# Lec_04 - Lecture 4: Sampling and Reconstruction Sampling...

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1 Lecture 4: Sampling and Reconstruction Sampling Data Reconstruction (Hold) Reading: Chapter 3 of the textbook A Typical Digital Control System In the above digital control system, sampling and data reconstruction of signals are needed to interface the digital computer with the physical world Continuous-time Signals Discrete-time Signals Sampling Data reconstruction discrete-time signal continuous-time signal digital signal Plant Computer A/D D/A y ( t ) Sensor T r ( t ) Data Hold continuous-time signal discrete-time signal continuous-time system discrete-time system Sampling Reconstruction + ¡

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2 A sampler obtains from a continuous-time signal a discrete- time signal by sampling every T seconds Input is a continuous-time signal Output is a discrete-time signal e ( kT ), k =0,1,… Both signals have identical values at sampling moments Not a lossless procedure in general Not a time-invariant procedure e ( t ) t T e ( kT ) e ( t ), t ¸ 0 0 2 T 3 T 4 T T e ( t ) e ( kT ) Sampler A/D Converter (ADC) (From Wikipedia: 4-channel stereo multiplexed analog-to-digital converter WM8775SEDS made by Wolfson Microelectronics placed on a X-Fi Fatal1ty) Pro sound card .
3 Reconstruction: Zero-Order Hold Data reconstruction devices are needed to convert the discrete-time signal output by the digital controller into a continuous-time signal suitable for driving the physical plant Ideally both signals should agree at the sampling moment: 0, T , 2 T , … The simplest data reconstruction is zero-order hold Given a discrete time signal e ( kT ), k =0,1,… A continuous-time signal is produced The value of at any time t is equal to the value of e ( kT ) in the closest sampling moment preceding t : t e ( kT ) 0 T 2 T 3 T 4 T ¢¢¢ ¹ e ( t ), t ¸ 0 ¹ e ( t ) ¹ e ( t ) ¹ e ( t ) = e (0)[ u ( t ) ¡ u ( t ¡ T )] + e ( T )[ u ( t ¡ T ) ¡ u ( t ¡ 2 T )] + ¢¢¢ D/A Converter (DAC) (from www.ikalogic.com) An 8-Digit DAC implemented by R/2R resistor network AK4396 DAC for Headphone Output From Wikipedia: 8-channel digital-to-analog converter Cirrus Logic CS4382 as used in a soundcard.

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4 Sampled-Data Control System Sampler Plant Digital controller Data hold e ( t ) y ( t ) e ( kT ) u ( kT ) u ( t ) Digital controller is designed together with sampler and data hold Does there exist a transfer function from e ( t ) to u ( t )? Input-output relation from
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## This note was uploaded on 04/15/2011 for the course ECE 483 taught by Professor Evens during the Spring '08 term at Purdue University-West Lafayette.

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Lec_04 - Lecture 4: Sampling and Reconstruction Sampling...

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