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Detecting Signals in Gaussian Noise

Detecting Signals in Gaussian Noise - Detection and...

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Detection and Estimation (Spring, 2009) Detecting Signals in Gaussian Noise NCTU EE P.1 Detecting Signals in Gaussian Noise ~ Matched Filters Detecting a known deterministic signal in white Gaussian noise. s[n]: known w[n]: WGN with variance σ 2 We’ll decide H 1 if γ ’ is chosen to satisfy P FA = α .
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Detection & Estimation (Spring, 2009) Detecting Signals in Gaussian Noise NCTU EE P.2 correlator (replica-correlator) Alternative Interpretation: matched filter For a linear, time-invariant filter, we have Let At n = N-1, Remark: For signals with unknown arrival times, we cannot use the matched filter in its present form.
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Detection & Estimation (Spring, 2009) Detecting Signals in Gaussian Noise NCTU EE P.3 Example: In the frequency domain, Remarks: 9 The matched filter emphasizes the bands where there is more signal power. 9 When the noise is absent or X(f) = S(f), the matched filter output is just the signal energy.
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Detection & Estimation (Spring, 2009) Detecting Signals in Gaussian Noise NCTU EE P.4 Maximum SNR Property Consider all possible FIR filter detectors. The output SNR is defined as s = [s[0] s[1] … s[N-1]] T h = [h[N-1] h[N-2] … h[0]] T w = [w[0] w[1] … w[N-1]] T By the Cauchy-Schwarz inequality, with equality if and only if h = c s . Hence, and the maximum output SNR is attained for Remarks: 9 For the detection problem of a known deterministic signal in WGN, the NP criterion and the maximum SNR criterion both lead to the matched filter detector. 9 For nonGaussian noise, the matched filter is not optimal in the NP sense but may still be said to maximize the SNR at the output of a linear FIR filter. (In the nonGaussian noise case, the NP detector is not linear!) 9 The matched filter maximizes the SNR at the output of any linear filter.
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Detection & Estimation (Spring, 2009) Detecting Signals in Gaussian Noise NCTU EE P.5 Performance of Matched Filter Let
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Detection & Estimation (Spring, 2009) Detecting Signals in Gaussian Noise NCTU EE P.6 Remarks: 9
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