lecture4-f08

lecture4-f08 - EE264 Digital Signal Processing Lecture 4...

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EE264 Digital Signal Processing Lecture 4 Sampling, Reconstruction, and Filtering October 1, 2008 Ronald W. Schafer Department of Electrical Engineering Stanford University Stanford University, EE264 Administrative • HW 1 due Wednesday 1 October • HW 2 and subsequent HWs released on Mondays (or Tuesdays). • HW 2 and subsequent HWs due on Tuesdays by 5pm in EE264 drawer on 2 nd floor Packard. • Review Sessions: Thursdays 4:15 - 5pm?? • Office Hours: – RWS: Mon./Weds 10-11, and 12:15-12:45 – Raunaq: Monday evenings, Tuesdays 2-4?? – Rahim: Friday?? Stanford University, EE264 Overview of Lecture • Review of Ideal Continuous-to-Discrete conversion (sampling) – Oversampling – Aliasing distortion • Review of reconstruction from samples – Polynomial reconstruction filters • Discrete-time processing of continuous-time signals Stanford University, EE264 DSP Theory and Practice A-to-D conversion --> C-to-D conversion Finite precision arithmetic --> real numbers D-to-A conversion --> D-to-C conversion DSP Chip D-toA A-to-D x c ( t ) x [ n ] y [ n ] y c ( t ) Algorithm D-to-C C-to-D x c ( t ) x [ n ] y [ n ] y c ( t )
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Stanford University, EE264 Sampling (C-to-D Conversion) • Discrete-time Fourier transform: • Frequency-domain relation: • Sampling frequency: • Normalized frequency: C-to-D Converter x c ( t ) x [ n ] = x c ( nT ) X c ( j Ω ) X ( e j ω ), X ( e j Ω T ) X ( e j Ω T ) = x [ n ] n =−∞ e j Ω Tn = 1 T X c ( j ( Ω− k Ω s )) k T X ( e j ) = x [ n ] n e j n Ω s = 2 π / T Stanford University, EE264 Derivation of Basic FT Formula - I s ( t ) = δ ( t nT ) n x s ( t ) = x c ( t ) ( t nT ) n = x c ( t ) ( t nT ) n x s ( t ) = x c ( nT ) ( t nT ) n = x [ n ] ( t nT ) n C-to-D Converter Stanford University, EE264 Illustration of C-to-D Conversion ) ( ) ( ) ( t s t x t x c s = ) ( ] [ nT x n x c = Stanford University, EE264 Derivation of Basic FT Formula - II s ( t ) = ( t nT ) n S ( j Ω ) = 2 T k ( k 2 T ) x s ( t ) = x c ( t ) s ( t ) = x c ( t ) ( t nT ) n X s ( j Ω ) = 1 2 X c ( j Ω ) S ( j Ω ) X s ( j Ω ) = 1 T X c j k 2 T k X s ( j Ω ) = 1 2 X c ( j Ω ) 2 T k ( k 2 T )
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Stanford University, EE264 Derivation of Basic FT Formula - III x s ( t ) = x c ( t ) δ ( t nT ) n =−∞ = x [ n ] ( t nT ) n X s ( j Ω ) = x [ n ] e j Ω nT n X s ( j Ω ) = x [ n ] e j ( Ω T ) n n = X ( e j Ω T ) X ( e j Ω T ) = 1 T X c j Ω− k 2 π T k Stanford University, EE264 Oversampling X ( e j Ω T ) = 1 T X c ( j ( k Ω s )) k Ω s = 2 / T 0 Ω Ω N −Ω N X c ( j Ω ) = 0, Ω ≥Ω N “Typical” bandlimited signal 0 Ω Ω s s 2 Ω s 2 Ω s X ( e j Ω T ) Fourier transform of samples Ω s /2 A / T X c ( j Ω )/ T X c ( j ( Ω−Ω s ))/ T Stanford University, EE264 Undersampling (Aliasing Distortion) •I f Ω s < 2 N , the copies of X c ( j ) overlap, and we have aliasing distortion .
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lecture4-f08 - EE264 Digital Signal Processing Lecture 4...

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