Convolution Interview Questions

Q1. What is convolution and what is its physical meaning in LTI systems?

Convolution is the integral y(t) = ∫x(τ)h(t−τ)dτ that computes the output of an LTI system by summing the system's response to each infinitesimal piece of the input, scaled and time-shifted to the appropriate moment. For an RC low-pass filter with h(t) = (1/RC)e^(−t/RC), convolving with a step input x(t) = u(t) produces y(t) = (1 − e^(−t/RC))u(t), the familiar capacitor charging curve. Convolution captures the fact that the present output of any LTI system depends on the entire history of past inputs, weighted by how much the system 'remembers' each past input.

Follow-up: What happens to the convolution output if h(t) has infinite memory (i.e., never decays to zero)?

Q2. How do you graphically compute the convolution of two rectangular pulses?

Flip one pulse in time, slide it across the other, and at each position compute the area of overlap between the two pulses. For two unit rectangular pulses of width 1, as the flipped pulse slides from left to right, the overlap area first increases linearly to 1, then decreases linearly back to zero, producing a triangular output of width 2 centered at the sum of the pulse centers. This graphical method is the physical basis for understanding how convolution broadens signals and is directly used to visualize how a moving average filter smooths a noisy step edge.

Follow-up: What is the width of the convolution output of two rectangles of widths A and B?

Q3. What is the convolution theorem and how does it simplify filter computations?

The convolution theorem states that convolution in the time domain equals multiplication in the frequency domain: FT{x(t)h(t)} = X(jω)×H(jω). Instead of computing a 1000-tap FIR convolution directly (requiring 1000 multiplications per output sample), you take the FFT of both signals, multiply pointwise, and inverse FFT — reducing complexity from O(N²) to O(N log N). The overlap-add algorithm in the ADSP-21489 DSP processor implements exactly this approach for real-time audio convolution reverb engines.

Follow-up: What is the overlap-add method and when is it preferred over direct convolution?

Q4. What is the difference between linear convolution and circular convolution?

Linear convolution of an N-point and an M-point sequence produces an (N+M−1)-point output where boundary effects appear only at the edges, while circular convolution of two N-point sequences computes a periodic convolution over exactly N points, causing time-aliasing where the linear convolution tails wrap around and add to the beginning. A linear convolution of a 100-sample signal with a 10-sample filter produces 109 samples, but circular convolution of the same padded to 100 samples wraps the last 9 samples, corrupting the first 9 output values. FFT-based fast convolution always computes circular convolution, so zero-padding to at least N+M−1 points is required to get the correct linear convolution result.

Follow-up: How much zero-padding is required before FFT multiplication to ensure circular convolution equals linear convolution?

Q5. What is the convolution of any signal with a unit impulse δ(t)?

Convolving any signal x(t) with the unit impulse δ(t) returns x(t) unchanged: x(t) * δ(t) = x(t). This is because the impulse sifts out the value of x at every instant, effectively acting as the identity element of convolution. This property defines how every LTI system's impulse response h(t) is measured — applying δ(t) as input and recording the output directly gives h(t) without any further processing.

Follow-up: What is the result of convolving x(t) with a delayed impulse δ(t − t0)?

Q6. What is the result of convolving a signal with a unit step u(t)?

Convolving x(t) with u(t) computes the running integral of x(t): y(t) = ∫₋∞^t x(τ)dτ. For a rectangular pulse convolved with u(t), the result is a ramp that starts at the pulse onset and stops growing at the pulse end, forming a staircase that integrates the pulse area. In a capacitor circuit driven by a current source, the capacitor voltage is the time integral of the current — exactly this convolution with the unit step, scaled by 1/C.

Follow-up: What is the result of convolving u(t) with itself?

Q7. What is the associativity property of convolution and how is it used in cascaded filter analysis?

Convolution is associative: (x * h1) * h2 = x * (h1 * h2), meaning a signal passed through two cascaded LTI systems with impulse responses h1 and h2 produces the same output as passing through a single system whose impulse response is h1 * h2. Two cascaded first-order RC filters each with h(t) = 100e^(−100t)u(t) can be analyzed as a single second-order system with impulse response h1 * h2 = 10000t×e^(−100t)u(t). This property lets filter designers characterize an entire multi-stage signal chain with one equivalent transfer function.

Follow-up: Does the order of cascaded LTI systems matter for the overall input-output behavior?

Q8. What is the commutativity of convolution and what does it imply physically?

Convolution is commutative: x(t) * h(t) = h(t) * x(t), meaning it does not matter which function is flipped and which is slid in the graphical computation. Physically, it means the output of an LTI system with input x and impulse response h is the same as the output of a system with input h and impulse response x — both functions play symmetric roles in determining the output. This symmetry is exploited in matched filter receivers for radar and communications, where the filter's impulse response is set equal to the time-reversed transmitted pulse to maximize SNR at the sampling instant.

