167 Consider a causal discrete time LTI system described by the difference

167 consider a causal discrete time lti system

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1.6.7 Consider a causal discrete-time LTI system described by the difference equation y ( n ) - 5 6 y ( n - 1) + 1 6 y ( n - 2) = 2 x ( n ) + 2 3 x ( n - 1) . 20
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(a) Find the transfer function H ( z ). (b) Find the impulse response h ( n ). You may use MATLAB to do the partial fraction expansion. See the command residue . Make a stem plot of h ( n ) with MATLAB. (c) Plot the magnitude of the frequency response | H ( e ) | of the system. Use the MATLAB command freqz . 1.6.8 A room where echos are present can be modeled as an LTI system that has the following rule: y ( n ) = X k =0 2 - k x ( n - 10 k ) The output y ( n ) is made up of delayed versions of the input x ( n ) of decaying amplitude. (a) Sketch the impulse response h ( n ). (b) What is transfer function H ( z )? (c) Write the corresponding finite-order difference equation. 1.6.9 Echo Canceler. A recorded discrete-time signal r ( n ) is distorted due to an echo. The echo has a lag of 10 samples and an amplitude of 2 / 3. That means r ( n ) = x ( n ) + 2 3 x ( n - 10) where x ( n ) is the original signal. Design an LTI system with impulse response g ( n ) that removes the echo from the recorded signal. That means, the system you design should recover the original signal x ( n ) from the signal r ( n ). (a) Find the impulse response g ( n ). (b) Find a difference equation that can be used to implement the system. (c) Is the system you designed both causal and stable? 1.6.10 Consider a causal discrete-time LTI system with the impulse response h ( n ) = 3 2 3 4 n u ( n ) + 2 δ ( n - 4) (a) Make a stem plot of h ( n ) with MATLAB. (b) Find the transfer function H ( z ). (c) Find the difference equation that describes this system. (d) Plot the magnitude of the frequency response | H ( e ) | of the system. Use the MATLAB command freqz . 1.6.11 Consider a stable discrete-time LTI system described by the difference equation y ( n ) = x ( n ) - x ( n - 1) - 2 y ( n - 1) . (a) Find the transfer function H ( z ) and its ROC. (b) Find the impulse response h ( n ). 1.6.12 Two LTI systems are connected in series: - SYS 1 - SYS 2 - 21
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The system SYS 1 is described by the difference equation y ( n ) = x ( n ) + 2 x ( n - 1) + x ( n - 2) where x ( n ) represents the input into SYS 1 and y ( n ) represents the output of SYS 1. The system SYS 2 is described by the difference equation y ( n ) = x ( n ) + x ( n - 1) + x ( n - 2) where x ( n ) represents the input into SYS 2 and y ( n ) represents the output of SYS 2. (a) What difference equation describes the total system? (b) Sketch the impulse response of the total system. 1.6.13 Three causal discrete-time LTI systems are used to create the a single LTI system. x ( n ) H 1 H 2 H 3 + y ( n ) r ( n ) f ( n ) g ( n ) The difference equations used to implement the systems are: H 1 : r ( n ) = 2 x ( n ) - 1 2 r ( n - 1) H 2 : f ( n ) = r ( n ) - 1 3 f ( n - 1) H 3 : g ( n ) = r ( n ) - 1 4 r ( n - 1) What is the transfer function H tot ( z ) for the total system? 1.6.14 Given the difference equation... 22
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1.6.15 Difference equations in MATLAB Suppose a system is implemented with the difference equation: y ( n ) = x ( n ) + 2 x ( n - 1) - 0 . 95 y ( n - 1) Write your own MATLAB function, mydiffeq , to implement this difference equation using a for loop. If the input signal is N -samples long (0 n N - 1), your program should find the first N samples of the output y ( n ) (0 n N - 1). Remember that MATLAB indexing starts with 1, not 0, but don’t let this confuse you.
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