Digital Signal Processing By Assoc Prof Dr Erhan

Digital Signal Processing By: Assoc. Prof. Dr. Erhan A. İnce Electrical and Electronic Engineering Dept. SPRING 2016 e-mail: erhan. ince@emu. edu. tr http: //faraday. ee. emu. edu. tr/eeng 420 1

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Difference Equations An important subclass of LTI systems are defined by an Nth-order linear constant-coefficient difference equation: Often the leading coefficient a 0 = 1. Then the output y[n] can be computed recursively from A causal LTI system of this form can be simulated in MATLAB using the function filter y = filter(a, b, x); 4

Total Solution Calculation The output sequence y[n] consists of a homogeneous solution yh[n] and a particular solution yp[n]. where the homogenous solution yh[n] is obtained from the homogeneous equation: Some textbooks use the term complementary solution instead of homogeneous solution 5

Homogeneous Solution Given the homogeneous equation: Assume that the homogeneous solution is of the form then defines an Nth order characteristic polynomial with roots l 1, l 2 … l. N The general solution is then a sequence yh[n] (if the roots are all distinct) The coefficients Am may be found from the initial conditions. 6

Particular Solution The particular solution is required to satisfy the difference equation for a specific input signal x[n], n ≥ 0. To find the particular solution we assume for the solution yp[n] a form that depends on the form of the specific input signal x[n]. 7

General Form of Particular Solution Input Signal x[n] Particular Solution yp[n] A (constant) K AMn KMn An. M K 0 n. M+K 1 n. M-1+…+KM A nn M An(K 0 n. M+K 1 n. M-1+…+KM) 8

Example #1(a) Determine the homogeneous solution for Substitute Homogeneous solution is then 9

Example #1(b) Determine the particular solution for with and y[-1] = 1 and y[-2] = -1 The particular solution has the form which is satisfied by b = -2 10

Example #1 Determine the total solution for with and y[-1] = 1 and y[-2] = -1 The total solution has the form then

Initial-Rest Conditions

Example: Suppose we have a DT system characterized by the difference equation below 13

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Impulse Response The impulse response h[n] of a causal system is the output observed with input x[n] = d[n]. For such a system, x[n] = 0 for n >0, so the particular solution is zero, yp[n]=0. Thus the impulse response can be generated from the homogeneous solution by determining the coefficients Am to satisfy the zero initial conditions (for a causal system). 15

Example #1 Determine the impulse response for the DT system characterized by difference eq. The impulse response is obtained from the homogenous solution: For n=0 For n=1

Example #2 17

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