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Fourier Analysis S. Awad, Ph.D. M. Corless, M.S.E.E. D. Cinpinski E.C.E. Department University of Michigan-Dearborn Math Review with Matlab: Fourier Series

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Fourier Analysis:Fourier Series 2 n Periodic Signal Definition Periodic Signal Definition n Parseval’s Theorem Parseval’s Theorem Fourier Series n Complex Exponential Representation Complex Exponential Representation n Magnitude and Phase Spectra of Fourier Series Magnitude and Phase Spectra of Fourier Series n Fourier Series Representation of Periodic Signals Fourier Series Representation of Periodic Signals n Fourier Series Coefficients Fourier Series Coefficients n Orthogonal Signals Orthogonal Signals n Example: Full Wave Rectifier Example: Full Wave Rectifier n Example: Finding Complex Coefficients Example: Finding Complex Coefficients n Example: Orthogonal Signals Example: Orthogonal Signals

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Fourier Analysis:Fourier Series 3 n For example, the normal U.S. AC from wall outlet has a sine wave with a peak voltage of 170 V (110 Vrms) n The Period of a signal is the amount of time it takes for a given signal to complete one cycle. What is a Periodic Signal ? n A Periodic Signal is a signal that repeats itself every period

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Fourier Analysis:Fourier Series 4 General Sinusoid A general cosine wave, v(t), has the form: =Phase Shift, angular offset in radians F=Frequency in Hz T=Period in seconds (T=1/F) t=Time in seconds M=Magnitude, amplitude, maximum value =Angular Frequency in radians/sec ( =2 F)

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Fourier Analysis:Fourier Series 5 General Sinusoid Plot in Blue: Plot in Red: 1 Period = 1/60 sec. = 16.67 ms. /2 Phase Shift Amplitude = 5

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Fourier Analysis:Fourier Series 6 AC Wall Voltage Sine Wave 1 Period

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Fourier Analysis:Fourier Series 7 Represent Periodic Signals n For a general periodic signal x(t) shown to the right: x(t+nT) = x(t) for all where n is any integer, i.e. n = 0, ± 1, ± 2,… T x(t) -T/2T/2 t...

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Fourier Analysis:Fourier Series 8 Frequency of Periodic Signals n The frequency of a signal is defined as the inverse of the period and has the unit “number of cycles/sec.” is the fundamental frequency. n The frequency of a US standard outlet is 1/T = 60 Hz n T is the period and

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Fourier Analysis:Fourier Series 9 What is Fourier Series ? n Fourier Series is a technique developed by J. Fourier. n This technique (studied by Fourier) allows us to represent periodic signals as a summation of sine functions of different frequency, amplitude, and phase shift.

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Fourier Analysis:Fourier Series 10 Represent a Square Wave n Represent the Square Wave at the right using Fourier Series n Notice that as more and more terms are summed, the approximation becomes better

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Fourier Analysis:Fourier Series 11 Fourier Series Representation of Periodic Signals n Any periodic function can be represented in terms of sine and cosine functions: n This can also be written as:

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Fourier Analysis:Fourier Series 12 Fourier Series Coefficients n The above u a 0, a n, and b n are known as the Fourier Series Coefficients. u These coefficients are calculated as follows.

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Fourier Analysis:Fourier Series 13 Calculating the a 0 Coefficient n a o, the coefficient outside the summation, is known as the average value or the dc component n a o is calculated as follows:

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Fourier Analysis:Fourier Series 14 Calculating the a n and b n Coefficients n = 1, 2,… The a n and b n coefficients are calculated as follows:

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Fourier Analysis:Fourier Series 15 Orthogonal Signals Two periodic signals g 1 (t) and g 2 (t) are said to be “Orthogonal” if the the integral of their product over one period is equal to zero.

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Fourier Analysis:Fourier Series 16 Example: Orthogonal Signals n Show that the following signals are orthogonal:

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Fourier Analysis:Fourier Series 17 Orthogonal Signals

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Fourier Analysis:Fourier Series 18 n Note that the rectified wave has a period equal to one-half of the source wave period. Example: Full Wave Rectifier y(t)=|sin( o t)| t y=|x| x y x(t) t T/2 one period T n Consider the output of a full-wave rectifier:

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Fourier Analysis:Fourier Series 19 Function Characteristics The period of y(t) = T/2 and the fundamental frequency of y(t) is 2 o (rad/sec). n Thus, n Now b n =0 since y(t) is an even function.

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Fourier Analysis:Fourier Series 20 Finding a o

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Fourier Analysis:Fourier Series 21 Finding a o * Use o = 2pi/T

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Fourier Analysis:Fourier Series 22 Finding a o

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Fourier Analysis:Fourier Series 23 Finding a n, n = 1, 2, ….

