Discrete Fourier Transform

1. Discrete Fourier Transform Prof. Siripong Potisuk

2. Summary of Spectral Representations

3. Computation of DTFT • Computer implementation can be accomplished by: • Truncate the summation so that it ranges over finite limits •  x[n] is a finite-length sequence. • Discretize w to wk • evaluate DTFT at a finite number of discrete frequencies For an N-point sequence, only N values of frequency samples of X(ejw) at N distinct frequency points, are sufficient to determine x[n] and X(ejw) uniquely.

4. Nth Root of Unity

5. Im(z) z2 z3 z1 z4 z0 Re(z) 1 z7 z5 z6 N = 8 Uniform Frequency Sampling

6. Discrete Fourier Transform Let x[n] be an N-point signal, and WN be the Nth root of unity. The N-point discrete Fourier Transform of x[n], denoted X(k) = DFT{x[n]}, is defined as

7. Inverse Discrete Fourier Transform Let X(k) be an N-point DFT sequence, and WN be the Nth root of unity. The N-point inverse discrete Fourier Transform of X(k), denoted x[n] = IDFT{X(k)}, is defined as

8. Matrix Formulation

9. Matrix Formulation

10. DFT Computation Using MATLAB fft(x) - Computes the N-point DFT of a vector x of length N fft(x, M) - Computes the M-point DFT of a vector x of length N If N < M, x is zero-padded at the end to make it into a vector of length M If N > M, x is truncated to the first M samples ifft(X) - Computes the N-point IDFT of a vector X of length N ifft(X, M) - Computes the M-point IDFT of a vector X of length N If N < M, X is zero-padded at the end to make it into a vector of length M If N > M, X is truncated to the first M samples

11. DFT Interpretation DFT sample X(k) specifies the magnitude and phase angle of the kth spectral component of x[n]. The amount of power that x[n] contains at a normalized frequency, fk, can be determined from the power density spectrum defined as