The convolution theorem

Any signal may be understood as consisting of a sequence of impulses. This is obvious in the case of sampled signals, but can be generalized to continuous signals by representing the signal as a continuous sequence of Dirac impulses. We may construct the response of a linear system to an arbitrary input signal as a sum over suitably delayed and scaled impulse responses. This process is called a convolution:

$$g(t) = \int_{0}^{\infty} h(t') \ f(t-t') \ dt' = \int_{0}^{\infty} h(t-t') \ f(t') \ dt'$$ (15)

Here f(t) is the input signal and g(t) the output signal; h(t) characterizes the system. We assume that the signals are causal (i.e. zero at negative time), otherwise the integration would have to start at $t'=-\infty$ . Taking $f(t)=\delta(t)$ , i.e. using a single Dirac impulse as the input, we get g(t)=h(t), so h(t) is in fact the impulse response of the system.

The response of a linear system to an arbitrary input signal can thus be computed either by convolution with the impulse response in time domain, or by multiplication with the transfer function in the Laplace domain, or by multiplication with the complex frequency response in frequency domain.

A reason for choosing the FFT method is that responses are often specified in the frequency domain (this is, as a function of frequency), so one would anyhow need a Fourier transformation to determine the impulse response. Moreover, the impulse response has an infinite duration, so it can never be used in full length. The FFT method, on the other hand, assumes all signals to be periodic, which introduces certain inaccuracies as well; the signals must in general be tapered to avoid spurious results. Fig 1 illustrates the interrelations between signal processing in the time and frequency domains.

In digital processing, these methods translate into convolving discrete time series or transforming them with the FFT method and multiplying the transforms. For impulse responses with more than 100 samples, the FFT method is usually more efficient. The convolution method is also known as a FIR (finite impulse response) filtration. A third method, the recursive or IIR (infinite impulse response) filtration, is only applicable to digital signals; it is often preferred for its flexibility and efficiency although its accuracy requires special attention.

  

Figure 1: Pathways of signal processing in the time and frequency domains. The asterisk between h(t) and f(t) indicates a convolution.