What does a Gabor filter "filter" out?

Question

Almost all of the searches online have returned results related to image-processing, but I am looking for a more rudimentary example with some physical intuition (eg; a high-pass filter only lets the high frequency components pass, thus filtering out the low frequency components).

The specific filter that I found and am wanting to understand is:

$$g(t) = \sin\left(\frac{2 \pi t}{\lambda} + \phi\right)\exp\left(\frac{-t^2}{2 \sigma^2}\right)$$

1. Is this a general expression? Would it matter if I used $\cos$ instead of $\sin$?

2. What would the O/P look like if I applied the above 1D Gabor filters to a sinusoid? I am not exactly sure what a good sinusoid would be to demonstrate this, but I am guessing a simple sine/cosine won't suffice, since it has a single frequency?

3. How would changing the values of $\lambda$, $\sigma$, $\phi$ affect the O/P?

Answer 1

Multiplication in time-domain is convolution in Fourier domain. You have $\mathscr{F}\{\sin\}$ (assuming $\phi=0$) which is two Dirac-Delta impulses in the Fourier domain. The FT of a Gaussian (the exponential term) is also a Gaussian, proof.

So, in the Fourier domain you have a Gaussian convolved with two shifted impulses which will just place the Gaussian at those impulses. With that out of the way here are answers to your specific questions:

1. $\cos$ and $\sin$ have slightly different Fourier Transforms: $$\cos(2 \pi f_0 t)\overset{\mathscr{F}}{\longleftrightarrow}\tfrac12\big(\delta(f-f_{0})+\delta(f+f_{0})\big)$$ and $$\sin(2 \pi f_0 t)\overset{\mathscr{F}}{\longleftrightarrow}\tfrac1{2j}\big(\delta(f-f_{0})-\delta(f+f_{0})\big)$$

2. Depends at which frequency your sinusoidal is. You can use a chirp to test it.

3. $\lambda$ controls the frequency of your $\sin$, which determines the resonant frequency of the filter, $\phi$ changes the phase shift which adds an exponential term (magnitude only) in the Fourier Domain (verify this by writing the sinusoidal in terms of complex exponentials). $\sigma$ controls the width of your Gaussian. It is the variance. Essentially, changing these parameters changes the shape of your filter which will lead to filtering (attenuating) different frequencies depending on the shape.