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

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

What is a Periodic Signal ? 4/14/2017 What is a Periodic Signal ? A Periodic Signal is a signal that repeats itself every period The 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)

General Sinusoid A general cosine wave, v(t), has the form: 4/14/2017 General Sinusoid A general cosine wave, v(t), has the form: M = Magnitude, amplitude, maximum value w = Angular Frequency in radians/sec (w=2pF) F = Frequency in Hz T = Period in seconds (T=1/F) t = Time in seconds q = Phase Shift, angular offset in radians

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

AC Wall Voltage Sine Wave 1 Period 1 Period

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

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 and is the fundamental frequency. The frequency of a US standard outlet is 1/T = 60 Hz

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.

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

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:

Fourier Series Coefficients The above a0, an, and bn are known as the Fourier Series Coefficients. These coefficients are calculated as follows.

Calculating the a0 Coefficient 4/14/2017 Calculating the a0 Coefficient ao, the coefficient outside the summation, is known as the average value or the dc component ao is calculated as follows:

Calculating the an and bn Coefficients The an and bn coefficients are calculated as follows: n = 1, 2,… n = 1, 2,…

Orthogonal Signals Two 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.

Example: Orthogonal Signals Show that the following signals are orthogonal:

Orthogonal Signals

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

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,

Finding ao

Finding ao * Use o = 2pi/T

Finding ao

Finding an, n = 1, 2, ….

Solution for an, n = 1, 2, ….

Solution for an, n = 1, 2, …. So: Thus: 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

Euler’s Identity We could also say: and ...

Representing Sin  and Cos  with Complex Exponentials Subtract the equations: Add the equations:

Complex Exponential Representation The Sine and Cosine functions can be written in terms of complex exponentials.

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

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

Fourier Series with Complex Exponentials Make the following substitutions:

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

Line Spectra Line Spectra refers the plotting of discrete coefficients corresponding to their frequencies For 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).

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

Magnitude of Cn Recall that the magnitude for a complex number a+jb is calculated as follows:

Phase of Cn Recall 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:

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

Phase Spectrum of Cn Note: If x(t) is real then the Phase of Cn is odd

Example: Finding Complex Coefficients x(t) t -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 Consider the periodic signal x(t) with period T = 2 sec. Thus:

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

Calculating Cn

Factor Evaluation Now it can be shown that: sin(np/2) = 0 for n = ±2, ±4, …  Cn = 0 sin(2p/2) = sin(p) = 0 sin(-4p/2) = sin(-2p) = 0 etc . 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) = -1 sin(-7p/2) = 1 etc .

Factor Evaluation Recall: Co(avg) = 0.5

Summary of Results Note: Cn=p if Cn is negative Therefore: and 4/14/2017 Summary of Results Note: Cn=p if Cn is negative Therefore: and

Plot the Magnitude Response 4/14/2017 Plot the Magnitude Response

Plot the Phase Response 4/14/2017 Plot the Phase Response

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

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