Adv DSP Spring-2015 Lecture#11 Spectrum Estimation Parametric Methods

Introduction  One limitation of non-parametric methods is that they cant incorporate the a-priori information about the process in estimation.  For example: In speech processing, an acoustic tube model for the vocal tract imposes an AR model on speech waveform.  If it is possible to incorporate a model for the process directly in to spectrum estimation, then a more accurate and high resolution estimate can be found.

Parametric Method  First step is to select an appropriate model for the process.  This selection is based upon: A-priori knowledge about how the process is generated Experimental results indicate that a particular model “works well”.  Models used are Autoregressive (AR) Model Moving Average (MA) Model Autoregressive Moving Average (ARMA) Model Harmonic Model (Complex exponential in noise)

Parametric Methods  Once the model is selected, the next step is to estimate the model parameters from the given data.  The final step is the estimate the power spectrum by incorporating the estimated parameters into the parametric form for the spectrum.  Example: An ARMA(p,q) model with a p (k) and b q (k) estimated, the spectrum estimate would be

Autoregressive Spectrum Estimation  An AR process, x[n], may be represented as the output of an all-pole filter that is driven by unit variance white noise.  The power spectrum of a pth-order AR process is:  Therefore, if b(0) and a p (k) can be estimated from the data then an estimate of the power spectrum may be formed using:

Autoregressive Spectrum Estimation  The accuracy of this estimate will depend on how accurately the model parameters may be estimated.  Also, whether or not an AR model is consistent with the way in which data is generated.  AR spectrum estimation requires that an all-pole model may be found.  Variety of techniques are available for all-pole modeling.

AR: The Autocorrelation Method  The Autocorrelation Method: AR coefficients are found by solving following normal equations: Yule-Walker Method

AR: The Autocorrelation Method  The autocorrelation method effectively applies a rectangular window to the data when estimating the autocorrelation sequence, hence the data is effectively extrapolated with zeros  Due to this, the autocorrelation method generally produces a lower estimation estimate.  For short data records the autocorrelation method is not generally used.

AR: The Autocorrelation Method  An artifact that may be observed with the autocorrelation method is “Spectral Line Splitting”. Involves the splitting of single spectral line into two separate and distinct peaks  Spectral line splitting occur when x[n] is over- modeled, i.e. when model order ‘p’ is too large.

AR: The Covariance Method  The covariance method requires finding the solution to the set of linear equations:  The advantage is that no windowing of data is required.  For short data record this method produces a high resolution estimate.

AR Method: Example  Consider the AR process that is generated with the difference equation, where w[n] is unit variance white Gaussian noise:  Data records of length 128, and an ensemble of 50 estimates were computed using Yule-Walker and the covariance method.

AR Method: Example zplane([1],[ ]) Pair of Complex Poles at:

AR: The Autocorrelation Method

AR: The Covariance Method

Model Order Selection  How to select the model order ‘p’ of the AR process.  If the model order is too small, then the resulting spectrum will be smoothed and will have poor resolution.  If the model order is too large, then the spectrum may contain spurious peaks, and may lead to spectral line splitting.  It is useful to have criteria that indicates the appropriate model order.

Model Order Selection  One approach would be to increase the model order until the modeling error is minimized.  Several criteria of the following form has been proposed: ‘p’ is the model order N is the data record length ε p is the modeling error F(N) is a constant that depend upon N

Model Order Selection  Akaike Information Criteria (AIC)  Minimum Description Length (MDL)  Akaike’s Final Prediction Error (FPE)  Criterion Autoregressive Transfer Function (CAT)

Moving Average (MA) Spectrum Estimation  A MA process may be generated by filtering unit variance white noise,w[n], with an FIR filter as follows:  The relationship between the power spectrum of a MA process and the coefficients b q [k] is,

Moving Average (MA) Spectrum Estimation  Equivalently, the power spectrum may be written in terms of the autocorrelation sequence r x [k] as:  Where r x [k] is related to the filter coefficients b q [k] through Yule-Walker equations:

ARMA Spectrum Estimation  An ARMA process has a power spectrum of the form:  This process can be generated by filtering unit variance white noise with filter having both poles and zeros:

ARMA Spectrum Estimation  Following the approach used for AR(p) and MA(q) spectrum estimation, the spectrum of ARMA(p,q) process may be estimated by using the estimates of model parameters (Ch:4)