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The Z-transform of a discrete signal x[n] is X(z) = 4z / ((z − 1/5)(z − 2/3)(z − 3)) with ROC = R. Which one of the following statements is true?

1. Discrete-time Fourier transform of x[n] converges if R is |z| > 3
2. Discrete-time Fourier transform of x[n] converges if R is 2/3 < |z| < 3
3. Discrete-time Fourier transform of x[n] converges if R is such that x[n] is a left-sided sequence
4. Discrete-time Fourier transform of x[n] converges if R is such that x[n] is a right-sided sequence

Correct answer: Discrete-time Fourier transform of x[n] converges if R is 2/3 < |z| < 3

Solution

The DTFT converges iff the ROC includes the unit circle. Poles are at 1/5, 2/3, and 3; the ROC 2/3<|z|<3 contains |z|=1, so the DTFT converges. A right-sided sequence would have ROC |z|>3, which excludes the unit circle, so that statement is false.

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