In a previous blog post, we discussed how linear systems can be characterized by their impulse response. However, it is often difficult to stimulate a linear system with perfect pulses. An alternative—and often insightful—way of characterizing a linear system is through its frequency response.

Characterization of linear systems is closely linked to the Fourier theorem:

Under "natural" conditions, we can represent a signal \(x=(x_t)_{t=1}^N\) as a superposition of complex sinusoids, i.e.

$$ x = \sum_k X_k\exp(itk) = \sum_k C_k \cos(tk) + iS_k \sin(tk) = \sum_k A_k\cos(k(t-\phi_k)), $$

where \(i=\sqrt{-1}\).

Here we used three identities. Most technical expositions present the Fourier theorem only in terms of the first term, which makes use of complex numbers, i.e. numbers that can be written as \(z=x+iy\), with \(i=\sqrt{-1}\). Although these numbers are called "complex", a better name would probably be simplifying numbers, because the simplify many difficult calculations and they certainly do so for the Fourier transform.

I included the other two identities here, to make the connection to sine waves explicit. The second identity directly uses Euler's identity to emphasize that the terms in the sum are really cosine and sine terms. The last identity is meant to illustrate that the combinations of cosine and sine terms can also be combined into a single shifted cosine function. In this case, we refer to \(A_k\) as the amplitude of the cosine and \(\phi_k\) as the phase of the cosine.

So, why is the Fourier theorem important to understand linear time invariant systems? The main reason is that the \(\exp(itk)\) are so called eigenvectors of these systems. That means the operation of a linear time invariant system to \(\exp(itk)\) is simple a (complex) multiplication,

With a complex number \(H_k\). In other words, no matter how complex the system is, if it is linear and time invariant, its operation on sinusoids is simply a multiplication by a complex number. In particular, we have that \(f(\cos(tk)) = H_k^* \cos(k(t-H_k^\phi))\), i.e. the operation of the system is a change in the amplitude and the phase of the cosine, but not (for example) the shape or the frequency.

In a previous blog post, we observed that a linear time invariant system is fully characterized by its impulse response. In face, the same is true for a system's frequency response \(H=(..., H_k, H_{k+1}, ...)\). To see this, we simply take an arbitrary input signal \(x=(x_t)_{t=1}^N\) and we expand it using the Fourier theorem:

Because \(f\) is linear, we can move it into the sum to get

Now \(f\) is only operating on the \(\exp(ikt)\) terms and for those, we know that it can be expressed as multiplication with the respective \(H_k\) factor. Thus, in total, we have

Note, that in the Fourier domain, the convolution operation that was needed with the impulse response characterization turns into a simple, pointwise multiplication.

Note that the frequency response consists of complex numbers. That is, every \(H_k\) has a real and an imaginary part. Although complex numbers simplify many calculations, they don't just simply give us an intuitive insight into how the system operates. It is therefore common to separate the sequence of \(H_k\) values into \(H_k^*\) and \(H_k^\phi\) sequences. The amplitude response \(H_k^*\) describes how much signal components with the frequency \(k\) are attenuated, while the phase response \(H_k^\phi\) describes how much signal components with the frequency \(k\) are shifted.

It turns out that measuring the phase response of the visual system is not easy, because humans tend to move their eyes when viewing images. However, in a seminal study Campbell and Robson (1968) have measured the amplitude response function of the visual system using sine gratings. They found that \(H_k\) starts low for very low spatial frequency \(k\), increases up to spatial frequencies of \(k\approx 5\) cycles per degree visual angle (cpd, this number depends a bit on the viewing conditions) and then quickly goes down to zero for frequencies higher than \(k=10\) or \(k=12\)cpd. Knowing that the amplitude is essentially 0 for high frequencies means that it is actually sufficient to test the visual system only for frequencies up to about 10cpd. That's a managable number of measurements.