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Fourier Analysis:Fourier Series 24 Solution for a n, n = 1, 2, ….

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Fourier Analysis:Fourier Series 25 Solution for a n, n = 1, 2, …. n So: n Thus: n Note: We can only obtain an output signal with a nonzero average value by using a nonlinear system with our zero average value input signal

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Fourier Analysis:Fourier Series 26 Euler’s Identity n We could also say: n and...

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Fourier Analysis:Fourier Series 27 Representing Sin and Cos with Complex Exponentials Add the equations: Subtract the equations:

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Fourier Analysis:Fourier Series 28 Complex Exponential Representation n The Sine and Cosine functions can be written in terms of complex exponentials.

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Fourier Analysis:Fourier Series 29 Complex Exponential Fourier Series n From previous slides… n Using the Complex Exponential representation of Sine and Cosine, the Fourier series can be written as:

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Fourier Analysis:Fourier Series 30 Fourier Series with Complex Exponentials n Noting that 1/j = -j, we can write:

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Fourier Analysis:Fourier Series 31 Fourier Series with Complex Exponentials n Make the following substitutions:

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Fourier Analysis:Fourier Series 32 Fourier Series with Complex Exponentials n The Complex Fourier series can be written as: where: n Complex cn n *Complex conjugate n Note: if x(t) is real, c -n = c n *

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Fourier Analysis:Fourier Series 33 Line Spectra n Line Spectra refers the plotting of discrete coefficients corresponding to their frequencies n For a periodic signal x(t), c n, n = 0, ±1, ± 2,… are uniquely determined from x(t). n The set c n uniquely determines x(t) Because c n appears only at discrete frequencies, n( n = 0, ± 1, ± 2,… the set c n is called the discrete frequency spectrum or line spectrum of x(t).

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Fourier Analysis:Fourier Series 34 n The Cn coefficients are in general complex. Line Spectra n The standard practice is to make 2 2D plots. u Plot 1: Magnitude of Coefficient vs. frequency n The standard practice is to make 2 2D plots. u Plot 1: Magnitude of Coefficient vs. frequency u Plot 2: Phase of Coefficient vs. frequency

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Fourier Analysis:Fourier Series 35 Magnitude of Cn n Recall that the magnitude for a complex number a+jb is calculated as follows:

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Fourier Analysis:Fourier Series 36 Phase of Cn n Recall that the phase for a complex number a+jb depends on the quadrant that the angle lies in. Quadrant 1:Quadrant 2: Quadrant 3:Quadrant 4: Angle(a+jb) =

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Fourier Analysis:Fourier Series 37 Amplitude Spectrum of Cn Note: If x(t) is real then |Cn| is of even symmetry.

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Fourier Analysis:Fourier Series 38 Phase Spectrum of Cn n Note: If x(t) is real then the Phase of Cn is odd

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Fourier Analysis:Fourier Series 39 Example: Finding Complex Coefficients n Consider the periodic signal x(t) with period T = 2 sec. Thus: x(t) t -2.5-2-1.5-0.50.511.522.5 0 1

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Fourier Analysis:Fourier Series 40 Finding C o(avg) C o(avg) = 0.5 00 The area under x(t) from -1 to -.5 and from.5 to 1 is zero.

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Fourier Analysis:Fourier Series 41 Calculating C n

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Fourier Analysis:Fourier Series 42 n Now it can be shown that: sin(n /2) = 0 for n = ±2, ±4, … C n = 0 sin(2 /2) = sin( ) = 0 sin(-4 /2) = sin(-2 ) = 0 F etc. n It can be also be shown that: sin(n /2) = -1 for n = 3, 7, 11,… sin(n /2) = 1 for n = 1, 5, 9,… sin(3 /2) = -1 sin(-7 /2) = 1 F etc. Factor Evaluation

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Fourier Analysis:Fourier Series 43 Recall: Factor Evaluation C o(avg) = 0.5

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Fourier Analysis:Fourier Series 44 Note: C n = if C n is negative n Therefore: and Summary of Results

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Fourier Analysis:Fourier Series 45 Plot the Magnitude Response

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Fourier Analysis:Fourier Series 46 Plot the Phase Response

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Fourier Analysis:Fourier Series 47 What is Parseval’s Theorem ? n Parseval’s Theorem states that the average power of a periodic signal x(t) is equal to the sum of the squared amplitudes of all the harmonic components of the signal x(t). n This theorem is excellent for determining the power contribution of each harmonic in terms of its coefficients

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Fourier Analysis:Fourier Series 48 Parseval’s Theorem n Average power of x(t) is calculated from the time or frequency domain by: n Time Domain: n Frequency Domain:

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