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Significance of convolution; compute convolution

Short answer

Convolution models LTI output as sum of scaled shifted impulses, essential for time-domain filtering. For h(n)={1,0,1}, x(n)={1,-2,-2,3,4}, y(n)=x(n)+x(n-2)={1,-2,-1,1,2,3,4} for n=0 to 6.

Illustrate the significance of convolution summation in digital signal analysis. Compute the convolution of the following signals: h(n) = {1,0,1} and x(n) = {1,−2,−2,3,4}.

Significance of convolution

  • The output of any LTI system is the convolution of the input with the impulse response: y[n] = x[n] * h[n] = Σ x[k] h[n − k]. So the single sequence h[n] fully characterises the system.
  • Any input is a sum of scaled, shifted impulses; the output is the sum of correspondingly scaled, shifted impulse responses (superposition).
  • It is how digital filtering is done in the time domain (FIR filtering is convolution with the filter coefficients).
  • Its properties (commutative, associative, distributive) let us combine cascaded and parallel systems: cascade h = h₁ * h₂, parallel h = h₁ + h₂.
  • Convolution in time becomes multiplication in the z or frequency domain: Y(z) = X(z)H(z). This is the basis of fast convolution using the FFT.
  • System properties such as causality and stability can be read directly from h[n].

Computation

h(n) = {1, 0, 1}, x(n) = {1, −2, −2, 3, 4}, both starting at n = 0. Output length = 3 + 5 − 1 = 7.

Since h has taps 1, 0, 1: y(n) = x(n) + x(n − 2).

n0123456
x(n)1−2−23400
x(n−2)001−2−234
y(n)1−2−11234

Answer: y(n) = {1, −2, −1, 1, 2, 3, 4}, n = 0 … 6.

Check: Σy = 8 = (Σx)(Σh) = 4 × 2. ✓

✓ Answer checked by Claude Opus 5.5 (high effort)

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