Small-Signal Detection

Module by: Don Johnson

Before we introduce truly non-Gaussian detection strategies, we need to discuss the structure of the detector when the noise amplitude distribution is known. From the results presented while solving the general model evaluation problem, we find that the likelihood ratio test for statistically independent noise values is ∑l=0L-1lnpnrl- s 1 l-lnpnrl- s 0 l ≷ ℳ 0 ℳ 1 γ l L 1 0 p n r l s 1 l p n r l s 0 l ≷ ℳ 0 ℳ 1 γ The term in braces is ϒ l ϒ l , the sufficient statistic at the l th l th sample. As LL is usually large and each term in the summation is statistically independent of the others, the threshold γγ is found by approximating the distribution of the sufficient statistic by a Gaussian (the Central Limit Theorem again). Computing the mean and variance of each term, the false-alarm probability is found to be P F =Qlnη-∑E ϒ l | ℳ 0 ∑σ ϒ l | ℳ 0 2 P F Q η ℳ 0 ϒ l ϒ l | ℳ 0 This general result applies to any assumed noise density, including the Gaussian. The matched filter answer derived earlier is contained in the decision rule given above.

A matched-filter-like answer also results when small signal-to-noise ratio problems are considered for any amplitude distribution Kassam pp. 5-8, Spaulding, Spaulding and Middleton). Assume that lnpnrl-sl p n r l s l , considered as a function of the signal value sl s l , can be expressed in a Taylor series centered at rl r l . lnpnrl-sl=lnpnrl-ddnlnpnn|n=rlsl+12d2dn2lnpnn|n=rls2l+… p n r l s l p n r l n r l n p n n s l 1 2 n r l n 2 p n n s l 2 … In the small signal-to-noise ratio case, second and higher order terms are neglected, leaving the decision rule ∑l=0L-1-ddnlnpnn|n=rl s 1 l+ddnlnpnn|n=rl s 0 l ≷ ℳ 0 ℳ 1 γ l L 1 0 n r l n p n n s 1 l n r l n p n n s 0 l ≷ ℳ 0 ℳ 1 γ This rule says that in the small signal case, the sufficient statistic for each signal is the result of the match-filtering the result of passing the observations through a front-end memoryless non-linearity -ddxlnpnx x p n x , which depends only on noise amplitude characteristics (see Figure 1).

Figure 1: The decision rule for the non-Gaussian case when the signal is small can be expressed as a matched filter where the observations are first passes through a memoryless nonlinearity.

Because of the presence of the matched filter and the relationship of the matched filter to Gaussian noise problems, one might presume that the nonlinearity serves to transform each observation into a Gaussian random variable. In the case of the zero-mean Gaussian noise, -ddxlnpnx=xσ2 x p n x x σ 2 . Interestingly, this Gaussian case is the only instance where the "non-linearity" is indeed linear and yields a Gaussian output. Now consider the case where the noise is Laplacian ( pnn=12σ2ⅇ-|n|σ22 p n n 1 2 σ 2 n σ 2 2 ). The detector's front-end transformations is -ddxlnpnx=signxσ22 x p n x sign x σ 2 2 . This non-linearity is known as an infinite clipper and the output consists only of the values ±1σ22 ± 1 σ 2 2 , not a very Gaussian quantity.

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This work is licensed by Don Johnson under a Creative Commons Attribution License (CC-BY 1.0).

Last edited by Charlet Reedstrom on Sep 15, 2003 3:22 pm GMT-5.