Commentary, Technical insights

Rician Fading – a Channel Model Often Misunderstood

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Line-of-sight channels normally contain many propagation paths, whereof one is the direct path and the others are paths were the signals are scattered on different objects. The interaction between these paths lead to fading phenomena, which is often modeled statistically using Rician fading (sometimes written as Ricean fading). The main assumption is that the complex-valued channel coefficient h \in \mathbb{C} in the complex baseband can be divided into two parts:

h = m e^{j\theta}+ s,

where m \geq 0 is the magnitude of the direct path between the transmitter and receiver and \theta \in [0,2\pi] is the corresponding phase shift. The second part, s, represents all the scattered paths. This part is separated from the direct path since it consists of many paths, each being of roughly the same strength but substantially weaker than the direct path. It is modeled by Rayleigh fading, which implies s\sim \mathcal{CN}(0,\sigma^2). The complex Gaussian distribution is motivated by the central limit theorem, which says that the sum of many independent and identically distributed random variables is approximately Gaussian.

Under these assumptions, the magnitude |h|=|m e^{j\theta} +s| of the channel coefficient is Rice/Rician distributed, which is why it is called Rician fading. More precisely, |h| \sim \mathrm{Rice}(m,\sqrt{\sigma^2/2}), which depends on the magnitude m and the variance \sigma^2 of the scattering.

Interestingly, the distribution does not depend on the phase \theta, because the magnitude removes phases and s and s e^{j\theta} are equally distributed. Hence, it is common to omit \theta in the performance analysis of Rician fading channels. As long as the channel is perfectly known at the receiver, it will not make any difference when quantifying the SNR or capacity.

The common misunderstanding

We cannot neglect the phase \theta when analyzing practical systems where the receiver needs to estimate the channel. The value of \theta varies at the same pace as s, and for exactly the same reason: The transmitter or receiver moves, which induces small phase shifts in every path. Since s contains a large number of paths with approximately the same magnitude but random phases, the sum of the many terms with random phases give rise to the Gaussian distribution. The phase-shift of the direct path must be treated separately since this path is substantially stronger.

Unfortunately, my experience is that the vast majority of paper on Rician fading channels ignores this fact by simply treating m e^{j\theta} as a deterministic constant that is perfectly known at the receiver. I have done this myself in several papers, including this one from 2010 that has received 200+ citations. Unfortunately, the results obtained with that simplified model are practically questionable. If we don’t know s in advance, how can we know \theta? At best, the results obtained with a perfectly known \theta can be interpreted as an upper bound on what is practically achievable.

We analyzed the importance of correctly modeled random phases in a recent paper on cell-free massive MIMO. We compared the performance when using an ideal phase-aware MMSE estimator and a phase-unaware LMMSE estimator. The spectral efficiency loss due to a lack of knowing \theta ranges from 2% to 50% in different simulations, depending on the pilot length and interference situation. Hence, there are cases where it is very important to know the phase correctly.

13 thoughts on “Rician Fading – a Channel Model Often Misunderstood”

    1. Yes, this is a good point. If the transmitter and receiver are fixed, but the scatterers move, then theta will be constant even if s changes. But I don’t think this is the main cause of small-scale fading in practice.

      Reply

  2. Hi Emil,

    I am wondering that in the context of cell-free OFDM, should we assume the frequency domain channel or time-domain channel to be Rician distributed.

    If the time domain channel is Rician, what will be the distibution of frequency domain channel?

    Thank you very much and look forward to reply from you.

    Tony

    Reply

    1. The time-domain channel is described by, say, L taps. The first tap includes the line-of-sight path (which has the shortest time delay) while the remaining L-1 taps contain scattered paths. The channel at a particular frequency subcarrier is a linear combination of the L taps, which is computed using an FFT matrix. Hence, it is the channel at a particular subcarrier that is subject to Rician fading.

      Reply

  3. With Rayleigh fading, we tend to separate the channel gain in small-scale and large-scale fading contributions: g= sqrt(b) * h, where b is real and related with the large-scale fading and h is complex and related with small-scale fading. It was h that was Rayleigh distributed. Furthermore, due to channel hardening, it would disappear from closed-form expressions.

