Lecture 8: Signal Detection and Noise Assumption


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1 ECE 83 Fall Statistical Signal Processing instructor: R. Nowak, scribe: Feng Ju Lecture 8: Signal Detection and Noise Assumption Signal Detection : X = W H : X = S + W where W N(, σ I n n and S = [s, s,..., s n ] T is the known signal waveform. P (X = P (X = (πσ n (πσ n exp( σ XT X exp[ σ (X ST (X S] The second equation holds because under hypothesis, W = X S. The Log Likelihood Ratio test is log Λ(x = log After simplifying it, we can get P W (X P W (X S = σ [(X ST (X S X T X] = σ [ XT + S T S] H X T S H σ γ + ST S In this case, X T S is the sufficient statistics t(x for the parameter θ =,. Note that S T S = is the signal energy. The LR detector filters data by projecting them onto signal subspace.. Example Suppose we want to control the probability of false alarm. For example, choose γ so that P(X T S > γ.5. The test statistic X T S is usually called matched filter. In particular, projection onto subspace spanned by S is = γ P S = SST S T S = S ST γ
2 Lecture 8: Signal Detection and Noise Assumption X PsX S Figure : Projection of X onto subspace S P S X = SST X = S (XT S where XT S is just a number. Geometrically, suppose the horizontal line is the subspace S and X is some other vector. The projection of vector X into subspace S can be expressed in the figure.. Example Suppose the signal value S k is sinusoid. S k = cos(πf k + θ, k =,..., n The match filter in this case is to compute the value in the specific frequency. So P S in this example is a bandpass filter..3 Performance Analysis Next problem what we want to know is what s the probability density of X T S, which is the sufficient statistics of this test. : X N(, σ I H : X N(S, σ I X T S = n k= X ks k is also Gaussian distributed. Recall if X N(µ, Σ, then Y = AX N(Aµ, AΣA T, where A is a matrix. Since Y = X T S = S T X, Y is a scalar. So we can get : X T S N( T S, S T σ IS = N(, σ H : X T S N(S T S, S T σ IS = N(, σ The probability of false alarm is P F A = Q( γ σ, and the probability of detection is = Q( γ σ = γ Q( σ σ. Since Q function is invertible, we can get γ σ = Q (P F A. Therefore, = Q(Q (P F A. In the equation, is the square root of Signal Noise Ratio( SNR. σ σ
3 Lecture 8: Signal Detection and Noise Assumption 3 S Figure : Distribution of P and P = = 3 = SNR(dB AWGN Assumption Figure 3: Relation between probability of detection and false alarm Is realworld noise really additive, white and Gaussian? Well, here are a few observations. Noise in many applications (e.g. communication and radar arose from several independent sources, all adding together at sensors and combining additively to the measurement. AWGN is gaussian distributed as the following formula. W N(, σ I CLT(Central Limit Theorem: If x,..., x n are independent random variables with means µ i and variances σi <,then Z n = n x i µ i n i= σ i N(, in distribution quite quickly. Thus, it is quite reasonable to model noise as additive and Gaussian list in many applications. However, whiteness is not always a good assumption.. Example 3 Suppose W = S + S + + S k, where S, S,... S k are inteferencing signals that are not of interest. But each of them is structured/correlated in time. Therefore, we need a more generalized form of noise, which is Colored Gaussian Noise.
4 Lecture 8: Signal Detection and Noise Assumption Increasing SNR Colored Gaussian Noise Figure 4: Relation between probability of detection and SNR W N(, Σ is called correlated or colored noise, where Σ is a structured covariance matrix. Consider the binary hypothesis test in this case. : X = S + W H : X = S + W where W N(, Σ and S and S are know signal waveforms. So we can rewrite the hypothesis as : X N(S, Σ H : X N(S, Σ The probability density of each hypothesis is P i (X = exp[ (π n (Σ (X S i T Σ (X S i ], i =, The log likelihood ratio is log( P (X P (X = [(X S T Σ(X S (X S T Σ (X S ] = X T Σ (S S ST Σ S + ST Σ H S γ Let t(x = (S S Σ X, we can get (S S Σ X H γ + ST Σ S ST Σ S = γ The probability of false alarm is : t N((S S Σ S, (S S T Σ (S S H : t N((S S Σ S, (S S T Σ (S S
5 Lecture 8: Signal Detection and Noise Assumption 5 In this case it is natural to define γ (S S T Σ S P F A = Q( [(S S T Σ (S S ] SNR = (S S T Σ (S S 3. Example 4 [ ] [ ] ρ ρ S = [, ], S = [, ], Σ =, Σ ρ = ρ. ρ The test statistics is [ ] [ ] y = (S S Σ ρ x X = [, ] ρ ρ x = + ρ (x + x The probability of false alarm is : y N( + ρ, + ρ H : y N(+ + ρ, + ρ The probability of detection is +ρ +ρ P F A = Q( γ + +ρ +ρ = Q( γ p=.5 p= p= Figure 5: ROC curve at different ρ
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