Presentation on theme: "Math Review with Matlab:"— Presentation transcript:
1 Math Review with Matlab: Fourier AnalysisFourier SeriesS. Awad, Ph.D.M. Corless, M.S.E.E.D. CinpinskiE.C.E. DepartmentUniversity of Michigan-Dearborn
2 Fourier Series Periodic Signal Definition Fourier Series Representation of Periodic SignalsFourier Series CoefficientsOrthogonal SignalsExample: Orthogonal SignalsExample: Full Wave RectifierComplex Exponential RepresentationExample: Finding Complex CoefficientsMagnitude and Phase Spectra of Fourier SeriesParseval’s Theorem
3 What is a Periodic Signal ? 4/14/2017What is a Periodic Signal ?A Periodic Signal is a signal that repeats itself every periodThe Period of a signal is the amount of time it takes for a given signal to complete one cycle.For example, the normal U.S. AC from wall outlet has a sine wave with a peak voltage of 170 V (110 Vrms)
4 General Sinusoid A general cosine wave, v(t), has the form: 4/14/2017General SinusoidA general cosine wave, v(t), has the form:M = Magnitude, amplitude, maximum valuew = Angular Frequency in radians/sec (w=2pF)F = Frequency in HzT = Period in seconds (T=1/F)t = Time in secondsq = Phase Shift, angular offset in radians
5 General Sinusoid Amplitude = 5 Plot in Blue: /2 Phase Shift 1 Period = 1/60 sec.= ms.Plot in Red:
7 Represent Periodic Signals For a general periodic signal x(t) shown to the right:Tx(t)-T/2T/2t...x(t+nT) = x(t) for allwhere n is any integer, i.e. n = 0, ± 1, ± 2,…
8 Frequency of Periodic Signals The frequency of a signal is defined as the inverse of the period and has the unit “number of cycles/sec.”T is the period andis the fundamental frequency.The frequency of a US standard outlet is 1/T = 60 Hz
9 What is Fourier Series ?Fourier Series is a technique developed by J. Fourier.This technique (studied by Fourier) allows us to represent periodic signals as a summation of sine functions of different frequency, amplitude, and phase shift.
10 Represent a Square Wave Represent the Square Wave at the right using Fourier SeriesNotice that as more and more terms are summed, the approximation becomes better
11 Fourier Series Representation of Periodic Signals Any periodic function can be represented in terms of sine and cosine functions:This can also be written as:
12 Fourier Series Coefficients The abovea0, an, and bn are known as the Fourier Series Coefficients.These coefficients are calculated as follows.
13 Calculating the a0 Coefficient 4/14/2017Calculating the a0 Coefficientao, the coefficient outside the summation, is known as the average value or the dc componentao is calculated as follows:
14 Calculating the an and bn Coefficients The an and bn coefficients are calculated as follows:n = 1, 2,…n = 1, 2,…
15 Orthogonal SignalsTwo periodic signals g1(t) and g2(t) are said to be “Orthogonal” if the the integral of their product over one period is equal to zero.
16 Example: Orthogonal Signals Show that the following signals are orthogonal:
18 Example: Full Wave Rectifier Consider the output of a full-wave rectifier:y(t)=|sin(wot)|ty=|x|xyx(t)T/2one periodTNote that the rectified wave has a period equal to one-half of the source wave period.
19 Function Characteristics The period of y(t) = T/2 and the fundamental frequency of y(t) is 2wo (rad/sec).Now bn=0 since y(t) is an even function.Thus,
27 Representing Sin and Cos with Complex Exponentials Subtract the equations:Add the equations:
28 Complex Exponential Representation The Sine and Cosine functions can be written in terms of complex exponentials.
29 Complex Exponential Fourier Series From previous slides…Using the Complex Exponential representation of Sine and Cosine, the Fourier series can be written as:
30 Fourier Series with Complex Exponentials Noting that 1/j = -j, we can write:
31 Fourier Series with Complex Exponentials Make the following substitutions:
32 Fourier Series with Complex Exponentials The Complex Fourier series can be written as:where:Complex cn*Complex conjugateNote: if x(t) is real, c-n = cn*
33 Line SpectraLine Spectra refers the plotting of discrete coefficients corresponding to their frequenciesFor a periodic signal x(t), cn, n = 0, ±1, ± 2,… are uniquely determined from x(t).The set cn uniquely determines x(t)Because cn appears only at discrete frequencies, n(wo), n = 0, ± 1, ± 2,… the set cn is called the discrete frequency spectrum or line spectrum of x(t).
34 Line Spectra The Cn coefficients are in general complex. The standard practice is to make 2 2D plots.Plot 1: Magnitude of Coefficient vs. frequencyPlot 2: Phase of Coefficient vs. frequencyThe standard practice is to make 2 2D plots.Plot 1: Magnitude of Coefficient vs. frequency
35 Magnitude of CnRecall that the magnitude for a complex number a+jb is calculated as follows:
36 Phase of CnRecall that the phase for a complex number a+jb depends on the quadrant that the angle lies in.Angle(a+jb) =Quadrant 1:Quadrant 2:Quadrant 3:Quadrant 4:
37 Amplitude Spectrum of Cn Note: If x(t) is real then |Cn| is of even symmetry.
38 Phase Spectrum of CnNote: If x(t) is real then the Phase of Cn is odd
39 Example: Finding Complex Coefficients x(t)t-2.5-2-1.5-1-0.50.511.522.5Consider the periodic signal x(t) with period T = 2 sec. Thus:
40 The area under x(t) from -1 to -.5 and from .5 to 1 is zero. Finding Co(avg)The area under x(t) from -1 to -.5 and from .5 to 1 is zero.Co(avg) = 0.5
42 Factor Evaluation Now it can be shown that: sin(np/2) = 0 for n = ±2, ±4, … Cn = 0sin(2p/2) = sin(p) = 0sin(-4p/2) = sin(-2p) = 0etc .It can be also be shown that:sin(np/2) = -1 for n = 3, 7, 11,…sin(np/2) = 1 for n = 1, 5, 9,…sin(3p/2) = -1sin(-7p/2) = 1etc .
44 Summary of Results Note: Cn=p if Cn is negative Therefore: and 4/14/2017Summary of ResultsNote: Cn=p if Cn is negativeTherefore:and
45 Plot the Magnitude Response 4/14/2017Plot the Magnitude Response
46 Plot the Phase Response 4/14/2017Plot the Phase Response
47 What is Parseval’s Theorem ? 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).This theorem is excellent for determining the power contribution of each harmonic in terms of its coefficients
48 Parseval’s TheoremAverage power of x(t) is calculated from the time or frequency domain by:Time Domain:Frequency Domain: