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Consider a wireless channel $h=e^{j\theta}$, where $\theta$ is a uniform random variable in $[0,2\pi]$ independent of the input messages and the independent of the noise. The channel randomly rotates the phase of the signal. For each new symbol (channel use), we have independent realization of $\theta$. Let average signal power be $S$ and noise power be $N$. Assuming that $\theta$ is not known at the transmitter and receiver, and remains unchanged during one symbol transmission, what is the capacity of this channel?

Note that we have an i.i.d. zero-mean complex Gaussian noise.

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  • $\begingroup$ Is there any Gaussian noise? Without any Gaussian noise, the capacity seems to be $\infty$. Can you write the channel equation properly? $\endgroup$
    – Campello
    Dec 10, 2015 at 20:30
  • $\begingroup$ yes @Campello, we have i.i.d. Gaussian noise, as usual. $\endgroup$
    – Jeff
    Dec 13, 2015 at 4:49
  • $\begingroup$ I still don't get it. For a general fading channel of the form $y = h x + n$, the (ergodic) capacity is given by $E_h[log(1+|h|^2SNR)]$. Hence in your case |h| = 1, and you recover the capacity of the Gaussian channel.. $\endgroup$
    – Campello
    Dec 13, 2015 at 14:58
  • $\begingroup$ I am not sure, but I guess $E_h[\log_2(1+|h|^2\mathrm{SNR})]$ is the ergodic capacity with transmitter and receiver side information. In my case, I do not know $\theta$, so I do not know the channel neither at the transmitter nor at the receiver. $\endgroup$
    – Jeff
    Dec 14, 2015 at 4:37
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    $\begingroup$ ok u r right....... sounds like capacity is same as PAM $\endgroup$
    – user76479
    Dec 16, 2015 at 17:57

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$Y=e^{i\theta}X+W, \ \theta\sim U(0,2\pi),\ W\sim \mathcal{CN}(0, 1)$ with $\theta \perp W \perp X$ and input condition $E[|X|^2]\leq \mathsf{SNR}.$ Shannon's theorem gives $$C=\max_{X:E|X|^2\leq \mathsf{SNR}} I(X;Y).$$ Since the channel completely randomizes phase then $I(X;Y)=I(|X|^2;|Y|^2)$, and
$P_{\left. |Y|^2 \middle| |X|^2=r^2\right.}$ is a noncentral chi-squared distribution with 2 degrees of freedom and noncentrality $r^2$.

The maximizing distribution over $r$ was finally characterized in The Capacity of Discrete-Time Memoryless Rayleigh-Fading Channels by Abou-Faycal, Trott, and Shamai here: https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=923716

Surprisingly it is concentrated over a finite number of points.

But the number of points and their positions for a general SNR aren't known, so you have to numerically optimize the mutual information over these quantities.

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