SP_III_P03_4p - Introduction to Signal Processing 80 Introduction to Signal Processing 81 In general the Fourier series of a periodic signal x t C

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Introduction to Signal Processing 8 0 Copyright © 2005-2009 – Hayder Radha C. Role of the Amplitude and Phase Spectra In general, the Fourier series of a periodic signal ( ) x t decomposes () x t into an infinite series of sinusoids. An important question is the following: How many of these sinusoids are “really” needed to reconstruct the original signal x t ? Introduction to Signal Processing 8 1 Copyright © 2005-2009 – Hayder Radha ¾ In general, the signal x t can be reconstructed “perfectly” from all of the Fourier series’ harmonic component sinusoids (an infinite number of them). ¾ However, sometimes, we only need a small number (much smaller than the infinite number in the series) to reconstruct the signal x t with “reasonable quality”. ¾ By increasing the number of harmonic sinusoid terms (from the Fourier series), our approximation of () x t should be improved. Introduction to Signal Processing 8 2 Copyright © 2005-2009 – Hayder Radha Example The Fourier series of the pulse train function shown is, -6 -4 -2 0 2 4 6 -0.2 0 0.2 0.4 0.6 0.8 1 t x(t) -6 -4 -2 0 2 4 6 -0.2 0 0.2 0.4 0.6 0.8 1 t Introduction to Signal Processing 8 3 Copyright © 2005-2009 – Hayder Radha This signal, with a period 0 T π = and 0 0.5 a = , has an even symmetry and hence all of the sine amplitudes are zeros: 0 n b = . The cosine amplitudes are: 02 , 4 , 6 2 1, 5, 9, 1 1 2 3,7,11, n n n n an n = =+ ⋅ = −⋅ = " " "
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Introduction to Signal Processing 8 4 Copyright © 2005-2009 – Hayder Radha Hence: 12 1 1 1 ( ) cos cos3 cos5 cos7 23 5 7 xt t t t t π =+ + + We will now reconstruct it by starting with an approximation that uses only the lowest (zero) frequency component (which is the DC component), 0 1 2 a = : -6 -4 -2 0 2 4 6 0 0.5 1 t x n (t) -6 -4 -2 0 2 4 6 0 0.5 1 t Introduction to Signal Processing 8 5 Copyright © 2005-2009 – Hayder Radha This is a “coarse” approximation of () x t by its mean. Now we add the lowest frequency component, the 1 st harmonic, i.e. cos 2 t + . -6 -4 -2 0 2 4 6 0 0.5 1 t -6 -4 -2 0 2 4 6 0 0.5 1 t Introduction to Signal Processing 8 6 Copyright © 2005-2009 – Hayder Radha Adding successive harmonics yields closer approx. If we use the 3 rd harmonic, 1 cos cos3 tt +− -6 -4 -2 0 2 4 6 0 0.5 1 t -6 -4 -2 0 2 4 6 0 0.5 1 t Introduction to Signal Processing 8 7 Copyright © 2005-2009 – Hayder Radha And the 5 th , 7 th and 9 th harmonics … -6 -4 -2 0 2 4 6 0 0.5 1 t -6 -4 -2 0 2 4 6 0 0.5 1 t -6 -4 -2 0 2 4 6 0 0.5 1 t -6 -4 -2 0 2 4 6 0 0.5 1 t -6 -4 -2 0 2 4 6 0 0.5 1 t -6 -4 -2 0 2 4 6 0 0.5 1 t
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Introduction to Signal Processing 8 8 Copyright © 2005-2009 – Hayder Radha In the above example, the role of additional (higher frequency) harmonic terms diminish at the rate 1 n .
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This note was uploaded on 06/08/2009 for the course ECE 366 taught by Professor Staff during the Spring '08 term at Michigan State University.

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SP_III_P03_4p - Introduction to Signal Processing 80 Introduction to Signal Processing 81 In general the Fourier series of a periodic signal x t C

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