    Will happen the same to Rician Fading?

    Reply

    1. It is common to also separate between large-scale and small-scale fading when dealing with the Rician fading model. You can basically take the expression in (1) and multiply with a large-scale fading coefficient. Moreover, people often define what the “k-factor” to specify which fraction of the large-scale fading that goes into the LOS path and which fraction goes into the NLOS component.

      Channel hardening appears also in Rician fading channels.

      Reply

      1. Hi, Prof. Emil,
        Continuing with João’s question, if we separate between large-scale (for simplicity, only consider pass-loss) and small-scale Rician fading, then one may try to define the total channel model in the following two ways:

        1) using the equation in this post:
        h1 = m*exp(1j*theta) + s,
        where m = PL_Los = lambda^2/(4*pi)^2/d^2 ([Rappaport, Wireless communications principles and practice, 2002, Page 107], assuming omnidirectional antenna and no system loss factor), lambda is the wavelength, d is T -R separation distance; and s =PL_NLos*CN(0,1), where PL_NLos may following some existing pass-loss models (keeping carrier frequency consistency with lambda defined before), such as 3GPP pico-cell pass-loss model PL=38 + 30*log10(d) [3GPP TR 36.931], CN(.,.) denotes CSCG distributions.

        2) using the concept of K-factor:
        h2 = PL*( sqrt(K/(K+1))exp(1j*theta) + sqrt(1/(K+1))CN(0,1) ).
        then how to choose a proper pass-loss model for PL?

        To make the above two definitions consistent, the total “power” of the two models should be the same, but choosing a pass-loss model for the second definition seems to be cumbersome. Or, do I have some misunderstanding of the Rician fading model?

        Reply

        1. This is a good question.

          Since there are two different types of paths, we need one pathloss value for each of them.

          The simplest (?) thing would be to define a separate formula for LOS and NLOS, as you did in Option 1.

          However, it has become popular to instead define a common pathloss value for both paths and then let the k-factor determine how this power is divided between the paths. If you check the 3GPP document, you will likely find a LOS formula as well. It is meant to be used as the common PL factor in Option 2 and one then needs a k-factor as well, which is also stated in the document.

          It is not so easy to make these two modeling approaches entirely consistent, but they both use two parameters to determine two quantities. So there is a 1-to-1 mapping for a constant propagation distance, but not for a range of d-values.

          Reply

          1. Thank you, Prof. Emil!

            And there are some typos in my question, and they should be corrected as follows:
            “pass-loss” –> path-loss;
            m = PL_Los = lambda^2/(4*pi)^2/d^2 –> m = sqrt(PL_Los), PL_Los = lambda^2/(4*pi)^2/d^2;
            s =PL_NLos*CN(0,1) –> s =sqrt(PL_Nlos)*CN(0,1);
            h2 = PL*( sqrt(K/(K+1))exp(1j*theta) + sqrt(1/(K+1))CN(0,1) ) –> h2 = sqrt(PL)*( sqrt(K/(K+1))exp(1j*theta) + sqrt(1/(K+1))CN(0,1) ).

  5. Hi
    I have a question about Rician channel with L taps. The frequencyselective
    Rician fading channels with delay spread of L = 6
    taps are considered for both direct link and reflecting link,
    where the first tap is set as the deterministic line-of-sight (LoS)
    component and the remaining taps are non-LoS components
    following the Rayleigh fading distribution, with η being the
    ratio of the total power of non-LoS components to that of
    LoS component.

    Is this a correct matlab code for the above definition?

    h = zeros(L, 1);
    h(1) = sqrt(eta/(1 + eta)); % LoS component
    h(2:end) = sqrt(1/(2*(1 + eta))) * (randn(L-1, 1) + 1i * randn(L-1, 1)); % Non-LoS components

    Reply

    1. Yes, I think that is correct, but to be fully general, the LoS component should also have a random phase:

      h(1) = sqrt(eta/(1 + eta))*exp(1i*2*pi*rand(1,1));

      Reply

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