6 003 Signals and Systems Relations among Fourier Representations November 19 2009 Fourier Representations We ve seen a variety of Fourier representations CT Fourier series CT Fourier transform DT Fourier series One more today DT Fourier transform and relations among all four representations DT Fourier transform Representing aperiodic DT signals as sums of complex exponentials DT Fourier transform X ej X x n e j n analysis equation n Z 1 x n X ej ej n d 2 2 synthesis equation Comparison to DT Fourier Series From periodic to aperiodic DT Fourier Series ak ak N 1 N X x n e j 0 kn 0 n N X x n x n N ak ej 0 kn 2 N analysis equation synthesis equation k N DT Fourier transform X ej X x n e j n analysis equation n Z 1 x n X ej ej n d 2 2 synthesis equation Comparison to DT Fourier Series Sum over an infinite number of time samples instead of N DT Fourier Series ak ak N 1 N X x n e j 0 kn 0 n N X x n x n N ak ej 0 kn 2 N analysis equation synthesis equation k N DT Fourier transform X ej X x n e j n analysis equation n Z 1 x n X ej ej n d 2 2 synthesis equation Comparison to DT Fourier Series Sum over an infinite number of frequency components P R DT Fourier Series ak ak N 1 N X x n e j 0 kn 0 n N X x n x n N ak ej 0 kn 2 N analysis equation synthesis equation k N DT Fourier transform X e j X x n e j n analysis equation n Z 1 x n X ej ej n d 2 2 synthesis equation DT Fourier Series and Transform Example of series Let x n represent the following periodic DT signal x n x n 8 n 8 1 1 8 1 X 2 x n e j 0 kn 0 N N n N 2 1 cos k 1 1 1 2 4 1 e j 8 k e j 8 k 8 2 2 8 ak ak 1 4 8 1 1 k 8 DT Fourier Series and Transform Example of transform Let x n represent the aperiodic base of the previous signal x n n 1 1 H ej X x n e j n n 1 1 1 e j e j 2 2 1 cos X ej 2 2 2 DT Fourier Series and Transform Similarities DT Fourier series ak 1 4 8 1 1 k 8 DT Fourier transform X ej 2 2 2 Relations among Fourier Representations Different Fourier representations are related because they apply to signals that are related DTFS discrete time Fourier series DTFT discrete time Fourier transform CTFS continuous time Fourier series CTFT continuous time Fourier transform periodic DT DTFS interpolate N periodic extension sample periodic CT CTFS periodic DT aperiodic DT periodic CT aperiodic CT aperiodic DT DTFT interpolate T periodic extension sample aperiodic CT CTFT Relation between Fourier Series and Transform A periodic signal can be represented by a Fourier series or by an equivalent Fourier transform x t x t T X ak ej 0 kt k 0 2 T Because the Fourier transform of ej 0 kt is 2 k 0 X j X 2 ak k 0 k This expression shows the relation between the Fourier Series and Fourier transform for a periodic signal Relation between Fourier Series and Transform A periodic signal can be represented by a Fourier series or by an equivalent Fourier transform Fourier Series a a 2 1a0 a1 a2 a3 a4 a 4a 3 k 0 1 ak ej 0 kt k Fourier Transform 2 a 4 2 a 3 2 a 2 2 a 1 2 a0 2 a1 2 a2 2 a3 2 a4 x t x t T X 0 0 Relations among Fourier Representations Start with an aperiodic CT signal Determine its Fourier transform Convert the signal so that it can be represented by alternate Fourier representations and compare periodic DT DTFS interpolate N periodic extension sample periodic CT CTFS aperiodic DT DTFT interpolate T periodic extension sample aperiodic CT CTFT Start with the CT Fourier Transform Determine the Fourier transform of the following signal x t 1 1 0 1 t Could calculate Fourier transform from the definition Z X j x t ej t dt Easier to calculate x t by convolution of two square pulses y t y t 1 1 12 12 t 12 12 t Start with the CT Fourier Transform If the transform of y t is sin 2 y t 1 Y j 1 t 12 12 2 2 then the transform of x t y y t is X j Y j Y j x t 1 1 1 t X j 1 2 2 Relation between Fourier Transform and Series What is the effect of making a signal periodic in time Find Fourier transform of periodic extension of x t to period T 4 z t X x t 4k k 1 t 4 1 1 4 Could calculate Z j for the definition ugly Relation between Fourier Transform and Series Easier to calculate z t by convolving x t with an impulse train z t X x t 4k k 1 t 4 z t X 1 1 4 x t 4k x p t k where p t X t 4k k Then Z j X j P j We already know P j it s also an impulse train Relation between Fourier Transform and Series Convolving in time corresponds to multiply in frequency X j 1 2 2 P j 2 2 2 Z j 2 2 2 Relation between Fourier Transform and Series The Fourier transform of a periodically extended function is a discrete function of frequency z t X x t 4k k 1 t 4 1 1 4 Z j 2 2 2 Relation between Fourier Transform and Series The weight area of each impulse in the Fourier transform of a periodically extended function is 2 times the corresponding Fourier series coefficient Z j 2 2 2 ak 1 4 1 1 k Relation between Fourier Transform and Series The effect of periodic extension of x t to z t is to sample the frequency representation X j 1 2 2 Z j 2 2 2 ak 1 4 1 1 k Relation between Fourier Transform and Series Periodic extension of a CT signal produces a discrete function of frequency Periodic extension convolving with impulse train in time multiplying by impulse train in frequency sampling in frequency N periodic DT DTFS interpolate periodic extension sample periodic CT CTFS aperiodic DT DTFT interpolate T periodic extension sampling in frequency sample aperiodic CT CTFT Relations between CT and DT transforms Sampling a CT signal generates a DT signal x n x nT x t 1 1 0 1 t Take T 12 x n n 1 1 What is the effect on the frequency representation …
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