Random Processes Handout IV

Transcription

1 RP-IV.1 Random Processes Handout IV CALCULATION OF MEAN AND AUTOCORRELATION FUNCTIONS FOR WSS RPS IN LTI SYSTEMS In the last classes, we calculated R Y (τ) using an intermediate function f(τ) (h h)(τ) We showed that Y (t) is WSS if X(t) is WSS, but we did not prove that X(t) and Y (t) are jointly WSS An alternative approach to finding R Y (τ) answers the second question: Consider the cross-correlation function R Y X (t + τ, t): R Y X (t + τ, t) E[Y (t + τ)x(t)] E[Y (t)x(t τ)] [ ] E h(u)x(t u)dux(t τ) (h R X )(τ) h(u)e [X(t u)x(t τ)] du h(u)r X (τ α)du Then from our previous work, R Y (τ) ( h R Y X )(τ) Note that R Y X (t + τ, t) R Y X (τ) implies that X(t) and Y (t) are also jointly WSS POWER SPECTRAL DENSITY DEFN If X(t) is a WSS RP, then the power spectral density (PSD) of X(t) is S X (f) F {R X (τ)}.

2 RP-IV. 1. S X (f) S X ( f), f (even) PROPERTIES OF PSD FOR REAL X(t). S X (f) [S X (f)], f (real) 3. S X (f) 0, f (non-negative) 4. If then S X (f) is a continous function of f. R X (τ) dτ < DEFN If X(t) and Y (t) are jointly WSS RPs, then the cross-power spectral density (PSD) of X(t) is S X (f) F {R XY (τ)}. Note that the cross-psd is generally complex-valued and does not satisfy the properties of a PSD For WGN, R X (τ) ( /)δ(τ) Then S X (ω) F{R X (τ)} / PSD OF WHITE GAUSSIAN NOISE I.e., white noise has equal power density at every frequency (like white light, which has equal power at every wavelength) The origin of the factor of (1/) comes from the two-sided nature of the Fourier transform.if we have an ideal filter of bandwidth B such that H(f) 1, f 0 B/ f f 0 + B/, then we will see that the power at the output of the filter is B The PSD can be an easy way to characterize the output process of a BIBO LTI system: Let H(f) F {h(t)}, then S Y (f) F {h h } R X H(f)H (f)s X (f) H(f) S X (f).

3 RP-IV.3 The output power is given by E[Y (t)] R Y (0) H(f) S X (f)df. SUFFICIENT CONDITION FOR A FUNCTION TO BE A PSD Suppose F (ω) is a real-valued function such that F (ω) 0 ω (1) Then let H(ω) F (ω) and input WGN with PSD 1 to a filter with frequency response H(ω) Then the output Y (t) has PSD S Y (ω) F (ω) Thus any function that satisfies (1) is a valid PSD Moreover, the condition that an autocorrelation function be positive semidefinite is equivalent to (1) Thus, a necessary and sufficient condition for a function f(τ) to be a valid autocorrelation function is for f(τ) to be positive semidefinite It is easier to determine if a function is a valid autocorrelation function by transforming it into the frequency domain

4 RP-IV.4 OPTIMAL DETECTION OF A DETERMINISTIC SIGNAL IN NOISE Consider detecting whether a signal s(t) is present under additive white Gaussian noise (AWGN) N(t) with S N (ω) / Let X(t) s(t) + N(t) when s(t) is present, and X(t) N(t) otherwise X(t) is input to a LTI filter h(t), Y (t) (X h)(t) Y (t) is sampled at some time T 0 to form a decision statistic Y (T 0 ) We decide that s(t) is present if Y (T 0 ) is greater than a threshold γ Problem: Design a LTI filter h(t) to maximize the probability of detecting whether s(t) is present Given whether s(t) is present, Y (T 0 ) is a Gaussian random variable, and thus is completely characterized by its mean and variance If s(t) is present, E[Y (T 0 )] µ Y (T 0 ) E[(X h)(t 0 )] E[(s 0 h)(t 0 )] + E[(N h)(t 0 )] E[(s 0 h)(t 0 )] because WGN can be treated as if it has mean zero Similarly if s(t) is not present, E[Y (T 0 )] E[(N h)(t 0 )] 0 Note that we can always choose h(t) such that the conditional mean of Y (T 0 ) given that s(t) is present is greater than zero, and the threshold γ will be between that value and zero The conditional variance of Y (T 0 ) given either that s(t) is present or not present is the same, Let s T0 (t) s(t 0 t) Var[Y (T 0 )] Var[N(T 0 )] (R N h h)(0) Then we can rewrite the conditional mean of Y (T 0 ) given s(t) is present at (s T0, h) Then the probability of miss, i.e., the probability of not detecting s(t) given that it is present is Q (s T 0, h) γ

5 RP-IV.5 The probability of false alarm, i.e., the probability of detecting s(t) given that it is not present is Q γ Thus, we can choose h to minimize the probability of miss By applying Schwartz Inequality, (s T0, h) s T0 with equality if and only if h(t) λs T0 (t) The probability of miss is minimized when (s T0, h) is maximized, which occurs if h(t) λs(t 0 t) Can take λ 1 without loss of generality I.e., h(t) is a time-reversed and shifted version of s(t) This h(t) is known as the matched filter for s(t) at sampling time T 0 Note that if s(t) is a causal signal with finite duration T, then the minimum sampling time for a causal h(t) is T 0 T, and h(t) s(t t) Let the signal-to-noise ratio (SNR) be defined by SNR (s T 0, h) Then in the frequency domain, the SNR is given by SNR 1 π S(ω)H(ω)ejωT 0 dω H(ω) dω By applying the complex form of the Schwartz Inequality, we can find that the SNR is maximized when H(ω) λs (ω)e jωt 0 F{s(T 0 t)}