Lesson_80 - EEL 3135: Signals and Systems Dr. Fred J....

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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Lesson Title: Reconstruction Lesson Number: 08 (section 4-4 to 4-6) Background: One of the most important scientific advancements in the first half of the 20 th century was Claude Shannon’s celebrated sampling theorem . Shannon formally established the rules that govern sampling as well as defining a reconstruction procedure. Most scholars simply remember the sampling rate condition that f s > 2f max , where f max is the highest frequency in the signal to be sampled. It is the reconstruction part that makes the theory useful in practice. While the author’s explores several approximate discrete-to-continuous time (D-C) conversion before studying the genius of Shannon, the study is reversed in this note. Sampling Theorem Revisited Suppose that the highest frequency contained in a continuous-time signal x ( t ) is f max = B Hz. Then, if x ( t ) is sampled periodically at a rate f s > 2 B , the original signal, x ( t ), can be exactly recovered (reconstructed) from the sample values x [ k ] using the interpolation rule -∞ = - = k s kT t h k x t x ) ( ] [ ) ( , 1 where h ( t ) is called the impulse response of Shannon’s interpolating filter. The process described by Equation 1 is called discrete to continuous time conversion and the mathematical means by which the discrete-time times series x[k] is converted into a continuous time signal x(t) is called i nterpolation . Shannon defined the the theoretically ideal interpolating filter h(t) to have a sin( x )/ x = sinc( x ) shape (Equation 4.22 in text). Formally, the interpolating filter’s impulse-response h(t) is mathematically given by = = s s s T t T t T t t h π sinc sin ) ( . 2 The terms denoted h ( t kT s ) found in Equation 1 are delayed or advanced versions of h ( t ) , where the delay or advancement is kT s seconds. It is important to remember that the Sampling Theorem requires that the sampling frequency be strictly greater than the Nyquist sample rate, that is f s >2 B . If f s >> 2 B the system is said to be oversampled . If f s < 2 B, the system is said to be undersampled . If f s is set to, or near the Nyquist sample rate, the system is said to be critically sampled . The frequency located at f N = f s /2 is likewise important and is called the Nyquist frequency or, as it is sometimes referred to, the folding frequency . 1
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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor The sampling process is motivated in Figure 1 where ideally the reconstruction filter has a sinc(x) shape (Equation 2). Figure 1: Example of sampling process ideal sampler and reconstruction filter. Equation 1 will later be interpreted as a discrete-time
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This note was uploaded on 08/21/2010 for the course EEL 3135 taught by Professor ? during the Spring '08 term at University of Florida.

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Lesson_80 - EEL 3135: Signals and Systems Dr. Fred J....

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