and shall lay stress on CORRELATION In this lecture, we shall learn about the correlation function, and shall differentiate between correlation and convolution processes and shall lay stress on CORRELATION

Correlation Provides the measure of similarity or ________between two functions at a given lag of time. If the two function originate from one single function, it is called ______-correlation. If they originate independently, the measure is called ______-correlation.

CONVOLUTION & CORRELATION basic mathematical model. Let x(t) and h(t) be the two real TIME functions, The convolution integral of the two aperiodic functions, y(t) = x(t)*h(t) is: While the correlation Rxh(t) = x(t) y(t) is given by the equation: The difference is: in correlation, the time function is not reversed. And that, Rxh(t) represent energy/power. Then, does y(t) also represent energy?

Convolution & Correlation The two signals are real. The ____under the curve of two integrals one due to convolution and other due to correlation is ______. Their nature is ______ same. The length of the time-duration after integration is also the same. If one or both of the functions are symmetrical, it results into ______ nature of the integration curve.

Convolution & Correlation Each of the two functions, x(t) and h(t) may be represented by power series. Multiplication of them yields convolution. While multiplication with one series _______ yields Correlation.

Convolution & Correlation In both the cases, x(t) and h(t) are real time functions. Convolutions can be had for the functions in domains other than time. Correlation is meaningful exclusively in ________domain. Both can can be transformed in __________ domain. Frequency domain output can be __________. If complex, it should have _________ term also. Correlation is a special case of convolution with one function time __________.

Auto and Cross correlations If the functions originate from the same source, the resulting summation or, integration is termed as ______ Correlation. Should they belong to different independent sources, the resulting summation or, integration is _______-correlation. The signals can be power (_________) signal or, energy (_________) signal.

Correlation of Periodic signals If x(t) and h(t) are periodic, the correlation represents power and is worked out to be: The integral at different time delay , represent the power developed by the two signals at that delay. If the signals are aperiodic, the above integral, with the term 1/T set to 1, would represent energy

Properties of Correlation functions Rxh(t) represent power and power is always a _______ quantity. It has Rxh(t)=Rhx(-t) symmetry. It can be splitted in even + odd. In auto-correlation since ______, is inherently an even symmetric. Its maximum value rests at t=0. In general Rxh(t)  Rhx(t), does not commutate. It has maximum value at t=0. Therefore Rxh ()  Rxh (0).

More Properties If x(t) and h(t) are periodic, then Rxh (t) is also ________. Rxx (t) Rhh (t)>Rxh (t)2. Since geometric mean ________ exceed their arithmetic mean, Therefore [Rxx (t) + Rhh (t)]/2  Rxx (t) Rhh (t) and also  Rxh (t)

Properties of correlations The Fourier Transform of correlation function is called _________________PSD; V2/Hz.. If x(t) and h(t) are statistically independent random processes, then Rxh (t) = Rhx (t), with _______, they follow the rule of orthogonal functions, Rxh (t) = Rhx (t) = 0. That is, resultant output power in the duration of composite periodicity is ______. It is the effective way of matching two functions. Here we match x(t) with delayed version of y(t).

Example: To work out convolution and correlation of the sequences: x1 [n]=[1 2 3 4] and x2 [n]=[0 1 2 3] Convolution:x1 [n]*x2[n] =x2 [n]*x1[n] = [0 1 4 10 16 17 12] Area= sum of all numbers=60. Cross Correlation: X2 [n]**x1[n] = conv. X2[n]*x1[-n] = [0 1 2 3]* [4 3 2 1] = [0 4 11 20 14 8 3] Area = sum of all numbers = 60. unlike in convolution: X2[n]**x1[n]  x1[n]**x2[n]. = [ ] The output sequence is _______ !!

Cross correlation should have same length. Or, append zero The sequences. The maximum is always at center of the sequence, denoted as n=0. The sequence can be represented as even and odd part.

Even and Odd sequences of correlation function The correlation function is: z[n] = [1 4 11 20 14 8 3] z[-n]= [_____________] Even sequence: {z[n] +z[-n]}/2 Odd sequence: {z[n]-z[-n]}/2 Zev[n] = [____________________] Zodd[n]= [____________________] Even function is symmetrical about __________. Sequence has __________at Center. Sum is that of original sequence. odd function is skew symmetrical (mirror image) about _____ and its value is ______. Sum of odd function sequence is _______.

Autocorrelation Auto correlation is the correlation of a sequence with itself. Let the sequence be: x[n] [1 2 3 4] It’s autocorrelation is = x[n]**x[n]= [1 2 3 4][4 3 2 1] = [4 11 20 40 20 11 4] always symmetrical and maximum at center .

Correlation and Regression s/n correlation regression 01 Karl Pearson Method xy = [byx bxy ]1/2 Bxy = cov(x,y)/var(x); y on x. Byx = cov(x,y)/var(y); x on y. 02 Standard error of estimate: Syx = y [1-xy2 ] and Syx /Sxy = y /x var(x) =x , var(y) =y. 03 Coefficients of correlation provides the degree of relationship between variables. Coefficient of regression provides the nature of relationship between the variables. 04 It does not employ cause-effect relationship, the transfer function. It does imply the cause-effect relationship, that is, transfer function 05 Relationship may be arbitrary. Relationship is founded. 06 Coefficient is independent of: origin and the scale. Coefficient is independent of : origin but not 07 Prediction is not possible. Prediction is possible.