple of 1 / L ) is observed in additive colored Gaussian noise. Signal amplitudes are also unknown by the receiver. In addition, the power spectrum of the noise is uncertain: what is known is that the noise power spectrum is broadband, and varies gently across frequency. Find a detector that performs well for this problem. What notable properties (if any) does your receiver have?
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