Question

The frequency response magnitude of a DT-LTI system tends to infinity when

• A. Input is zero
• B. Poles lie on the unit circle
• C. Zeros lie on the unit circle
• D. System is causal and stable

Hint:

Poles act as infinite peaks in the $z$-plane, while zeros act as valley bottoms. If a peak (pole) sits directly on the unit circle ($|z|=1$), the frequency response magnitude evaluated at that specific angular location goes to infinity.

Answer 1

The Correct Option is B

Concept: The frequency response $H(e^{j\omega})$ of a Discrete-Time Linear Time-Invariant (DT-LTI) system is obtained by evaluating its system transfer function $H(z)$ directly along the unit circle in the $z$-complex plane, meaning we substitute $z = e^{j\omega}$. \[ H(e^{j\omega}) = H(z) \Big|_{z = e^{j\omega}} \] The rational expression of a transfer function is represented in terms of its poles ($p_k$) and zeros ($z_m$): \[ H(z) = A \cdot \frac{\prod (z - z_m)}{\prod (z - p_k)} \]

Step 1: Understand the geometric interpretation of poles and zeros.
In the complex $z$-plane: - Zeros ($z_m$): Points where the magnitude of the transfer function drops precisely to zero ($|H(z)| = 0$). - Poles ($p_k$): Points where the denominator becomes zero, causing the magnitude of the transfer function to approach infinity ($|H(z)| \rightarrow \infty$).

Step 2: Evaluating the behavior along the unit circle.
When the system frequency response is evaluated, the variable $z$ traces the path defined by $|z| = 1$. If a pole $p_k$ happens to lie directly on this unit circle, then at the specific frequency $\omega_0$ where $e^{j\omega_0} = p_k$, the distance from the point on the unit circle to that pole drops to zero: \[ |z - p_k| = |e^{j\omega_0} - e^{j\omega_0}| = 0 \] As this term resides in the denominator of $H(z)$, division by zero causes the overall frequency response magnitude to surge to infinity: \[ |H(e^{j\omega_0})| \rightarrow \infty \]