In the z-transform, what do poles represent in a discrete-time system?

• A. Points where the transfer function is undefined due to division by zero, indicating potential instability.
• B. Frequencies at which the gain is zero.
• C. The time-domain impulse response length.
• D. The energy of the signal in the z-domain.

Answer: (A)

Poles are the z-values where the transfer function becomes unbounded because the denominator equals zero. They reflect the natural modes of the discrete-time system, shaping the impulse response as exponential-type terms in time. The location of these poles in the z-plane governs stability: poles inside the unit circle lead to a decaying, BIBO-stable response; on the circle give sustained oscillations (marginal stability); outside cause growing, unstable behavior. This is why poles indicate potential instability and how the system naturally responds over time.

The other statements mix up concepts: frequencies where the gain is zero describe zeros, not poles; the impulse response length is tied to system order and pole/zero placement but not solely determined by poles; and the energy of the signal in the z-domain is not what poles measure.