Zion Wheeler

Zion Wheeler

Answered

2022-06-26

The problem is this:
The impulse response of a system is the output from this system when excited by an input signal δ ( k ) that is zero everywhere, except at k = 0, where it is equal to 1. Using this definition and the general form of the solution of a difference equation, write the output of a linear system described by:
y ( k ) 3 y ( k 1 ) 4 y ( k 2 ) = δ ( k ) + 2 δ ( k 1 )
The initial conditions are: y ( 2 ) = y ( 1 ) = 0.
My question is: How can the particular solution be found using the method of undetermined coefficients if the non-homogeneous equation is also a difference equation?

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Recalculate according to your conditions!

Answer & Explanation

Daniel Valdez

Daniel Valdez

Expert

2022-06-27Added 19 answers

OK. So, I suppose you want to solve your difference equation for k 0. If you set k = 0 we easily get y ( 0 ) = 1 and you could find y ( k ) recursively for k > 0, right? that's the point. To obtain the impulse response, you can see it as a response of an homogeneous equation with initial conditions different from zero. Then, check it out that the impulse response h ( k ) to your problem is the same of this one:
y ( k ) 3 y ( k 1 ) 4 y ( k 2 ) = 0 , y ( 2 ) = 1 / 4 , y ( 1 ) = 0
Why? simply because they have the same solution for k 0. Now its easy to find h ( k ) right? The final solution is then h ( k ) + 2 h ( k 1 ).

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