The Discrete Fourier Transform 主講人：虞台文

Content Introduction Representation of Periodic Sequences – DFS (Discrete Fourier Series) Properties of DFS The Fourier Transform of Periodic Signals Sampling of Fourier Transform Representation of Finite-Duration Sequences – DFT (Discrete Fourier Transform) Properties of the DFT Linear Convolution Using the DFT

The Discrete Fourier Transform Introduction

Signal Processing Methods Time Continuous Discrete Periodicity PeriodicAperiodic Fourier Series Continuous-Time Fourier Transform Continuous-Time Fourier Transform DFS Duration Finite Infinite Discrete-Time Fourier Transform and z-Transform Discrete-Time Fourier Transform and z-Transform

Frequency-Domain Properties Time Continuous Discrete Periodicity PeriodicAperiodic Fourier Series Continuous-Time Fourier Transform Continuous-Time Fourier Transform DFS Duration Finite Infinite Discrete-Time Fourier Transform and z-Transform Discrete-Time Fourier Transform and z-Transform Discrete & Aperiodic Discrete & Aperiodic Continuous & Aperiodic Continuous & Aperiodic Continuous & Periodic (2  ) Continuous & Periodic (2  ) ? ?

Frequency-Domain Properties Time Continuous Discrete Periodicity PeriodicAperiodic Fourier Series Continuous-Time Fourier Transform Continuous-Time Fourier Transform DFT Duration Finite Infinite Discrete-Time Fourier Transform and z-Transform Discrete-Time Fourier Transform and z-Transform Discrete & Aperiodic Discrete & Aperiodic Continuous & Aperiodic Continuous & Aperiodic Continuous & Periodic (2  ) Continuous & Periodic (2  ) ? ? Relation?

Frequency-Domain Properties Time Continuous Discrete Periodicity PeriodicAperiodic Fourier Series Continuous-Time Fourier Transform Continuous-Time Fourier Transform DFT Duration Finite Infinite Discrete-Time Fourier Transform and z-Transform Discrete-Time Fourier Transform and z-Transform Discrete & Aperiodic Discrete & Aperiodic Continuous & Aperiodic Continuous & Aperiodic Continuous & Periodic (2  ) Continuous & Periodic (2  ) Discrete & Periodic Discrete & Periodic

The Discrete Fourier Transform Representation of Periodic Sequences --- DFS

Periodic Sequences Notation: a sequence with period N where r is any integer.

Harmonics Facts: Each has Periodic N. N distinct harmonics e 0 (n), e 1 (n),…, e N  1 (n).

Synthesis and Analysis Notation Synthesis Analysis Both have Period N

Example A periodic impulse train with period N.

Example n

n

n

DFS vs. FT 0N n NN 0 n

Example n n

n n

n n

The Discrete Fourier Transform Properties of DFS

Linearity

Shift of A Sequence Change Phase (delay)

Shift of Fourier Coefficient Modulation

Duality

Periodic Convolution Both have Period N

Periodic Convolution Both have Period N

The Discrete Fourier Transform The Fourier Transform of Periodic Signals

Fourier Transforms of Periodic Signals Time Continuous Discrete Periodicity PeriodicAperiodic Fourier Series Continuous-Time Fourier Transform Continuous-Time Fourier Transform DFT Duration Finite Infinite Discrete-Time Fourier Transform and z-Transform Discrete-Time Fourier Transform and z-Transform Discrete & Aperiodic Discrete & Aperiodic Continuous & Aperiodic Continuous & Aperiodic Continuous & Periodic (2  ) Continuous & Periodic (2  ) DFS Sampling

Fourier Transforms of Periodic Signals 0N n NN 0 n

The Discrete Fourier Transform Sampling of Fourier Transform

Equal Space Sampling of Fourier Transform z-plane 0 n N’1N’1 NN’N’ >=<>=<

Equal Space Sampling of Fourier Transform

z-plane

Example n 08 N=12 12 n 08 N’=9 n 08 N=7 12

n 08 N=12 12 n 08 N’=9 n 08 N=7 12 Example Time-Domain Aliasing

Time-Domain Aliasing vs. Frequency-Domain Aliasing To avoid frequency-domain aliasing – Signal is bandlimited – Sampling rate in time-domain is high enough To avoid time-domain aliasing – Sequence is finite – Sampling interval (2  /N ) in frequency- domain is small enough

DFT vs. DFS Use DFS to represent a finite-length sequence is call the DFT (Discrete Fourier Transform). So, we represent the finite-duration sequence by a periodic sequence. One period of which is the finite-duration sequence that we wish to represent.