Follow-up: How does the matched filter use the commutativity of convolution to maximize SNR?

Q9. How is convolution used in image processing?

Image convolution applies a kernel (small matrix) by sliding it across each pixel of the image and computing the weighted sum of the neighborhood, implementing spatial filtering operations like blurring (Gaussian kernel), sharpening (Laplacian kernel), or edge detection (Sobel kernel). Applying a 3×3 Sobel kernel to a grayscale camera frame in an STM32H7 running OpenCV detects vertical edges in real-time at 30 fps using the 2D convolution theorem and FFT acceleration. Convolutional neural networks (CNNs) use hundreds of learned convolutional kernels to extract hierarchical image features, making convolution the fundamental operation of modern computer vision.

Follow-up: Why does a Gaussian kernel applied through convolution blur an image, and how does kernel size affect blur radius?

Q10. What is deconvolution and where is it used?

Deconvolution is the inverse operation of convolution: given the output y = x * h and the system impulse response h, it recovers the input x, equivalent to dividing Y(jω) by H(jω) in the frequency domain. In ultrasound medical imaging, the received signal is the convolution of the tissue reflectivity with the transducer's pulse response; deconvolution sharpens the image by inverting this blurring effect. Direct deconvolution by frequency-domain division is noise-sensitive — small values in H(jω) amplify noise enormously — which is why Wiener deconvolution is used in practice to balance noise amplification against deblurring.

Follow-up: Why does direct frequency-domain deconvolution fail badly in the presence of noise?

Q11. What is the length of the linear convolution of two finite sequences of lengths N and M?

The linear convolution output has length N + M − 1 samples, because the shorter sequence takes M − 1 extra samples to fully clear the longer sequence when sliding through it. Convolving a 512-sample audio frame with a 64-tap FIR filter produces 512 + 64 − 1 = 575 output samples, of which the first and last 63 samples are transition effects. The overlap-add FFT convolution algorithm processes the 512-sample block as a 1024-point FFT (next power of 2 above 575) to handle this length correctly.

Follow-up: In the overlap-save method of block convolution, how many samples are saved from each block for the next processing frame?

Q12. How does an FIR filter implement convolution in a DSP processor?

An FIR filter computes y[n] = Σh[k]×x[n−k] for k=0 to N−1, multiplying each stored input sample in a delay buffer (shift register) by the corresponding filter coefficient and accumulating the products — this is direct-form convolution implemented as a multiply-accumulate (MAC) loop. A 64-tap FIR filter on a TMS320C5505 DSP runs its entire MAC loop in 64 clock cycles using the processor's hardware loop and dual-MAC capability, operating at the audio sample rate of 48 kHz. The filter coefficients are the sampled impulse response values h[k] computed offline by the Parks-McClellan algorithm or window method.

Follow-up: What is the computational complexity in MACs per output sample for a direct-form FIR filter with N coefficients?

Q13. What is the convolution of an exponential decay with itself?

The convolution of h(t) = e^(−at)u(t) with itself is t×e^(−at)u(t) — a critically damped second-order impulse response. This result arises naturally when two identical first-order systems are cascaded, such as two identical RC stages connected in series (with a buffer between them to prevent loading). The t×e^(−at) shape shows the output ramps up before decaying, which is the characteristic impulse response of a critically damped RLC circuit with repeated poles at s = −a.

Follow-up: How would the shape of the convolution output change if the two exponentials had different decay rates a and b?

Q14. What is cross-correlation and how does it differ from convolution?

Cross-correlation slides one signal across another without flipping, computing Rxy(τ) = ∫x(t)y(t+τ)dt, measuring the similarity between two signals as a function of lag τ. Convolution flips one function before sliding: y(t) = ∫x(τ)h(t−τ)dτ. In GPS receivers, the acquisition correlator cross-correlates the received satellite signal with locally generated pseudorandom code replicas to detect code phase — an operation implemented as FFT-based cross-correlation on a Qualcomm Atheros GPS chipset. Autocorrelation (signal cross-correlated with itself) reveals periodicity in noisy signals.

Follow-up: How is cross-correlation used to measure the time delay between two sensor signals in an acoustic localization system?

Q15. What is the distributive property of convolution and how does it apply to parallel filter banks?

Convolution distributes over addition: x * (h1 + h2) = (x * h1) + (x * h2), meaning convolving an input with a sum of impulse responses equals summing the individual convolution outputs. A graphic equalizer with 10 bandpass filters operating in parallel computes the output as the sum of the input convolved with each band's impulse response, and by the distributive property this equals convolving the input with the sum of all band impulse responses — a single equivalent filter. This distributive property justifies combining multiple parallel FIR filter paths into a single merged FIR filter for computational efficiency.

Follow-up: How does the distributive property allow you to combine a parametric equalizer's multiple filter bands into one equivalent FIR filter?