1 ECE310 – Lecture 20 Relationship between Signals - Correlation 04/11/01

2 Signal Relationship Orthogonality No relationship Correlation How much alike two signals are?

3 Orthogonality Orthogonality: the dot product of the two vectors is zero Continuous-time signals orthogonality - over some time range from t 0 to t 0 +T If signals x 1 (t), x 2 (t), … x n (t) are mutually orthogonal in some interval, then all the signals are linearly independent in the interval, or none of the signal can be represented as a linear combination of others.

4 Correlation An analysis of whether two signals tend to “move together” Classification Highly correlated Highly positive correlated – over a long period of time, the signals tend to move in the same direction at the same time Highly negative correlated – over a long period of time, the signals tend to move in the opposite directions at the same time Partially correlated Uncorrelated – over a long period of time, the two signals tend to move in the same direction about half the time and in opposite directions the other half of the time Correlogram (with the two signals as the two axes) Correlation is a number

5 Correlogram Examples

6 % Function: Demonstrate the correlation between signals and the corresponding correlograms, correlation number % Author: Hairong Qi % Date: 04/08/01 clear all; clf; t = 0:0.01:1; % uncorrelated y1 = sin(2*pi*t); y2 = cos(2*pi*t); subplot(341); plot(t, y1); title('sin(2*pi*t)'); % signal 1 subplot(345); plot(t, y2); title('cos(2*pi*t)'); % signal 2 subplot(349); plot(y1, y2, '*'); xlabel('y1'); ylabel('y2'); % the correlogram % sum(y1.* y2) is the correlation number. sprintf() is the output reformat, and corrnr is a string corrnr = sprintf('UC: %5.2f', sum(y1.* y2)); title(corrnr); % highly negative correlated y1 = sin(2*pi*(t-1/4)); % signal 1 shifted to the right by 1/4 y2 = cos(2*pi*t); subplot(342); plot(t, y1); title('sin(2*pi*(t-1/4))'); subplot(346); plot(t, y2); title('cos(2*pi*t)'); subplot(3,4,10); plot(y1, y2, '*'); xlabel('y1'); ylabel('y2'); corrnr = sprintf('HNC: %5.2f', sum(y1.* y2)); title(corrnr); % highly positive correlated y1 = sin(2*pi*(t+1/4)); % signal 1 shifted to the left by 1/4 y2 = cos(2*pi*t); subplot(343); plot(t, y1); title('sin(2*pi*(t+1/4))'); subplot(347); plot(t, y2); title('cos(2*pi*t)'); subplot(3,4,11); plot(y1, y2, '*'); xlabel('y1'); ylabel('y2'); corrnr = sprintf('HPC: %5.2f', sum(y1.* y2)); title(corrnr); % partially correlated y1 = sin(2*pi*(t-1/3)); % signal 1 shifted to the right by 1/3 y2 = cos(2*pi*t); subplot(344); plot(t, y1); title('sin(2*pi*(t-1/2))'); subplot(348); plot(t, y2); title('sin(2*pi*t)'); subplot(3,4,12); plot(y1, y2, '*'); xlabel('y1'); ylabel('y2'); corrnr = sprintf('PC: %5.2f', sum(y1.* y2)); title(corrnr); Posted on web

7 Correlation Function Autocorrelation and crosscorrelation Correlation function is the correlation (number) when one signal is shifted across the other

8 Autocorrelation Function Correlation of a function with itself When shift t=0 When shift  != 0 Therefore, ???

9 Correlation Function of Energy and Power Signals Energy signal Power signal

10 Correlation Function of Two Periodic Signals T 0 is the period of the product, i.e. the LCM of T x and T y

11 Discussion The representation of a signal by a Fourier series can be seen as a process of correlating the signal with the sinusoids or complex exponentials to find out whether any particular sinusoid or complex exponential is present in the signal, and if so, how much of it is there. In another word, the coefficients calculate a “weight”