Transform-Based Interview Questions

Introduction

Transform-based questions are among the most frequently asked in technical interviews for signal processing, communications, and control systems positions. They test your understanding of fundamental concepts — not just mechanical computation but the ability to reason about signals in the frequency domain, assess stability from pole locations, and apply transform properties to solve problems efficiently. This section presents the most common question types with solution approaches.

Category 1: Fourier Transform Conceptual Questions

Q1: Why do we need the Fourier Transform when we have the Fourier Series?

Answer: The Fourier Series applies only to periodic signals. The Fourier Transform generalizes this to aperiodic signals by treating them as periodic with period approaching infinity. As $T \to \infty$, the discrete harmonics $n\omega_0$ become a continuous frequency variable $\omega$, and the coefficient sum becomes an integral.

Q2: A real signal has magnitude spectrum $|X(j\omega)| = 1$ for all $\omega$. What is the signal?

Answer: A flat magnitude spectrum across all frequencies corresponds to $x(t) = \delta(t)$ (unit impulse). The impulse is the only signal with equal energy at every frequency. Alternatively, white noise has flat PSD, but that describes power rather than a deterministic transform.

Q3: If $x(t)$ is real and even, what can you say about $X(j\omega)$?

Answer: $X(j\omega)$ is real and even. For any real signal, $X(-j\omega) = X^*(j\omega)$. If additionally $x(t) = x(-t)$ (even), then $X(j\omega) = X(-j\omega)$. Combining: $X = X^*$, meaning $X$ is real. Since $X(-j\omega) = X(j\omega)$ (from evenness of $x$), $X$ is also even.

Q4: What is the bandwidth of a rectangular pulse of width $T$?

Answer: Depends on the definition:

• First-null bandwidth: $1/T$ Hz (or $2/T$ for two-sided)
• 3-dB bandwidth: approximately $0.44/T$ Hz
• Essential (99%) bandwidth: approximately $10/T$ Hz

The key insight: shorter pulses have wider bandwidth (time-bandwidth tradeoff).

Category 2: Laplace Transform and System Analysis

Q5: Given $H(s) = \frac{s+2}{s^2+5s+6}$, determine stability and find the impulse response.

Solution: Factor: $H(s) = \frac{s+2}{(s+2)(s+3)} = \frac{1}{s+3}$ (pole-zero cancellation at $s=-2$)

Remaining pole: $s = -3$ (left half-plane) → Stable ✓

Impulse response: $h(t) = e^{-3t}u(t)$

Key point: Pole-zero cancellation removes a mode but the cancelled pole still affects the system\'s response to initial conditions.

Q6: A system has poles at $s = -1 \pm j2$. Describe the impulse response qualitatively.

Answer: Complex conjugate poles in the left half-plane produce a damped sinusoidal response: $$h(t) = Ae^{-t}\cos(2t + \phi)u(t)$$

• Oscillation frequency: 2 rad/s (from imaginary part)
• Decay rate: $e^{-t}$ (from real part = -1)
• The system is stable (poles in LHP) and exhibits ringing at approximately 0.32 Hz

Q7: What is the significance of the ROC?

Answer: The ROC determines:

1. Uniqueness: Same $X(s)$ can represent different signals depending on ROC
2. Causality: Causal signal has ROC extending right to $\text{Re}(s) = +\infty$
3. Stability: Stable signal has ROC including the $j\omega$ axis
4. Existence of Fourier Transform: FT exists only if $j\omega$ axis is in ROC

Category 3: Z-Transform and Digital Systems

Q8: A digital filter has $H(z) = \frac{1-z^{-1}}{1-0.9z^{-1}}$. Is it stable? What type of filter is it?

Solution: Pole at $z = 0.9$ (inside unit circle) → Stable ✓ Zero at $z = 1$ (on unit circle) → $H(e^{j0}) = 0$, so DC is completely blocked

This is a high-pass filter — it blocks DC (frequency 0) and passes higher frequencies.

Q9: Explain the relationship $z = e^{sT_s}$.

Answer: This mapping connects continuous and discrete domains:

• Left half s-plane ($\text{Re}(s)<0$) maps inside the unit circle ($|z|<1$) → stable
• Right half s-plane maps outside unit circle → unstable
• $j\omega$ axis maps to unit circle → frequency response
• The mapping is many-to-one (periodic in $\omega$ with period $\omega_s$), which is why digital systems exhibit aliasing

Q10: Given $x[n] = (0.5)^n u[n] + (0.25)^n u[n]$, find $X(z)$.

Solution: By linearity: $$X(z) = \frac{z}{z-0.5} + \frac{z}{z-0.25} = \frac{z(z-0.25)+z(z-0.5)}{(z-0.5)(z-0.25)} = \frac{2z^2-0.75z}{z^2-0.75z+0.125}$$

ROC: $|z| > 0.5$ (outside the outermost pole).

Category 4: Property Application Questions

Q11: If $\mathcal{F}\{x(t)\} = X(j\omega)$, find $\mathcal{F}\{x(2t-3)\}$.

Solution: Apply scaling and shifting properties: $$x(2t-3) = x(2(t-3/2))$$

First apply time scaling ($a=2$): $\mathcal{F}\{x(2t)\} = \frac{1}{2}X(j\omega/2)$

Then apply time shift ($t_0 = 3/2$): multiply by $e^{-j\omega(3/2)}$

$$\mathcal{F}\{x(2t-3)\} = \frac{1}{2}X(j\omega/2) \cdot e^{-j3\omega/2}$$

Q12: Prove that the convolution of two Gaussian pulses is another Gaussian.

Approach: In the frequency domain, both Gaussians have Gaussian spectra. Convolution becomes multiplication: the product of two Gaussians is a Gaussian. Inverse transforming gives a Gaussian in time.

Interview Strategy Tips

1. Start with the approach: Before computing, state which property or method you will use
2. Check dimensions: Verify units match throughout your calculation
3. Verify with special cases: Test your answer at $\omega = 0$, $t = 0$, or limits
4. Draw pole-zero diagrams: Visual reasoning impresses interviewers and catches errors
5. Know the physical meaning: Be ready to explain what a transfer function means for the actual system
6. Stability first: When given any system, immediately assess stability

Key Takeaways

• Transform questions test conceptual understanding, not just computation
• Always specify ROC with Laplace/Z-transforms — interviewers specifically check for this
• Know the physical interpretation: what poles mean, what the frequency response shows
• Practice property-based shortcuts — they save time and demonstrate mastery
• Pole-zero analysis is the fastest way to assess stability and frequency response characteristics