1. 1 Signal Processing Mike Doggett Staffordshire University

2. 2 CORRELATION n Introduction n Correlation Function – Continuous-Time Functions n Auto Correlation and Cross Correlation Functions n Correlation Coefficient n Correlation – Discrete-Time Signals n Correlation of Digital Signals

3. 3 INTRODUCTION Correlation techniques are widely used in signal processing with many applications in telecommunications, radar, medical electronics, physics, astronomy, geophysics etc.....

4. 4 Correlation has many useful properties, giving for example the ability to: n Detect a wanted signal in the presence of noise or other unwanted signals. n Recognise patterns within analogue, discrete- time or digital signals. n Allow the determination of time delays through various media, eg free space, various materials, solids, liquids, gases etc...

5. 5 n Correlation is a comparison process. n The correlation betweeen two functions is a measure of their similarity. n The two ‘functions’ could be very varied. For example fingerprints: a fingerprint expert can measure the correlation between two sets of fingerprints.

6. 6 n This section will consider the correlation of signals expressed as functions of time. The signals could be continuous, discrete time or digital. n When measuring the correlation between two functions, the result is often expressed as a correlation coefficient, , with  in the range –1 to +1.

7. 7 n Correlation involves multiplying, ‘sliding’ and integrating

8. 8 n Consider 2 functions

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10. 10

11. 11 n Consider 2 more functions

12. 12 n Consider 2 more functions

13. 13 CONVOLUTION

14. 14 n CORRELATION FUNCTION – CONTINUOUS TIME FUNCTIONS n Consider two continuous functions of time, v 1 (t) and v 2 (t). The functions may be random or deterministic. n The correlation or similarity between these two functions measured over the interval T is given by:

15. 15 n The functions may be deterministic or random. n R12(  ) is the correlation function and is a measure of the similarity between the functions v1(t) and v2(t). n The measure of correlation is a function of a new variable, , which represents a time delay or time shift between the two functions.

16. 16 n Note that correlation is determined by multiplying one signal, v1(t), by another signal shifted in time, v2(t-τ), and then finding the integral of the product, n Thus correlation involves multiplication, time shifting (or delay) and integration.

17. 17 n The integral finds the average value of the product of the two functions, averaged over a long time (T   ) for non-periodic functions. n For periodic functions, with period T, the correlation function is given by:

18. 18 n The correlation process is illustrated below: n As previously stated:

19. 19 n The output R 12 (τ) is the correlation between the two functions as a function of the delay τ. n The correlation at a particular value of τ would be solved by solving R 12 (τ),

20. 20 AUTO CORRELATION AND CROSS CORRELATION FUNCTIONS n Auto Correlation n In auto correlation a signal is compared to a time delayed version of itself. This results in the Auto Correlation Function or ACF. n Consider the function v(t), (which in general may be random or deterministic). n The ACF, R(  ), is given by

21. 21 n Of particular interest is the ACF when  = 0, and v(t) represents a voltage signal: n R(0) represents the mean square value or normalised average power in the signal v(t)

22. 22 n Cross Correlation n In cross correlation, two ‘separate’ signals are compared, eg the functions v1(t) and v2(t) previously discussed. n The CCF is

23. 23 n Diagrams for ACF and CCF n Auto Correlation Function, ACF n Note, if the input is v1(t) the output is R11(  ) n if the input is v2(t) the output is R22(  )

24. 24 n Cross Correlation Function, CCF

25. 25 n CORRELATION COEFFICIENT n The correlation coefficient, , is the normalised correlation function. n For cross correlation (ie the comparison of two separate signals), the correlation coefficient is given by: n Note that R 11 (0) and R 22 (0) are the mean square values of the functions v 1 (t) and v 2 (t) respectively.

26. 26 n For auto correlation (ie the comparison of a signal with a time delayed version of itself), the correlation coefficient is given by: n For signals with a zero mean value,  is in the range –1    +1

27. 27 n If  = +1 then the are equal (Positive correlation). n If  = 0, then there is no correlation, the signals are considered to be orthogonal. n If  = -1, then the signals are equal and opposite (negative correlation)

28. 28 n EXAMPLES OF CORRELATION – CONTINUOUS TIME FUNCTIONS

29. 29 n The above may be used for the demodulation of PSK/PRK (Phase Shift Keying / Phase Reversal Keying) signals. n For PSK/PRK, the input signal is v1(t) = d(t)cos  t, d(t) = +V for data 1’s and d(t) = -V for data 0’s. n The second function, v2(t) = cos  t, is the carrier signal. n Analyse the above process to determine the output R12(0) for the inputs given.

30. 30 n CORRELATION – DISCRETE TIME SIGNALS

31. 31 n CORRELATION OF DIGITAL SIGNALS n Searching for Synchronisation Pattern 01111110, in Data Bit Stream