Determine the Z-Transform of the sequence x[n] = delta [n] - 0.95 delta (n - 6]

Question

Determine the Z-Transform of the sequence x[n] = delta [n] - 0.95 delta (n - 6] Sketch the pole zero pattern of X(z) By considering the behavior of the pole and zeros vectors as the unit circle is traversed, develop and approximate sketch of the Magnitude of the Frequency Response of x[n]. You may need an expression for finding roots of a polynomial of the form zN - aN = 0 Roots are of the form zk = aej(2 pi k)/N; k = 0,1,... N - 1

Explanation / Answer

X(z)= 1 - 0.95 z^-6 (delay) X(z)= (z^6-0.95)/ z^6 All the poles are at z=0 zeros are given by z^6=0.95 cos(2npi) {n is an integer => z= (0.95)^1/6 cos(2npi/6) for n=0 to 5 mark these in the x-y plot. Mark zeros using a small circle. mark the poles using a cross.

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