Interview questions & answers
Q1. What is the difference between a continuous-time and a discrete-time signal?
A continuous-time signal is defined for every value of time t in a continuous range, such as the voltage output of a microphone preamplifier, while a discrete-time signal is defined only at specific time instants separated by a sampling interval T, such as the output of an ADC sampling at 44.1 kHz. A 16-bit audio codec operating at 44.1 kHz produces a discrete-time sequence x[n] from the continuous microphone signal x(t), with n being the sample index. The distinction determines which transform tools (Fourier integral vs. DTFT/DFT) and which filter design methods apply.
Follow-up: What information is lost when a continuous-time signal is sampled to produce a discrete-time signal?
Q2. What is a Linear Time-Invariant (LTI) system and why is this classification important?
An LTI system satisfies both linearity (superposition: output to a sum of inputs equals the sum of individual outputs) and time-invariance (a time shift in input produces the same time shift in output without any other change). A resistor-capacitor low-pass filter is an LTI system because doubling the input voltage doubles the output and delaying the input by 1 ms delays the output by exactly 1 ms. LTI systems are characterized completely by their impulse response h(t), which allows the output to any input to be computed as a convolution — a powerful result that underlies all filter design.
Follow-up: Give an example of a system that is linear but not time-invariant, and explain why it fails time-invariance.
Q3. What is the impulse response of a system and how is it used to find the output?
The impulse response h(t) is the output of a system when the input is a Dirac delta impulse δ(t); once known, the output to any input x(t) is the convolution y(t) = x(t) * h(t). For a first-order RC low-pass filter with R=10kΩ, C=1µF, h(t) = (1/RC)e^(−t/RC)u(t) = 100e^(−100t)u(t). Every digital filter in DSP is defined by its impulse response: an FIR filter's coefficient array is literally its impulse response stored as a lookup table.
Follow-up: How does the shape of h(t) reveal whether a system is low-pass, high-pass, or band-pass?
Q4. What is causality in a system and how do you verify it from the impulse response?
A causal system produces output only after or at the time the input is applied — it does not respond before the input arrives. A system is causal if its impulse response h(t) = 0 for all t < 0, meaning the response is zero before the impulse occurs. A real-time audio compressor is causal (it can only act on audio already received), while an ideal Hilbert transform filter has a non-causal impulse response that extends to negative time and can only be approximated in practice with added delay.
Follow-up: Can a non-causal system be implemented in a real-time hardware circuit? If not, how is it approximated?
Q5. What is BIBO stability and how is it tested for an LTI system?
BIBO (Bounded Input Bounded Output) stability means that any bounded input produces a bounded output. For an LTI system, BIBO stability requires that the impulse response be absolutely integrable: ∫|h(t)|dt < ∞ for continuous-time or Σ|h[n]| < ∞ for discrete-time. An underdamped RLC circuit with poles in the right half of the s-plane is BIBO unstable because its impulse response grows exponentially — this is what causes oscillator startup and amplifier runaway.
Follow-up: What is the BIBO stability condition in terms of pole locations in the s-plane?
Q6. What is the difference between energy signals and power signals?
An energy signal has finite total energy (E = ∫|x(t)|²dt < ∞) and zero average power, while a power signal has finite average power (P = lim(T→∞) (1/2T)∫|x(t)|²dt > 0) and infinite total energy. A single pulse from a radar transmitter is an energy signal, while a continuous 1 kHz sine wave from a signal generator is a power signal because it lasts indefinitely with constant amplitude. Noise signals are classified as power signals because they extend indefinitely with finite power spectral density.
Follow-up: Is a decaying exponential like e^(−t)u(t) an energy signal or a power signal?
Q7. What is convolution and what does it represent in an LTI system?
Convolution is the integral y(t) = ∫h(τ)x(t−τ)dτ that computes the LTI system output by summing the weighted, time-shifted impulse responses at each point in time. For a 3-tap FIR filter with coefficients [0.25, 0.5, 0.25], convolving with input samples performs a weighted moving average that removes high-frequency noise from an accelerometer signal. Convolution in the time domain is equivalent to multiplication in the frequency domain, which is why digital filter frequency responses are designed in the frequency domain and then converted to coefficients.
Follow-up: What is circular convolution and how does it differ from linear convolution?
Q8. What is the frequency response of a system and how is it related to the impulse response?
The frequency response H(jω) is the Fourier transform of the impulse response h(t), describing how the system scales amplitude and shifts phase of each sinusoidal frequency component of the input. For a first-order RC low-pass filter with RC = 1 ms, H(jω) = 1/(1 + jω×0.001) shows magnitude dropping to −3 dB at ω = 1000 rad/s (f = 159 Hz). Plotting |H(jω)| versus frequency gives the Bode magnitude plot used to verify filter cutoff, passband ripple, and stopband attenuation in analog and digital filter designs.
Follow-up: How does the phase response H(jω) affect the output waveform of a system, separate from the magnitude response?
Q9. What is an even and odd signal and how is any signal decomposed into these components?
