Discrete Fourier Transform The Citadel, The Military ollege 离散傅里叶变换的堡垒，军事学院.pptVIP

Discrete Fourier Transform Prof. Siripong Potisuk Summary of Spectral Representations Signal type Transform Frequency 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: 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. Nth Root of Unity Uniform Frequency Sampling Re(z) Im(z) z0 z1 z2 z3 z4 z5 z6 z7 1 N = 8 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 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 Matrix Formulation Matrix Formulation 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 v