Convolution Properties

Convolution is the mathematical operation that computes the output of any LTI system given its impulse response and an arbitrary input. Engineers use convolution to predict how a filter reshapes audio, how a communication channel distorts a pulse, and how a feedback controller reacts to a disturbance.

Core Concept

Convolution answers the question: if a system responds to an impulse with h(t), what does it output for any other input x(t)? The answer is the integral of x(tau) times h(t - tau) over all tau. You are summing the system's response to every past input sample, each weighted by the input's amplitude at that sample time.

The operation has several important properties that make analysis tractable. Commutativity means x * h equals h * x. Associativity means you can regroup cascaded convolutions. Distributivity means convolution distributes over addition, which directly corresponds to parallel systems.

The convolution theorem ties time-domain convolution to multiplication in the frequency domain. Convolving two signals in time equals multiplying their Fourier or Laplace transforms. This is why filter design works: you shape H(jw) in frequency, and the time-domain filtering is automatically the convolution with h(t).

Key Formula

Continuous-time convolution: y(t) = integral from -inf to inf of x(tau) * h(t - tau) d(tau), written as y(t) = x(t) * h(t).

Discrete-time convolution: y[n] = sum over k from -inf to inf of x[k] * h[n - k].

tau or k: dummy variable of integration or summation. h(t-tau): impulse response flipped and shifted. The output at time t accumulates all past inputs weighted by the impulse response.

Problem: Convolve x[n] = {1, 2, 1} (n=0,1,2) with h[n] = {1, 1} (n=0,1).

Method: Tabular / shift-and-add

     x[n]:  1  2  1
     h[0]=1: 1  2  1
     h[1]=1:    1  2  1

Add column by column:
  n=0:  1
  n=1:  2 + 1 = 3
  n=2:  1 + 2 = 3
  n=3:  0 + 1 = 1

Final Answer: y[n] = {1, 3, 3, 1} for n = 0, 1, 2, 3
Length check: len(y) = len(x) + len(h) - 1 = 3 + 2 - 1 = 4. Correct.

Exam Tip: The length of a finite convolution is N1 + N2 - 1, where N1 and N2 are the lengths of the two sequences. GATE problems frequently ask for the value of y[n] at a specific index; compute only that column rather than the full table to save time. For causal LTI systems with causal input, the lower limit of the convolution sum shifts from -inf to 0, reducing the computation range significantly.

Properties Summary

• Commutativity: x(t) * h(t) = h(t) * x(t); order of operands does not matter.
• Associativity: [x(t) * h1(t)] * h2(t) = x(t) * [h1(t) * h2(t)]; cascades can be merged.
• Distributivity: x(t) * [h1(t) + h2(t)] = x(t)*h1(t) + x(t)*h2(t); parallel systems add.
• Convolution theorem: x(t)*h(t) <-> X(jw)*H(jw); convolution in time = multiplication in frequency.
• Time-shift: if x(t)*h(t)=y(t), then x(t-t1)*h(t-t2) = y(t-t1-t2).
• Identity element: convolving any signal with delta(t) returns the signal unchanged: x(t)*delta(t)=x(t).
• Duration: convolving a signal of duration T1 with one of duration T2 gives a signal of duration T1+T2.

Quick Revision

• Convolution is the fundamental input-output relationship for every LTI system.
• Flip h(tau) about the origin, shift it by t, multiply pointwise with x(tau), integrate.
• In frequency domain, convolution becomes multiplication: Y = X * H.
• Length rule for finite sequences: output length = N1 + N2 - 1.
• Associativity justifies replacing a cascade of filters with a single equivalent filter.
• Distributivity justifies replacing parallel filters with a single summed impulse response.
• Convolution with u(t) computes the running integral of the signal.
• Exam trap: students flip x instead of h when sliding; always flip and shift h(tau) to compute h(t-tau), not x.