Coherence Analysis Overview Definition Coherency Definition R xy - PowerPoint PPT Presentation

Coherence Analysis Overview Definition Coherency • Definition R xy (e jω ) • Properties G xy ( ω ) � � R x (e jω ) R y (e jω ) • Estimation • Correlation of complex-valued RVs Also known as the coherency spectrum (Weiner, 1930) or normalized cross-spectrum . Similar to the correlation coefficient in • Examples frequency. • Discussion Magnitude Squared Coherence (MSC) | R xy (e jω ) | 2 G 2 xy ( ω ) � R x (e jω ) R y (e jω ) Also known as the coherence function and simply coherence J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 1 J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 2 What is it Really? Example 1: Coherence and LTI Systems | R xy (e jω ) | 2 G 2 xy ( ω ) � x ( n ) H ( z ) y ( n ) R x (e jω ) R y (e jω ) • Coherence is a measure of correlation in frequency Consider the stochastic process above where x ( n ) and y ( n ) are jointly wide sense stationary. Solve for the coherence G 2 xy ( ω ) . Hint: recall • Usual assumptions apply (WSS) that if y ( n ) = h ( n ) ∗ x ( n ) , then R xy ( z ) = H ∗ ( z −∗ ) R x ( z ) . • For estimation, we require ergodicity J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 3 J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 4

Example 1: Workspace Example 2: Coherence and LTI Systems w ( n ) G ( z ) x ( n ) H ( z ) F ( z ) y ( n ) Σ Consider the stochastic process above where w ( n ) and x ( n ) are jointly wide sense stationary and uncorrelated and the LTI systems are all stable. Solve for the coherence G 2 xy ( ω ) . J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 5 J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 6 Example 2: Workspace Properties | R xy (e jω ) | 2 G 2 xy ( ω ) � R x (e jω ) R y (e jω ) Coherence has many interesting and useful properties • If y ( n ) = h ( n ) ∗ x ( n ) , then G 2 xy ( ω ) = 1 • If r xy ( ℓ ) = 0 , then • G 2 xy ( ω ) = 0 if – r xy ( ℓ ) = 0 – x ( n ) and y ( n ) are statistically independent • Bounded: 0 ≤ G 2 xy ≤ 1 • Symmetry: G 2 xy ( ω ) = G 2 yx ( ω ) • Can be applied to complex signals, though range must span − π < ω < π in that case J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 7 J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 8

Coherence as Correlation Complex Correlation | R xy (e jω ) | 2 ρ 2 = | cov[ x, y ] | 2 Let x and y be complex-valued zero-mean random variables. The G 2 xy ( ω ) = complex correlation coefficient is then defined as R x (e jω ) R y (e jω ) var[ x ] var[ y ] cov[ x, y ] E[ xy ∗ ] • Coherence is very similar to the coefficient of determination ρ = = (square of the correlation coefficient) between two RVs � � var[ x ] var[ y ] E[ xx ∗ ] E[ yy ∗ ] • It is, actually, the coherence between the random spectra • ρ will be complex-valued � π x ( n ) = 1 – Less intuitive than for real-valued variables �� � 2 � e jωn d X (e jω ) � d X (e jω ) � = R x (e jω )d ω E 2 π – Does not retain the sign to indicate positive or negative − π correlation � 2 � d X (e jω ) , d Y ∗ (e jω ) �� � � cov G 2 • Can’t look at scatter plot ( x, y span four dimensions) xy ( ω ) = var [d X (e jω )] var [d Y (e jω )] but this requires stochastic integrals, which are beyond the scope of this class (see [1, chapter 4]) • It is therefore worthwhile to understand the difference between correlation of real- and complex-valued RVs J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 9 J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 10 Understanding Complex Correlation Understanding Complex Correlation E[ xy ∗ ] E[ xy ∗ ] ρ = ρ = � E[ xx ∗ ] E[ yy ∗ ] � E[ xx ∗ ] E[ yy ∗ ] E[ xy ∗ ] = E[ a x e jθ x a y e − jθ y ] = E[ a x a y e j ( θ x − θ y ) ] E[ xy ∗ ] = E[ a x e jθ x a y e − jθ y ] = E[ a x a y e j ( θ x − θ y ) ] E[ xx ∗ ] = E[ a 2 x ] E[ xx ∗ ] = E[ a 2 x ] • ρ will be zero if ( θ x − θ y ) ∼ U [0 , 2 π ] • Strong correlation, say ρ = 0 . 90 , does not mean could accurately estimate x from y or vice versa • If a x and a y are constants, ρ is the phase correlation : ρ = E[e j ( θ x − θ y ) ] – Sufficient accuracy depends on the application – Also does not mean a linear estimate is best • If ( θ x − θ y ) is a constant, | ρ | is the un-normalized correlation coefficient of the amplitudes – Could possibly estimate more accurately with nonlinear estimate E[ a x a ∗ y ] e j ( θ x − θ y ) ρ = • Weak correlation, say ρ = 0 . 10 , does not mean x and y are � E[ a 2 x ] E[ a 2 y ] unrelated • Only measures degree of linear association of x and y • In general is a mixture of phase and amplitude correlation J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 11 J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 12

