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ps5-VectorSpace

# ps5-VectorSpace - 6.450 Principles of Digital Communication...

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6.450 Principles of Digital Communication Wednesday, October 2, 2002 MIT, Fall 2002 Handout #19 Due: Wednesday, October 9, 2002 Problem Set 5 Problem 5.1 One often approximates the value of an integral by a discrete sum; i.e. , Z -∞ g ( t ) dt δ X k g ( ) . (a) Show that if u ( t ) is a real finite-energy function, low-pass limited to W Hz, then the above approximation is exact for g ( t ) = u 2 ( t ) if δ 1 / (2 W ); i.e. , show that Z -∞ u 2 ( t ) dt = δ X k u 2 ( ) . (b) Show that if g ( t ) is a real finite-energy function, low-pass limited to W Hz, then for δ 1 / (2 W ), Z -∞ g ( t ) dt = δ X k g ( ) . (c) Show that if δ > 1 / 2 W , then there exists no such relation in general. Problem 5.2 Assume that u ( t ) is a finite-energy complex-valued function. Let { θ k ( t ); 1 < k < ∞} be a set of orthogonal waveforms and assume that u ( t ) has the orthogonal expansion u ( t ) = X k =1 u k θ k ( t ) Assume the set of orthogonal waveforms satisfy Z -∞ θ k ( t ) θ * j ( t ) dt = 0 for k 6 = j A j for k = j where { A j } is an arbitrary set of non-negative numbers.

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