Lecture 2 Outline “Fun” with Fourier Announcements: Poll for discussion section and OHs: please respond First HW posted 5pm tonight Duality Relationships Connections between Continuous/Discrete Time Filtering and Convolution Periodic Signals Energy, Power, and Parseval

Duality in Fourier Transforms

t.5T -.5T  x(t): Aperiodic Continuous CTFT: X(j  ): Aperiodic Continuous 0N0N0 2N 0 -N 0 N -N n x[n]: Periodic Discrete DTFS: a k : Periodic Discrete a0a0 a1a1 a2a2 a3a3 a4a4 a -2 a -1 a -3 a -4 Samples of X(e j  ) every 2  /N 0 n N -N x[n]: Aperiodic Discrete DTFT: X(e j  ): Periodic Continuous CONNECTIONS BETWEEN THE CTFT, DTFT, CTFS, DTFS Using rect  sinc example Aliased version of X(j  ) from sampling x(t) every T s =T/(2N+1), then scale  axis by 2  /T s to get  axis 0T0T0 2T 0 -T 0.5T -.5T t x(t): Periodic Continuous a2a2 a1a1 a0a0 a3a3 0  a4a4 a -2 a -3 a -4 a5a5 a6a6 a -5 a -6 CTFS: a k Aperiodic Discrete a -1 Samples of X(j  ) every 2  /T 0

Filtering and Convolution Filtering Example: Important convolution examples 0N0N0 -N 0 N -N n x[n]: Periodic Discrete a0a0 a1a1 a2a2 a3a3 a4a4 a -2 a -1 a -3 a -4 X( j  ) H( j  ) Y( j  )=H( j  )X( j  ) X( e j  ) H(e j  ) Y(e j  )=H( e j  )X( e j  ) W -W  1 X( e j  ) 2W T0 A * = 2T 0 A2TA2T T0 A * … 0N-1 … 0 = … 0 … 2N-1 A2A2 2A 2 3A 2 NA 2 A2A2 2A 2 3A 2 A

Periodic Signals Continuous time: x(t) periodic iff there exists a T 0 >0 such that x(t)=x(t+T 0 ) for all t T 0 is called a period of x(t); smallest such T 0 is fundamental period If x(t) periodic with period T 1, y(t) periodic with period T 2, then x(t)+y(t) is periodic with period k 1 T 1 =k 2 T 2 if such integers k i exist. Discrete time: x[n] periodic iff there exists a N 0 >0 such that x[n]=x[n+N 0 ] for all n N 0 is called a period of x(t); smallest such N 0 is fundamental period If x[n] periodic with period N 1, y[n] periodic with period N 2, then x[n]+y[n] is periodic with period k 1 N 1 =k 2 N 2 if such integers k i exist. Filtering periodic signals H( j  ) H( e j  ) 0N0N0 -N 0 N -N n 0T0T0 2T 0 -T 0.5T -.5T t

Energy, Power, and Parseval’s Relation Continuous Time Aperiodic Signals Discrete Time Aperiodic Signals Energy signals have zero power; Power signals have infinite energy Periodic:

a0a0 a1a1 a2a2 a3a3 a4a4 a -2 a -1 a -3 a -4 Examples Continuous-time Discrete-time AT .5T -.5T A t x(t) X(j  ) Which would you rather integrate? 0N2N-N N1N1 -N 1 n x[n]: Periodic Discrete Which would you rather sum?

Main Points Duality saves work! Only need to compute half of the Fourier Series and Transform pairs, use duality to get the other half. Fourier transforms of continuous aperiodic signals yield their discrete- time counterparts by sampling and appropriate scaling of  axis. Fourier transforms of continuous or discrete periodic signals obtained by sampling their aperiodic Fourier transforms Filtering convolves input signal with impulse response of filter. This becomes multiplication of corresponding Fourier Transforms One domain typically easier to compute than the other Sum of 2 periodic signals has period equal to the LCM of the two periods. Filtering a periodic signal preserves Fourier Series components in filter BW Energy and power can be computed in time or frequency domain