Spectra (2A) Young Won Lim 11/8/12

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1 Spectra (A) /8/

2 Copyright (c) 0 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover exts, and no Back-Cover exts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. his document was produced by using OpenOffice and Octave. /8/

3 Summary () Non-periodic signals Periodic signals Random signals Energy Signal Power Signal Power Signal E x = lim + / / x (t) d t P x = + 0 / 0 0 / x (t) dt P x = lim + / / x (t) dt E x < P x < P x < Energy Spectral Density Power Spectral Density Power Spectral Density Ψ( f ) = X ( f ) G x ( f ) = c n δ(f n f 0 ) n= G x ( f ) = lim otal Energy X ( f ) Ψ( f ) d f Gx ( f ) d f Gx ( f ) d f Spectra 3 /8/

4 Summary () Energy Signal Autocorrelation Power Signal Autocorrelation Random Signal Autocorrelation R x ( τ) = R x ( τ) = x(t) x(t +τ) dt lim + / / x (t)x (t + τ) dt E {X (t) X (t + τ)} R x ( τ) = Non-periodic signals Periodic signals Random signals for a Periodic Signal for a Ergodic Signal R x ( τ) = R x ( τ) = R x ( τ) = x(t) x(t +τ) dt + 0 / 0 / x (t) x(t + τ) dt 0 lim + / / X (t) X (t+τ) dt Spectra 4 /8/

5 Single-Sided Spectrum x(t) = a 0 + k= ( 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 ) k= a 0 = 0 x(t) dt a k = 0 x(t) cos (k ω0 t) dt b k = 0 x(t ) sin (k ω0 t) dt g 0 = a 0 g k = a k + b k ϕ k = tan ( b k a k ) k = +, +,... 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 = g k cos(ϕ k ) b k = g k sin(ϕ k ) a k + b k = g k b k a k = tan (ϕ k ) Spectra 5 /8/

6 wo-sided Spectrum x(t) = C k e + j k ω 0 t k= x(t) = C k e + j k ω 0 t k= C k = 0 x(t ) e j k ω 0 t dt C k = 0 x(t ) e j k ω 0 t dt k =...,,, 0, +, +,... k =...,,, 0, +, +,... C k = a 0 (k = 0) k jb k ) (k > 0) k + jb k ) (k < 0) C k = a 0 (k = 0) k e + j ϕ k (k > 0) k e j ϕ k (k < 0) C k = a 0 (k = 0) a k + b k (k 0) Arg (C k ) = tan ( b k /a k ) (k > 0) tan (+b k /a k ) (k < 0) C k = a 0 (k = 0) g k ( k 0) Arg (C k ) = +ϕ k (k > 0) ϕ k (k < 0) Power Spectrum C k + C k wo-sided Periodogram = g k = (a k + b k ) C k = g k = a k + b k One-Sided Spectra 6 /8/

7 Power Spectrum () x(t) = a 0 + ( a k cos(k ω 0 t) + b k sin (k ω 0 t )) n= x(t) = C k e + j kω 0 t k= a 0 = 0 x(t) dt C k = 0 x(t ) e j k ω 0 t dt a k = 0 x(t) cos (k ω0 t) dt k =,, 0, +, +,... b k = 0 x(t ) sin (k ω0 t) dt k =,,... C k = a 0 (k = 0) k jb k ) (k > 0) k + jb k ) (k < 0) P = 0 x(t) dt P = 0 x(t) dt P = a 0 + k = Power Spectrum P k = a 0 ( a k + b k ( a k + b k ) ) (k = 0) (k > 0) P = C 0 + k= ( C k + C k ) Power Spectrum P k = C 0 ( C k + C k ) (k = 0) (k > 0) Spectra 7 /8/

8 Power Spectrum () x t = a 0 a n cos n 0 t b n sin n 0 t n= x(t) = C k e + j kω 0 t k= P = 0 x(t) dt P = 0 x(t) dt P = a 0 + k = ( a k + b k ) P = C 0 + k= ( C k + C k ) C k + C k = (a k + b k ) C k = 4 ( a k + b k ) C k = 4 ( a k + b k ) C k = a 0 (k = 0) k jb k ) (k > 0) k + jb k ) (k < 0) Power Spectrum Power Spectrum P k = a 0 ( a k + b k ) (k = 0) (k > 0) P k = C 0 ( C k + C k ) (k = 0) (k > 0) Spectra 8 /8/

