Fourier Analysis Overview (0B)

Transcription

1 CTFS: Continuous Time Fourier Series CTFT: Continuous Time Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform

2 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave.

3 Methods Frequency View Discrete Frequency Continuous Frequency Continuous Time CTFS CTFT C k X ( j ω) e j k ω 0 t k ω 0 Aperiodic in freq e j ω t ω Aperiodic in freq DTFS / DFT DTFT γ k e j k ^ω 0 n x[n] k ^ω 0 Periodic in freq X ( j ^ω) e j ^ω n x[n] ^ω Periodic in freq Normalized Discrete Frequency Normalized Continuous Frequency 3

4 Methods Time View Discrete Frequency Continuous Frequency Continuous Time CTFS CTFT x(t ) Periodic in time x(t ) k ω 0 Aperiodic in time DTFS / DFT DTFT x[n] k ^ω 0 Periodic in time x[n] Aperiodic in time Normalized Discrete Frequency Normalized Continuous Frequency 4

5 Methods Discrete Frequency Continuous Frequency Continuous Time CTFS CTFT C k x(t ) Periodic in time X ( j ω) x(t ) e j k ω 0 t k ω 0 Aperiodic in freq e j ω t ω Aperiodic in time Aperiodic in freq DTFS / DFT DTFT γ k e j k ^ω 0 n x[n] k ^ω 0 Periodic in time Periodic in freq X ( j ^ω) e j ^ω n x[n] ^ω Aperiodic in time Periodic in freq Normalized Discrete Frequency Normalized Continuous Frequency 5

6 Time and Frequency Domain Resolutions Discrete Frequency Continuous Frequency Continuous Time CTFS 0 CTFT ω 0 ω 0 DTFS / DFT 0 π DTFT T 0 s 0 +π π 0 +π ^ω 0 ^ω 0 Normalized Discrete Frequency Normalized Continuous Frequency 6

7 Time Domain Resolutions Discrete Frequency Continuous Frequency Continuous Time CTFS x(t ) Signal Period T 0 CTFT x(t ) Practical Signal Duration T 0 DTFS / DFT x[n] Sample Period Sample Counts N 0 DTFT x[n] Sample Period Sample Counts N 0 Normalized Discrete Frequency Normalized Continuous Frequency 7

8 Frequency Domain Resoltuions Discrete Frequency Continuous Frequency Continuous Time CTFS C k CTFT X ( j ω) Frequency Resolution + + ω 0 ω 0 DTFS / DFT γ k / X [k ] Frequency Resolution π +π DTFT π X ( j ^ω) +π ^ω 0 ^ω 0 Normalized Discrete Frequency Normalized Continuous Frequency 8

9 and Periodic Frequency Periodic in discrete freq Periodic in continuous freq ^ω s ^ω s DTFS / DFT DTFT Normalized Discrete Frequency Normalized Continuous Frequency 9

10 Periodic Time and Discrete Frequency Discrete Frequency Continuous Time CTFS Discrete Frequency + ω 0 ω 0 Periodic in continuous time T 0 DTFS / DFT π Discrete Frequency +π Periodic in discrete time ^ω 0 ^ω 0 N 0 Normalized Discrete Frequency 10

11 Resolution ω s : replication frequency ^ω s : normalized replication frequency ω s : replication frequency ^ω s : normalized replication frequency ω s = 2π ^ω s = 1 ω s = 2π ^ω s = 2π 1 DTFS / DFT DTFT Periodic in discrete freq Periodic in continuous freq Normalized Discrete Frequency Normalized Continuous Frequency 11

12 Discrete Frequency Resolutions ω 0, ^ω 0 Discrete Frequency Continuous Time CTFS Periodic in continuous time Discrete Frequency + ω 0 ω 0 ω 0 = 2π T 0 period : T 0 seconds DTFS / DFT π Periodic in discrete time Discrete Frequency +π ^ω 0 = 2π N 0 period : N 0 samples ^ω 0 ^ω 0 Normalized Discrete Frequency 12

