A channel under narrowband fading is called a frequency-flat channel.

Applies when channel delay spread is small relative to inverse signal bandwidth. This state implies that the delay of each multipath component is approx lower than the delay spread.

The following is a mathematical description of the features of narrowband fading. Simpler computational models like the FSMC.

Our received signal appears as a single pulse, as if there was no multipath at all.

This means that recieved signal is

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

Do note that the exp on the left is the carrier.

The whole sum tends to be very close to a single complex number, and may very quickly because of phase fluctuation. There exist models for this sum value, sometimes called $$\beta(t)$$, such as rayleigh fading model or the rician fading model.

The scale factor from the sum is now constant (of the transmitted signal).

If the signal, $$s(t)$$ is an unmodulated carrier with phase offset $$\phi_0$$,

\begin{equation} s(t) = \Re\left( e^{j(2\pi f_c t + \phi_0)} \right) = \cos(2\pi f_c t - \phi_0) \end{equation}

now,

\begin{equation} r(t) = r_I(t) \cos(2\pi f_c t) + r_Q(t) \sin{2\pi f_c t} \end{equation}

where $$r_I$$ is the in-phase component,

\begin{equation} r_I = \sum_{n = 1}^{N} \alpha(t) \cos{\phi_n} \end{equation}

And $$r_Q$$ is the quadrature component of a quadrature signal:

\begin{equation} r_Q = \sum_{n = 1}^{N} \alpha(t) \cos{\phi_n} \end{equation}

Note that $$r_I$$ and $$r_Q$$ are treated as independent Gaussian processes, following from the fact that they are uncorrelated since each is multiplied by sine or cosine.

The phase is

\begin{equation} \phi_n(t) = 2\pi f_c \tau_n (t) - \phi_D - \phi_0 \end{equation}

## 1. Correlation results

Under some assumptions, namely

• No domininant LOS path
• $$\alpha_n$$, $$\tau_n(t)$$ and $$f_D_n(t)$$ are constant enough over time intervals of interest
• $$2\pi f_c \tau_n$$ changes rapidly compared to every other term in the expression for the $n$the multipath.

Cross correlation of $$r_I$$ and $$r_Q$$ is simply

\begin{equation} -\mathbb{E}[r_Q(t)r_I(t + \Delta t)] \end{equation}

## 2. Simulation technique

Common method for simulating the complex envelope of the narrowband fading process is to pass two gaussian noise sources with PSD $$N/2$$ through lowpass filters with frequency response $$H(f)$$ such that $$S_r_I(f) = S_r_Q(f) = \frac{N}{2}|H(f)|^2$$