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Problem 5.1 Determine which of the following sequences have Fourier transforms which converge:

Problem 5.2 (a) For each of the following sequences sketch the pole-zero pattern of their z-transforms. Include an indication of the region of convergence.

(b) From your results in part (a) determine which of those sequences could correspond to the unit sample response of a stable system.

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Problem 5.3 Consider a sequence x(n) with z-transform X(z). The pole zero pattern for X(z) is shown below:

Figure P5.3-1 (a) Determine the region of convergence of X(z) if it is known that the Fourier transform of the sequence converges. For this case determine also whether the sequence is right-sided, left-sided or two-sided. (b) If it is not known whether the Fourier transform of x(n) converges but the sequence is known to be two-sided how many different possible sequences could the pole-zero pattern of figure P5.3-1 correspond to? For each of these possibilities indicate the associated region of convergence.

Problem 5.4 Let X(z) denote the z-transform of x(n). Determine in terms of X(z) the ztransform of x(n + n0 ). If you have difficulty deciding how to proceed you may find it helpful to first consider an example such as x(n) =( )n u(n) and n0 1. * Problem 5.5 Determine the z-transform of each of the following. Include with your answer the region of convergence in the z-plane and a sketch of the pole-zero pattern. Express all sums in closed form. a can be complex. (a) x(n) = a r 0 < |al < 1 (b) x(n) = Arn cos (W0 n + f) u(n), 0 < r < 1

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[Hint (easy way): First express x(n) in terms of the x(n) in part (c).] * Problem 5.6* For a sequence x(n) which is zero for n < 0, show that lim X(z) = x(O). z o What is the corresponding theorem if the sequence is zero for n > 0? * Problem 5.7* Let x(n) denote a causal sequence; i.e., x(n) = 0, n < 0. Furthermore, assume that x(0) ≠ 0. (a) Show that there are no poles or zeros of X(z) at z = ꚙ4 (b) Show that the number of poles in the finite z-plane equals the number of zeros in the finite z-plane. (The finite z-plane excludes z=ꚙ.)

Solution 5.1 matlabassignmentexperts.com For convergence of the Fourier transform, the sequence must be absolutely summable or square summable, (i.e. have finite energy) depending on the type of convergence to be considered.

In this problem, sequences (i) and (iv) are neither absolutely summable nor square summable, and thus their Fourier transforms do not converge. Sequences (ii) and (iii) are both absolutely summable and square summable.

Solution 5.2

We obtain the z-transforms of the remaining sequences in a similar manner:

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Figure S5.2-5 (b) In order that a sequence correspond to the unit sample response of a stable system the region of convergence must include the unit circle. Thus only sequence (ii) would not correspond to a stable system.

Solution 5.3 (a) If the Fourier transform converges then the region of convergence includes the unit circle. It is bounded by two circles extending outward and inward to the nearest poles. Thus the region of convergence is < Izj < 2. Since it does not extend to zero or to infinity, the sequence is twosided.

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(b) If the sequence is two-sided then the region of convergence must be bounded by poles. Thus, the given pole-zero pattern with the region of convergence either as < |z < 2 or as 2 < |z| < 3 will correspond to two-sided sequences.

Solution 5.4

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Figure S5.5-3 (d) xd (n + 1) = x (n) * x (n) where xd (n) and x (n) are the sequences in part (d) and (c) respectively. Therefore Xd (z) = z~ Xc (z) Xc (z) Therefore the zeros in the pole-zero plot of part (c) become double zeros and there is now a (2N - 1)st order pole at z = 0.

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For z + all of the terms in-this sum approach zero with the exception of the first term. Thus lim X(z) = x(0) for x(n) causal.

If x(n) = 0 n > 0 then the corresponding relation is lim X(z) = x(0) z+ 0 Solution 5.7* (a) From problem 5.7 we know that lim X(z) is (b) x(0). Assuming that Z+ -)-0 x(0) is finite, then, lim X(z) is finite and consequently there are no Z+ -*.0 (c) poles of X(z) at z = x. Furthermore, since x(0) 3 0 there are no zeros of X(z) at z = (b) Since x(n) = 0; n < 0 X(z) is a rational function of z~ , i.e. it is of the form

The numerator factor introduces M zeros and also an M=- order pole th at z = 0. The denominator factor introduces N poles and an Norder zero at z = 0. Thus there are a total of (M + N) zeros and (M + N) poles i.e. the total number of poles and zeros in the entire z-plane (including z = o) is the same.

From part (a) of this problem we know that there are no poles or zeros at z = w. Thus the number of poles and zeros in the finite z-plane is the same.