# Flat fading In practice, the channel is considered flat when the signal BW is less than the [coherence BW](Coherence%20bandwidth.md). All signal frequencies undergo the same fading when BW of Tx signal < coherence bandwidth. This means the channel has a single impulse frequency response and $y(t)=\alpha x(t)+n(t)$ ## Modelling  %%[🖋 Edit in Excalidraw](Public%20Extras/Doodles/Flat%20fading-attachment-drawing.md), and the [dark exported image](Public%20Extras/Doodles/Flat%20fading-attachment-drawing.dark.svg#invert)%% $\alpha = \sum Re(a_ie^{j\omega_1\tau_1})+j\sum Im(a_ie^{j\omega_1\tau_1})$ $\begin{align}\alpha&=\alpha_I+j\alpha_Q\\&=re^{j\theta}\end{align}$ The real and imaginary parts of alpha (the channel response) can be modelled as independent zero mean Gaussian random variables (by [CLT](central%20limit%20theorem.md), assuming there is no line of sight). $r=\sqrt{\alpha_I^2+\alpha_Q^2}\in [0,\infty)$ is [rayleigh](Rayleigh%20distribution.md) distributed and $\theta\in (-\pi,\pi]$ is uniformly distributed. ## Distributions The Rayleigh distribution is not symmetric!  [^1] It seems that the likelihood of the channel boosting your signal strength is lower than the probability it will attenuate your channel. ### Assumptions Rayleigh channel model assumes all multipath components have zero mean value (non dominant component). But there might be a direct line of sight! ### Channel Power Gain Distribution $f_p(p)=\frac{1}{P_0}exp\left(\frac{-p}{P_0}\right)$ where $p=r^2$ and $P_0=2\sigma^2$ (the mean channel gain)  ### How can the channel boost the signal? Possibly we get constructive interference at the receiver. For a given delay, we may have contructive interference and improve the signal magnitude! ## Is fading bad? Generally fading is bad, unless you can estimate the channel and if you know the channel gain is higher than 1 (i.e. constructive interference is happening). ### Counteracting fading Originally, the fading was counteracted. This is done by decreasing the variance until the channel becomes almost deterministic with variance equal to zero. You can reduce observation variance using [diversity](diversity%20scheme.md) schemes. ### Exploiting Fading The chance for gains to be higher than 1 allows clever designers to exploit it and achieve better system performance. # References 1. [TELE4652-lecture-04](../../Spaces/University/TELE4652/Lectures/TELE4652-lecture-04.pdf) # Footnotes [^1]: https://www.mathworks.com/help/stats/raylpdf.html