hw3Sol - 1.4.15 Consider the parallel combination of two LTI systems h1(n l x(n y(n 6 h2(n You are told that the impulse responses of the two systems

hw3Sol - 1.4.15 Consider the parallel combination of two...

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1.4.15 Consider the parallel combination of two LTI systems. - h 2 ( n ) - h 1 ( n ) x ( n ) ? 6 l + - y ( n ) You are told that the impulse responses of the two systems are h 1 ( n ) = 3 1 2 n u ( n ) and h 2 ( n ) = 2 1 3 n u ( n ) (a) Find the impulse response h ( n ) of the total system. (b) You want to implement the total system as a cascade of two first order systems g 1 ( n ) and g 2 ( n ). Find g 1 ( n ) and g 2 ( n ), each with a single pole, such that when they are connected in cascade, they give the same system as h 1 ( n ) and h 2 ( n ) connected in parallel. x ( n ) - g 1 ( n ) - g 2 ( n ) - y ( n ) 1.4.15) Solution h tot ( n ) = h 1 ( n ) + h 2 ( n ) = 3 1 2 n u ( n ) + 2 1 3 n u ( n ) H tot ( z ) = 3 z z - 1 / 2 + 2 z z - 1 / 3 = 5 z 2 - 2 z ( z - 1 / 2)( z - 1 / 3) = z (5 z - 2) ( z - 1 / 2)( z - 1 / 3) so we can set G 1 ( z ) = z z - 1 / 2 G 2 ( z ) = 5 z - 2 z - 1 / 3 so g 1 ( n ) = 1 2 n u ( n ) To find g 2 ( n ) we can perform partial fraction expansion on G 2 ( z ) /z . G 2 ( z ) z = 5 z - 2 z ( z - 1 / 3) = A z + B z - 1 / 3 where we can find that A = 6 and B = - 1, so G 2 ( z ) = 6 - z z - 1 / 3 61
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and so g 2 ( n ) = 6 δ ( n ) - 1 3 n u ( n ) Other answers are also possible. We can check this answer in MATLAB using the following code fragment. >> n = 0:20; >> h1 = 3 * (1/2).^n .* (n >= 0); >> h2 = 2 * (1/3).^n .* (n >= 0); >> htot = h1 + h2; >> >> g1 = (1/2).^n .* (n >= 0); >> g2 = 6 * (n == 0) - (1/3).^n .* (n >= 0); >> gtot = conv(g1,g2); >> >> % check that htot is the same as gtot for n = 1:10 >> [htot(1:10); gtot(1:10)]’ ans = 5.0000 5.0000 2.1667 2.1667 0.9722 0.9722 0.4491 0.4491 0.2122 0.2122 0.1020 0.1020 0.0496 0.0496 0.0244 0.0244 0.0120 0.0120 0.0060 0.0060 62
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1.5.1 The impulse response of a discrete-time LTI system is h ( n ) = - δ ( n ) + 2 1 2 n u ( n ) . (a) Find the impulse response of the stable inverse of this system. (b) Use MATLAB to numerically verify the correctness of your answer by computing the convolution of h ( n ) and the impulse response of the inverse system. You should get δ ( n ). Include your program and plots with your solution.
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