Fourier Transformation

Fourier transformation

It transforms between time and frequency domains.

h (t) \circ - ∙ H (f)

Transfer function

H (f) = \int h (t) e^{- i 2 π f t} d t

Impulse response

h (t) = \int H (f) e^{i 2 π f t} d f

For general functions, equivalent descriptions in time and frequency domains are based on Fourier and inverse Fourier transformations.

Fourier transformation

S (f) = \int s (t) e^{- i 2 π f t} d t = F T {s (t)}

Inverse Fourier transformation

s (t) = \int S (f) e^{i 2 π f t} d f = F T^{- 1} {S (f)}

(notice that the $e x p$ with different numbers!)

Use in Computational Neuroscience

the impulse response is important in experimental approach to system identification: Apply sine stimuli and measure $S (f)$ ,
i.e., gain and phase, for all frequencies $f$ , then you can get $s (t)$ .

Special Cases

FT of sine & cosine

the FT of cosine functions are real-valued
the FT of sine functions are purely imaginary

Proof

ω = 2 π f

F T {δ (t - t_{0})} = \int_{- \infty}^{\infty} δ (t - t_{0}) e^{- i ω t} d t = e^{- i ω t_{0}}

F T {e^{- i ω_{0} t}} = \int_{- \infty}^{\infty} e^{- i ω_{0} t} e^{- i ω t} d t = \int_{- \infty}^{\infty} e^{- i (ω + ω_{0}) t} = 2 π δ (ω - ω_{0})

Euler´s formula:

\cos (ω_{0} t) = \frac{1}{2} (e^{i ω_{0} t} + e^{- i ω_{0} t})

\sin (ω_{0} t) = \frac{1}{2 i} (e^{i ω_{0} t} - e^{- i ω_{0} t})

then, $$FT{\cos(\omega_0 t)}
= \int_{-\infty}^{\infty} \cos(\omega_0 t)e^{-i\omega t}dt \
= \frac{1}{2} (( \int_{-\infty}^{\infty} e^{i\omega_0 t}e^{-i\omega t}dt+
\int_{-\infty}^{\infty} e^{-i\omega_0 t}e^{-i\omega t}dt) \
= \frac{1}{2} ( \int_{-\infty}^{\infty} e^{-i(\omega-\omega_0)t}dt+
\int_{-\infty}^{\infty} e^{-i(\omega+\omega_0) t}dt) \
= \pi (\delta (\omega+\omega_0)+\delta(\omega-\omega_0))$$
Similar for sine (purely imaginary):

F T {\sin (ω_{0} t)} = i π (δ (ω + ω_{0}) - δ (ω - ω_{0}))

Difference of Gaussians (DoG)

A classical model for a radially symmetrical center-surround receptive field is a Difference of Gaussians (DoG), where two Gaussians with different widths are subtracted from each other (Mexican hat model).

FT of Dirac Delta function

s (t) * δ (t) = \int s (τ) δ (t - τ) d τ = s (t)

δ (f - f_{0}) = \int e^{- i 2 π (f - f_{0}) t} d t

F T {δ (t - t_{0})} = \int δ (t - t_{0}) e^{i 2 π f t} d t = e^{- i 2 π f t_{0}}

F T {e^{- i 2 π f_{0} t}} = \int e^{- i 2 π f_{0} t} e^{i 2 π f t} d t = \int e^{- i 2 π (f - f_{0}) t} d t = δ (f - f_{0})

FT of a Gaussian kernel

With a Gaussian $G (x; σ) = \frac{1}{\sqrt{2 π} σ} e x p (- \frac{x^{2}}{2 σ^{2}})$ :

F T {G (x; σ)} = \frac{1}{\sqrt{2 π}} e x p (- \frac{σ^{2} ω^{2}}{2})

So the Fourier transform of the Gaussian function is again a Gaussian function, but now of the frequency $ω$ .
A smaller kernel in the spatial domain gives a wider kernel in the Fourier domain, and vice versa.

FT and Convolution

Convolution in time domain can be converted to multiplication in frequency domain by Fourier transformation.

r (t) = s (t) * h (t) \circ - ∙ R (f) = S (f) H (f)

\int | s (t) |^{2} d t \circ - ∙ \int | S (f) |^{2} d f