1. Lecture#10 Spectrum Estimation
Adv DSP Spring-2015
Lecture#10
Spectrum Estimation

2. Introduction
Spectrum Estimation: To estimate the power spectral density (PSD) of WSS random process.
PSD is the DTFT of the autocorrelation sequence.
Estimating the power spectrum is equivalent to estimating the autocorrelation.

3. Introduction The autocorrelation is generally estimated by using
If x[n] is known for all ‘n’, then power spectrum estimation is straight forward
Determine autocorrelation sequence above and then compute its Fourier transform.
Two problems with this:
The amount of data is never unlimited , in many cases very small
Data is often corrupted by noise

4. Introduction
Spectrum estimation is the problem that involves estimating Px(ejw) from a finite number of noisy measurement of x[n].
Spectrum estimation is a problem that is important in a variety of different fields and applications.
Applications: Optimum Filter design, Signal Detection & Tracking, Harmonic Analysis & Prediction, Spectral Smoothing, Beamforming & Direction Finding,

5. Techniques for Spectrum Estimation
Classical or Non-parametric Approach
Estimating the autocorrelation sequence from given set of data.
Power Spectrum is then obtained by Fourier transforming the estimated autocorrelation.
Non-Classical or Parametric Approach
Based on using a model for the process to estimate the power spectrum.
AR, MA or ARMA Models

6. Techniques for Spectrum Estimation

7. Power Spectral Density

8. Non-Parametric Methods
Based on the idea of estimating the autocorrelation sequence of a random process from a set of measured data.
Take the Fourier transform to obtain an estimate of the power spectrum.
Periodogram (1898)
Easy to compute.
Limited in its ability to produce an accurate estimate of the power spectrum, particularly for short data records.
Modification to Periodogram to improve its statistical properties
Modified periodogram method, Bartlett’s method, Welch’s method, Blackman-Tukey method.

9. The Periodogram
The power spectrum of WSS random process is the Fourier transform of the autocorrelation sequence:
Spectrum estimation  Autocorrelation estimation problem
Autocorrelation sequence for an unlimited amount of data can be estimated using the time average:

10. The Periodogram
If x[n] is only measured over a finite interval, say n=0,1,….,N-1 , then we can estimate the autocorrelation sequence using:
In order to ensure that the values of x[n] that fall outside the interval [0,N-1] are excluded, we can write:

11. The Periodogram
Taking the Fourier transform of the autocorrelation sequence gave us the Periodogram.
Next we will represent the Periodogram directly in terms of the process x[n].
Let xN[n] be the finite length signal of length ‘N’ that is equal to x[n] over the interval [0,N-1] and is zero otherwise:

12. The Periodogram
Thus, xN[n] is the product of x[n] with rectangular window wR[n],
In term of xN[n] , the estimated autocorrelation sequence may be written as follows:
Taking the Fourier transform and using convolution theorem, the Periodogram becomes:

13. The Periodogram
XN(ejw) is the DTFT of the N-point data sequence xN[n],
Thus, the Periodogram is proportional to the squared magnitude of DTFT of xN[n], and may be easily computed using a DFT (FFT) as follows,

14. Example 8.2.1: Periodogram of White Noise
If x[n] is the white noise with variance σx2
Thus the power spectrum should be constant.
Consider a sample realization of white noise of unite variance and N=32

15. Example 8.2.1: Periodogram of a White Noise
Sample realization
Estimated autocorrelation
sequence
Periodogram

16. White Noise in Matlab
rand( ) : Uniformly distributed white noise with unit variance.
randn( ): Gaussian Distributed white noise with unit variance.

17. Example: The Periodogram

18. Matlab Code clear all; close all; A=1; N=100; fs=0.3; n=1:N;
rn=randn(1,N);
phi=2*pi*rand(1,N);
x=A.*exp(j*2*pi*fs*n+phi)+rn;
Px=abs(fft(x,1024)).^2/length(n);
w=2.*pi.*[0:1023]./1024;
plot(w./(2*pi),Px)
xlabel('\bf Normalized Frequency')
ylabel('\bf Periodogram')
Title('\bf [A=1,N=100]')
Matlab Function can also be used directly: periodogram ( )

19. A=1
N=100
N=1000
N=10000
A=0.001

20. Modified Periodogram
The Periodogram is proportional to the squared magnitude of the Fourier transform of the windowed signal, i.e. xN[n]=x[n]wR[n]
Instead of applying a rectangular window , the above equation suggests the possibility of using other data windows.

21. Modified Periodogram
Amount of smoothing in the Periodogram is determined by the window that is applied to the data.
A rectangular window has a narrow main lobe compared to other windows, and therefore produces the least amount of spectral smoothing, it has relatively large side lobes that may lead to masking of weak narrowband components.

22. Rectangular Window DTFT amplitude ‘M’ is the length

23. Example: Two sinusoid in White Noise
Rectangular Window (Narrow main lobe High Resolution, large side lobes)
Hamming Window (Wide main lobe, Less Resolution, Small side lobes)

24. Example: Two sinusoid in White Noise

25. Example: Two sinusoid in White Noise
Rectangular window: Sinusoid at frequency w1 is almost completely masked by the side lobes of the window at frequency w2.
Hamming window: The sinusoid at w1 is clearly visible. This is due to smaller side lobes of a Hamming window
Reduction in the side lobe amplitude comes at the expense of an increase in the width of main lobe, which in turn, effects the resolution.

26. Modified Periodogram
The Periodogram of a process that is windowed with a general window w[n] is called a “Modified Periodogram”.
“N” is the length of window
“U” is the constant given by

27. Modified Periodogram
Matlab Function : periodogram ( ) has a window as an input parameter
Matlab has functions to generate the above mentioned windows

28. Matlab Window Functions
Rectangular: rectwin ( n)
Hanning: hanning (n )
Hamming: hamming (n )
Blackman: blackman (n)
Bartlett: bartlett(n )

29. Rectangular window has the best resolution