Fourier Analysis Overview (0A)

Transcription

1 CTFS: Fourier Series CTFT: 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 CTFS Correlation Process 1 cycle x(t ) 2 cycles 3 cycles Measure the degree of correlation with these cosine and sine waves whose frequencies are the integer multiples of the fundamental frequency 4 cycles C n = 1 T 0 T x t e j n 0 t dt x(t) = + n= C n e + j nω 0 t 3

4 DTFS Correlation Process 1 cycle x[n] 2 cycles 3 cycles Measure the degree of correlation with these cosine and sine waves whose frequencies are the integer multiples of the fundamental frequency 4 cycles N 1 γ k = 1 N n=0 x[n] e j ( 2π N ) k n x[n] = + M k = M γ k e + j ( 2 π N ) k n 4

5 CTFS CTFT k ω 0 0ω 0 1 ω 0 2ω 0 3 ω 0 4 ω 0 5 ω 0 6ω 0 7ω 0 ω0 x T 0 (t ) = + C k e + j ω 0k t 1 k= = ( + C k e + j ω k t T ) 0 0 k= 2 π ( 2 π ) T 0 = π k= C k T 0 e + j ω 0k t ( 2 π T 0 ) T 0 ω 0 0 x T 0 (t ) = π k= C k T 0 e + j ω 0k t ω 0 ω d ω x (t) = π X ( j ω) e + j ω t d ω 5

6 DTFS DTFT 0 2 π k ^ω 0 0ω^ 0 1 ^ω 0 2 ^ω 0 3 ^ω 0 4 ^ω 0 5 ^ω 0 6ω^ 0 7 ^ω 0 ω0 N 0 x N 0 [n] = k =0 γ k e + j ( 2π N 0 ) k n 1 N 0 = k=0 = 1 2 π N 0 k=0 γ k e + j ( 2 π N 0 ) k n ( N ) 0 2π ( 2π ) N 0 γ k N 0 e + j ( 2 π N 0 ) k n ( 2 π ) N 0 N 0 ^ω 0 0 x N 0 [n] = 1 N 0 2 π k=0 γ k N 0 e + j ( 2 π N 0 ) k n ( 2 π ) N 0 ^ω x [n] = 1 +π 2π π X (e j ^ω ) e + j ^ω n d ^ω d ω 0 2 π 6

7 CTFS & DTFS Correlation Processes x(t ) e + j ( 2 π T ) k t 1 T jk ω T 0 x(t ) e 0 t dt = C k R T R I x[n] e + j ( 2 π N ) k n 1 N 1 x[ n] e j ( 2π N ) n k N n=0 = γ k R N samples R I 7

8 DTFS and DFT Discrete Fourier Transform N 1 X [k] = n = 0 N 1 x [n] e j 2 / N k n x[n] = 1 N k = 0 X [ k ] e + j (2π/N ) k n Fourier Series DTFS N 1 X [ k ] = 1 N n = 0 N 1 x[n] e j (2 π / N ) k n x[n] = k = 0 + j(2π/ N )k n X [ k ] e DTFS ( x[n]) = 1 N DFT (x[n]) 8

9 Fourier Transform Types Fourier Series C k = 1 T 0 N 1 γ[k ] = 1 N n = 0 T x(t ) e j k ω 0 t Fourier Series dt Fourier Transform x(t ) = + k= N 1 x[n] e j k ^ω n 0 x[n] = k = 0 C k e + j k ω 0 t γ[k] e + j k ^ω 0 n X j = x t e j t d t x t = 1 2 X j e j t d Fourier Transform X ( j ^ω) = + n = x[n] e j ^ω n x[n] = 1 +π 2π π X ( j ^ω) e + j ^ω n d ^ω 9

