1. Signals, Fourier Series
Autumn
Sep 09, 2005
Signals, Fourier Series
Analog and Digital Communications
Autumn
Sep 09, 2005
CS477: Analog and Digital Communications
CS477: Analog and Digital Communications

2. Review of Signals and Spectra
Autumn
Sep 09, 2005
Review of Signals and Spectra
A generic sinusoidal signal
Phasor representation
Frequency domain representation
Rotating phasors
Frequency plots
Amplitude
Phase
Amplitude
Phase
Sep 09, 2005
CS477: Analog and Digital Communications
CS477: Analog and Digital Communications

3. Review of Signals and Spectra
Two sided spectra can be seen from
This represents two rotating phasors
Amplitude and phase spectrum (two sided)
Amplitude
Phase
Sep 09, 2005
CS477: Analog and Digital Communications

4. CS477: Analog and Digital Communications
Periodic Signals
A signal is periodic if there exists such that
Smallest such is called fundamental period
Any integer multiple of is also a period
Sep 09, 2005
CS477: Analog and Digital Communications

5. Average signal and Power
Average power
For periodic signals, limit goes away
Sep 09, 2005
CS477: Analog and Digital Communications

6. Fourier Series Representation
Projection of periodic signals onto basis functions
Periodic signal is a weighted sum of these basis functions
Exponentials are used as basis functions for writing Fourier series
Any periodic signal can be expressed as a sum of infinite number of exponentials (or sinusoids for real signals)
Sep 09, 2005
CS477: Analog and Digital Communications

7. CS477: Analog and Digital Communications
Autumn
Sep 09, 2005
Fourier Series
(Each term represents one frequency)
(fundamental frequency)
Line spectra at frequencies that are integer
multiple of fundamental frequency
Sep 09, 2005
CS477: Analog and Digital Communications
CS477: Analog and Digital Communications

8. CS477: Analog and Digital Communications
Sinc Function
Note the points where this
function becomes zero
Where does this appear?
Sep 09, 2005
CS477: Analog and Digital Communications

9. Fourier Series: Example
Autumn
Sep 09, 2005
Fourier Series: Example
Amplitude normalized by A/4
Sep 09, 2005
CS477: Analog and Digital Communications
CS477: Analog and Digital Communications

10. Sinusoidal Fourier Series
For real signals
and can be expressed in terms of
Sep 09, 2005
CS477: Analog and Digital Communications