Lesson 03-Sampling Theorem

Lesson 03-Sampling Theorem - Sampling Theorem Challenge 02...

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Lesson 03 Sampling Theorem Challenge 02 Lesson 03: Review of the Sampling Theorem 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). Comment: What is this thing called compressed sampling?
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Lesson 03 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 ordinary differential equation What is the equivalent discrete-time model of the circuit’s impulse response based on a sample rate of f s =1 k Sa/s, and RC=10 -2 ? ) ( 1 = ) ( 1 ) ( 0 0 t v RC t v RC dt t dv +
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Lesson 03 Challenge 02 It immediately follows that the system’s impulse response is given by: If the sampling period is  T s , the resulting discrete-time impulse response is: or, for  k 0, ) ( 1 = ) ( / - t u e RC t h RC t ) ( ] [ s s d kT h T k h k s RC kT s d RC T e RC T k h s α = / - = ] [ RC T s e / ; - = α
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Lesson 03 Challenge 02 For RC =10 -2 , and f s =1000 Hz, the following results. 1 . 0 10 / 10 / ; 904837 . 0 2 3 1 . 0 10 10 * 10 1 3 2 = = = = = = - - - - - - - RC T e e e s α ( 29 k k s s s d e RC T kT h T k h ) . ( . ) ( / . 904837 0 1 0 ) ( ] [ 1 0 = = = -
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Lesson 03 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 y[k] = 0.904837y[k-1] + 0.1x[k] x[k] T 0.904837 y[k] If x[k]= δ [k], then y[k]: y[0] = 0.1 y[1] = (0.1)(0.9) y[2]= (0.1)(0.9)(0.9)= =(0.1)(0.9) 2 0.1
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Lesson 03 >> 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 >> plot(h) h[k]
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Lesson 03 DSP – a gift from Claude Shannon What were his accomplishments? Who was his patron? (MS (MIT) Sampling Theorem, Ph.D. (MIT) Information Theory The telephone company (Bell Labs) – does this explain the interest in sampling and information theory?
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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 ADC, DAC, digital or DSP processor, plus analog signal conditioning filters ( i.e ., anti-aliasing and reconstruction filter). Typical signal processing stream.
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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 mathematically proven by Shannon in 1949 (MS Thesis). Some use the term "Nyquist Sampling Theorem", and others use "Shannon Sampling Theorem" to refer to the underlying samplilng theory.
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Lesson 03 Shannon Sampling Theorem (just like you heard time and time again) If x(t) is baseband limited from above by some frequency f max , If x(t) is periodically sampled at some rate f s , where f s > 2·f max ; T s =1/ f s Then x(t) can be reconstructed (interpolated) from the sample values x[kT s ].
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