9 Periodogram x t = a 0 a n cos n 0 t b n sin n 0 t n= An estimate of the spectral density a 0 = 0 x(t) dt a n = 0 x(t) cos (n ω0 t) dt b n = 0 x(t) sin (n ω0 t) dt n =,,... Positive Frequency Only f = n n =,,... a n t + = t x(t ) cos (k t) dt b n t + = t x(t ) sin (k t ) dt : integer multiple = π k n k = π n = n ω 0 π k Single-sided Spectrum a n + b n sometimes (a n + b n ) abscissa ordinates π k r = a n + b n Spectra 9 /8/

10 Periodogram and Power Spectrum Power Spectrum Single-Sided Power Spectrum wo-sided C k = (a 4 k + b k ) P avg = (V rms ) (a k + b k ) ( C k + C k ) C k + C k = g k = (a k + b k ) C k = a k + b k Periodogram V peak a k + b k ( C k + C k ) C k + C k = g k = a k + b k Spectra 0 /8/

11 Root Mean Square Continuous ime Discrete ime g [k ] Δ Δ N Δ mean squared amplitude mean squared amplitude 0 g (t) dt N g [ k ] Δ = N g [ k ] N Δ k=0 N k=0 root mean squared g rms = 0 g (t) dt root mean squared N g rms = g [ k ] N k=0 Spectra /8/

12 CFS and CF Continuous ime Fourier Series C k = 0 x(t ) e j k ω 0 t dt x(t) = k=0 C k e + j k ω 0t Continuous ime Fourier ransform X ( jω) = x(t) e j ωt d t x (t) = +j π X ( j ω) e ωt d ω X (f ) = x (t) e j π f t d t x (t) = X (f ) e + j π f t d f Spectra /8/

13 Continuous Periodic Signal + instantaneous power x (t) Ω v(t) = x(t) average power 0 : x (t) dt period v(t) = x(t) Continuous Periodic CFS (Fourier Series) Parseval s heorem x(t) = C k e + j k ω 0 t k= 0 x (t) dt = n= C n C k = j k ω 0 x(t) e 0 t dt k =...,,, 0, +, +,... average power sum of power spectrum C n Spectra 3 /8/

14 Continuous Aperiodic Signal + instantaneous power x (t) Ω v(t) = x(t) total energy x (t) dt v(t) = x(t) Continuous Aperiodic CF (Fourier Integral) Parseval s heorem x(t) = X ( f ) e + j π f t d f x (t) dt = X (f ) df X ( f ) = x(t) e j π f t d t total energy integral of energy spectral density X ( f ) Spectra 4 /8/

15 Parseval s heorem CFS E [ x (t ) ] = CFS x (t ) dt x(t ) = C k e + jk ω t k= x (t ) = x(t) x (t) = ( t)( t) C k e + jk ω C jl ω l e k= l= x (t ) = k= C k C + j( k l)ω t l e l = x (t ) dt = k= ( C k C l l= e dt) + j (k l )ωt x (t ) dt = k= C k Power Spectrum C k power associated with individual frequency components Spectra 5 /8/

16 Parseval s heorem CF otal Energy CF x (t) dt x(t ) = X ( f ) e + jπ ft df x (t ) = x(t) x (t) = x (t ) = x (t) dt ( X ( f )e + j π ft df ) ( X ( f ) e + j π( f ν)t df d ν = X ( f ) e + j π( f ν)t dt df d ν X (ν)e jπ νt d ν) otal Energy x (t) dt = X ( f ) df = π X (ω) d ω Energy Spectral Density X ( f ) energy associated with individual frequency components Spectra 6 /8/

17 Parseval s heorem DF otal Energy DF x (t) dt otal energy over all discrete time n otal energy in fundamental period of frequency +0.5 x[n] = 0.5 X (e j f ) e + j π f n d f x [n] = x[n] x [n] = x [n] = n= x [n] dt ( X (e j f ) e + j π( f ν) n df d ν = X (e j f ) X (e j f ) e + j π f n d f ) ( 0.5 n= +0.5 e + j π( f ν)n df d ν X (e j ν ) e + j π ν n d ν) otal Energy Energy Spectral Density n= X ( f ) x[n] +0.5 = 0.5 X ( f ) d f = π +π π energy associated with individual frequency components X ( ω) d ω Spectra 7 /8/

18 Parseval s heorem DF N N n=0 x [n] DF N x[n] = N k=0 X [k ]e + j ( π N ) kn x [n] = x[n] x [n] = x [n] = N N n=0 x [n] dt N N N k=0 l=0 ( N N k=0 ( X [k ] e + j π N ) kn ) ( N ( X [l ]e j π N ) n) l l=0 X [k ] X [l ]e + j ( π N ) ( k l)n = N N N 3 k=0 l=0 N ( X [ k ] X [l ] e + j π N )(k l )n n=0 N x[n] N n=0 = N X [k ] N k=0 N x[n] = N X [k ] N n=0 k=0 Periodogram X [k ] N Spectra 8 /8/

