Discrete Fourier Transform Prof. Siripong Potisuk

Summary of Spectral Representations Signal type TransformFrequency Domain CT, Periodic Continuous-time Fourier Series (CTFS) Discrete Spectrum CT, Aperiodic Continuous-time Fourier Transform (CTFT) Continuous Spectrum DT, Aperiodic Discrete-time Fourier Transform (DTFT) Continuous Spectrum, periodic DT, Periodic Discrete-time Fourier Series (DTFS) Discrete Fourier Transform (DFT) Discrete Spectrum, periodic

Computation of DTFT Computer implementation can be accomplished by: 1.Truncate the summation so that it ranges over finite limits  x[n] is a finite-length sequence. 2.Discretize  to  k  evaluate DTFT at a finite number of discrete frequencies For an N-point sequence, only N values of frequency samples of X(e j  ) at N distinct frequency points, are sufficient to determine x[n] and X(e j  ) uniquely.

Sequence Truncation

Uniform Frequency Sampling Re(z) Im(z) z0z0 z1z1 z2z2 z3z3 z4z4 z5z5 z6z6 z7z7 1 N = 8

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

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

N th Root of Unity

Matrix Formulation

Example 4.2 Define a sequence x[n] = 1, 2, 3, 4 when n = 0, 1, 2, 3, respectively. Evaluate its DFT, X(k).

Example 4.3 Using the result from example 4.2, evaluate the IDFT to obtain the time-domain sequence, x(n).

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

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

Example 4.3 Consider the sequence given below. Compute and sketch the magnitude, phase, and power density spectra.