An even signal satisfies x(t) = x(−t) (symmetric about the vertical axis), while an odd signal satisfies x(t) = −x(−t) (antisymmetric). Any signal decomposes as x(t) = x_e(t) + x_o(t), where x_e(t) = [x(t) + x(−t)]/2 and x_o(t) = [x(t) − x(−t)]/2. The Fourier series of a full-wave rectified sine is a purely even function, which is why its Fourier coefficients include only cosine terms — recognizing even/odd symmetry halves the computation of Fourier series.
Follow-up: What does it mean for the Fourier transform if a signal is purely even or purely odd?
Q10. What is a periodic signal and how does its Fourier representation differ from that of a non-periodic signal?
A periodic signal satisfies x(t) = x(t + T0) for all t and is represented by a discrete-frequency Fourier series with harmonically related components at f0, 2f0, 3f0, ..., while a non-periodic signal has a continuous frequency spectrum represented by the Fourier transform integral. The 50 Hz power supply voltage is periodic and has a Fourier series dominated by the fundamental at 50 Hz with odd harmonics from transformer core saturation. A speech phoneme is non-periodic and requires the Fourier transform, with energy spread across a continuous band of frequencies.
Follow-up: What happens to the Fourier series coefficients as the period T0 of a signal is made infinitely large?
Q11. What is the unit step function and how is it related to the impulse function?
The unit step function u(t) equals 1 for t ≥ 0 and 0 for t < 0, representing the sudden switching of a circuit or system. The impulse δ(t) is the derivative of u(t), and conversely u(t) is the integral of δ(t). The step response of an LTI system (output to a unit step input) is the integral of its impulse response, so measuring a circuit's step response with a function generator and oscilloscope and differentiating gives the impulse response without requiring an ideal impulse signal.
Follow-up: What practical signal approximates an impulse when testing the impulse response of an audio amplifier?
Q12. What is the difference between deterministic and random signals?
A deterministic signal has a precisely known value at every time instant described by a mathematical function, while a random signal can only be described statistically through probability distributions, mean, variance, and power spectral density. The output of a 10 MHz crystal oscillator is deterministic (x(t) = A·sin(2π×10^7·t)), while thermal noise from a 1 kΩ resistor at 300 K is a random signal with power spectral density 4kTR = 1.66×10^−20 W/Hz. Filter design must account for random noise signals using spectral density techniques rather than Fourier transforms of specific waveforms.
Follow-up: What is the autocorrelation function of white noise and what does it tell you about the signal?
Q13. What is system bandwidth and how is it defined for a low-pass system?
Bandwidth is the range of frequencies over which a system transmits signals with acceptable amplitude, commonly defined as the frequency range from DC to the −3 dB cutoff frequency for a low-pass system where |H(jω)| = (1/√2)|H(0)|. A TLV271 op-amp configured as a low-pass filter with 10 kΩ and 10 nF gives a bandwidth of 1/(2π×10k×10n) = 1.59 kHz. The −3 dB bandwidth is the point where signal power is halved and amplitude reduced by about 30%, chosen as a standard because it marks the corner of the filter's pass and transition band.
Follow-up: How is bandwidth defined for a band-pass system and what is the Q-factor?
Q14. What is the transfer function of a system and how is it related to the frequency response?
The transfer function H(s) is the Laplace transform of the impulse response, expressing the system's input-output relationship as a ratio of polynomials in s that directly reveals poles and zeros governing stability and frequency behavior. For a first-order RC low-pass filter with R=1 kΩ, C=1 µF, H(s) = 1/(1 + s×10^−3) with a pole at s = −1000 rad/s. The frequency response is obtained by substituting s = jω, so H(jω) = H(s)|{s=jω}, connecting Laplace and Fourier analysis for stable systems.
Follow-up: What do the locations of poles and zeros on the s-plane tell you about the system's frequency response shape?
Q15. What is the difference between IIR and FIR digital filters in terms of systems theory?
An FIR (Finite Impulse Response) filter has an impulse response of finite duration — its output depends only on a finite number of past inputs — making it always BIBO stable, while an IIR (Infinite Impulse Response) filter has feedback and an impulse response that extends to infinity, potentially becoming unstable if poles fall outside the unit circle. A 64-tap FIR low-pass filter built in MATLAB with windowing has exactly 64 non-zero coefficients and is unconditionally stable. An IIR Butterworth filter achieves the same frequency selectivity with only 6 coefficients but requires checking its poles remain inside the unit circle to guarantee stability.
Follow-up: What is the advantage of FIR filters for audio applications specifically related to their phase response?
Common misconceptions
Misconception: A stable system always produces a small output for any input.
Correct: BIBO stability only guarantees that a bounded input produces a bounded output; the output can still be large in amplitude as long as it remains finite when the input is finite.
Misconception: Convolution and correlation are the same operation.
Correct: Convolution flips one signal in time before sliding and integrating, while cross-correlation slides one signal without flipping; they are equal only when one signal is even-symmetric.
Misconception: Any real-world system can be made causal by adding a time delay.
Correct: Adding a time delay makes a non-causal system's impulse response shift forward in time, which approximates causality for finite-support filters but cannot perfectly implement a truly non-causal ideal filter.
Misconception: The Fourier transform and the Fourier series are the same tool applied differently.
Correct: The Fourier series applies to periodic signals and produces a discrete set of frequency coefficients, while the Fourier transform applies to non-periodic signals and produces a continuous frequency spectrum.