This document discusses various techniques for data compression, correlation, convolution, and power spectrum estimation. It defines correlation as the degree of relationship between two or more variables. Convolution is described as a way to combine two signals into a resultant signal. Power spectrum estimation involves determining the energy distribution of a time series in the frequency domain without phase information. Nonparametric and parametric methods are discussed for power spectrum estimation.

1. Data Compression
Techniques
Module 3

2. • Definition: Correlation relates to what degree a relationship exist
between 2 or more variables.
Two types:
• Cross correlation
• Auto correlation
• Correlation index is 1 or -1

4. Data sets showing different directions and
degrees of correlation

5. Purpose of correlation

6. Correlation Coefficient

7. Correlation
• Correlation addresses “to what degree is signal A is similar to Signal
B”.
• By inspection A is “Correlated” with
B2 but B1 is uncorrelated with both
and B2 and A.

8. • Correlation XCORR
• Xcorr(a,b);
• Xcorr(a,a);
• Where a=1,2,3,4;
• b= 2,4,1,3 ;

9. Convolution
• Convolution is a mathematical way of combining two signals to form a
resultant signal.
• Convolution of two Continuous-time signal is x(t) and h(t) is

10. Discrete Convolution
• For LTI system, if the input sequence is x(n) and impulse response
h(n), the output y(n) can be found
• This is known as convolution sum and is represented as
• Y(n)=x(n)*h(n) = h(n)* x(n)

11. Properties of Convolution
• Commutative
It states that order of convolution does not matter, which can be shown mathematically as
x1(t)∗x2(t)=x2(t)∗x1(t)
• Associative
It states that order of convolution involving three signals, can be anything. Mathematically, it
can be shown as;
x1(t)∗[x2(t)∗x3(t)]=[x1(t)∗x2(t)]∗x3(t)
• Distributive
x1(t)∗[x2(t)+x3(t)]=[x1(t)∗x2(t)+x1(t)∗x3(t)]

12. Example of discrete linear Convolution

13. Example of discrete linear Convolution

14. Periodic Convolution

15. Example of periodic convolution

16. Applications of Convolution
• These two applications are:
1. Characterizing a linear time-invariant (LTI) system in terms of its
transfer function
2. Determining the output of an LTI system when its input is known
3. FIR filter design

17. Power Spectrum Estimation
• A power spectrum describes the energy distribution of a time series
in the frequency domain.
• Energy is a real-valued quantity, so the power spectrum does not
contain phase information.
• Because a time series may contain non-periodic or asynchronously-
sampled periodic signal components, the power spectrum of a time
series typically is considered to be a continuous function of
frequency.

18. • WHAT IS A SPECTRUM?
• A spectrum is a relationship typically represented by a plot of the
magnitude or relative value of some parameter against frequency.
• Every physical phenomenon, whether it be an electromagnetic,
thermal, mechanical, hydraulic or any other system, has a unique
spectrum associated with it.
• A study of relationships between the time domain and its
corresponding frequency domain representation is the subject of
Fourier analysis and Fourier transforms.

20. • The PSD measures the signal power per unit bandwidth for a time
series.
• PSD estimation methods are classified as follows:
• Parametric methods
• Nonparametric methods

21. • Parametric methods—These methods are based on parametric
models of a time series, such as AR models, moving average (MA)
models, and autoregressive-moving average (ARMA) models.
• Therefore, parametric methods also are known as model based
methods.
• Nonparametric methods—These methods, which include
the periodogram method, Welch method, and Capon method, are
based on the DFT

22. • Autoregressive spectrum estimation:
• A. The autocorrelation method
• B. The covariance method
• C. Modified covariance method
• D. Burg algorithm:
• E. Selection of the model order:

23. SPECTRAL ESTIMATION BY AVERAGING PERIODOGRAMS:
• It was shown in the last section that the periodogram was not a
consistent estimate of the power spectral density function.
• A technique introduced by Bartlett, however, allows the use of the
periodogram and, in fact, produces a consistent spectral estimation
by averaging periodograms.
• A periodogram is used to identify the dominant periods (or
frequencies) of a time series. This can be a helpful tool for
identifying the dominant cyclical behavior in a series,
particularly when the cycles are not related to the commonly
encountered monthly or quarterly seasonality.

25. Applications
The need for power spectrum estimation arises in a variety of contexts,
including the
• measurement of noise spectra for the design of optimal linear filters,
• the detection of narrow-band signals in wide-band noise,
• and the estimation of parameters of a linear system by using a noise
excitation.