The convolution theorem converts the tedious integral of convolution into simple multiplication in the frequency domain. Every linear filter in audio, image processing, and communications is designed and analyzed using this theorem.

Convolution theorem: two equivalent approaches to finding system output

Core Concept

Convolution in time computes the output of a linear time-invariant system given an input and impulse response. Computing it directly requires a sliding integral — expensive for long signals. The convolution theorem replaces that integral with pointwise multiplication of two spectra.

The physical meaning is direct: H(f) is the system's frequency response. Multiplying X(f) by H(f) scales each frequency component of the input by the system's gain at that frequency. High-pass, low-pass, and band-pass filters all work exactly this way.

The dual holds too: multiplication in time becomes convolution in frequency. This is used in amplitude modulation analysis, where a carrier multiplied by a baseband signal produces a spectrum that is the convolution of the two individual spectra.

Key Formula

The convolution theorem states: if y(t) = x(t) * h(t) then Y(f) = X(f)·H(f). The asterisk denotes convolution: y(t) = ∫ x(τ)h(t-τ)dτ. Each of X(f), H(f), Y(f) is a complex-valued function of frequency f in hertz. The dual (multiplication theorem): if y(t) = x(t)·h(t) then Y(f) = X(f)*H(f), where the asterisk on the right now means convolution of the two spectra.

Example

Problem: x(t) = e^(-t)u(t), h(t) = e^(-2t)u(t). Find output y(t) = x(t)*h(t) using FT.

Step 1 — Transform each signal:
  X(f) = 1/(1 + j2πf)    [standard pair for e^(-at)u(t), a=1]
  H(f) = 1/(2 + j2πf)    [a=2]

Step 2 — Multiply in frequency domain:
  Y(f) = X(f)·H(f)
       = 1 / [(1 + j2πf)(2 + j2πf)]

Step 3 — Partial fractions:
  Y(f) = A/(1+j2πf) + B/(2+j2πf)
  A(2+j2πf) + B(1+j2πf) = 1
  Set j2πf = -1: A(1) = 1 → A = 1
  Set j2πf = -2: B(-1) = 1 → B = -1
  Y(f) = 1/(1+j2πf) - 1/(2+j2πf)

Step 4 — Inverse transform each term:
  y(t) = e^(-t)u(t) - e^(-2t)u(t)
       = (e^(-t) - e^(-2t))u(t)

Final Answer: y(t) = (e^(-t) - e^(-2t))u(t)

Exam Tip: The convolution theorem requires both signals to have Fourier transforms — they must be absolutely integrable or have transforms in the generalised sense. In GATE, questions often give H(f) directly as a rectangular function (ideal low-pass filter). Then Y(f) = X(f)·rect(f/2W), and you inverse transform using the sinc function. Always check whether the result is causal before writing u(t) in the final answer.

Properties Summary

Convolution theorem: x(t)*h(t) ↔ X(f)·H(f) — time convolution equals frequency multiplication.
Dual (multiplication theorem): x(t)·h(t) ↔ X(f)*H(f) — time product equals frequency convolution.
LTI output: Y(f) = H(f)·X(f), where H(f) is the transfer function.
Filtering: H(f) acts as a pointwise gain/phase modifier at each frequency.
Commutativity: x(t)*h(t) = h(t)*x(t) — order does not matter.
Associativity: cascaded systems multiply their transfer functions: H₁(f)·H₂(f).
Parseval link: energy of convolution relates to product of individual energy spectra.

Quick Revision

Convolution in time ↔ multiplication in frequency.
Multiplication in time ↔ convolution in frequency.
LTI frequency response H(f) shapes the input spectrum X(f).
Cascade of two systems: H(f) = H₁(f)·H₂(f).
Partial fractions in frequency domain invert easily using standard transform pairs.
Ideal LPF with cutoff W has H(f) = rect(f/2W).
Convolution is commutative and associative — system order can be swapped.
Exam trap: applying the dual incorrectly — multiplying in frequency should give convolution in time, not another multiplication.

#Signals & Systems