1. 1 BIEN425 – Lecture 8 By the end of the lecture, you should be able to: –Compute cross- /auto-correlation using matrix multiplication –Compute cross- /auto-correlation using DFT and Matlab –Relate auto-correlation of white noise with its average power –Extract periodic signal from noise

2. 2 Recall: Convolution Is a filter operation that computes the zero-state output corresponding to an input x(k) Matlab code: conv.m As denoted by * If h(i) is a L-point signal, and x(k) is a M-point signal then the convolution has length L+M-1

3. 3 Circular convolution Let h(k) and x(k) be N-point signals and x p (k) be the periodic extension of x(k). Circular convolution is defined as Very simply y c (k) = C(x)h, where C(x) is defined as:

4. 4 Note that circular convolution is more efficient (faster) However, it cannot be used directly to compute zero- state response Also needs to assume h(k) and x(k) have the same length. Another problem is with periodicity We can solve this using zero-padding. But the zero-state response is not periodic

5. 5 Zero padding Pad both h(k) and x(k) to be length N = L+M-1, now denoted as h z (k) and x z (k) Now, y = C(x z )h z y is now M+L-1 in length

6. 6 Example Matlab example

7. 7 Fast convolution Convolution in the time domain maps into multiplication in the frequency domain using Z- transform. We may substitute circular for linear convolution; DFT for Z-transform

8. 8 Now if enough zeros are padded to h(k) and x(k) so that they both have length N≥L+M-1 where N = 2 r This is called fast linear convolution We get H z (i) and X z (i) Zero pad FFT IFFT h(k) x(k) y(k) x z (k) h z (k)

9. 9 Cross-correlation Measures the degree to which a pattern of L-point signal y(k) is similar to the pattern of an M-point signal x(k) This definition works fine for deterministic signals. To generalize it to include stochastic signals: For example: on next slide

10. 10

11. 11 Again, it can be computed using matrix formulation: Example 4.8, the index corresponding to the largest r xy (k) represents the optimal delay.

12. 12 This shows optimal delay for y is 3 samples

13. 13 Normalized linear cross-correlation So that it takes on values [-1, 1] This allows us to measure the amount of “significance” that is independent on the scaling or amplitude of the signal

14. 14 We can have a fixed threshold to determine if two signals are significantly similar.

15. 15 Fast correlation Again, similar to fast convolution, we could utilize properties of FFT to achieve a fast version of correlation Block diagram: Zero pad FFT FFT * IFFT /L x(k) y(k) r xy (k) y z (k) x z (k)

16. 16 Auto-correlation For special case where y(k) = x(k) It can be expressed in terms of expected values For the special case of white noise v(k),

17. 17 Relating auto-correlation and PSD If we apply the same principle of circular convolution to correlation, circular-correlation c xy (k) can be obtained Notice here (–k)

18. 18 Extracting periodic signals from noise Let’s say y(k) = x(k) + v(k), 0≤ k <N and the period M<N. Therefore, for k≠0, c yy (k) = c xx (k) Motivation? x(k) is periodic M (but we didn’t know that from the beginning Because correlation of noise with any other signal is 0!

19. 19 Auto-correlation averages the noise out! Also note that c xx (k) is periodic Therefore, the period of x(k) can be estimated by examining the peaks and valleys in c yy (k)

20. 20 Once the period of a noise-corrupted signal is determined, we can estimate the original periodic signal Let L = floor (N/M) We define delta function of period M 0 ≤ k < M By definition Because L = floor of (N/M)

21. 21 Now the second term approaches 0 for large L Therefore Because x is period M

22. 22 Block diagram Circular cross correlation y(k) x(k) v(k)  (k) N/L