Fourier Series & Transforms Chapter 4 Fourier Series & Transforms

Basic Idea notes

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

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

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

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:

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

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)

Euler’s Relationship Review  Euler formulas notes

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

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

Examples Find Fourier Series Coefficients for Remember:

Examples notes textbook Find the Complex Exponential Fourier Series Coefficients notes textbook

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

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

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

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 Filter @ [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

Demo Ck corresponds to frequency components In the signal.

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.

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

Fourier Series - Applet http://www.falstad.com/fourier/

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

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

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

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!

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

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

Example

Example

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

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