CTFS: Fourier Series CTFT: Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform
Copyright (c) 2011-2016 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.
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
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
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 = 1 + 2 π k= C k T 0 e + j ω 0k t ( 2 π T 0 ) T 0 ω 0 0 x T 0 (t ) = 1 + 2 π k= C k T 0 e + j ω 0k t ω 0 ω d ω x (t) = 1 + 2 π X ( j ω) e + j ω t d ω 5
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
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
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
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
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
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
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
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
Types of Fourier Transforms Fourier Series CTFS Fourier Transform CTFT Fourier Series DTFS / DFT Fourier Transform DTFT 14
1. CTFS CTFT Fourier Series CTFS Time Aperiodic Fourier Transform CTFT 15
2. DTFS DTFT Fourier Series DTFS / DFT Time Aperiodic Fourier Transform DTFT T= 16
3. CTFS CTFT Fourier Series CTFS Time Periodic Frequency Sampling Fourier Transform CTFT T= 17
4. DTFS DTFT Fourier Series DTFS / DFT Time Periodic Frequency Sampling Fourier Transform DTFT T= 18
5. CTFT DTFT Fourier Transform CTFT Time Sampling Frequency Periodic Fourier Transform DTFT 19
6. CTFS DTFS Fourier Series CTFS Time Sampling Frequency Periodic Fourier Series DTFS / DFT 20
7. CTFT DTFT Fourier Transform CTFT Frequency Aperiodic Fourier Transform DTFT 21
8. CTFS DTFS Fourier Series CTFS Frequency Aperiodic Fourier Series DTFS / DFT 22
**** Time Aperiodic Time Periodic Frequency Sampling Time Sampling Frequency Periodic Frequency Aperiodic 23
***** CTFS Time Aperiodic Frequency Sampling CTFT DTFS / DFT Time Aperiodic Frequency Sampling DTFT 24
***** CTFS Time Sampling Frequency Aperiodic CTFT DTFS / DFT Time Sampling Making Aperiodic Frequency Aperiodic DTFT 25
9 CTFS DTFT Fourier Series CTFS Time Aperiodic Time Sampling Frequency Periodic Fourier Transform DTFT 26
10 CTFS DTFT Fourier Series CTFS Time Periodic Frequency Sampling Frequency Aperiodic Fourier Transform DTFT 27
11. CTFT DTFS Fourier Transform CTFT Time Periodic Time Sampling Frequency Sampling Frequency Periodic Fourier Series DTFS / DFT 28
12. CTFT DTFS Fourier Transform CTFT Time Aperiodic Frequency Aperiodic Fourier Series DTFS / DFT 29
CTFS & DTFT Fourier Series CTFS T Periodic in time Sampled in frequency Sampled in time Periodic in frequency Fourier Transform DTFT 30
CTFT & DTFS Fourier Transform CTFT Sampling in time Periodic in time Periodic in frequency Sampled in frequency Fourier Series DTFS / DFT 31
References [1] http://en.wikipedia.org/ [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