Transfer Function

The transfer function of a linear, shift-invariant system characterizes the system’s response in the frequency domain.

Definition

For a system with impulse response $h(x, y)$, the transfer function $H(f_x, f_y)$ is the Fourier transform of the impulse response:

$$ H(f_x, f_y) = \iint_{-\infty}^{\infty} h(x, y) \, e^{-j2\pi(f_x x + f_y y)} \, dx \, dy $$

Input-Output Relationship

For an input $g_1(x, y)$ with Fourier transform $G_1(f_x, f_y)$, the output $g_2(x, y)$ has Fourier transform:

$$ G_2(f_x, f_y) = H(f_x, f_y) \cdot G_1(f_x, f_y) $$

This follows from the convolution theorem: multiplication in the frequency domain corresponds to convolution in the spatial domain.

Properties

• Magnitude $|H(f_x, f_y)|$: describes amplitude scaling at each frequency
• Phase $\arg H(f_x, f_y)$: describes phase shift at each frequency
• A system is stable if the transfer function is bounded

Optical Transfer Function (OTF)

In optics, the transfer function of an imaging system is called the Optical Transfer Function. Its magnitude is the Modulation Transfer Function (MTF), and its phase is the Phase Transfer Function (PTF).