Understanding Sampling and Discrete Fourier Transform (DFT) in Signal Processing

1. Lesson 4 Discrete Fourier Transform

2. How to sample the frequency axis?

3. Sampling of Analog Signals Aliasing Formula

5. How to sample the frequency axis?

6. Fourier Series of a Periodic Sequence

7. Fourier Series of a Periodic Sequence

8. Fourier Series of a Periodic Sequence

9. Matlab Implementation Analysis or DFS equation Synthesis or inverse DFS equation

10. Matlab Implementation

11. Relation to DTFT

12. sinc function

13. Periodic Signals Are Completely Discrete: • Discrete rather than continuous frequencies. • Discrete rather than continuous times. • Summations instead of integrals.

14. Aperiodic Discrete-Time Signals

15. Aperiodic Discrete-Time Signals • Sampling in frequency generates a periodic signal in time

16. DFT of an Aperiodic Discrete-Time Signal of length N • Choose an integer L larger than or equal to N to be the period of a periodic extension of the aperiodic signal x(n). Pad zeros to x(n) if necessary. • Find the normalized Fourier Series representation of the periodic extension through DFS. • Then the DFT of x(n) is given by the DFS of the periodic extension for k on [0 L-1] and the IDFT is given by the IDFS with n on [0 L-1].

17. Computation of DFT via FFT • FFT (Fast Fourier Transform) is not another transformation but an algorithm to efficiently compute DFT. • Causal aperiodic signals: {x(n), n = 0, 1, … N-1}: proceed using FFT to obtain {X(k), k=0,1,…, N-1}. To compute for L>N, we simply attach L-N zeros at the end of the x(n) sequence and then FFT to obtain L values.

18. Computation of DFT via FFT • Non-Causal aperiodic signals: {x(n), n=-n0, …, 0, 1, …, N-n0-1}: • Move the non-causal samples to the causal side: {x(0), x(1), …, x(N-n0-1), x(-n0), x(-n0+1),…,x(-1)} • To improve frequency resolution, attach zeros between the causal and non-causal samples: {x(0), x(1), …, x(N-n0-1), 0, 0, 0, …, 0, 0, 0, x(-n0), x(-n0+1),…,x(-1)}

19. Example • Consider the DFT computation via FFT of a causal signal x(n) = (sin(πn/32))(u(n)-u(n-34)) and its shifted version x(n+16). To improve its frequency resolution, compute FFTs of length N = 512.

20. Convolution

21. Convolution

22. Convolution • DTFT of y

23. Convolution • We can obtain y(n) through the inverse Fourier transform • The L-length DFT of x(n) and h(n) are obtained by padding zeros. • Pad x(n) with L-M zeros • Pad h(n) with L-K zeros

24. Convolution • Given x(n) and h(n) of lengths M and K, the convolution y(n) of length N=M+K-1 can be found by the following 3 steps: • Compute DFTs X(k) and H(k) of length L>=N for x(n) and h(n). • Multiply them to get Y(k)=X(k)H(k) • Find the inverse DFT of Y(k) of length L to obtain y(n)

25. Example • x(n) = u(n) – u(n-21) of length 20, convolve with itself for different values of its length.