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SPFirst-L26 - Signal Processing First LECTURE OBJECTIVES...

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8/22/2003 © 2003, JH McClellan & RW Schafer 1 Signal Processing First Lecture 26 Review: Digital Filtering of Analog Signals 8/22/2003 © 2003, JH McClellan & RW Schafer 3 LECTURE OBJECTIVES Sampling Theorem Revisited GENERAL: in the FREQUENCY DOMAIN Fourier transform of sampled signal Reconstruction from samples Effective Frequency Response Important FT properties Convolution ÅÆ multiplication Frequency shifting 8/22/2003 © 2003, JH McClellan & RW Schafer 4 Sampling: Freq. Domain EXPECT FREQUENCY SHIFTING !!! −∞ = −∞ = = = k t jk k n s s e a nT t t p ω δ ) ( ) ( −∞ = = k t jk k s e a t p ω ) ( 8/22/2003 © 2003, JH McClellan & RW Schafer 5 Frequency-Domain Representation of Sampling X s ( j ω ) = 1 T s X ( j ( ω k =−∞ k ω s )) “Typical” bandlimited signal
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8/22/2003 © 2003, JH McClellan & RW Schafer 6 Aliasing Distortion If ω s < 2 ω b , the copies of X ( j ω ) overlap, and we have aliasing distortion . “Typical” bandlimited signal 8/22/2003 © 2003, JH McClellan & RW Schafer 7 Reconstruction of x ( t ) x s ( t ) = x ( nT s ) δ ( t nT s ) n =−∞ X s ( j ω ) = 1 T s X ( j ( ω k =−∞ k ω s )) X r ( j ω ) = H r ( j ω ) X s ( j ω ) 8/22/2003 © 2003, JH McClellan & RW Schafer 8 Reconstruction: Frequency-Domain ) ( ) ( ) ( so overlap, not do ) ( of copies the , 2 If ω ω ω ω ω ω j X j H j X j X s r r b s = > H r ( j ω ) 8/22/2003 © 2003, JH McClellan & RW Schafer 9 Ideal Reconstruction Filter h r ( t ) = sin π T s t π T s t H r ( j ω ) = T s ω < π T s 0 ω > π T s
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