113_1_Week09_print

113_1_Week09_print - 1 EE 113: Digital Signal Processing...

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1 EE 113: Digital Signal Processing Week 9 Sampling and Reconstruction Nyquist sampling theorem Understanding Aliasing
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2 1. Sampling & Reconstruction ± DSP must interact with an analog world: DSP Anti- alias filter Sample and hold A to D Reconstruction filter D to A Sensor Actuator WORLD x ( t ) x [ n ] y [ n ] y ( t ) ADC DAC
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3 Sampling ± Sampling: CT signal DT signal by recording values at ‘ sampling instants ’: ± What is the relationship of the spectra ? ± i.e. relate and g n [ ] = g a nT ( ) Discrete Continuous Sampling period T samp.freq. Ω T = 2 π / T rad/sec G a j Ω () = g a t e j Ω t dt −∞ G ( e j ω ) = gn [] e j n −∞ CTFT DTFT Ω in rad/second in rad/sample
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4 Sampling ± DT signals have same ‘content’ as CT signals gated by an impulse train: ± g p ( t ) = g a ( t p ( t ) is a CT signal with the same information as DT sequence g [ n ] × t t t p ( t ) g a ( t ) g p ( t ) ‘sampled’ signal: still continuous - but discrete values T3T = δ t nT ( ) n =−∞ CT delta fn
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5 Spectra of sampled signals ± Also, note that is periodic , thus has Fourier Series : ± Given CT ± Spectrum pt ( ) = δ t nT ( ) n () = 1 T
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113_1_Week09_print - 1 EE 113: Digital Signal Processing...

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