19 of Random Signals A truncated sample function x (t) = x(t ) < t < + = 0 otherwise Fourier ransform X (ω) = X (f ) = Parseval s herorem x (t) dt = x (t) e jω t dt x (t) e j π f t dt X ( f ) df x (t) = π x (t) = total energy X (ω) e + jω t d ω X (f ) e + j π f t df x (t) dt X = ( f ) dt total energy / Spectra 9 /8/

20 Power Spectral Density of Random Signals x (t) dt X = (f ) df Ŝ xx (f ) df Raw Power Spectral Density X ( f ) = Ŝ xx ( f ) Power Spectral Density lim E [ X ( f ) ] = S xx ( f ) lim x (t) dt = lim X ( f ) df not converge E [ lim x (t) dt ] = E [ lim X (f ) df ] random signal Var (x(t)) = σ x = lim E [ X ( f ) ] df S xx (f ) df Spectra 0 /8/

21 Power and Power Density Spectra () E [ x (t )] = lim / / x (t) dt x[n] of N sample of x(t) N N 0 N N x (t) dt x [n] = N n=0 N k=0 X k N N n=0 N x [n] = N k=0 X [ k ] P xx (k ) = N X [k ] Periodogram: k = 0,,, N otal Energy N n=0 N x [n] = N k=0 P xx (k ) N n=0 N x [n] = k=0 P xx (k ) N 0 x (t ) dt Spectra /8/

22 Power and Power Density Spectra () Periodogram: x[n] P xx (k ) = N X [k ] k = 0,,, N N N n=0 N x [n] = N k=0 P xx (k ) if P xx (k) is constant P N n= x [n] = 0.5 P xx ( f )df P xx ( f ) = P N n=0 x [n] = +/ / P xx ( f )df P xx ( f ) = P N n=0 x [n] = +π / π / P xx (ω)d ω P xx (ω) = P/ π Spectra /8/

23 Correlation and Power Spectrum () Periodogram: Autocorrelation Function P xx (k ) = N X [k ] N R xx [m] = N l=0 x[l ] x [l+m] k = 0,,, N m = 0,,, N Autocorrelation DF Periodogram N m=0 ( N l=0 N x[l ] x[l+m]) e j ( π N ) mk N N n=0 x [n] = R xx [0] N = m=0 ( N N l=0 = N N N 3 α=0 β=0 = N N N 3 α=0 β=0 ( N N α=0 N N X α X β m=0 l=0 N N X α X β m=0 l=0 ( X α e + j π N ) l α )( N N β=0 X β e j ( π N ) (l+m)β)) e j ( π N ) mk ( e + j( π N ) l α ) ( e j( π N ) (l+m)β ) e j( π N ) mk + ( e j( π N ) ) (l (α β) m(β+k)) = N N N α=0 β=0 X α X β Spectra 3 /8/

24 Periodic Signals Aperiodic Signals Random Signals Frequency Spacing Δ f = N Δ t Δ f = N Δ t S Δ f = N Δ t S N Δ t x Δ t wo Sided N X (k ) Δ t N X (k ) S (k)= Δ t N X (k ) P= N k=0 S(k)Δ f One Sided k=0, N N X (k ) Δ t N X (k ) S (k )=S(k) N P= / k=0 S (k )Δ f k=,, N N X (k ) Δt N X (k ) S (k )=S (k ) Frequenc y Scale k Δ f k Δ f k Δ f Spectra 4 /8/

25 References [] [] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 003 [3] K.H. Shin, J.K. Hammond, Fundamentals of Signal Processing for Sound and Vibration Engineers, Wiley, 008 [4] Brian D. Storey, Computing Fourier Series and Power Spectrum with MALAB [5] Univ. of Rhode Island, ELE436, FF utorial [6] Guy Beale, Signal & Systems Fourier Series Example # [7] National Instrument, he Fundamentals of FF-Based Signal Analysis and Measurement in LabVIEW and LabWindows/CVI [8] Cambridge University Press, Numerical Recipes [9] E. Overman, A MALAB utorial [0] S. D. Stearns, Digital Signal Processing with examples in MALAB [] S. D. Stearns and R. A. David, Signal Processing Algorithms in MALAB [] L. an, Digital Signal Processing: Fundamentals and Applications /8/