ECE 830 Fall 0 Statistical Signal Processing instructor: R. Nowak Lecture 8: Signal Detection and Noise Assumption Signal Detection : X = W H : X = S + W where W N(0, σ I n n and S = [s, s,..., s n ] T is the known signal waveform. P 0 (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 = 0,. 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 0.05. 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 P S X = SST X = S (XT S γ
Lecture 8: Signal Detection and Noise Assumption Figure : Projection of X onto subspace 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 0 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(0, 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(0 T S, S T σ IS = N(0, 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 0 σ, and the probability of detection is P D = Q( γ σ = γ Q( σ σ. Since Q function is invertible, we can get γ σ = Q (P F A. Therefore, P D = Q(Q (P F A. In the equation, is the square root of Signal Noise Ratio( SNR. σ AWGN Assumption σ Is real-world 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
Lecture 8: Signal Detection and Noise Assumption 3 Figure : Distribution of P 0 and P Figure 3: Relation between probability of detection and false alarm at sensors and combining additively to the measurement. AWGN is gaussian distributed as the following formula. W N(0, 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(0, 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. 3 Colored Gaussian Noise W N(0, Σ is called correlated or colored noise, where Σ is a structured covariance matrix. Consider the binary hypothesis test in this case.
Lecture 8: Signal Detection and Noise Assumption 4 Figure 4: Relation between probability of detection and SNR : X = S 0 + W H : X = S + W where W N(0, Σ and S 0 and S are know signal waveforms. So we can rewrite the hypothesis as : X N(S 0, 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 = 0, The log likelihood ratio is log( P (X P (X = [(X S T Σ(X S (X S 0 T Σ (X S 0 ] = X T Σ (S S 0 ST Σ S + ST 0 Σ H S 0 γ Let t(x = (S S 0 Σ X, we can get (S S 0 Σ X H γ + ST Σ S ST 0 Σ S 0 = γ The probability of false alarm is : t N((S S 0 Σ S 0, (S S 0 T Σ (S S 0 H : t N((S S 0 Σ S, (S S 0 T Σ (S S 0 γ (S S 0 T Σ S 0 P F A = Q( [(S S 0 T Σ (S S 0 ] In this case it is natural to define SNR = (S S 0 T Σ (S S 0
Lecture 8: Signal Detection and Noise Assumption 5 3. Example 4 [ ] [ ] ρ ρ S = [, ], S 0 = [, ], Σ =, Σ ρ = ρ. ρ The test statistics is [ ] [ ] y = (S S 0 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 P D = Q( γ Figure 5: ROC curve at different ρ