Understanding Complex Correlation Continued Example 3: Complex Correlation Scatter Plots Create 500 pairs of IID, zero-mean, unit-variance complex-valued y (e jω ) = R y (e jω ) 1 − G 2 � � R ˜ xy ( ω ) Gaussian RVs, x and y , with a true correlation of ρ = 0 . 9 and ρ = 0 . 1 . y (e jω ) = R y (e jω ) G 2 R ˆ xy ( ω ) Estimate the absolute value of the correlation coefficient. Plot scatter plots of the real and imaginary components of x and y . Also create a R y (e jω ) = R ˆ y (e jω ) + R ˜ y ( ω ) scatter plot of the ratio of x/y . Discuss the results. • Let x ( n ) and y ( n ) be zero-mean jointly WSS random processes – ˆ y ( n ) = h ( n ) ∗ x ( n ) be the best linear estimate of y ( n ) given x ( n ) – ˜ y ( n ) = y ( n ) − ˆ y ( n ) • Then G 2 xy ( ω ) can be interpreted as the fraction of the variance explained by an optimal linear model • Alternatively the MSC is the fraction of R y (e jω ) that can be estimated from x ( n ) J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 13 J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 14 Example 3: Complex Correlation Scatter Plots Example 3: Complex Correlation Ratio Scatter Plot N:500 True Correlation:0.900 Estimated Correlation:0.895 Ratio y/x N:500 True Correlation:0.900 Estimated Correlation:0.895 4 0.5 0.5 3 Imag(x) Real(x) 0 0 2 −0.5 −0.5 1 Imaginary −1 0 1 −1 0 1 Real(y) Real(y) 0 −1 0.5 0.5 Imag(x) Real(x) −2 0 0 −3 −0.5 −0.5 −4 −1 0 1 −1 0 1 −6 −4 −2 0 2 4 6 Imag(y) Imag(y) Real J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 15 J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 16

Example 3: Complex Correlation Scatter Plots Example 3: Complex Correlation Ratio Scatter Plot N:500 True Correlation:0.100 Estimated Correlation:0.042 Ratio y/x N:500 True Correlation:0.100 Estimated Correlation:0.042 4 1 1 3 Imag(x) Real(x) 0 0 2 1 −1 −1 Imaginary −1 0 1 −1 0 1 Real(y) Real(y) 0 1 1 −1 0.5 0.5 Imag(x) Real(x) −2 0 0 −3 −0.5 −0.5 −1 −1 −4 −1 0 1 −1 0 1 −6 −4 −2 0 2 4 6 Imag(y) Imag(y) Real J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 17 J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 18 Example 3: MATLAB Code Estimation Let x ( n ) be a WSS process N = 500; rho = 0.90; % Desired correlation • It seems that most people use Welch’s nonparametric PSD and am = sqrt(rho^2/(1-rho^2)); a = am*exp(-j*pi/4); CPSD estimates (why?) x = 1/sqrt(2)*(randn(N,1) + j*randn(N,1)); w = 1/sqrt(2)*(randn(N,1) + j*randn(N,1)); • Most of the statistical properties are apparently only known for y = (w + a*x)./sqrt(1+abs(a)^2); Bartlett’s estimate yc = conj(y); xc = conj(x); • Usual (unrealistic) assumptions apply – Independent segments cyx = sum(y.*xc); cyx2 = cyx * conj(cyx); – Gaussian process sxx2 = sum(x.*xc); syy2 = sum(y.*yc); – No spectral leakage Cyx = sqrt(cyx2/(sxx2*syy2)); % Estimated correlation coefficient J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 19 J. McNames Portland State University ECE 538/638 Coherence Analysis Ver. 1.01 20