## The Secret Life of Modern RF Signals - Part 5

In the first installment of this blog I said I’d adjust the topic based on reader feedback. As we prepare for 4G, certain RF testing topics keep coming up, so I’m going to derail our topic thread.

By the way, while I do appreciate your emails (most of them, anyway… as to the others, you know who you are...), some good questions and comments might have stimulated useful conversations had they been shared publicly. We ask for contact info when you comment, but we won’t release that to anyone. Also, unless I have your clear permission, I will not publicly share anything that’s emailed to me. So in the interest of sharing information and questions, I’d urge you to please use the Comments section below.

Today’s topic is based on some questions I’ve been asked. The common thread in these questions seems to be, “If fading is all based on reasonable tractable math, why would one fader (emulator) work better than another?” The answer is the same as it is for any endeavor in communications engineering: there is theoretical perfection, and then there is the balance between theory and practical implementation.

**So what are the pitfalls in fading emulation?**

Why do we need expertise to do fading modeling? After all, most of us took statistics and EE courses. We can understand what needs to be done, so what’s the big deal? The big deal is in the way fading is normally done. Usually, a “classical Doppler spread” (shown in Figure 1) is used to describe the way Doppler shifts statistically spread frequencies across a band.

Figure 1 - Classical Doppler spread spectrum

In the old days we used sinusoids to create this spectrum. We still believe M. Fourier when he said that sinusoids can be used to create any signal, but unless you can create an infinite number of sinusoids you’re missing something when you’re attempting to create a completely randomized thing. *Autocorrelation* is a measure of “randomness”, where an autocorrelation of 0 would be perfectly random and a value of 1 means perfect self-correlation. The key is that you’re using a finite number of sinusoids to create an environment that is theoretically uncorrelated with itself (autocorrelation = 0), but using a finite number of sinusoids means that at some frequencies the sinusoids sum up to contribute a significantly non-zero value. So a classical fading spectrum cannot be accurately modeled this way.

Another measure of how realistic fading can look is by measuring Level Crossing Rates (LCRs). If you were one of the few who read the last installment of this blog (aptly named The Secret Life of Modern RF Signals - Part 4) you know a little something about them. While it is not evident in an informal discussion like ours, proper LCRs are critical in determining the statistical validity of a channel model. Without digging in too deeply, only a very careful implementation using sinusoids comes close to creating valid LCRs.

For these reasons, my employer and most channel emulator manufacturers abandoned the sinusoidal approach for classical fading emulation many years ago. That doesn’t mean everyone has… you can buy a brand-new fader today which still uses this obsolete approach.

If you want to find out what fading method your emulator is using, a simple test is to input a sinusoid and use a single classically-faded path. Run the output into a spectrum analyzer. If you see the classical Doppler shape on the right-hand side of Figure 2, breathe a sigh of relief. If you see the spectrum on the left-hand side looking like Bart Simpson’s head, throw out your test results and start looking for a better fader.

Disclaimer: before you ask, yes the fader I’d recommend is the one my employer sells. But no, I’m not using this blog as an advertising space. There are competing faders of varying quality that use modern and reliable fading methods to achieve classical fading.

Figure 2 - Poor (left) and better (right) classical Doppler spectra.

**So is M. Fourier Dead?**

Yes, unfortunately he passed in 1830 at the age of 62. But the use of discrete sinusoids to simulate fading is still mandated in certain areas. For example, in areas like Geometrical Channel Modeling (GCMs, which include Spatial Channel Models [SCMs]) sum-of-sinusoids methods are specifically called out. There have been (and still are) valid arguments for using a “filtered noise” approach in SCMs, but for the time being the industry demands sinusoidal approaches, and results are evaluated based on that assumption. If they’re not perfect, what’s the difference between a good implementation and a bad one?

A lot of modern wireless testing requires a realistic implementation of a directional spread of received signals and reflections. In Figure 3, the spectrum on the right differs drastically from the classical spectrum. It is the result of distributing the received signals along a narrow angle of arrival. This is intuitively critical when thinking about spatial channel models… when we are in the spatial domain, the model must have some spatial fidelity, or else you’re missing the point of the testing.

Implementing this kind of modeling using the required sinusoidal approach is extremely difficult, but it has been done and done well. Unfortunately, the journeyman fader user will never know whether this kind of modeling is being created by the fader he or she has. Many would be shocked to find that some so-called “spatial channel model”-capable generators completely ignore this critical factor… not because it’s they don’t know about it, but because they know you’ll never find out the truth.

Figure 3 - Classical Doppler spectrum (left) and Laplacian Spectrum (right)

If you decide to fire up some lab equipment and look for a Laplacian spread, be aware that a Laplacian spectrum is not necessarily symmetrical. Set your spectrum analyzer resolution bandwidth to a very low setting and average a couple of hundred readings. If you see something that roughly resembles a single concentration of sinusoids (see the right-hand side of Figure 3), your fader-maker was probably on the right track. If you see an SCM representation that looks more like a Doppler spread (at low resolution bandwidth it will look like the left-hand side of Figure 3), something is amiss.

Unfortunately, an implementation that is done right is disappointingly rare. Worse, some companies use weak modeling as a marketing tool, as in, “Hey, your design passes pretty easily using our models.” In all too many cases, poor fading implementations allow a poor design to squeak through inadequate testing processes. If you believe, as I do, that misleading results are worse than no results at all, using this model for testing can cause more harm than good.

Rather than try to prove this in this little blog, or asking you to take my word for it, I’d urge you to read some excellent work by experts in the field. To wit:

M. F. Pop, N. C. Beaulieu (2001). “Limitations of Sum-of-Sinusoids Fading Channel Simulators.” *IEEE Transactions on Communications, Vol 49, No. 4.*

Y. Li, Y.L. Guan (2000). “The generation of Independent Rayleigh Faders.”

C. Xiao, Y. Zheng, N. Beaulieu (2002). “Second-order Statistics of an Improved Jakes’ Fading Simulator.” VTC2002.