Spectrum Representation (5A) Young Won Lim 11/3/16

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1 Spectrum (5A)

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 Fourier Series with real coefficients 1 x(t ) = a 0 + ( a k cos(k ω 0 t) + b k sin(k ω 0 t )) k =1 a x(t ) = g 0 + k =1 g k cos(k ω 0 t + ϕ k ) b x(t ) = X 0 + R {X k e + j k ω 0 t } X k = g k e + jϕ k 3

4 Phasor X k via g k, ϕ k x(t ) = g 0 x(t ) = g g k cos(k ω 0 t + ϕ k ) g k R {e + j(k ω 0t + ϕ k ) } x(t ) = g 0 + R {g k e + j ϕ k e + j k ω 0 t } X k = g k e + jϕ k 4

5 Fourier Series with complex coefficients 4 x(t ) = + C k e + j k ω 0t k= a x(t ) = g 0 + k =1 g k cos(k ω 0 t + ϕ k ) b x(t ) = X 0 + R {X k e + j k ω 0 t } C k = 1 2 g +k e + j ϕ k (k > 0) C k = 1 2 g k e j ϕ k (k < 0) 5

6 Complex Coefficient C k via g k, ϕ k 4 x(t ) = + C k e + j k ω 0t k=, C 1, C 0, C +1, x (t) = g 0 + = g 0 + = g 0 + g k cos(k ω 0 t + ϕ k ) g k 1 2 (e+ j(k ω 0t + ϕ k ) + e j(k ω 0t + ϕ k) ) ([ 1 k] 2 g k e+ j ϕ e+ j k ω0t [ + 1 k] 0t) 2 g k e j ϕ e j k ω = g 0 + ( [ C k ] e + j k ω t 0 + [C k ] e j k ω t 0 ) C k = 1 2 g +k e + j ϕ k (k > 0) C k = 1 2 g k e j ϕ k (k < 0) g 0, g 1, g 2, 6

7 Fourier Coefficients Relationship 1 k = 1, 2,... a a 0 = 1 T T 0 x(t) dt a k = 2 T T 0 x(t) cos (k ω0 t) dt b k = 2 T T 0 x(t) sin (k ω0 t) dt k = 1, 2,... g 0 = a 0 g k = a k 2 + b k 2 ϕ k = tan 1 ( b k a k ) 4 k = 0, ±1, ±2,... b k = 1, 2,... C k = 1 2 g +k e + j ϕ k (k > 0) X 0 = g 0 C k = 1 2 g k e jϕ k (k < 0) X k = g k e + j ϕ k 7

8 Single-Sided Spectrum x(t) = a 0 + ( a k cos(k ω 0 t) + b k sin(k ω 0 t)) x(t) = g 0 + g k cos(k ω 0 t + ϕ k ) a 0 = 1 T 0 a k = 2 T 0 b k = 2 T 0 T x(t) dt T x(t) cos (k ω0 t) dt T x(t ) sin (k ω0 t) dt a 0, a 1, a 2,... b 0, b 1, b 2,... g 0 = a 0 g k = a k 2 + b k 2 ϕ k = tan 1 ( b k a k ) g 0, g 1, g 2,... ϕ 0, ϕ 1, ϕ 2,... 8

9 Single-Sided Spectrum x(t) = a 0 + ( a k cos(k ω 0 t) + b k sin(k ω 0 t)) x(t) = g 0 + g k cos(k ω 0 t + ϕ k ) cos(α +β) = cos(α) cos(β) sin(α) sin(β) g k cos(k ω 0 t + ϕ k ) = g k cos (ϕ k ) cos(k ω 0 t) g k sin(ϕ k ) sin (k ω 0 t) a k cos(k ω 0 t ) + b k sin(k ω 0 t ) a k 2 + b k 2 = g k 2 b k a k = tan (ϕ k ) a 0 = g 0 a k = g k cos(ϕ k ) b k = g k sin(ϕ k ) g 0 = a 0 g k = a k 2 + b k 2 ϕ k = tan 1 ( b k a k ) 9

10 Periodogram a x(t ) = g 0 + k =1 g k cos(k ω 0 t + ϕ k ) Periodogram One-Sided g k = a k 2 + b k 2 g 0, g 1, g 2, 10

11 Power Spectrum 4 x(t ) = + C k e + j k ω 0t k= Power Spectrum Two-Sided C k 2 = 1 (a 2 4 k + b 2 k ), C 1, C 0, C +1, 11

