Lecture 3

1. Lecture 3 Linear filtering based on the DFT Advanced Digital Signal Processing Dr Dennis Deng Department of Electronic Engineering, La Trobe University 1998,1999

2. Contents • Linear filter and circular convolution • Overlap and save method • Overlap and add method • Self test exercises

3. Linear Filtering • A sequence x(n) of length L filtered by an FIR filter h(n) of length M • The length of y(n) is L+M-1 • Example x(n) =[1 1 1] , h(n) =[-1 1]

4. Linear Filtering • In the frequency domain • Sampling both side by an interval • The kth sample is given by • Using the DFT notation X(k) and H(k) are the DFT (with zero padding) of x(n) and h(n), respectively.

5. Linear Filtering • Performing the inverse DFT • Is the same as y(n) ? YES. Because is the Fourier transform of y(n), to recover y(n), an N-point sampling of and an N-point IDFT is necessary. • An equivalent question is under what condition circular convolution becomes linear convolution ? • A 4-point circular convolution of the previous example in the following slide

6. Linear Filtering • This example shows that by zero padding such that the length of the circular convolution is greater than or equal to L+M-1, circular convolution is the same as linear convolution.

7. Linear Filtering • The multiplication in DFT domain corresponds to circular convolution in the time domain. • By zero padding the two sequences, circular convolution becomes linear convolution • To perform linear convolution using the DFT (1) zero padding x(n) and h(n) such that the length of them is N=L+M-1 (2) Calculate an N-point DFT for both sequences and multiply them point-by-point (3) perform an N-point inverse DFT

8. Linear Filtering • The following example shows a 3-point circular convolution

9. Linear Filtering • The following example shows a 5-point circular convolution • These examples show that when circular convolution is the same as linear convolution

10. Overlap-save method • When the DFT is used to implement linear filtering, a signal is processed in blocks. Due to the real-time requirement (low delay) and the limitation of physical memory, the size of the block can not be arbitrarily large. • The length of the FIR filter is M and the length of on block of data is L (L>M) • Each time a block of data of length L+M-1 is filtered by using the DFT method.

11. Overlap-save method • The method is shown in the following diagram

12. Overlap-save method • Each step consists: (1) calculating N-point DFT of x(n) and multiply it with the N-point DFT of h(n). (2) calculating N-point IDFT and discarding the first M-1 output sample. The last L samples are the desired filter output (3) append the last M-1 samples to the beginning of the new block of signal (4) goto (1) until reach the end of the signal.

13. Overlap-save method • The reason for discarding the first M-1 output samples is that they are the result of circular convolution. Linear convolution starts at the Mth sample. • For the same reason, the last M-1 samples of the processed block should be appended to the beginning of the new block

14. Overlap-save method • This is can be easily seen from a simple example: x(n)=[1 2 2 1]; h(n)=[1 -1]. (L=4, M=2). A 6-point circular convolution (x(n) represents a block of data)

15. Overlap-add method • This method is shown in the following diagram

16. Overlap-add method • In each step: (1) perform the L+M-1 point DFT, multiplication and IDFT (2) the last M-1 samples of the previous output is added to the first M-1 samples of the current output

17. Summary • The DFT can be used to implement linear filtering • A key point is to choose the N-point DFT such that circular convolution is the same as linear convolution • The overlap-save and overlap-add methods can be used in real time signal processing.

18. Self test exercises • For a signal x(n)=[1 2 3 2 1] and a filter h(n)=[1 -2 1], (1) Calculate its circular convolution. (2) Verify your result by using Matlab (hint: real(ifft(fft(x).*fft(h,6)))) (3) Calculate its N-point (N=8,9,10,128) circular convolution. Which one is the same as linear convolution • For a signal x(n)=[1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4] and a filter h(n)=[1 2 1], calculate the filter output using (1) Overlap-save (2) Overlap-add