Function File: [a, v, k] = arburg (x, poles) ¶

Function File: [a, v, k] = arburg (x, poles, criterion) ¶

Calculate coefficients of an autoregressive (AR) model of complex data x using the whitening lattice-filter method of Burg (1968). The inverse of the model is a moving-average filter which reduces x to white noise. The power spectrum of the AR model is an estimate of the maximum entropy power spectrum of the data. The function ar_psd calculates the power spectrum of the AR model.

ARGUMENTS:

• x sampled data
• poles number of poles in the AR model or limit to the number of poles if a valid criterion is provided.
• criterion model-selection criterion. Limits the number of poles so that spurious poles are not added when the whitened data has no more information in it (see Kay & Marple, 1981). Recognized values are ’AKICc’ – approximate corrected Kullback information criterion (recommended), ’KIC’ – Kullback information criterion ’AICc’ – corrected Akaike information criterion ’AIC’ – Akaike information criterion ’FPE’ – final prediction error" criterion The default is to NOT use a model-selection criterion

RETURNED VALUES:

• a list of (P+1) autoregression coefficients; for data input x(n) and white noise e(n), the model is

                        P+1
  x(n) = sqrt(v).e(n) + SUM a(k).x(n-k)
                        k=1
  

  v mean square of residual noise from the whitening operation of the Burg lattice filter.

• k reflection coefficients defining the lattice-filter embodiment of the model

HINTS:

(1) arburg does not remove the mean from the data. You should remove the mean from the data if you want a power spectrum. A non-zero mean can produce large errors in a power-spectrum estimate. See "help detrend". (2) If you don’t know what the value of "poles" should be, choose the largest (reasonable) value you could want and use the recommended value, criterion=’AKICc’, so that arburg can find it. E.g. arburg(x,64,’AKICc’) The AKICc has the least bias and best resolution of the available model-selection criteria. (3) Autoregressive and moving-average filters are stored as polynomials which, in matlab, are row vectors.

NOTE ON SELECTION CRITERION:

AIC, AICc, KIC and AKICc are based on information theory. They attempt to balance the complexity (or length) of the model against how well the model fits the data. AIC and KIC are biased estimates of the asymmetric and the symmetric Kullback-Leibler divergence respectively. AICc and AKICc attempt to correct the bias. See reference [4].

REFERENCES:

[1] John Parker Burg (1968) "A new analysis technique for time series data", NATO advanced study Institute on Signal Processing with Emphasis on Underwater Acoustics, Enschede, Netherlands, Aug. 12-23, 1968.

[2] Steven M. Kay and Stanley Lawrence Marple Jr.: "Spectrum analysis – a modern perspective", Proceedings of the IEEE, Vol 69, pp 1380-1419, Nov., 1981

[3] William H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery "Numerical recipes in C, The art of scientific computing", 2nd edition, Cambridge University Press, 2002 — Section 13.7.

[4] Abd-Krim Seghouane and Maiza Bekara "A small sample model selection criterion based on Kullback’s symmetric divergence", IEEE Transactions on Signal Processing, Vol. 52(12), pp 3314-3323, Dec. 2004

See also: ar_psd.

Package: signal