Chapter 4Fourier Series & Transforms

Basic Ideanotes

Taylor SeriesComplex 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 (x) is the power series

http://upload.wikimedia.org/wikipedia/commons/6/62/Exp_series.gif

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

Square WaveFrequency Components of Square WaveK=1,3,5K=1,3,5, 7K=1,3,5, 7, 9, ..Fourier Expansion

Periodic SignalsA Periodic signal/function can be approximated by a sum (possibly infinite) sinusoidal signals. Consider a periodic signal with period TA periodic signal can be Real or ComplexThe fundamental frequency: woExample:

Fourier SeriesWe can represent all periodic signals as harmonic series of the formCk are the Fourier Series Coefficients; k is real k=0 gives the DC signal k=+/-1 indicates the fundamental frequency or the first harmonic w0 |k|>=2 harmonics

Fourier Series CoefficientsFourier Series Pair

We have

For k=0, we can obtain the DC value which is the average value of x(t) over one periodSeries of complex numbers Defined over a period of x(t)

Eulers RelationshipReview Euler formulas notes

ExamplesFind Fourier Series Coefficients for

Find Fourier Series Coefficients for

Find Fourier Series Coefficients for

Find Fourier Series Coefficients for

notesC1=1/2; C-1=1/2; No DCC1=1/2j; C-1=-1/2j; No DC

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

ExamplesFind Fourier Series Coefficients for

Find Fourier Series Coefficients for

Remember:

Examples

Find the Complex Exponential Fourier Series Coefficientsnotestextbook

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

ExampleConsider the following periodic square waveExpress x(t) as a piecewise functionFind 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

VTo/2-VTonotesX(t)2|Ck||4V/p||4V/5p||4V/3p|w03w05w0Use aLow Pass Filter to pick any tone you want!!

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

Practical ApplicationUsing a XTL oscillator which produces positive 1Vp-p how can you generate a sinusoidal waveforms with different frequencies? Square Signal@ woFilter @ [kwo]Level ShifterSinusoidal waveform1To/2ToX(t)0.5To/2ToX(t)-0.5@ [kwo]kwoB changes depending on k value

DemoCk corresponds to frequency componentsIn the signal.

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

Use the Fourier Series Table (Table 4.3)Consider the following periodic square waveFind the Exponential Fourier Series of representations of x(t)

X0V

VTo/2-VToX(t)2|Ck||4V/p||4V/5p||4V/3p|w03w05w0

Fourier Series - Applethttp://www.falstad.com/fourier/

Using Fourier Series TableGiven the following periodic square wave, find the Fourier Series representations and plot Ck as a function of k. (Rectangular wave) X01C0=T/ToT/2=T1T=2T1Ck=T/T0 sinc (Tkw0/2)

Same as beforeNote: sinc (infinity) 1 & Max value of sinc(x)1/x

Using Fourier Series TableExpress 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.

ToXo

Fourier Series TransformationExpress 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.

ToXoToXo/2-Xo/2From the table:

Fourier Series TransformationExpress 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.

ToXoToXo/2-Xo/2Only DC value changed!From the table:

Fourier Series TransformationExpress the Fourier Series for a sawtooth waveform?

Express the Fourier Series for this sawtooth waveform?ToXoToXoFrom the table:-31

Fourier Series TransformationExpress the Fourier Series for a sawtooth waveform?

Express the Fourier Series for this sawtooth waveform?We are using amplitude transferRemember Ax(t) + B Amplitude reversal A

Example

Example

Fourier Series and Frequency SpectraWe can plot the frequency spectrum or line spectrum of a signalIn 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 fk The lines |Ck| are called line spectra because we indicate the values by lines

Schaums Outline Problems Schaums Outline Chapter 5 Problems: 4,5 6, 7, 8, 9, 10Do all the problems in chapter 4 of the textbookSkip the following Sections in the text: 4.5 Read the following Sections in the textbook on your own4.4

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