Autumn 2005-2006 Analog and Digital Communications Autumn 2005-2006 Sep 13, 2005 Fourier Transforms Analog and Digital Communications Autumn 2005-2006 Sep 13, 2005 CS477: Analog and Digital Communications CS477: Analog and Digital Communications

Fourier Series: Example Autumn 2005-2006 Sep 13, 2005 Fourier Series: Example Find Fourier Series of ? Fourier series expansion is: Amplitude normalized by A/4 is sum of rotating phasors Sep 13, 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 13, 2005 CS477: Analog and Digital Communications

CS477: Analog and Digital Communications Parseval’s theorem Average power of a periodic signal can be obtained from its Fourier coefficients To prove, write the Fourier series expansion Sep 13, 2005 CS477: Analog and Digital Communications

CS477: Analog and Digital Communications Fourier Transform Amplitude normalized by A/4 What if the x-axis in the Fourier series represents frequency? The spectral lines appear at integer multiples of fundamental frequency! Separation between two consecutive spectral lines is equal to How does increasing fundamental period affect the separation between two consecutive spectral lines?  Fourier transform Sep 13, 2005 CS477: Analog and Digital Communications

CS477: Analog and Digital Communications Fourier Transform Fourier series in the limit Represents spectral components of a signal Signal uniquely represented in time or frequency domain Sep 13, 2005 CS477: Analog and Digital Communications

Shifting and Scaling Review Sep 13, 2005 CS477: Analog and Digital Communications

Fourier Transform: Example Sep 13, 2005 CS477: Analog and Digital Communications

CS477: Analog and Digital Communications Properties of FT Linearity Scaling Duality Time shifting Frequency shifting (modulation) Sep 13, 2005 CS477: Analog and Digital Communications

CS477: Analog and Digital Communications Properties of FT Modulation Product and convolution More on this under LTI systems Sep 13, 2005 CS477: Analog and Digital Communications

CS477: Analog and Digital Communications Properties of FT Differentiation Integration Sep 13, 2005 CS477: Analog and Digital Communications

The Dirac Delta Function A generalized function Sep 13, 2005 CS477: Analog and Digital Communications

CS477: Analog and Digital Communications FT of 1 and exponentials Fourier transform of RF Pulse: Sep 13, 2005 CS477: Analog and Digital Communications

CS477: Analog and Digital Communications Other Functions Signum Function: Unit step Function: Sep 13, 2005 CS477: Analog and Digital Communications

CS477: Analog and Digital Communications FT of Periodic Signals Sep 13, 2005 CS477: Analog and Digital Communications

CS477: Analog and Digital Communications FT of Periodic Signals Sep 13, 2005 CS477: Analog and Digital Communications