17 x n SYS 1 SYS 2 y n The impulse response of SYS 1 is h 1 n \u03b4 n 0 5 \u03b4 n 1 5 \u03b4

17 x n sys 1 sys 2 y n the impulse response of sys 1

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x ( n ) - SYS 1 - SYS 2 - y ( n ) The impulse response of SYS 1 is h 1 ( n ) = δ ( n ) + 0 . 5 δ ( n - 1) - 0 . 5 δ ( n - 2) and the transfer function of SYS 2 is H 2 ( z ) = z - 1 + 2 z - 2 + 2 z - 3 . (a) Sketch the impulse response of the total system. (b) What is the transfer function of the total system? 1.5 Inverse Systems 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. 1.5.2 A discrete-time LTI system x ( n ) - h ( n ) - y ( n ) has the impulse response h ( n ) = δ ( n ) + 3 . 5 δ ( n - 1) + 1 . 5 δ ( n - 2) . (a) Find the transfer function of the system h ( n ). (b) Find the impulse response of the stable inverse of this system. (c) 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. 1.5.3 Consider a discrete-time LTI system with the impulse response h ( n ) = δ ( n + 1) - 10 3 δ ( n ) + δ ( n - 1) . (a) Find the impulse response g ( n ) 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. 1.5.4 A causal discrete-time LTI system x ( n ) - H ( z ) - y ( n ) 18
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is described by the difference equation y ( n ) - 1 3 y ( n - 1) = x ( n ) - 2 x ( n - 1) . What is the impulse response of the stable inverse of this system? 1.6 Difference Equations 1.6.1 A causal discrete-time system is described by the difference equation, y ( n ) = x ( n ) + 3 x ( n - 1) + 2 x ( n - 4) (a) What is the transfer function of the system? (b) Sketch the impulse response of the system. 1.6.2 Given the impulse response . . . 1.6.3 A causal discrete-time LTI system is implemented using the difference equation y ( n ) = x ( n ) + x ( n - 1) + 0 . 5 y ( n - 1) where x is the input signal, and y the output signal. Find and sketch the impulse response of the system. 1.6.4 Given the impulse response . . . 19
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1.6.5 Given two discrete-time LTI systems described by the difference equations T 1 : y ( n ) + 1 3 y ( n - 1) = x ( n ) + 2 x ( n - 1) T 2 : y ( n ) + 1 3 y ( n - 1) = x ( n ) - 2 x ( n - 1) let T be the cascade of T 1 and T 2 in series. What is the difference equation describing T ? Suppose T 1 and T 2 are causal systems. Is T causal? Is T stable? 1.6.6 Consider the parallel combination of two causal discrete-time LTI systems. - SYS 1 - SYS 2 x ( n ) ? 6 l + - r ( n ) You are told that System 1 is described by the difference equation y ( n ) - 0 . 1 y ( n - 1) = x ( n ) + x ( n - 2) and that System 2 is described by the difference equation y ( n ) - 0 . 1 y ( n - 1) = x ( n ) + x ( n - 1) . Find the difference equation of the total system.
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