System Functions of the Channel

The four system functions and their Fourier relationships:

1. Time-variant impulse response: $h(\tau; t)$ — response at time $t$ to impulse at delay $\tau$

2. Time-variant transfer function: $H(f; t)$ — frequency response at time $t$ $$H(f; t) = \int h(\tau; t)\, e^{-j2\pi f\tau}\, d\tau$$ (FT over $\tau$)

3. Doppler-spread function: $S(\tau; \nu)$ — spreading in delay $\tau$ and Doppler $\nu$ $$S(\tau; \nu) = \int h(\tau; t)\, e^{-j2\pi \nu t}\, dt$$ (FT over $t$)

4. Output Doppler-spread function: $B(f; \nu)$ $$B(f; \nu) = \int H(f; t)\, e^{-j2\pi \nu t}\, dt$$ (FT over $t$), or equivalently FT of $S(\tau; \nu)$ over $\tau$.

These form a 2D Fourier transform pair: $h(\tau; t) \xleftrightarrow{\tau \leftrightarrow f} H(f; t) \xleftrightarrow{t \leftrightarrow \nu} B(f; \nu)$ and $h(\tau; t) \xleftrightarrow{t \leftrightarrow \nu} S(\tau; \nu) \xleftrightarrow{\tau \leftrightarrow f} B(f; \nu)$.

Under the WSSUS assumption, the scattering function is

$$C_S(\tau; \nu) = E[S(\tau; \nu)\, S^*(\tau; \nu)]$$

It describes how the channel's power is distributed in the delay-Doppler plane. The scattering function provides the most complete second-order description of a WSSUS channel.

Its marginals are:

• Integrating over $\nu$: $\int C_S(\tau; \nu)\, d\nu = P(\tau)$ (power delay profile)
• Integrating over $\tau$: $\int C_S(\tau; \nu)\, d\tau = S_H(\nu)$ (Doppler spectrum)