# Practicality of the assumption of i.i.d. for Rayleigh channels

I would like to understand how far the assumption of having i.i.d. is accurate/valid (from the practical point of view) when an OFDM system is working in Rayleigh channels. Is this means the channel has to encounter flat and slow fading? If not, under what conditions, the assumption of i.i.d. can be considered practically acceptable?

Any hints?

• The flat fading assumption is generally not true for a mobile wireless channel. But OFDM divides the spectrum in many, narrow channels which are approximately flat. This interactive simulation demonstrates that nicely: webdemo.inue.uni-stuttgart.de/webdemos/03_theses/OFDM/…
– Deve
Oct 29, 2015 at 17:14
• @Deve So the i.i.d. assumption for OFDM channels is practically acceptable (especially as number of sub-bands increases) due to the fact that OFDM divides the spectrum into narrow sub-bands. Is my understanding is accurate?
– Noor
Oct 29, 2015 at 17:23
• I think you should define what should be independent and identically distributed (i.i.d.) before we can answer this question.
– Deve
Oct 29, 2015 at 18:01
• @Deve I mean by i.i.d. is that all subchannel gains have the same PDF and all subchannels are mutually independent.
– Noor
Oct 29, 2015 at 18:57

I am a little late but I post my answer anyway so that someone having the same question will find it interesting and discuss.

The discrete baseband multipath channel can be modeled as a FIR, i.e. $$y[n] = \sum_{l=0}^{L-1} x[n-l] h_l + w[n]$$ where $L$ is the number of channel taps. $L$ depends on the relation between the bandwidth of basis waveform and the delay spread of channel.

The term "Rayleigh fading" channel implies that the channel taps $h_l$ can be modeled as i.i.d circular symmetric Gaussian complex random variables because :

• $h_l$ is the sum of a large number of small independent circular symmetric random variables, each random variable is channel of a physical path. This the rich scattering assumption, which is typically vaid for urban environment.
• there is not a particular path having gain much significant than others. Otherwise, we have Rician fading.

Let me call this random variables "Rayleigh".

With a sufficient cyclic prefix ("sufficient" means greater than delay spread, thus OFDM receiver captures all delayed versions of OFDM symbol, proof can be found at single-tap OFDM regardless subcarrier spacing), the demodulated data at subcarrier $k$ is $$z[k] = x[k] \times \sum_{l=0}^{L-1}h_l e^{-j2\pi \frac{l}{N} k} = x[k] \times H[k]$$ where $N$ is DFT size.

The channel taps $h_l$ are i.i.d circular symmetric Gaussian random variables, $H[k]$ are circular symmetric Gaussian random variables, but they are in general not iid.

As pointed out by Maximilian Matthé in the comment, the covariance matrix is $F \mathrm{diag}(\vec{p}_0) F^H$ where $\vec{p}_0$ is Power Delay Profile zero-padded to size $N$. The frequency bins at $k = u \times N/L, u \in \mathbb{N}$ are independent, if $N/L$ is integer. Other ones are sinc-interpolated thus they are correlated. Note that $N/L \times \Delta f = 1 / L T_s \approx 1/\tau_m$ can be seen as coherence bandwidth.

• The property that $h_l$ are i.i.d. does not imply that $H[k]$ are i.i.d. In fact, this only holds, if $l=N$. Otherwise (i.e. $l<N$) the $H[k]$ are correlated Gaussians with correlation matrix $F_N\text{diag}(\vec{p}_0)F_N^H$ and $\vec{p}_0$ is the power delay profile zero-padded to the block length $N$. Here you see, if $\vec{p}_0=\vec{1}$, i.e. consisting of $N$ ones, only then $H[k]$ are uncorrelated. Apr 28, 2017 at 10:57
• @MaximilianMatthé it is true. Thanks for pointing out my error. Apr 28, 2017 at 11:12
• I will update my answer to take into account your correction. Apr 28, 2017 at 11:17
• @MaximilianMatthé if you have time, could you take a look to see if you agree with my update ? Thanks. Apr 28, 2017 at 11:39
• I would add that the variables at $uN/L$ are uncorrelated, if $uN/L$ is an integer value. Otherwise, no variable is uncorrelated. Apr 28, 2017 at 12:56