CTFS: Fourier Series CTFT: Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform
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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 ( 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= ( ) T 0 = 1 k= C k T 0 e + j ω 0k t ( T 0 ) T 0 ω 0 0 x T 0 (t ) = 1 k= C k T 0 e + j ω 0k t ω 0 ω d ω x (t) = 1 X ( j ω) e + j ω t d ω 5
DTFS DTFT 0 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 N 0 k=0 γ k e + j ( N 0 ) k n ( N ) 0 2π ( 2π ) N 0 γ k N 0 e + j ( N 0 ) k n ( ) N 0 N 0 ^ω 0 0 x N 0 [n] = 1 N 0 k=0 γ k N 0 e + j ( N 0 ) k n ( ) N 0 ^ω x [n] = 1 +π 2π π X (e j ^ω ) e + j ^ω n d ^ω d ω 0 6
CTFS & DTFS Correlation Processes x(t ) e + j ( 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 ( 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 Fourier Series DTFS N 1 γ[k ] = 1 N n = 0 N 1 x[n] e j(/ N )k n x[n] = k = 0 + j(2π/ N )k n γ[k] e 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 DTFS(x[n]) = 1 N DFT (x[n]) ^ω 0 = ( N ) γ[n] = 1 N 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 x[n] e j k ^ω 0 n Fourier Transform X ( j ω) = Fourier Transform X ( j ^ω) = x(t) e j ωt dt n = x[n] e j ^ω n x(t ) = k= N 1 x[n] = k = 0 C k e + j k ω 0 t γ[k] e + j k ^ω 0 n x(t ) = 1 + j X ( j ω) e ωt d ω x[n] = 1 +π 2π π X ( j ^ω) e + j ^ω n d ^ω 9
Frequency Resolution Fourier Series C k = 1 T 0 T x(t ) e j k ω 0 t dt x(t ) = k= C k e + j k ω 0 t Frequency Resolution ω 0 = ( T 0 ) Signal Period T 0 Fourier Series N 1 γ[k ] = 1 N n = 0 x[n] e j k ^ω 0 n N 1 x[n] = k = 0 γ[k] e + j k ^ω 0 n Frequency Resolution ^ω 0 = ( N 0 ) Sample Counts N 0 10
Frequency Variable Notations Fourier Transform X ( j ω) = x(t) e j ωt dt x(t ) = 1 + j X ( j ω) e ωt d ω j ω e j ω X (e j ω ) j ω always appears as e j ω Fourier Transform X ( j ^ω) = n = x[n] e j ^ω n x[n] = 1 +π + j ^ω 2π π X ( j ^ω) e n d ^ω j ^ω e j ^ω X (e j ^ω ) j ^ω always appears as e j ^ω 11
*** 1 T CT x(t ) FS dt C T 0 e j k ω t 0 DF k AF N 1 n = 0 k= N 1 DT x[n] FS e j k ^ω n 0 DF γ[k ] PF k = 0 CT x(t ) FT dt e j ω t CF X ( j ω) AF 1 d ω DT x[n] FT n = e j ^ω n CF X ( j ^ω) PF 1 +π π d ^ω Fourier Series Fourier Transform Continuous Freq Discrete Freq Aperiodic Freq Periodic Freq 12
*** 1 T CT x(t ) PT dt C T 0 e j k ω t 0 DF k AF N 1 n = 0 k= N 1 DT x[n] PT e j k ^ω n 0 DF γ[k ] PF k = 0 CT x(t ) AT dt e j ω t CF X ( j ω) AF 1 d ω DT x[n] AT n = e j ^ω n CF X ( j ^ω) PF 1 +π π d ^ω Fourier Series Fourier Transform Continuous Freq Discrete Freq Aperiodic Freq Periodic Freq 13
*** PT 1 T T 0 dt DF C k PT N 1 n = 0 DF γ[k ] AT dt CF X ( j ω) AT n = CF X ( j ^ω) Fourier Series Fourier Transform Continuous Freq Discrete Freq Aperiodic Freq Periodic Freq 14
*** CT x(t ) AF k= DT x[n] PF N 1 k = 0 CT x(t ) AF 1 d ω DT x[n] PF 1 +π π d ^ω Fourier Series Fourier Transform Continuous Freq Discrete Freq Aperiodic Freq Periodic Freq 15
A. CTFS Fourier Series C k = 1 T 0 T x(t ) e j k ω 0 t dt x(t ) = k= C k e + j k ω 0 t Aperiodic Discrete Frequency Spectrum Periodic Signal k = C k 1 T T 0 x(t ) dt 16
B. DTFS Fourier Series N 1 γ[k ] = 1 N n = 0 x[n] e j k ^ω 0 n N 1 x[n] = k = 0 γ[k] e + j k ^ω 0 n Periodic Discrete Frequency Spectrum N 1 k = 0 γ[k ] Periodic Signal N 1 n = 0 x[n] 17
C. CTFT Fourier Transform X ( j ω) = x(t) e j ωt dt x(t ) = 1 + j X ( j ω) e ωt d ω Aperiodic Discrete Frequency Spectrum Aperiodic Signal 1 d ω dt X ( j ω) x(t ) 18
D. DTFT Fourier Transform X ( j ^ω) = n = x[n] e j ^ω n x[n] = 1 +π + j ^ω 2π π X ( j ^ω) e n d ^ω Periodic Spectrum 1 +π π X ( j ^ω) d ^ω Aperiodic Signal n = x[n] 19