12 Power Spectrum 4 x(t ) = + C k e + j k ω 0t k= Power Spectrum P = 1 T T x (t ) 2 dt C k 2 = 1 (a 2 4 k + b 2 k ) = 1 T T x(t)x (t )dt T C k = 1 T 0 T C k = 1 T 0 x(t) e j k ω 0 t dt x (t) e + j k ω 0 t dt = 1 T T x + (t) C k e + j k ω 0t dt k= = 1 + C T k x (t) e + j k ω 0t dt k= T = 1 + C T k T C k k= = C k 2 12

13 Two-Sided Spectrum x(t) = a 0 + ( a k cos(k ω 0 t) + b k sin(k ω 0 t )) x(t) = g 0 + gk cos(k ω 0 t + ϕ k ) x(t) = + C k e + j k ω 0 t k= x(t) = + C k e + j k ω 0 t k= C k = a 0 (k = 0) 1 2 k j b k ) (k > 0) 1 2 k + j b k ) (k < 0) C k = g 0 (k = 0) 1 2 +k e + jϕ k (k > 0) 1 2 k e j ϕ k (k < 0) C k = a 0 (k = 0) 1 2 a k 2 + b k 2 (k 0) C k = g 0 (k = 0) 1 2 g k (k 0) Arg(C k ) = tan 1 ( b k /a k ) (k > 0) tan 1 (+b k / a k ) (k < 0) Arg(C k ) = +ϕ k (k > 0) ϕ k (k < 0) Power Spectrum Two-Sided Periodogram One-Sided C k 2 = C k 2 = 1 4 g k 2 = 1 (a 2 4 k + b 2 k ) g k = 2 C k = a 2 2 k + b k 13

14 Spectrum Real Single-tone Sinusoidal Signal x(t) = A cos(ω 0 t + ϕ) = R{X e j ω 0 t } = { X 2 e+ j ω 0t + X 2 e j ω 0t } A cos(ω 0 t + ϕ) = A 2 (e+ j(ω 0t + ϕ) + e j(ω 0 t + ϕ) ) = A 2 (e+ j ϕ e + j ω 0t + e j ϕ e j ω 0t ) = {A e+ j ϕ } 2 e + j ω 0 t + {A e+ j ϕ } e j ω 0 t 2 X = A e j ϕ 14

15 Spectrum Real Single-tone Sinusoidal Signal x(t) = A cos(ω 0 t + ϕ) = R{X e j ω 0 t } Real Multi-tone Sinusoidal Signal N x(t) = A 0 + k =1 N = X 0 + R{ A k cos(ω k t + ϕ k ) X k e j ω k t } = { X 2 e+ j ω 0t + X 2 e j ω 0t } N = X 0 + { X k 2 e+ j ω t k + X k 2 e jω k t} X = A e j ϕ X k = A k e j ϕ k the phasor of angular frequency ω k X 0 = A 0 15

16 Spectrum X k 2 e j ω k t X k 2 ω k X k 2 e+ j ω kt X k 2 +ω k Real Multi-tone Sinusoidal Signal N x(t) = A 0 + = X 0 + R{ N = X 0 + N A k cos(ω k t + ϕ k ) X k e j ω k t } { X k 2 e+ j ω t k + X k 2 e jω k t} only magnitude is display X k = A k e j ϕ k the phasor of angular frequency ω k X 0 = A 0 16

17 Frequency Resolution ω 0, ^ω 0 CTFS 0 T 0 + Continuous Time period : T 0 seconds DTFS / DFT 0 + ω 0 = 2π Discrete Frequency ω 0 ω 0 T 0 0 T s T s Discrete Time period : N 0 samples π 0 ^ω 0 ^ω 0 +π Normalized Discrete Frequency ^ω 0 = 2π N 0 17

18 T 0 period and N 0 samples x(t ) Continuous Time T 0 period frequency resolution ω 0 = 2 π T 0 T 0 = N 0 T s ω 0 = 2 π N 0 T s T s ω 0 1/T s = 2 π N 0 T s T s f s x[n] T s Discrete Time N 0 samples ^ω 0 = 2 π N 0 normalized frequency resolution 18

19 Replication Frequencies and Frequency Resolutions ω = 2 π T Continuous Time ω s = 2 π T s replication frequency ω 0 = 2 π T 0 frequency resolution ω = ^ω T s Discrete Time ^ω s = 2 π 1 replication frequency T 0 = N 0 T s ^ω 0 = 2 π N 0 frequency resolution 19

