ELEC212 Chapter 4 Sampling of Continuous-Time Signals part 1

ELEC212 Chapter 4 Sampling of Continuous-Time Signals part...

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1 The Hong Kong University of Science and Technology Department of Electronic and Computer Engineering Prof. Matthew R. McKay ELEC 212 Chapter 4: Sampling of Continuous-Time Signals (Part I) 2 09-10 Fall ELEC 212 Sampling of CT Signals Typical DSP System 3 09-10 Fall ELEC 212 Sampling of CT Signals Periodic Sampling ± A typical method of obtaining a discrete-time representation of a continuous-time signal ± Discrete-time signal x [ n ] is obtained by extracting the information from a continuous-time signal x c ( t ) every T seconds: T is called sampling period (in seconds) and f s = 1/ T is called sampling frequency (in Hz OR samples per second) Commonly express the sampling frequency as (in radians per second) ± Notation: = frequency of continuous-time signal ω = frequency of discrete-time signal T s / 2 π = 4 09-10 Fall ELEC 212 Sampling of CT Signals Periodic Sampling
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5 09-10 Fall ELEC 212 Sampling of CT Signals Periodic Sampling 6 09-10 Fall ELEC 212 Sampling of CT Signals Periodic Sampling ± Is there any information lost in the sampling process? ± Can we retrieve the original analog signal from the sampled signal? ± The answer is YES if the following conditions are met : 1. x c ( t ) is bandlimited with bandwidth 2. Sampling frequency f s (in Hz ) or s is sufficiently high ± The answer is NO when either condition fail N N N j X > = | | for 0 ) ( i.e. , 7 09-10 Fall ELEC 212 Sampling of CT Signals Periodic Sampling ± High sampling frequency = more discrete-time sample data – High computation power for signal processing – More required memory (if needed) ± Desirable to find minimum sampling frequency which allows original analog signal x c ( t ) to be reconstructed perfectly from sampled signal x s ( t ) ? ± But what is this minimum sampling frequency? – We will answer this question using frequency domain representations 8 09-10 Fall ELEC 212 Sampling of CT Signals Frequency-Domain Representation of Sampling ± Model sampling operation as multiplication of continuous-time signal with periodic impulse train: ± Frequency-domain representation: −∞ = = = n c c s nT t t x t s t x t x ) ( ) ( ) ( ) ( ) ( δ −∞ = −∞ = −∞ = = = = = n s c n s c n s c c s n j X T n j X T n T j X j S j X j X )) ( ( 1 ) ( ) ( 1 ) ( 2 ) ( 2 1 ) ( ) ( 2 1 ) ( π Spectrum of sampled signal is periodic and scaled version of spectrum of original analog signal (with period s and scaling factor 1/ T).
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9 09-10 Fall ELEC 212 Sampling of CT Signals Frequency-Domain Representation of Sampling ± Graphical representation: Spectrum of sampled output signal: ± Case 1: Sampling rate s is large enough such that the replicas in the output spectrum do not overlap Spectrum of bandlimited analog signal Spectrum of impulse train Condition: N s N s N >
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ELEC212 Chapter 4 Sampling of Continuous-Time Signals part...

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