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# rlevinson() - Signal Processing

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The reverse Levinson-Durbin recursion implements the step-down algorithm for solving the following symmetric Toeplitz system of linear equations for r, where r = [r(1) … r(p + 1)] and r(i)* denotes the complex conjugate of r(i).[r(1)r(2)∗⋯r(p)∗r(2)r(1)⋯r(p−1)∗⋮⋱⋱⋮r(p)⋯r(2)r(1)][a(2)a(3)⋮a(p+1)]=[−r(2)−r(3)⋮−r(p+1)]r = rlevinson(a,efinal) solves the above system of equations for r given vector a, where a = [1 a(2) … a(p + 1)]. In linear prediction applications, r represents the autocorrelation sequence of the input to the prediction error filter, where r(1) is the zero-lag element. The figure below shows the typical filter of this type, where H(z) is the optimal linear predictor, x(n) is the input signal, x^(n) is the predicted signal, and e(n) is the prediction error. Input vector a represents the polynomial coefficients of this prediction error filter in descending powers of z.A(z)=1+a(2)z−1+⋯+a(n+1)z−pThe filter must be minimum-phase to generate a valid autocorrelation sequence. efinal is the scalar prediction error power, which is equal to the variance of the prediction error signal, σ2(e).[r,u] = rlevinson(a,efinal) returns upper triangular matrix U from the UDU* decomposition R−1=UE−1U∗where R=[r(1)r(2)∗⋯r(p)∗r(2)r(1)⋯r(p−1)∗⋮⋱⋱⋮r(p)⋯r(2)r(1)]and E is a diagonal matrix with elements returned in output e (see below). This decomposition permits the efficient evaluation of the inverse of the autocorrelation matrix, R−1.Output matrix u contains the prediction filter polynomial, a, from each iteration of the reverse Levinson-Durbin recursionU=[a1(1)∗a2(2)∗⋯ap+1(p+1)∗0a2(1)∗⋱ap+1(p)∗00⋱ap+1(p−1)∗⋮⋱⋱⋮0⋯0ap+1(1)∗]where ai(j) is the jth coefficient of the ith order prediction filter polynomial (i.e., step i in the recursion). For example, the 5th order prediction filter polynomial is a5 = u(5:-1:1,5)' Note that u(p+1:-1:1,p+1)' is the input polynomial coefficient vector a.[r,u,k] = rlevinson(a,efinal) returns a vector k of length p + 1 containing the reflection coefficients. The reflection coefficients are the conjugates of the values in the first row of u.k = conj(u(1,2:end)) [r,u,k,e] = rlevinson(a,efinal) returns a vector of length p + 1 containing the prediction errors from each iteration of the reverse Levinson-Durbin recursion: e(1) is the prediction error from the first-order model, e(2) is the prediction error from the second-order model, and so on. These prediction error values form the diagonal of the matrix E in the UDU* decomposition of R−1.R−1=UE−1U∗

r = rlevinson(a,efinal)[r,u] = rlevinson(a,efinal)[r,u,k] = rlevinson(a,efinal)[r,u,k,e] = rlevinson(a,efinal)

Optimum Autoregressive Model OrderOpen This Example Estimate the spectrum of two sine waves in noise using an autoregressive model. Choose the best model order from a group of models returned by the reverse Levinson-Durbin recursion. Generate the signal. Specify a sample rate of 1 kHz and a signal duration of 50 seconds. The sinusoids have frequencies of 50 Hz and 55 Hz. The noise has a variance of 0.22.Fs = 1000; t = (0:50e3-1)'/Fs; x = sin(2*pi*50*t) + sin(2*pi*55*t) + 0.2*randn(50e3,1); Estimate the autoregressive model parameters.[a,e] = arcov(x,100); [r,u,k] = rlevinson(a,e); Estimate the power spectral density for orders 1, 5, 25, 50, and 100.N = [1 5 25 50 100]; nFFT = 8096; P = zeros(nFFT,5); for idx = 1:numel(N) order = N(idx); ARtest = flipud(u(:,order)); P(:,idx) = 1./abs(fft(ARtest,nFFT)).^2; end Plot the PSD estimates.F = (0:1/nFFT:1/2-1/nFFT)*Fs; plot(F, 10*log10(P(1:length(P)/2,:))) grid legend([repmat('Order = ',[5 1]) num2str(N')]) xlabel('Frequency (Hz)') ylabel('dB') xlim([35 70])