20 Different Sampling Periods x(t ) T 0 period ω s = 2 π T s ω = ^ω T s ω T s = ^ω T 1 N 1 = T 2 N 2 x 1 [n] T s = T 1 N 1 samples replication frequency ω s1 = 2 π T 1 ^ω s1 = 2π x 2 [n] N 2 samples replication frequency ω s2 = 2π T 2 ^ω s2 = 2 π T s = T 2 frequency resolutions T 1 > T 2 coarse ^ω 0 1 = ω 0 T 1 > ω 0 T 2 = ^ω 02 fine N 1 < N 2 ω 1 < ω 2 20

21 Different Sampling Periods ω s = 2 π T s ω = ^ω T s ω T s = ^ω ω s1 = 2π T 1 ^ω s1 = 2π ω 0 = 2π N 1 T 1 ^ω 01 = ω 0 T 1 ω s2 = 2 π T 2 ^ω s2 = 2 π ω 0 = 2π N 2 T 2 ^ω 0 2 = ω 0 T 2 frequency resolutions T 1 > T 2 coarse ^ω 0 1 = ω 0 T 1 > ω 0 T 2 = ^ω 02 fine N 1 < N 2 ω 1 < ω 2 21

22 Normalized Frequency Resolution ω 0 = 2π N 1 T 1 ω s1 = 2 π T 1 replication frequency ^ω 0 1 = ω 0 T 1 coarse frequency resolution ^ω 02 = ω 0 T 2 fine frequency resolution ω 0 = 2π N 2 T 2 ω s2 = 2 π T 2 replication frequency 22

23 Frequency and Digital Frequency Continuous Time x (t) = cos(ω 0 t ) ω 0 = 2π T 0 T 0 t T s x [n] = x(nt s ) = cos(n ω 0 T s ) = cos(n ^ω 0 ) Discrete Time ^ω 0 = 2π N 0 ^ω = ω T s = ω f s ω 0 = 2 π N 0 T s 23

24 Frequency and Digital Frequency Frequency ω (rad /sec) ω s ω s 2 + ω s 2 +ω s ω = ^ω T s ω 0 ω s ω 0 +ω 0 +ω 0 +ω s e j(ω 0+ω s )t e j ω 0 t e + jω 0 t e + j(ω 0+ω s )t Digital Frequency ^ω (rad / sample) 2 π π +π +2π ^ω = ω T s ^ω 0 2 π ^ω 0 + ^ω 0 + ^ω 0 +2π e j( ^ω 0+2 π)n e j ^ω 0 n e + j ^ω 0 n e + j( ^ω 0+2 π )n 24

25 Fourier Transform Types Continuous Time Fourier Series C k = 1 T 0 N 1 γ[k ] = 1 N n = 0 T x(t ) e j k ω 0 t Discrete Time Fourier Series dt x[n] e j k ^ω 0 n Continuous Time Fourier Transform X ( j ω) = Discrete Time 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 ) = j 2 π X ( j ω) e ωt d ω x[n] = 1 +π 2π π X ( j ^ω) e + j ^ω n d ^ω 25

26 Continuous Time Continuous Time 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 Continuous Time Fourier Transform X ( j ω) = + x(t) e j ωt dt x(t ) = j 2 π X ( j ω) e ωt d ω 26

27 Discrete Time Discrete Time 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 Discrete Time 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 27

28 Continuous Time Signal Spectrum CTFS CTFT C k = 1 T 0 T x(t ) e j k ω 0 t dt X ( j ω) = + x(t) e j ωt dt C k e jk ω 0t C k e + j k ω 0t X ( j ω) e j ωt X (+ j ω)e + jω t C k C k X ( j ω) X (+ j ω) k ω 0 + k ω 0 ω + ω only magnitude is display only magnitude is display 28

29 Discrete Time Signal Spectrum DTFS N 1 γ[k ] = 1 N n = 0 DTFT + x[n] e j k ^ω n 0 X ( j ^ω) = n = x[n] e j ^ω n γ [ k ]e j k ^ω 0n γ [+k] e + jk ^ω 0n X (e j ω )e j ω n X (e + j ω )e + j ω n γ [ k ] γ [+k] X ( j ^ω) X (+ j ^ω) π k ^ω 0 +k ^ω 0 + π π ω + ω + π only magnitude is display only mag is display 29

30 30

31 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 [6] A graphical interpretation of the DFT and FFT, by Steve Mann

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Sampling Basics (1B) Young Won Lim 9/21/13 Sampling Basics (1B) Copyright (c) 2009-2013 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