Problem2 The system function of a linear time-invariant filter is given by the f

Question

Problem2 The system function of a linear time-invariant filter is given by the formula (a) Write the difference equation that gives the relation between the input r [n] and the output y[n]. (Hint: First multiply the factors to get a polynomial.) (b) What is the output if the input is x[n]-[n]? (e) Use multiplication of -transform polynomials to find the output when the input is (d) If the input to the system is of the form for what values ofo, will the output be zero for all "? we cannot use-transforms directly to solve this problem, but we can find the frequency response from H(E) and then solve the problem. Note that the factored form will tell you the answer and so will the pole zero plot for HE).

Explanation / Answer

1 h(z)=1+z^-2-z^-1-z^-3

Y(z)=x(z)(1-z^-1+z^-2-z^-3)

2 y(n)=$(n) +$(n-1)+$(n-2)+$(n-3)

3fir input (1-2z^-1+2z^-3+z^-4)

We will get

Y(z)=1-3z^-1+3z^-2-z^-3+z^-4+z^-5-z^-6-z^-7

Y$n)=1-3$(n-1)+3$(n-2)-$(n-3)+$(n-4)+$(n-5)-$(n-6)-$(n-7)

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