EE3TP4_21_SamplingofCTSignals_v4_Lecture 23

# EE3TP4_21_SamplingofCTSignals_v4_Lecture 23 - The Sampling...

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The Sampling Theorem The Connection Between Continuous Time and Discrete Time This is also called the Nyquist–Shannon Sampling Theorem or the Nyquist Sampling Theorem (implied by Harry Nyquist in 1928 and proven by Claude Shannon in 1949.) These overheads were originally developed by Mark Fowler at Binghamton University, State University of New York.

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• Sampled & Digitized music on a Compact Disc What ensures that we can “perfectly” reconstruct the music signal from its samples? 0 0 .2 0 .4 0 .6 0 .8 1 -2 -1 0 1 2 T ime Signal Value 1 2 Amp Code “Burn” bits into CD Record Creates a sequence of samples (i.e., a sequence of numbers) Play Laser Sensor Decode Reconstruct Amp Microphone Speaker = Original? Time Sample & Digitize Sampling is Key Part of CD Scheme
• Systems that use Digital Signal Proc. (DSP) generally – get a continuous-time signal from a sensor – a cont.-time system modifies the signal – an “analog-to-digital converter” (ADC) samples the signal to create a discrete-time signal – A discrete-time system to do the Digital Signal Processing – and then (if desired) convert back to analog using a “digital- to-analog converter (DAC) Analog Electronics ADC DSP Computer Sampling is Key Part of Many Systems C-T Signal C-T System C-T Signal D-T Signal D-T System D-T Signal DAC C-T Signal

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ADC If Sampling is “Valid”… We Should be Able to Perfectly” Reconstruct from Samples C-T Signal D-T Signal DAC C-T Signal x ( t ) x [ n ] Clock x t x t = x t ? Can we make If we can then we can process the samples x [ n ] as an alternative to processing x ( t )!
0 0 .2 0 .4 0 .6 0 .8 1 -2 -1 0 1 2 T ime Signal Value 0 0 0 0 0 1 0 1 2 T Practical Sampling-Reconstruction Set-Up Pulse Gen CT LPF Digital-to-Analog Converter (DAC) x t x ( t ) x [ n ] = x ( nT ) “Hold” Sample at t = nT Analog-to-Digital Converter (ADC) T = Sampling Interval F s = 1/ T = Sampling Rate x t Clock at t = nT

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x [ n ]= x t ∣ t = nT = x nT Note: the book uses an “impulse sampling” model for the ADC… but that has no connection to a physical ADC… we’ll see later that it does have a physical connection to the physical DAC ! Math Model for Sampling (ADC) Math Modeling the ADC is easy …. x [ n ] = x ( nT ) , so the n th sample is the value of x ( t ) at t = nT
Model for Reconstruction (DAC) Pulse Gen h ( t ) H (ω) x t x [ n ] x t The model for the DAC consists of two parts: converting a DT sequence (of numbers) into a CT pulse train

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## This note was uploaded on 02/01/2011 for the course ECE 3TP4 taught by Professor Drterrencetodd during the Fall '10 term at McMaster University.

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EE3TP4_21_SamplingofCTSignals_v4_Lecture 23 - The Sampling...

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