Notes9_correct_page_numbering

Notes9_correct_page_numbering - Notes#9, ECE594I, Fall...

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Notes#9, ECE594I, Fall 2009, E.R. Brown 134 Optimum Pre-Detection Signal Processing (the “matched filter” concept) Maximum Signal-to-Noise Ratio (Intuitive Derivation) Intuitively, detection in the presence of noise has limits imposed by physics (especially thermodynamics). To understand these limits, suppose we are attempting to detect the RF pulse plotted in Fig. 1 of waveform x(t) = A x sin( ω t + φ ), carrier frequency ω = 2 πν , and pulse duration T P . Assume that we are trying to detect this pulse in the presence of AWGN having power spectral density S P such that <( P) 2 > = S P ∆ν . In this case, the RF signal-to-noise ratio is () max 2 pp P pP p PT U P SNR Sv T Sv T P == ⋅∆ <∆ > (1) Where U P is the electrical pulse energy. If we assume that the pulse is sampled consistent with the Nyquist condition, then the sample rate f S should be matched to the pulse width f s = 1/T P , ( 2 ) and should be twice the twice the RF instantaneous bandwidth f S = 2 ∆ν . ( 3 ) Substitution of these last two into the SNR expression yields max 0 22 P UU SNR SN ≈≡ (4) where N 0 is another way of writing the power spectral density (following the convention in communications theory). This is a factor of two higher than might be expected intuitively because it assumes implicitly that we have precise knowledge of the phase and amplitude of a signal, not just one or the other.
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Notes9_correct_page_numbering - Notes#9, ECE594I, Fall...

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