The Discrete Fourier Transform Representation of Finite-Duration Sequences --- DFT

Definition of DFT Synthesis Analysis

Example

The Discrete Fourier Transform Properties of the DFT

Linearity n 0 N11N11 Duration N 1 n 0 N21N21 Duration N 2

Circular Shift of a Sequence 0 n N n 0 N n 0 N

Duality

Example Choose N=10 Re [X(k)] Im [X(k)] Re [x 1 (n)] = Re [X(n)] Im [x 1 (n)] = Im [X(n)] X 1 (k) = 10x((  k)) 10

Linear Convolution (Review)

Circular Convolution both of length N

Example 00N 0 n0n0 N 0 n 0 =2, N=5

Example 0 0N 0 n0n0 N n 0 =2, N=7 0

Example L=N=6 0 L0 L 0 L N

Example L=2N=12 0L N 0L N 0N N

The Discrete Fourier Transform Linear Convolution Using the DFT

Why Using DFT for Linear Convolution? FFT (Fast Fourier Transform) exists. But., we have to ensure that circular convolving nature of DFT gives the linear convolving result.

The Procedure Let x 1 (n) and x 2 (n) be two sequences of length L and P, respectively. 1. Compute N-point (N = ?) DFTs X 1 (k) and X 2 (k). 2. Let X 3 (k) = X 1 (k) X 2 (k), 0  k  N  Let x 3 (n) = DFT  1 [X 3 (k)] = x 1 (n)  x 2 (n). x 1 (n) * x 2 (n) = x 1 (n)  x 2 (n)?

Linear Convolution of Two Finite-Length Sequences 0 L x1(m)x1(m) m 0 PL x2(m)x2(m) m 0 PP 11 L m x2(1 m)x2(1 m) L 0 n  P+1 n m x2(n m)x2(n m) 0 n+P1n+P1 n L m x 2 (L+P  1  m) x 3 (  1) = 0 x 3 (n)  0 x 3 (L+P  1) = 0 n = 0, 1, , L+P  2

Linear Convolution of Two Finite-Length Sequences 0 L x1(m)x1(m) m 0 PL x2(m)x2(m) m 0 PP 11 L m x2(1 m)x2(1 m) L 0 n  P+1 n m x2(n m)x2(n m) 0 n+P1n+P1 n L m x 2 (L+P  1  m) x 3 (  1) = 0 x 3 (n)  0 x 3 (L+P  1) = 0 n = 0, 1, , L+P  2 Length of x 3 (n) = x 1 (n)*x 2 (n) = L+P  1

Circular Convolution as Linear Convolution with Time Aliasing x 1 (n)  Length L x 2 (n)  Length P x(n) = x 1 (n)*x 2 (n)  Length L+P  1

Circular Convolution as Linear Convolution with Time Aliasing L 0 x1(n)x1(n) n P x2(n)x2(n) 0 n L 0 P n L+P  1 x (n)x (n) L

N = L+P  1 0 n L+P  1 x (n)x (n) L0 n L 0 n L

For Finite Sequences L 0 x1(n)x1(n) n P x2(n)x2(n) 0 n x p (n) = x 1 (n)*x 2 (n) = x 1 (n)  x 2 (n), 0  n  L+P  2 Zero padding to length L+P  1 0 n L+P2L+P2

N = L 0 P1P1 n L+P  1 x (n)x (n) L 0 P1P1 n L  L L 0 P1P1 n L

N = L 0 P1P1 n L+P  1 x (n)x (n) L 0 P1P1 n L  L L 0 P1P1 n L Corrupted (P  1) points Uncorrupted (L  P+1) points

FIR Filter for Indefinite-Length Signals Overlap-Add Method Overlap-Save Method x (n)x (n) h (n)h (n) Block Convolution

Overlay-Add Method x (n)x (n) h (n)h (n)

Set N = L+P  1 for each block convolution

Overlay-Save Method Each block is of length L. P  1 samples are overlaid btw. adjacent blocks. Set N = L+P  1 for each block convolution. Save the last L  P+1 values for each block convolution.