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# lect9 - THE SAMPLING THEOREM A function f(t is called σ...

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THE SAMPLING THEOREM A function, ) ( t f is called s - bandlimited if 0 ) ( = w F for s w . The spectrum below illustrates the situation. In this diagram both the magnitude ( triangle) and phase (smooth curve) of ) ( w F are shown for convenience. Note that the phase function ends abruptly at s w ± = because when the magnitude of a complex number is zero the phase is meaningless. The Nyquist Sampling Theorem , proved below, tells us that if ) ( t f is s - bandlimited, then it can be reconstructed from its samples, provided the sampling interval satisfies s p T . To illustrate this, consider the diagram below in which ) ( t f is s - bandlimited. ) ( w F w s s - ) ( t f t

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In this diagram, the dots represent samples of ) ( t f taken each T seconds, where T is the sampling interval. According to the sampling theorem proved below, if ) ( t f is bandlimited and if T is small enough, then knowledge of the samples alone (shown below) is enough to completely and exactly reconstruct the entire continuous time signal. Note that there are certainly an infinite number of continuous time signals that could be constructed to fit through these samples. However, according to the sampling theorem, if the sampling interval is T , then there is exactly one signal bandlimited to T p s rad/sec that fits these samples, namely the original signal that generated them. In order to prove this remarkable theorem, we first prove the Poisson Sum formula for the fourier transform. The Poisson Sum Formula: If ) ( ) ( w F t f , then given any real number, T , the following identity holds samples t f ) ( t
-∞ = -∞ = - + = n n nT j T n F e nT f T p w w 2 ) ( . (9.1) Before proving this result, some discussion is in order. Note first that the left hand side is simply an approximation of the fourier transform integral, - - = dt e t f F t j w w ) ( ) ( , by a Riemann sum with partition increment

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lect9 - THE SAMPLING THEOREM A function f(t is called σ...

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