Signals, Fourier Series

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Presentation transcript:

Signals, Fourier Series Autumn 2005-2006 Sep 09, 2005 Signals, Fourier Series Analog and Digital Communications Autumn 2005-2006 Sep 09, 2005 CS477: Analog and Digital Communications CS477: Analog and Digital Communications

Review of Signals and Spectra Autumn 2005-2006 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

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

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

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

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

CS477: Analog and Digital Communications Autumn 2005-2006 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

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

Fourier Series: Example Autumn 2005-2006 Sep 09, 2005 Fourier Series: Example Amplitude normalized by A/4 Sep 09, 2005 CS477: Analog and Digital Communications CS477: Analog and Digital Communications

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