13 Normalized Frequency ^ω s = 1 = ( ) ^ω s = 1 = ( ) ^ω 0 = 2π N 0 = ( T 0 ) ^ω continuous variable DTFS / DFT γ k / X [k ] π +π DTFT π X ( j ^ω) +π ^ω 0 ^ω 0 Normalized Discrete Frequency Normalized Continuous Frequency 13

14 Normalized by 1/ Continuous Time CTFS 0 Discrete Frequency 0 ω 0 ω frequency resolution ω 0 = T 0 T 0 = N 0 ω 0 = N 0 DTFS / DFT 0 π 0 ^ω 0 ^ω 0 +π ω 0 1/ = N 0 ^ω 0 = N 0 f s Normalized Discrete Frequency normalized frequency resolution 14

15 CTFT pair of an impulse train p(t) = + δ(t n ) P ( j ω) = n= k= ( 2π ) T δ(ω kω ) s s 2π 1 ω s = 2π ω s ω s ω s 2 ω s Sampling Replication 15

16 Sampling and Replicating ω s = 2π 1 Sampling Replication T 0 ω 0 = 2π T 0 = N 1 = 2π N 2 Period Resolution 16

17 Normalization Relatively scaled figures ω s = ^ω s = 1 ω s = π Replication +π fine coarse = small = large ω 0 = 2π T 0 = N 1 ^ω 0 = 2π N 1 ^ω 0 = N 2 ω 0 = 2π T 0 = N 2 large N 1 small N 2 Resolution 17

18 Sampling Period and the Number of Samples x(t ) T 0 period fundamental period T 0 = N 1 = N 2 x[n] N 1 samples sampling period : number of samples : N 1 ( > ) (N 1 < N 2 ) x[n] N 2 samples sampling period : ( > ) number of samples : N 2 (N 1 < N 2 ) 18

19 Frequency Replication and Resolution (1) fundamental period T 0 T 0 = N 1 = N 2 frequency resolution ω 0 ω 0 = 2π T 0 = ω 01 = 2π N 1 = ω 0 2 = 2π N 2 N 1 < N 2 ^ω 0 1 = N 1 > ^ω 02 = N 2 ^ω 0 1 = ω 01 > ^ω 0 2 = ω 02 sampling period : replication period ω 1, ω 2 : > ω 1 = < ω 2 = 2π ^ω 1 = 2π = ^ω 2 = ^ω 1 = ω 1 = ^ω 2 = ω 2 19 coarse fine

20 Frequency Replication and Resolution (2) > ω s = ω = ^ω ω T = ^ω s N 1 < N 2 replication frequency frequency resolutions ω 1 = ^ω 1 = ω 0 1 = N 1 ^ω 01 = 2π N 1 ^ω 0 1 = ω 01 ω 2 = ^ω 2 = ω 0 2 = N 2 ^ω 02 = 2π N 2 ^ω 0 2 = ω 0 2 large bandwidth fine resolution 5A Spectrum Representation 20

21 Sampling Period and Replication Period T 0 period ω s = 2π ω 1 small BW ω 1 large BW ω 2 ω 2 21

22 & periods, ω 1 & ω 2 replication frequencies x 1 [n] ω 1 = small BW ^ω 1 = ^ω 2 = x 2 [n] ω 2 = large BW 5A Spectrum Representation 22

23 & periods, ω 01 & ω 02 frequency resolutions x 1 [n] ω 0 1 = N 1 N 01 ^ω 01 = ω 01 x(t ) T 0 period ^ω 02 = ω 0 2 N 02 ω 0 2 = 2π N 2 x 2 [n] 5A Spectrum Representation 23

24 Replication Frequency small number of samples N 1 small replication freq ω 1 large ω 1 = 2ω ω 1 +ω ω 1 large number of samples N 1 large replication freq ω 2 small ω 2 = 2π ω 2 0 +ω 2 24

25 Normalized Replication Frequencies ω 1 = replication freq ω 1 2π 2π ω 2 = 2π replication freq ω 2 25

26 Frequency Resolution the same signal period T 0 the same frequency resolution ω 0 ω 0 1 = 2π T 0 = 2π N 1 2ω ω 1 +ω ω 1 the same signal period T 0 the same frequency resolution ω 0 ω 0 2 = 2π T 0 = 2π N 2 ω 2 0 +ω 2 26

