1. Linear filtering based on the DFT
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)=[ ]; 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)=[ ] 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)=[ ] and a filter h(n)=[1 2 1], calculate the filter output using
(1) Overlap-save
(2) Overlap-add