= Solution
\b[<Convolution> describes the response of a <linear time-invariant system> to an arbitrary input.] If $h$ is its <impulse response> and $f$ its input,
$$
\boxed{g(t)=(h*f)(t)=\int_{-\infty}^{\infty}h(t-\tau)f(\tau)\,d\tau.}
$$
This is a weighted sum of responses to shifted <impulses>. For integrable <functions> the operation is commutative and associative. A causal system has $h(t)=0$ for $t<0$, restricting the <integral> to past inputs. With the Fourier convention $F(\omega)=\int f(t)e^{-i\omega t}dt$, the <convolution theorem> gives $G=HF$.
For example, an <RC low-pass frequency response> has $h(t)=\tau_0^{-1}e^{-t/\tau_0}\Theta(t)$ and $H(\omega)=(1+i\omega\tau_0)^{-1}$, where $\Theta$ is the <Heaviside step function>. Fast input variations are smoothed. The same mathematics describes optical blurring by a <point spread function> and many other <linear response> measurements.
<Deconvolution> tries to recover $f$ from measured $g$ and known $h$. In noise-free data one formally sets $F=G/H$ where $H\ne0$. Zeros of $H$ lose corresponding information; small values amplify measurement <noise>. If the measured spectrum is $G+N$, the direct estimate has error $N/H$, which can be very large at <frequencies> strongly suppressed by the instrument. An inverse is therefore not automatically unique or stable.
One <regularized spectral deconvolution> reconstruction is
$$
\boxed{F_\lambda(\omega)=\frac{H(\omega)^*G(\omega)}{|H(\omega)|^2+\lambda},\qquad\lambda>0.}
$$
This minimizes $|HF-G|^2+\lambda|F|^2$ at each <frequency> and limits <noise> amplification, at the cost of bias and lost resolution. Choosing the regularization requires assumptions about <noise> and the desired <signal>; <deconvolution> cannot restore components that the data do not determine without such additional information.
Back to article page