CTFS & CTFT Fourier Series C k = 1 T 0 T x(t ) e j k ω 0 t dt x(t ) = k= C k e + j k ω 0 t Discrete Frequency Fourier Transform X ( j ω) = x(t) e j ωt dt x(t ) = 1 + j X ( j ω) e ωt d ω 20
DTFS & DTFT Fourier Series N 1 γ[k ] = 1 N n = 0 x[n] e j k ^ω 0 n N 1 x[n] = k = 0 γ[k] e + j k ^ω 0 n 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 21
CTFS & DTFS Fourier Series C k = 1 T j k ω T 0 x(t ) e 0 t dt Fourier Series N 1 γ[k ] = 1 N n = 0 x[n] e j k ^ω 0 n x(t ) = k= N 1 x[n] = k = 0 C k e + j k ω 0 t γ[k] e + j k ^ω 0 n 22
CTFT & DTFT Fourier Transform X ( j ω) = Fourier Transform X ( j ^ω) = x(t) e j ωt dt n = x[n] e j ^ω n x(t ) = 1 + j X ( j ω) e ωt d ω x[n] = 1 +π 2π π X ( j ^ω) e + j ^ω n d ^ω 23
Methods Discrete Frequency CTFS CTFT C k X ( j ω) e j k ω 0 t k ω 0 Periodic in time Aperiodic in freq e j ω t ω Aperiodic in time Aperiodic in freq DTFS / DFT γ k / X [k ] e j k ^ω 0 n k ^ω 0 Periodic in time Periodic in freq DTFT X ( j ^ω) e j ^ω n ^ω Aperiodic in time Periodic in freq Normalized Discrete Frequency Normalized 24
Types of Fourier Transforms Fourier Series CTFS Discrete Frequency Fourier Series DTFS / DFT Discrete Frequency Fourier Transform CTFT Fourier Transform DTFT 25
1. CTFS CTFT Fourier Series CTFS Discrete Frequency CTFS CTFT DTFS DTFT Time Aperiodic Fourier Transform CTFT 26
2. DTFS DTFT Fourier Series DTFS / DFT Discrete Frequency CTFS CTFT DTFS DTFT Time Aperiodic Fourier Transform DTFT T= 27
3. CTFS CTFT Fourier Series CTFS Discrete Frequency CTFS CTFT DTFS DTFT Time Periodic Frequency Sampling Fourier Transform CTFT T= 28
4. DTFS DTFT Fourier Series DTFS / DFT Discrete Frequency CTFS CTFT DTFS DTFT Time Periodic Frequency Sampling Fourier Transform DTFT T= 29
5. CTFT DTFT Fourier Transform CTFT CTFS CTFT DTFS DTFT Time Sampling Frequency Periodic Fourier Transform DTFT 30
6. CTFS DTFS Fourier Series CTFS Discrete Frequency CTFS CTFT DTFS DTFT Time Sampling Frequency Periodic Fourier Series DTFS / DFT Discrete Frequency 31
7. CTFT DTFT Fourier Transform CTFT CTFS CTFT DTFS DTFT Frequency Aperiodic Fourier Transform DTFT 32
8. CTFS DTFS Fourier Series CTFS Discrete Frequency CTFS CTFT DTFS DTFT Frequency Aperiodic Fourier Series DTFS / DFT Discrete Frequency 33
**** CTFS CTFT DTFS DTFT Time Aperiodic CTFS CTFT DTFS DTFT Time Periodic Frequency Sampling CTFS CTFT DTFS DTFT Time Sampling Frequency Periodic CTFS CTFT DTFS DTFT Frequency Aperiodic 34
***** CTFS Time Aperiodic Frequency Sampling CTFT DTFS / DFT Time Aperiodic Frequency Sampling DTFT 35
***** CTFS Time Sampling Frequency Aperiodic CTFT DTFS / DFT Time Sampling Making Aperiodic Frequency Aperiodic DTFT 36
9 CTFS DTFT Fourier Series CTFS Discrete Frequency CTFS CTFT DTFS DTFT Time Aperiodic Time Sampling Frequency Periodic Fourier Transform DTFT 37
10 CTFS DTFT Fourier Series CTFS Discrete Frequency CTFS CTFT Time Periodic Frequency Sampling DTFS DTFT Frequency Aperiodic Fourier Transform DTFT 38
11. CTFT DTFS Fourier Transform CTFT CTFS CTFT DTFS DTFT Time Periodic Time Sampling Frequency Sampling Frequency Periodic Fourier Series DTFS / DFT Discrete Frequency 39
12. CTFT DTFS Fourier Transform CTFT CTFS CTFT DTFS DTFT Time Aperiodic Frequency Aperiodic Fourier Series DTFS / DFT Discrete Frequency 40
CTFS & DTFT Fourier Series CTFS Discrete Frequency T Periodic in time Sampled in frequency CTFS CTFT DTFS DTFT Sampled in time Periodic in frequency Fourier Transform DTFT 41
CTFT & DTFS Fourier Transform CTFT CTFS CTFT DTFS DTFT Sampling in time Periodic in time Periodic in frequency Sampled in frequency Fourier Series DTFS / DFT Discrete Frequency 42
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