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pcshw1_soln - THE STATE UNIVERSITY OF NEW JERSEY RUTGERS...

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THE STATE UNIVERSITY OF NEW JERSEY RUTGERS College of Engineering Department of Electrical and Computer Engineering 332:322 Principles of Communications Systems Spring 2003 Problem Set 1 1. Derive the convolution intergal from first principles (as outlined in class) given by y t x τ h t τ d τ , where h t is the impulse response of a LTI system, x t the input and y t the output. Convolution Integral: In time domain a linear system is described in terms of its impulse response which is defined as the response of the system (with zero initial conditions) to a unit impulse or delta funtion δ t applied to the input of a system. If the system is time invariant,then the shape of the impulse response is same no matter when the impulse is applied to the system. Let h(t) denote the impluse response of a LTI system. Let this system be subjected to an arbitrary excitation x(t). To determine the output y(t) we first approximate the input x(t) by staircase function composed of narrow rectangular pulses,each of duration ∆τ . The ap- proximation becomes better for smaller ∆τ . As ∆τ approaches zero, each pulse in the limit approaches a delta funtion weighed by a factor equal to the height of the pulse times ∆τ . Consider a pulse which occurs at t n ∆τ . By defenition,the response of the system to a unit impulse or a delta funtion δ t occuring at t 0 is h(t). Due to the time invariance property the response of the system to a delta funtion weighed by a factor x n ∆τ ∆τ occuring at t n ∆τ ,must be y t lim ∆τ 0 x n ∆τ h t n ∆τ ∆τ Above is the response to one component of input occuring at t n ∆τ . Since the system is linear we can apply superposition principle to find the response to the sum of the input
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