xMax meets Shannon once again …

I have received several inquires related to my post “xMax meets Shannon” which indicate that I may not have been sufficiently clear in my explanation of the relevance of the Shannon theorem to the claims made about xMax. So let me try one more time, intended to be my last.

It is now almost 60 years since Shannon published “A Mathematical Theory of Communications”. He actually presented a number of important results, not just one. Here we consider what is often referred to as the Shannon-Hartley theorem which establishes the channel capacity for a finite-bandwidth channel subject to Gaussian noise. The theorem states a relationship between the maximum data rate which can be reliably transmitted, the bandwidth occupied by the communication signals, and the ratio of the received signal power to the noise power, or SNR (for signal to noise ratio). The received signal power depends on the transmitted power, the path loss (representing how much power is lost during the trip from transmitter to receiver), and the gains of the antennas used by the transmitter and receiver. The noise power referred to here is the electronic noise generated in the circuitry of the receiver. The nature of this noise is very well understood and its power can be easily calculated. It turns out that this power is proportional to the bandwidth.

Consider a scenario where we want to communicate from point A to point B using a given set of antennas. In this case the received power is some fixed fraction of the transmitted power. If we want to double the received power, we must double the transmitted power. For simplicity assume we have fixed the bandwidth (for example, we are allocated a channel with a given bandwidth by the FCC). Then the noise power is also fixed. In other words the SNR depends entirely on the transmitted power.

Going back to the Shannon theorem – its implication can now be stated as follows: the maximum data rate which can be reliably transmitted is limited by the SNR at the receiver, or equivalently by the transmit power. For a given transmit power there is a maximum data rate you can achieve and no more. Conversely, for a given desired data rate there is a minimum transmit power you need, and you can not do with less.

It should be emphasized that this constraint is an absolute mathematical fact. It does not depend on the specific design of the communication system. This notion of an absolute limit seems to be a difficult for some people to grasp (or perhaps to accept), so let me try to illustrate it by an analogy.

Let us say that we want to travel from point A to point B using some vehicle. Assume that the shortest distance (as the crow flies) between the two point is D and that the vehicle has a maximum speed of V. It follows that the travel time must be larger than D/V. This is true regardless of what route you choose or what kind of vehicle you use. In other words D/V is an absolute lower limit on the travel time. In an analogous way, the Shannon theorem provides an absolute lower limit on the amount of transmit power in the scenario discussed above. Because we all have experience with traveling from point A to point B, it is easy to accept the minimum travel time constraint. However most people have not studied communication theory nor the Shannon theorem, and they often find it difficult to accept the minimum required power constraint.

Now here is where it gets relevant to the xMax discussion. Over the years engineers have developed increasingly sophisticated high performance communication systems. Current systems are power efficient, i.e. they operate close to the power constraint discussed above. In other words, to deliver a given data rate they require a transmit power which is close to the minimum required power. This does not mean that further power reductions are impossible, only that such improvements must be small, because there is not much room left. Reducing the required power by, say, a factor of 5, may be possible, albeit very difficult.

So when one evaluates a claim such as “xMax can use 1,000 to 100,000 times less power than comparable transmission technologies”, one can say with outmost certainty that this claim is false. One does not need to know a single thing about how xMax works in order to reach that definitive conclusion. To use the earlier analogy: if someone tells you they developed a new revolutionary vehicle which has a top speed of 60 MPH and can travel from San-Francisco to Los-Angeles in 2 hours, you can dismiss their claim out of hand even if you know nothing about the design of the vehicle or the route they plan to travel.

Well, off course they could load the vehicle on a 747 at SFO and land it LAX, but that is another story :=)


2 responses to “xMax meets Shannon once again …

  1. Perhaps xMax and other “gems” like VMSK are based on pre-Shannon communication theories? I always wondered if there were any “skeptiks” who doubted/opposed Shannon at the time.

  2. JimDeGries@gmail.com

    I don’t think this has anything to do with Shannon theory. It has to do with lack of understanding of the Fourier transform and how it applies to signals and systems.

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