SP_III_P01_4p - Introduction to Signal Processing 1 Introduction to Signal Processing 2 VI Frequency Domain Analysis of Continuous-Time Periodic

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Introduction to Signal Processing 1 Copyright © 2005-2009 – Hayder Radha VI. Frequency Domain Analysis of Continuous-Time Periodic Signals: The Fourier Series In this part of the course, we focus on the representation of periodic signals in terms of sums of sinusoids or exponentials. Introduction to Signal Processing 2 Copyright © 2005-2009 – Hayder Radha A. Periodic Signal Representation by Trigonometric Fourier Series A periodic signal () x t with period 0 T has the property, 0 ( ) x tT x t += for all t . The smallest value of 0 T for which this condition holds is the fundamental period of x t . Introduction to Signal Processing 3 Copyright © 2005-2009 – Hayder Radha 0 500 1000 1500 2000 -1 -0.5 0 0.5 1 1.5 t x(t) T 0 The area under x t over any interval of duration 0 T is always the same. For any real numbers a and b ; Introduction to Signal Processing 4 Copyright © 2005-2009 – Hayder Radha 00 aT bT ab xtd t t ++ = ∫∫ For example, for a periodic signal, we have the following equivalent integrals: 0 0 0 /2 3 /4 0/ 2/ 4 3 / 4 TT T T T t t t t −−− === = " The frequency in cycles per second or Hertz for a signal with period 0 T is denoted by 0 f .
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Introduction to Signal Processing 5 Copyright © 2005-2009 – Hayder Radha Then 00 2 f ω π = is the frequency in radians per second . So, 0 12 T f == and 2 T = A fundamental question regarding periodic signals is the following: can we represent an arbitrary periodic signal using a linear combination of “template” periodic signals? We begin by looking at such linear combination. Introduction to Signal Processing 6 Copyright © 2005-2009 – Hayder Radha Consider a signal () x t that is a linear combination of sinusoidal functions (sines and cosines) with a fundamental (radian) frequency 0 and all its harmonics ( 0 n , where 1, 2, , n = " ): 0 1 cos s in nn n x ta a n t b n t = =+ + Note that the above equation includes the zeroth harmonics which is represented by the term 0 a . Introduction to Signal Processing 7 Copyright © 2005-2009 – Hayder Radha This type of series is known as the Fourier Series (FS). An important set of questions regarding the FS representation of the signal ( ) x t are the following: ¾ Is ( ) x t periodic over all values of t ? ¾ Is it periodic for all values of the a and b parameters? We address these questions by examining if the condition 0 ( ) x tT x t += is true or not. Introduction to Signal Processing 8 Copyright © 2005-2009 – Hayder Radha 0 0 1 0 0 0 0 0 1 0 1 0 1 c o s s i n cos( ) sin( ) cos( 2 ) sin( 2 ) cos sin n n n n x tT a a n tT b n tT aa n t n T b n t n T n t n b n t n n t b n t xt ωω ωπ = = = = + ++ + + + + + + + + =
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Introduction to Signal Processing 9 Copyright © 2005-2009 – Hayder Radha Therefore, any linear combination of sinusoidal functions (sines and cosines) with (any) harmonic frequencies, 00 0 0, ,2 , , , f fk f …… and any amplitudes, 012 12 ,,, ,, aaa bb produces a periodic signal of frequency 0 f .
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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_P01_4p - Introduction to Signal Processing 1 Introduction to Signal Processing 2 VI Frequency Domain Analysis of Continuous-Time Periodic

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