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Chapter 4 Fourier Series & Transforms

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Basic Idea notes

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Taylor Series Complex signals are often broken into simple pieces Signal requirements –Can be expressed into simpler problems –The first few terms can approximate the signal Example: The Taylor series of a real or complex function ƒ(x) is the power series

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Square Wave S(t)=sin(2ft) S(t)=1/3[sin(2f)t)] S(t)= 4/{sin(2ft) +1/3[sin(2f)t)]} Fourier Expansion

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Square Wave Frequency Components of Square Wave K=1,3,5K=1,3,5, 7 K=1,3,5, 7, 9, ….. Fourier Expansion

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Periodic Signals A Periodic signal/function can be approximated by a sum (possibly infinite) sinusoidal signals. Consider a periodic signal with period T A periodic signal can be Real or Complex The fundamental frequency: o Example:

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Fourier Series We can represent all periodic signals as harmonic series of the form –C k are the Fourier Series Coefficients; k is real –k=0 gives the DC signal –k=+/-1 indicates the fundamental frequency or the first harmonic 0 –|k|>=2 harmonics

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Fourier Series Coefficients Fourier Series Pair We have For k=0, we can obtain the DC value which is the average value of x(t) over one period Series of complex numbers Defined over a period of x(t)

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Eulers Relationship –Review Euler formulas Euler formulas notes

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Examples Find Fourier Series Coefficients for notes C1=1/2; C-1=1/2; No DC C1=1/2j; C-1=-1/2j; No DC

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Different Forms of Fourier Series Fourier Series Representation has three different forms Also: Harmonic Also: Complex Exp. Which one is this? What is the DC component? What is the expression for Fourier Series Coefficients

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Examples Find Fourier Series Coefficients for Remember:

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Examples Find the Complex Exponential Fourier Series Coefficients notes textbook

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Example Find the average power of x(t) using Complex Exponential Fourier Series – assuming x(t) is periodic This is called the Parsevals Identity

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Example Consider the following periodic square wave Express x(t) as a piecewise function Find the Exponential Fourier Series of representations of x(t) Find the Combined Trigonometric Fourier Series of representations of x(t) Plot Ck as a function of k V To/2 -V To notes X(t) 2|Ck| |4V/ | |4V/5 | |4V/3 | Use a Low Pass Filter to pick any tone you want!!

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Practical Application Using a XTL oscillator which produces positive 1Vp-p how can you generate a sinusoidal waveforms with different frequencies?

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Practical Application Using a XTL oscillator which produces positive 1Vp-p how can you generate a sinusoidal waveforms with different frequencies? Square wo [k o ] Level Shifter Sinusoidal waveform 1 To/2 To X(t) 0.5 To/2 To X(t) [kwo] kwo B changes depending on k value

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Demo Ck corresponds to frequency components In the signal.

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Example Given the following periodic square wave, find the Fourier Series representations and plot Ck as a function of k. Sinc Function Only a function of freq. 1 Note: sinc (infinity) 1 & Max value of sinc(x) 1/x Note: First zero occurs at Sinc (+/-pi)

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Use the Fourier Series Table (Table 4.3) Consider the following periodic square wave Find the Exponential Fourier Series of representations of x(t) X0 V V To/2 -V To X(t) 2|Ck| |4V/ | |4V/5 | |4V/3 | 0 0 0

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Fourier Series - Applet

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Using Fourier Series Table Given the following periodic square wave, find the Fourier Series representations and plot Ck as a function of k. (Rectangular wave) X0 1 C0=T/To T/2=T1 T=2T1 Ck=T/T0 sinc (Tkw0/2) Same as before Note: sinc (infinity) 1 & Max value of sinc(x) 1/x

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Using Fourier Series Table Express the Fourier Series for a triangular waveform? Express the Fourier Series for a triangular waveform that is amplitude shifted down by –X0/2 ? Plot the signal. To Xo

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Fourier Series Transformation Express the Fourier Series for a triangular waveform? Express the Fourier Series for a triangular waveform that is amplitude shifted down by –X0/2 ? Plot the signal. To Xo To Xo/2 -Xo/2 From the table:

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Fourier Series Transformation Express the Fourier Series for a triangular waveform? Express the Fourier Series for a triangular waveform that is amplitude shifted down by –X0/2 ? Plot the signal. To Xo To Xo/2 -Xo/2 Only DC value changed! From the table:

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Fourier Series Transformation Express the Fourier Series for a sawtooth waveform? Express the Fourier Series for this sawtooth waveform? To Xo To Xo From the table: -3 1

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Fourier Series Transformation Express the Fourier Series for a sawtooth waveform? Express the Fourier Series for this sawtooth waveform? –We are using amplitude transfer –Remember Ax(t) + B Amplitude reversal A<0 Amplitude scaling |A|=4/Xo Amplitude shifting B=1 To Xo To Xo From the table: -3 1

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Example

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Fourier Series and Frequency Spectra We can plot the frequency spectrum or line spectrum of a signal –In Fourier Series k represent harmonics –Frequency spectrum is a graph that shows the amplitudes and/or phases of the Fourier Series coefficients Ck. Amplitude spectrum |Ck| Phase spectrum k The lines |Ck| are called line spectra because we indicate the values by lines

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Schaums Outline Problems Schaums Outline Chapter 5 Problems: –4,5 6, 7, 8, 9, 10 Do all the problems in chapter 4 of the textbook Skip the following Sections in the text: –4.5 Read the following Sections in the textbook on your own –4.4

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