27 Normalized Frequency Resolutions ω 01 = N 1 ^ω 0 1 = ω 01 ^ω 01 = coarse N 1 ^ω 0 2 = ω 02 fine ^ω 0 2 = 2π N 2 ω 0 1 = 2π N 2 27

28 Normalized ω 0 & ω s ω 0 1 = N 1 small bw ω 1 = replication frequency coarse ^ω 0 1 = ω 01 ^ω 1 = 2π frequency resolution ^ω 2 = fine ^ω 0 2 = ω 02 frequency resolution ω 0 2 = 2π N 2 large bw ω 2 = replication frequency 5A Spectrum Representation 28

29 Types of Fourier Transforms Continuous Time Fourier Series CTFS Continuous Time Discrete Frequency Fourier Series DTFS / DFT Discrete Frequency Continuous Time Fourier Transform CTFT Continuous Time Continuous Frequency Fourier Transform DTFT Continuous Frequency 29

30 1. CTFS CT FS Continuous Time Discrete Frequency ω 0 = T 0 x(t ) C k Signal Period T 0 Frequency Resolution + ω 0 ω 0 30

31 2. DTFS / DFT DT FS Discrete Frequency (Normalized) ^ω 0 = N 0 ^ω s = 1 ω s = 2π ^ω s ^ω s x[n] γ k / X [k ] Sample Period Sample Counts N 0 Frequency Resolution π +π ^ω 0 ^ω 0 31

32 3. CTFT CT FT Continuous Time Continuous Frequency x(t ) X ( j ω) Practical Signal Duration T

33 4. DTFT DT FT Continuous Frequency (Normalized) ^ω s = 1 ω s = 2π ^ω s ^ω s x[n] X ( j ^ω) Sample Period π +π Sample Counts N 0 33

34 Fourier Transform Types Continuous Time Fourier Series T C k = 1 T 0 N 1 γ[k ] = 1 N n = 0 x(t ) e j k ω 0 t dt Fourier Series x[n] e j k ^ω 0 n Continuous Time Fourier Transform + X ( j ω) = x(t) e j ω t dt x(t ) = + k= N 1 x[n] = k = 0 x(t ) = 1 + C k e + j k ω 0 t γ[k] e + j k ^ω 0 n X ( j ω) e + j ωt d ω Fourier Transform X ( j ^ω) = + n = x[n] e j ^ω n x[n] = 1 +π 2π π X ( j ^ω) e + j ^ωn d ^ω 5A Spectrum Representation 34

35 Multiplication with an Impulse Train x (t) p (t) 1 + n= δ(t n ) x (t) p(t) Multiplication with a dense impulse train + n= x(n ) δ(t n ) x [n] 35

36 Convolution with an Impulse Train x (t) p (t) 1 1 x (t) p(t) Multiplication with a sparse impulse train 36

37 Convolution & Multiplication Properties x(t ) y(t ) X ( j ω) Y ( j ω) x(t ) y(t) 1 X ( j ω) Y ( j ω) x(t ) y(t ) X (f ) Y (f ) x(t ) y(t) X (f ) Y (f ) 37

38 Types of Fourier Transforms Continuous Time Fourier Transform CTFT Continuous Time Continuous Frequency Fourier Transform DTFT Continuous Frequency 38

39 Multiplication & Convolution x (t) X ( j ω) A p (t) 1 P( j ω) x (t) p(t) Multiplication 1 X ( j ω) P( j ω) Convolution A 39

40 Convolution & Multiplication x (t) X ( j ω) A p (t) 1 P( j ω) x (t) p(t) X ( j ω) P( j ω) Convolution Multiplication 40

41 References [1] [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003 [3] M.J. Roberts, Fundamentals of Signals and Systems [4] S.J. Orfanidis, Introduction to Signal Processing [5] K. Shin, et al., Fundamentals of Signal Processing for Sound and Vibration Engineerings