[PPT].Lesson 03-Sampling Theorem

# [PPT].Lesson 03-Sampling Theorem - Challenge 02 02 Sampling...

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Sampling Theorem Challenge 02 Lesson 03: Review of the Sampling Theorem dditional Sample Theorem related topics Additional Sample Theorem related topics Challenge 03 What’s it all about! Shannon’s (Nyquist) Sampling Theorem. Signal reconstruction (interpolation). Practical interpolation. Sampling modalities (critical, over, under). pg ( , , ) Comment: What is this thing called compressed sampling? Lesson 03 hallenge 02 Challenge 02 1 st -Order Impulse Invariant System A simple first-order RC circuit is show. The relationship between the input forcing function v ( t ) and voltage developed across the capacitor, denoted v o ( t ), is defined by the 1 st order rdinary differential equation ordinary differential equation 1 = 1 ) ( 0 t t t dv What is the equivalent discrete-time model of the circuit’s impulse ) ( ) ( 0 RC RC dt response based on a sample rate of f s =1 k Sa/s, and RC=10 -2 ? Lesson 03 hallenge 02 Challenge 02 It immediately follows that the system’s impulse response is given by: f th li i d i th lti di t i l ) ( 1 = ) ( / - t u e RC t h RC t If the sampling period is T s , the resulting discrete-time impulse response is: or, for k 0, ) ( ] [ s s d kT h T k h RC kT T T - C k s C s d RC e RC k h s / = ] [ RC T e / ; α Lesson 03 hallenge 02 Challenge 02 For RC =10 -2 , and f s =1000 Hz, the following results. 0 0 ; 904837 . 0 2 3 1 . 0 10 10 * 10 1 3 2 C e e e 1 . 0 10 / 10 / RC T s  k s s s d e RC T kT h T k h ) ( / . ) ( ] [ 1 0 k ) . ( . 904837 0 1 0 Lesson 03 hallenge 02 Challenge 02 For RC =10 -2 , and f s =1000 Hz, the following difference equation results. y[k] - 0.904837y[k-1] = 0.1x[k] or [k] = 0 904837y[k- ] + 0 1x[k] y[k] = 0.904837y[k 1] + 0.1x[k] [k] If x[k]= [k], then y[k]: x[k] T y[k] y[0] = 0.1 y[1] = (0.1)(0.9) [2]= (0 1)(0 9)(0 9)= 0.1 0.904837 y[2]= (0.1)(0.9)(0.9)= =(0.1)(0.9) 2 Lesson 03 >> den=[1 - .9094837]; num=[0.1]; den [1 0.9094837]; num [0.1]; >> x=[1 0 0 0 0 0 0 0 0 0 0 0 0 ] ; % impulse >> h=filter(num,den,x); % impulse response (,, ) ; p p >> plot(h) [k] h[k] Lesson 03 SP gift from Claude Shannon DSP – a gift from Claude Shannon hat were his accomplishments? (MS (MIT) Sampling Theorem, What were his accomplishments? Who was his patron? Ph.D. (MIT) Information Theory The telephone company (Bell Labs) – does this explain the interest in sampling and information theory? Lesson 03 DSP System Architecture The Sampling Theorem is core to understanding DSP. The theorem both enables and constrains the performance of a DSP system consisting of an DC DAC di it l DSP l l i l diti i filt ADC, DAC, digital or DSP processor, plus analog signal conditioning filters ( i.e ., anti-aliasing and reconstruction filter). Typical signal processing stream. Lesson 03 Shannon Factoid: Some attribute the sampling theorem to Claude Shannon , and others to Harry Nyquist . Nyquist suggested the sampling theorem in 1928, which was th ti ll b Sh i 1949 (MS Th i ) mathematically proven by Shannon in 1949 (MS Thesis). ome use the term "Nyquist Sampling Theorem" and others use Some use the term Nyquist Sampling Theorem , and others use "Shannon Sampling Theorem" to refer to the underlying samplilng theory.

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## This note was uploaded on 09/18/2011 for the course EEL 5718 taught by Professor Janisemcnair during the Fall '11 term at University of Florida.

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[PPT].Lesson 03-Sampling Theorem - Challenge 02 02 Sampling...

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