CH 3 Intro DT Signals and Systems

CH 3 Intro DT Signals and Systems - EE 3120 Chapter 3...

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EE 3120 Chapter 3 Introduction to Discrete Time Signals and Systems
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Sampling We can use a D-T system to process a C-T signal by sampling the C-T signal and applying the resulting sequence to the D-T systems, then constructing the output C-T signal from the output sequence . () yt y [ n ] x ( t ) x [ n ] y [ n ] C/D D-T system D/C C-T signal is sampled to convert to a D-T signal Signal is processed by a D-T system. C-T signal constructed from the output sequence
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Why Sample? Digital signals and systems have many benefits over analog signals. Less sensitive to changes in component and signal variations caused by temperature, manufacture tolerance. Circuits can be easily duplicated, built in high volume for low cost and built on a single chip using VLSI circuits Hardware implementation of digital filters using microprocessors and LSI Digital filters can be programmed and changed easily, more variety of filtering Digital signals can be stored in memory, accessed remotely D sigs can be coded low bit errors and encrypted for privacy D sys can be time-shared allowing for multiple inputs simultaneously (Time division multiple access, TDMA) Reproduction of D sigs can be done with NO deterioration or noise added. (photocopying an image vs. scanning) CPU speeds allow for faster and more complex (real time) digital signal processing, DSP
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Sampling an Analog Signal Consider an analog f(t) signal defined for all t in (, ) −∞∞ sampling f(t) with sampling period or sampling rate T gives a discrete time (D-T) signal: . −∞ , where n is an integer ranging from f ( nT ) to Define: assumes T = 1 f [ n ]:= f ( nT ) is a sequence of numbers f [n] D-T signals are also called sequences.
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Example: is sampled every T = 0.1 seconds. The D-T sequence would be: () t xt e = 0.1 [ ] tn T x nT e e x n −− = == = t e = Plotted for t from -2 and 2 [] xn Plotted for n from -20 to 20 t = 0 nT = 0
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How often should we sample? The sampling rate should be selected so that the primary shape of the signal is maintained. T = .01 T = .02 T = .08 4 cycles 1 cycle
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Chapter 3 Frequencies of Analog and Digital Sinusoids () s in o f tt ω = 2 o P π = 0 T > [] : ( ) s i n o fk fkT kT == 0, 1, 2,. .. k = ±± Analog sinusoid is periodic with fundamental (smallest) period P where: (repeats itself every P seconds) If we sample f ( t ) with a sampling rate: This is a discrete-time sinusoid. for [ ] f k f kN =+ (, ) k →−∞∞ [ ] [ 2 ] [ 3 ] [ ] f kf kNf k n N = += + = + = + A D-T signal is periodic with period N , where N is a positive integer if:
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CH 3 Intro DT Signals and Systems - EE 3120 Chapter 3...

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