Signals are attenuated by echoes of themselves that are recieved at the same time as the reciever.

The form of a signal recieved through a multipath time varying channel is

\begin{equation} r(t) = \Re \left( \left[ \alpha_n(t)e^{-j\phi_n(t)}u(t-\tau_n(t)) \right] e^{-j2\pi f_c t} \right) \end{equation}
\begin{equation} \phi_n(t) = 2\pi f_c \tau_n (t) - \phi_D_n \end{equation}


Basically, the e exponent represents a phase shift that is a function of frequency, the delay of the path, and doppler phase shift. Alpha is a simple attenuation.

Note that these two factors, phase shift and attenuation are caused by two different processes (delay and doppler vs path loss and shadowing) and so can be treated as independent.

By letting

\begin{equation} c(\tau, t) = \sum_{n=0}^{N(t)} \alpha_n(t)e^{-j\phi_n(t)}\delta(\tau-\tau_n(t)) \end{equation}

be the channel, which essentially abstracts away the two factors of attenuation and phase shift. The \(\delta\) is the kronecker delta function.

Very important that \(c\), the channel has two parameters, this is what differentiates the time varying channel from the invariant one. \(t\) is when the impulse response(i.e, the channel) is recieved, and \(t - \tau\) is when the impulse is launched into the channel (transmission time). This means that \(\tau\) is basically time of flight.

If the channel is time invariant, then

\begin{equation} c(\tau) = \sum \alpha_n e^{-j\phi_n} \delta(\tau-\tau_n) \end{equation}

Now we can rewrite the recieved signal as a convolution as

\begin{equation} \Re \left( \left( \int_{-\infty}^{\infty} c(\tau, t) u(t - \tau)d\tau \right) e^{j2\pi f_c t} \right) \end{equation}

Typically, \(f_c \tau_n >> 1\). If this is true, then small change in \(\tau_n(t)\) can lead to very large phase changes in the nth mulipath component. This causes rapid variation in recieved signal strength, this is called fading.

If the channel delay spread is small then it is likely that multipath and LOS are nonresolvable. This leads to narrowband fading. If the delay spread is large, then this is wideband fading.

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