Review on Fourier

1
Review on Fourier
2

• Slides edited from
• Prof. Brian L. Evans and Mr. Dogu Arifler Dept.
  of Electrical and Computer Engineering The
  University of Texas at Austin course
• EE 313 Linear Systems and Signals Fall
  2003

3
Fourier Series
4
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
• Spectrogram representation
• Time on the horizontal axis
• Frequency on the vertical axis

5
Frequency Content Matters

• FM radio
• Single carrier at radio station frequency (e.g.
  94.7 MHz)
• Bandwidth of 165 kHz left audio channel, left
  right audio channels, pilot tone, and 1200 baud
  modem
• Station spacing of 200 kHz
• Modulator/Demodulator (Modem)

6
Demands for Broadband Access
Courtesy of Milos Milosevic (Schlumberger)
7
DSL Broadband Access Standards
Courtesy of Shawn McCaslin (Cicada Semiconductor,
Austin, TX)
8
Multicarrier Modulation

• Discrete Multitone (DMT) modulation
• ADSL (ANSI 1.413) and proposed for VDSL
• Orthogonal Freq. Division Multiplexing (OFDM)
• Digital audio/video broadcasting (ETSI
  DAB-T/DVB-T)

Courtesy of Güner Arslan (Cicada Semiconductor)
channel frequency response
magnitude
a carrier
a subchannel
frequency
Harmonically related carriers
9
Periodic Signals

• f(t) is periodic if, for some positive constant
  T0
• For all values of t, f(t) f(t T0)
• Smallest value of T0 is the period of f(t).
• sin(2pfot) sin(2pf0t 2p) sin(2pf0t 4p)
  period 2p.
• 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 f(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

10
Sinusoids

• f0(t) C0 cos(2 p f0 t q0)
• fn(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)

11
Fourier Series

• General representationof a periodic signal
• Fourier seriescoefficients
• Compact Fourierseries

12
Existence of the Fourier Series

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

13
Example 1

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

14
Example 2

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

15
Example 3

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

16
Fourier Analysis
17
Periodic Signals

• For all t, x(t T) x(t)
• x(t) is a period signal
• Periodic signals havea Fourier
  seriesrepresentation
• Cn computes the projection (components) of x(t)
  having a frequency that is a multiple of the
  fundamental frequency 1/T.

18
Fourier Integral

• Conditions for the Fourier transform of g(t) to
  exist (Dirichlet conditions)
• x(t) is single-valued with finite maxima and
  minima in any finite time interval
• x(t) is piecewise continuous i.e., it has a
  finite number of discontinuities in any finite
  time interval
• x(t) is absolutely integrable

19
Laplace Transform

• Generalized frequency variable s s j w
• Laplace transform consists of an algebraic
  expression and a region of convergence (ROC)
• For the substitution s j w or s j 2 p f to
  be valid, the ROC must contain the imaginary axis

20
Fourier Transform

• What system properties does it possess?
• Memoryless
• Causal
• Linear
• Time-invariant
• What does it tell you about a signal?
• Answer Measures frequency content
• What doesnt it tell you about a signal?
• Answer When those frequencies occurred in time

21
Useful Functions

• Unit gate function (a.k.a. unit pulse function)
• What does rect(x / a) look like?
• Unit triangle function

rect(x)
1
x
0
1/2
-1/2
D(x)
1
x
0
1/2
-1/2
22
Useful Functions

• Sinc function
• Even function
• Zero crossings at
• Amplitude decreases proportionally to 1/x

23
Fourier Transform Pairs
24
Fourier Transform Pairs
F(w) 2 p d(w)
(2p)
w
0
(2p) means that the area under the spike is (2p)
25
Fourier Transform Pairs
F(w)
f(t)
(p)
(p)
t
w
w0
-w0
0
0
26
Fourier Transform Pairs
sgn(t)
1
-1
27
Fourier Transform Properties
28
Fourier vs. Laplace Transform Pairs
Assuming that Rea gt 0
29
Duality

• Forward/inverse transforms are similar
• Example rect(t/t) ? t sinc(w t / 2)
• Apply duality t sinc(t t/2) ? 2 p
  rect(-w/t)
• rect() is even t sinc(t t /2) ? 2 p
  rect(w/t)

f(t)
1
t
0
t/2
-t/2
30
Scaling

• Same as Laplacetransform scaling property
• a gt 1 compress time axis, expand frequency
  axis
• a lt 1 expand time axis, compress frequency
  axis
• Effective extent in the time domain is inversely
  proportional to extent in the frequency domain
  (a.k.a bandwidth).
• f(t) is wider ? spectrum is narrower
• f(t) is narrower ? spectrum is wider

31
Time-shifting Property

• Shift in time
• Does not change magnitude of the Fourier
  transform
• Does shift the phase of the Fourier transform by
  -wt0 (so t0 is the slope of the linear phase)

32
Frequency-shifting Property
33
Modulation
34
Modulation

• Example y(t) f(t) cos(w0 t)
• f(t) is an ideal lowpass signal
• Assume w1 ltlt w0
• Demodulation is modulation followed by lowpass
  filtering
• Similar derivation for modulation with sin(w0 t)

Y(w)
1/2 F(ww0)
1/2 F(w-w0)
1/2
w
-w0 - w1
-w0 w1
w0 - w1
w0 w1
0
-w0
w0
35
Time Differentiation Property

• Conditions
• f(t) ? 0, when t ? ?
• f(t) is differentiable
• Derivation of propertyGiven f(t) ? F(w)

36
Time Integration Property
37
Summary

• Definition of Fourier Transform
• Two ways to find Fourier Transform
• Use definitions
• Use properties