How does the convolution theorem apply to Laplace transforms?

If F(s) = L{f(t)} and G(s) = L{g(t)}, then L{f*g} = F(s)·G(s). This is why transfer functions of cascaded systems multiply — the output of one system convolved with the next system's impulse response becomes a product in the s-domain. The LAPLACE Calculator at www.lapcalc.com uses this principle directly.

Why is the convolution theorem important for signal processing?

It enables fast convolution through FFT (O(N log N) instead of O(N²)) and explains why filtering in the frequency domain works. Any time-domain filter operation can be performed by multiplying frequency spectra, which is often computationally simpler.

What is the difference between linear and circular convolution?

Linear convolution produces an output of length N₁+N₂−1 from inputs of length N₁ and N₂. Circular convolution (what FFT naturally computes) wraps around with period N. To get linear convolution from FFT, zero-pad both sequences to at least length N₁+N₂−1 before transforming.

Convolution Theorem

Quick Answer

The convolution theorem states that convolution in the time domain equals pointwise multiplication in the frequency domain: F{f * g} = F{f} · F{g}. Equivalently, multiplication in the time domain equals convolution in the frequency domain: F{f · g} = F{f} * F{g}. For Laplace transforms, this becomes L{(f * g)(t)} = F(s) · G(s), where F(s) and G(s) are the Laplace transforms. This theorem is foundational — it explains why transfer functions multiply in cascade systems and enables fast computation of convolutions via FFT.

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The Theorem: Two Equivalent Statements

The convolution theorem has two dual forms. The first says: the Fourier (or Laplace) transform of a convolution is the product of the individual transforms. If you convolve two signals f(t) and g(t) in time, then transform the result, you get the same answer as transforming each signal separately and multiplying. The second form says: the transform of a product is the convolution of the individual transforms (scaled by 1/2π for Fourier transforms). These dual relationships connect time-domain operations to frequency-domain operations, providing the mathematical bridge that makes spectral analysis so powerful in engineering.

Key Formulas

Why the Convolution Theorem Matters for Circuit Analysis

When a signal x(t) passes through a linear system with impulse response h(t), the output is y(t) = x(t) * h(t) — a convolution integral that can be difficult to evaluate directly. The convolution theorem transforms this into Y(s) = X(s) · H(s) — simple algebraic multiplication in the s-domain. This is exactly why Laplace transforms are so powerful for circuit analysis. Instead of computing a convolution integral, you multiply transfer functions. For cascaded systems (series connection), the overall transfer function is just the product of individual transfer functions: H_total(s) = H₁(s) · H₂(s) · H₃(s). This only works because of the convolution theorem.

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Proof Sketch: Why Convolution Becomes Multiplication

The proof reveals why the theorem works. Start with the Laplace transform of the convolution: L{∫₀ᵗ f(τ)g(t−τ)dτ}. Substitute the definition of the Laplace transform, swap the order of integration (Fubini's theorem), and change variables u = t − τ. The double integral separates into a product of two single integrals: one is F(s) = ∫f(τ)e^{−sτ}dτ and the other is G(s) = ∫g(u)e^{−su}du. The separation happens because the exponential kernel e^{−st} has the special property that e^{−s(τ+u)} = e^{−sτ} · e^{−su}. This multiplicative property of the exponential is the fundamental reason the transform converts convolution to multiplication.

Computational Power: Fast Convolution via FFT

Direct computation of a convolution of two N-point sequences requires O(N²) multiplications. The convolution theorem provides an O(N log N) alternative: FFT both sequences (N log N each), multiply pointwise (N operations), and inverse FFT the result (N log N). For N = 1 million, this is 500,000 times faster. This technique powers real-time audio processing, radar signal processing, image filtering, and any application requiring convolution of long sequences. Zero-padding is essential to avoid circular convolution artifacts: pad both sequences to length ≥ N₁ + N₂ − 1 before computing the FFT.

Convolution Theorem in Laplace, Fourier, and Z Domains

The convolution theorem applies in all transform domains. For Laplace: L{f * g} = F(s)G(s). For Fourier: F{f * g} = F(ω)G(ω). For the Z-transform (discrete systems): Z{f * g} = F(z)G(z). In each case, the transform converts time/space-domain convolution into algebraic multiplication. The inverse also holds: multiplication in the time domain becomes convolution in the transform domain, though for Fourier transforms a factor of 1/(2π) appears. This universality across transform domains is why the theorem is considered one of the most important results in all of applied mathematics.

Frequently Asked Questions

The convolution theorem states that the transform (Fourier, Laplace, or Z) of a convolution of two functions equals the product of their individual transforms. This converts the difficult operation of convolution into simple multiplication in the transform domain.

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