The Fourier Series

1
The Fourier Series
ECE 2221/MCT 2210 Signals and Systems
(Analysis) Sem. I (03/04)

• Br. Shahrul Naim Sidek
• Department of Mechatronics Engineering
• International Islamic University Malaysia

Lecture Chapter 3
http//eng.iiu.edu.my/snaim
2
Introduction

• Previously, to find system response, we need to
  find impulse response, h(t), before performing
  the actual convolution integral. (which always
  not a simple task).
• Advantage of Fourier Series
• Some systems are easier to be analyzed in
  frequency domain representation, so we can see
  their bandwidth, spectral line and so on

3
Spectrogram Demo (DSP First)

• Sound clips
• Sinusoid with frequency of 660 Hz (no harmonics)
• Square wave with fundamental frequency of 660 Hz
• Sawtooth wave with fundamental frequency of 660
  Hz
• Beat frequencies at 660 Hz /- 12 Hz

4
Periodic Signals

• x(t) is periodic if, for some positive constant
  T0
• -For all values of t, x(t) x(t T0)
• -Smallest value of T0 is the period of x(t).
• -sin(2pfot) sin(2pf0t 2p) sin(2pf0t 4p)
  period T0
• A periodic signal f(t)
• -Unchanged when time-shifted by one period
• -Two-sided extent is t ? (-?, ?)
• -May be generated by periodically extending one
  period
• -Area under x(t) over any interval of duration
  equal to the period is the same e.g.,
  integrating from 0 to T0 would give the same
  value as integrating from T0/2 to T0 /2

5
Sinusoids

• x0(t) C0 cos(2 p f0 t q0)
• xn(t) Cn cos(2 p n f0 t qn)
• The frequency, n f0, is the nth harmonic of f0
• Fundamental frequency in Hertz is f0
• Fundamental frequency in rad/s is w 2 p f0
• Cn cos(n w0 t qn) Cn cos(qn) cos(n w0 t) -
  Cn sin(qn) sin(n w0 t) an cos(n w0 t) bn
  sin(n w0 t)

6
Trigonometric Fourier Series

• Trigonometric F.S. representationof a periodic
  signal
• Fourier seriescoefficients

7
Existence of the Fourier Series

• Existence
• Convergence for all t
• Finite number of maxima and minima in one period
  of x(t)

8
Example 1

• Fundamental period
• T0 p
• Fundamental frequency
• f0 1/T0 1/p Hz
• w0 2p/T0 2 rad/s

9
Example 2

• Fundamental period
• T0 2
• Fundamental frequency
• f0 1/T0 1/2 Hz
• w0 2p/T0 p rad/s

10
Complex Exponential Fourier Series

• Complex exponential F.S. representationof a
  periodic signals
• Fourier seriescoefficients

11
Example 3

• Fundamental period
• T0 2p
• Fundamental frequency
• f0 1/T0 1/2p Hz
• w0 2p/T0 1 rad/s

12
Useful Function

• Sinc function

13
Symmetry Properties of F.S. Coefficients
14
Parsevals Theorem

• To find average power of periodic signals

15
Line Spectra

• Ex. from previous example

16
Steady State Response of LTI System to a Periodic
Input Distortionless System
17
Distortionless System

• Definition The LTI system response got no
  effects to input except possibly amplitude
  scaling and delay.
• If the amplitude scaling is not constant
  amplitude distortion.
• If the delay is not constant phase distortion
• If both factors are not constant system is non
  linear