Chapter 8. The Discrete Fourier Transform 8.1 Laplace, z-, and Fourier Transforms 8.2 Fourier Transform 8.3 Fourier Series 8.4 Discrete Fourier Transform (DFT) 8.5 Properties of DFS/DFT 8.6 DFT and z-Transform 8.7 Linear Convolution vs. Circular Convolution 8.8 Discrete Cosine Transform(DCT) BGL/SNU

1. Laplace, z-, Fourier Transforms Analog systems (continuous time) Digital Systems (discrete time) H(s) H(z) BGL/SNU

Laplace transform -z-transform LHP inside u.c Fouier transforms BGL/SNU

2. Fourier Transform (1) continuous aperiodic signals conti aper aper conti x(t) 1 t BGL/SNU

(2) Discrete aperiodic signals conti per aper discr x(n) 1 t ω

3. Fourier Series (1) continuous periodic signals discrete aper per conti BGL/SNU

(2) discrete periodic signals (*Discrete Fourier Series) X(t) 1 k t T (2) discrete periodic signals (*Discrete Fourier Series) discrete per per discre BGL/SNU

x[n] 1 BGL/SNU

4. Discrete Fourier Transform (DFT) -For a numerical evaluation of Fourier transform and its inversion, (i.e,computer-aided computation), we need discrete expression of of both the time and the transform domain data. -For this,take the advantage of discrete Fourier series(DFS, on page 4), in which the data for both domain are discrete and periodic. discrete periodic periodic discrete -Therefore, given a time sequence x[n], which is aperiodic and discrete, take the following approach. BGL/SNU

Mip Top Top Mip DFS DFT Reminding that, in DFS BGL/SNU

Define DFT as (eq) X[k] x[n] 1 k n N N BGL/SNU

Graphical Development of DFT

DFS BGL/SNU

DFT BGL/SNU

5. Property of DFS/DFT (8.2 , 8.6) (1) Linearity (2) Time shift (3) Frequency shift BGL/SNU

(4) Periodic/circular convolution in time (5) Periodic/circular convolution in frequency BGL/SNU

(6) Symmetry DFS DFT BGL/SNU

6. DFT and Z-Transform (1) Evaluation of from ①If length limited in time, (I.e., x[n]=0, n<0, n>=N) then BGL/SNU

② What if x[n] is not length-limited? then aliasing unavoidable. … … … … … …

(2) Recovery of [or ] from (in the length-limited case) BGL/SNU

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7. Linear Convolution vs. Circular Convolution (1) Definition ① Linear convolution BGL/SNU

Rectangular window of length N ② Circular convolution N Rectangular window of length N Periodic convolution N BGL/SNU

(2) Comparison N H[n] 2N 2N Omit chap. 8.7

Test signal for computing DFT and DCT 8. Discrete cosine transform (DCT) Definition - Effects of Energy compaction BGL/SNU Test signal for computing DFT and DCT

(a) Real part of N-point DFT; (b) Imaginary part of N-point DFT; (c) N-point DCT-2 of the test signal BGL/SNU

Comparison of truncation errors for DFT and DCT-2 BGL/SNU

Appendix: Illustration of DFTs for Derived Signals BGL/SNU

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