10 Fourier Transform Types Fourier Series C n = 1 T 0 T x t e j n 0 t Fourier Transform dt x(t) = + n= C n e + j nω 0 t X j = x t e j t d t x t = 1 2 X j e j t d Fourier Transform X e j = n = Discrete Fourier Transform x [n] e j n x[n] = 1 + π 2π π X (e j ω ) e + j ω n N 1 X [k] = n = 0 N 1 x [n] e j 2 / N k n x [n] = 1 N k = 0 j 2 / N k n X [k ] e 10

11 Fourier Series C n = 1 T 0 T x t e j n 0 t dt x(t) = + n= C n e + j nω 0 t Aperiodic Spectrum Periodic Signal Fourier Transform X j = x t e j t d t x t = 1 2 X j e j t d Aperiodic Spectrum Aperiodic Signal 11

12 Fourier Transform X e j = n = x [n] e j n x[n] = 1 + π 2π π X (e j ω ) e + j ω n Periodic Spectrum Aperiodic Signal Discrete Fourier Transform N 1 X [k] = n = 0 N 1 x [n] e j 2 / N k n x [n] = 1 N k = 0 j 2 / N k n X [k ] e Periodic Spectrum Periodic Signal 12

13 Methods CTFS Periodic in time CTFT Aperiodic in freq C k X ( j ω) Aperiodic in time Aperiodic in freq DTFS / DFT Periodic in time DTFT Periodic in freq γ k / X [k ] X (e j ^ω ) Aperiodic in time Periodic in freq 13

14 Types of Fourier Transforms Fourier Series CTFS Fourier Transform CTFT Fourier Series DTFS / DFT Fourier Transform DTFT 14

15 1. CTFS CTFT Fourier Series CTFS Time Aperiodic Fourier Transform CTFT 15

16 2. DTFS DTFT Fourier Series DTFS / DFT Time Aperiodic Fourier Transform DTFT T= 16

17 3. CTFS CTFT Fourier Series CTFS Time Periodic Frequency Sampling Fourier Transform CTFT T= 17

18 4. DTFS DTFT Fourier Series DTFS / DFT Time Periodic Frequency Sampling Fourier Transform DTFT T= 18

19 5. CTFT DTFT Fourier Transform CTFT Time Sampling Frequency Periodic Fourier Transform DTFT 19

20 6. CTFS DTFS Fourier Series CTFS Time Sampling Frequency Periodic Fourier Series DTFS / DFT 20

21 7. CTFT DTFT Fourier Transform CTFT Frequency Aperiodic Fourier Transform DTFT 21

22 8. CTFS DTFS Fourier Series CTFS Frequency Aperiodic Fourier Series DTFS / DFT 22

23 **** Time Aperiodic Time Periodic Frequency Sampling Time Sampling Frequency Periodic Frequency Aperiodic 23

24 ***** CTFS Time Aperiodic Frequency Sampling CTFT DTFS / DFT Time Aperiodic Frequency Sampling DTFT 24

25 ***** CTFS Time Sampling Frequency Aperiodic CTFT DTFS / DFT Time Sampling Making Aperiodic Frequency Aperiodic DTFT 25

26 9 CTFS DTFT Fourier Series CTFS Time Aperiodic Time Sampling Frequency Periodic Fourier Transform DTFT 26

27 10 CTFS DTFT Fourier Series CTFS Time Periodic Frequency Sampling Frequency Aperiodic Fourier Transform DTFT 27

28 11. CTFT DTFS Fourier Transform CTFT Time Periodic Time Sampling Frequency Sampling Frequency Periodic Fourier Series DTFS / DFT 28

29 12. CTFT DTFS Fourier Transform CTFT Time Aperiodic Frequency Aperiodic Fourier Series DTFS / DFT 29

30 CTFS & DTFT Fourier Series CTFS T Periodic in time Sampled in frequency Sampled in time Periodic in frequency Fourier Transform DTFT 30

31 CTFT & DTFS Fourier Transform CTFT Sampling in time Periodic in time Periodic in frequency Sampled in frequency Fourier Series DTFS / DFT 31

32 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