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**Fourier Series & Transforms**

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(2pft) S(t)=1/3[sin(2p(3f)t)]**

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

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**Square Wave K=1,3,5 K=1,3,5, 7 Frequency Components of Square Wave**

Fourier Expansion K=1,3,5, 7, 9, …..

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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: wo Example:

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Fourier Series We can represent all periodic signals as harmonic series of the form Ck 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

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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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Euler’s Relationship Review Euler formulas notes

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

C1=1/2j; C-1=-1/2j; No DC notes

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**Different Forms of Fourier Series**

Fourier Series Representation has three different forms Also: Complex Exp. Also: Harmonic 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 notes textbook**

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 Parseval’s 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 X(t) V To/2 To -V Use a Low Pass Filter to pick any tone you want!! 2|Ck| |4V/p| |4V/3p| |4V/5p| notes w0 3w0 5w0

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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 Signal @ wo Level Shifter [kwo] Sinusoidal waveform X(t) 1 To/2 @ [kwo] To X(t) 0.5 To/2 To -0.5 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 Only a function of freq.**

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

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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) X0V X(t) V To/2 To -V 2|Ck| |4V/p| |4V/3p| |4V/5p| w0 3w0 5w0

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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) X01 C0=T/To T/2=T1T=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. Xo To

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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. Xo To From the table: Xo/2 -Xo/2 To

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

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**Fourier Series Transformation**

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

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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 Xo To From the table: Xo 1 To -3

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Example

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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 fk The lines |Ck| are called line spectra because we indicate the values by lines

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**Schaum’s Outline Problems**

Schaum’s 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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Complex Form of Fourier Series For a real periodic function f(t) with period T, fundamental frequency where is the “complex amplitude spectrum”.

Complex Form of Fourier Series For a real periodic function f(t) with period T, fundamental frequency where is the “complex amplitude spectrum”.

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