Signal and Systems Interview Questions

Introduction

Technical interviews for electronics, communications, and DSP roles frequently test signals and systems fundamentals. Interviewers want to assess not just your ability to solve textbook problems, but your conceptual understanding — can you explain WHY something works, not just HOW to compute it? This section covers the most commonly asked questions with answers that demonstrate deep understanding.

Category 1: System Properties

Q1: What is the difference between a linear system and a time-invariant system?

Answer: Linearity means the system satisfies superposition — scaling and additivity. If input $ax_1 + bx_2$ produces output $ay_1 + by_2$. Time-invariance means the system's behavior doesn't change over time — if input $x(t)$ produces $y(t)$, then $x(t-t_0)$ produces $y(t-t_0)$. A system can be linear but not TI (e.g., $y(t) = tx(t)$), or TI but not linear (e.g., $y(t) = x^2(t)$). LTI systems have both properties and are characterized by convolution with the impulse response.

Q2: How do you determine if a system is causal?

Answer: A system is causal if the output at any time depends only on present and past inputs, not future inputs. For LTI systems: causal iff $h(t) = 0$ for $t < 0$. Practically, all real-time physical systems must be causal (you can't use tomorrow's data today). Non-causal systems arise in offline processing (where the entire signal is available) — example: zero-phase filtering by processing forward and backward.

Q3: What is BIBO stability? How do you check it?

Answer: Bounded-Input Bounded-Output stability means every bounded input produces a bounded output. For LTI systems: BIBO stable iff $\int_{-\infty}^{\infty}|h(t)|dt < \infty$ (impulse response is absolutely integrable). Equivalently: all poles of $H(s)$ in the left half-plane (for causal systems). For discrete: all poles inside unit circle.

Category 2: Transforms

Q4: Why do we need the Laplace transform when we have the Fourier transform?

Answer: The Fourier transform requires absolute integrability, so it doesn't exist for many practical signals (growing exponentials, unstable system responses). The Laplace transform adds a real exponential weighting $e^{-\sigma t}$ that can make non-integrable signals integrable. It also enables analysis of system stability through pole locations in the complex s-plane, and provides a natural framework for initial-value problems (transient analysis). The Fourier transform is a special case: $s = j\omega$ (purely imaginary axis).

Q5: Explain the Region of Convergence (ROC).

Answer: The ROC is the set of complex values of $s$ (or $z$) for which the transform integral/sum converges. It's essential because two different signals can have the same algebraic expression for their transform but different ROCs. For example, $X(s) = 1/(s+1)$ could be $e^{-t}u(t)$ (ROC: Re(s) > -1) or $-e^{-t}u(-t)$ (ROC: Re(s) < -1). The ROC determines uniqueness and tells us about signal properties (causal, stable, etc.).

Q6: What is the physical meaning of the Fourier transform?

Answer: The Fourier transform decomposes a signal into its constituent frequencies. $X(\omega)$ at frequency $\omega$ tells you how much of that frequency is present (magnitude) and at what phase alignment (phase). It converts the "what happens over time" description into a "what frequencies are present" description. Physically: a prism decomposes white light into colors; the FT decomposes a signal into frequency components.

Category 3: Filtering and Convolution

Q7: Explain convolution in simple terms.

Answer: Convolution computes the output of an LTI system by summing the system's response to each individual input sample. The input is decomposed into shifted impulses, each producing a shifted, scaled copy of the impulse response. Convolution sums all these copies. Mathematically: $y(t) = \int x(\tau)h(t-\tau)d\tau$. Graphically: flip $h$, slide it across $x$, compute the overlap integral at each position.

Q8: What is the difference between FIR and IIR filters?

Answer: FIR (Finite Impulse Response) filters have no feedback — output depends only on current and past inputs. Impulse response is finite length. Always stable, can have exactly linear phase. IIR (Infinite Impulse Response) filters use feedback — output depends on past outputs too. Impulse response is infinite (decaying exponentials). More efficient (fewer coefficients for same selectivity) but can be unstable and cannot achieve linear phase. FIR for phase-critical applications (audio, data); IIR when computational efficiency is paramount.

Q9: How does the FFT differ from the DFT?

Answer: The FFT computes the exact same result as the DFT — it's not a different transform. It's an efficient ALGORITHM for computing the DFT that exploits symmetry and periodicity of the twiddle factors to reduce complexity from $O(N^2)$ to $O(N\log N)$. For N=1024, that's 200× fewer operations. The Cooley-Tukey algorithm recursively splits the DFT into smaller sub-DFTs using the butterfly computation structure.

Category 4: Sampling and Digital

Q10: What happens if you sample below the Nyquist rate?

Answer: Aliasing occurs — high-frequency components fold back into the baseband and appear as lower frequencies that are indistinguishable from actual low-frequency content. This is irreversible (you cannot un-alias after sampling). Example: sampling a 300 Hz tone at 500 Hz creates an alias at 200 Hz. Prevention: anti-aliasing low-pass filter before the sampler, cutting off above $f_s/2$.

Q11: Why can't we build an ideal low-pass filter?

Answer: An ideal LPF has impulse response $h(t) = \text{sinc}(t)$ which extends from $t = -\infty$ to $+\infty$. This makes it non-causal (requires future inputs) and has infinite delay. In practice, we approximate with causal filters that introduce some passband ripple, finite transition band, and finite stopband attenuation. The closer to ideal, the longer the filter (more delay and computation).

Category 5: Practical DSP

Q12: How would you remove 50 Hz power line noise from an ECG signal?

Answer: Use a notch (band-stop) filter centered at 50 Hz. A second-order IIR notch filter with transfer function $H(z) = \frac{1-2\cos(\omega_0)z^{-1}+z^{-2}}{1-2r\cos(\omega_0)z^{-1}+r^2z^{-2}}$ where $\omega_0 = 2\pi\times50/f_s$ and $r$ close to 1 (controls notch width). Alternatively, adaptive filtering if the interference varies. Must ensure the notch doesn't distort the QRS complex (which has energy near 50 Hz in its higher harmonics).

Key Interview Tips

• Always explain the physical intuition, not just the math
• Draw diagrams (pole-zero plots, frequency responses, block diagrams)
• Mention practical considerations (finite word-length, real-time constraints)
• Connect theory to applications you've worked on
• Be prepared to derive basic results from first principles