Fourier Series

1
Fourier Series
2
Organization of Class Material

• Recall lecture slides 1-15, 8-2 and 17-3

Complex Frequency Domain
Frequency Domain
Time Domain
Continuous Discrete Time Time
(t) k
Continuous Discrete Time Time
(s) z
Continuous Discrete Time Time
(w) W
Lectures 1-10
Lectures 11-18
Lectures 19-25
Discrete-Time Fourier Transform Chapter 9 not
covered
Fourier Series Chapter 6 and Fourier
Transform Chapter 7
Final exam will be comprehensive
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
• Spectrogram representation
• Time on the horizontal axis
• Frequency on the vertical axis

4
Frequency Content Matters
One-way communication

• FM radio broadcast
• 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 (in the US)
• Modulator/demodulator (modem)

Two-way communication
5
Demands for Broadband Access
Courtesy of Dr. Milos Milosevic (Schlumberger)
6
DSL Broadband Access Standards
Courtesy of Mr. Shawn McCaslin (Austin, TX)
7
Multicarrier Modulation

• Transmit parallel data streams at same time
• Each data stream in separate frequency band
  (subchannel)
• Digital subscriber line (DSL) for home and small
  office
• Terrestrial digital audio/video broadcasting
• IEEE 802.11a/g wireless LAN

Courtesy of Dr. Güner Arslan (ST-Ericsson,
Austin, TX)
channel frequency response
a carrier
magnitude
a subchannel
frequency
Harmonically related carrier frequencies as Dirac
deltas
8
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

9
Sinusoids

• Fundamental f0(t) C0 cos(2 p f0 t q0)
• Fundamental frequency in Hertz is f0
• Fundamental frequency in rad/s is w 2 p f0
• Harmonic fn(t) Cn cos(2 p n f0 t qn)
• Frequency, n f0, is nth harmonic of f0
• Magnitude/phase and Cartesian representations
• 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)

10
Fourier Series

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

11
Existence of the Fourier Series

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

12
Example 1

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

13
Example 2

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

14
Example 3

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