The Shannon capacity curve

They say that a picture is worth a thousand words. So here is the picture of the Shannon capacity curve – just click on it twice and you will see it in its full glory (the second click will magnify it further).

shannon31.jpg

Unfortunately it will take me a thousand words to explain it, but what can I do?

The horizontal axis of this plot is the energy-per-bit (Eb) divided by the noise power per Hz (N0). This ratio is called Eb/N0 and it represents the signal to noise ratio at the receiver in normalized form. See my earlier post “xMax performance claims can be easily tested. Why aren’t they?” for some more info. The vertical axis is the spectral efficiency, which is the capacity divided by the bandwidth. For our discussion think of this simply as the data rate in bits-per-second, divided by the bandwidth in Hz of the channel used by the communication system. Spectral efficiency therefore has units of bits per second per Hz. Note that every pair of values of spectral efficiency and Eb/N0 determines a point in this plane.

Now look at the blue curve sweeping majestically from the lower left corner towards the upper right. This is the Shannon capacity curve. It divides the plane into two parts: one part above the curve (marked by a kind of green fog), and a second part below. What Shannon has shown is that it is possible to design communication systems operating with combinations of spectral efficiency and Eb/N0 below the curve, and make them work with small bit error rates. Not so for any system attempting to operate with spectral efficiency and Eb/N0 values above the curve. Such systems can not operate with small bit error rates. In other words, any reliable communication system must operate below the curve. No reliable communications is possible in the green fog.

Not surprisingly, all known communication systems operate in the region below the curve. As an example I plotted the points corresponding to some conventional modulation techniques. The reddish squares correspond to Quadrature-Amplitude-Modulation (QAM) of orders 2,4,8,16, and 32 (Note that 2-QAM is really called BPSK). The blueish circles correspond to Frequency-Shift-Keying (FSK) of orders 2,4,8,16,32 and 64. The Eb/N0 values correspond to bit-error-rates (BER) of 0.00001. Other modulation techniques can be placed at different points in this plane. Note that these points follow the general shape of the capacity curve, but are some distance from it. We will refer to the horizontal distance from the capacity curve as the “distance from Shannon”. So for example 2-QAM is 10dB from Shannon – see the red horizontal line. Generally the various modulation techniques shown here are 8 – 10dB from Shannon.

Actually, conventional communication systems are much closer to Shannon than would seem from this graph. The reason is that the points shown on the graph were for uncoded modulation. All advanced communication systems used some form or another of coding. The points corresponding to coded modulation lie to the left of the points shown on this graph. Just how far left, depends on the coding. Typically the points will be midway between the ones shown and the capacity curve. In other words, advanced communication systems are typically within 4-5 dB of Shannon. However, we know how to design codes which will bring the point much closed to the Shannon curve. For the purpose of the discussion here we will stick with the figure of 4-5 dB mentioned above.

Note that moving the point horizontally to the left means that we can transmit the same data rate at a lower value of Eb/N0, and hence at a lower transmit power for a given communication scenario. This is of course highly desirable. Unfortunately, there is not much room to move anymore, because existing modulation techniques, especially when properly coded, are already close to the Shannon curve.

It is certainly possible to invent new and interesting modulation techniques. Each such technique (assuming that it works) must represent a point below the blue Shannon curve. Which brings us back to the xMax story. Based on published information we know that xMax has a spectral efficiency on the order of 0.4 – 0.6. So the point corresponding to the xMax system should be in the region indicated by the green cloud in the figure below for the uncoded case, or in the blue cloud for the coded case. This estimate assumes an efficient implementation of xMax which would make its performance similar to that of FSK. An inefficient implementation will require a much higher Eb/N0.

shannon4.jpg

The exact position of the xMax operating point can of course be easily established by a BER vs. Eb/N0 measurement, which unfortunately is not publicly available at this time. However, one thing we can say with absolute certainty is that the claim that “xMax can use 1,000 to 100,000 times less power than comparable transmission technologies” is false. A power reduction of even 10, let alone 1,000 – 100,000, takes us far to the left of the Shannon capacity curve, into the the land of the impossible ….

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One response to “The Shannon capacity curve

  1. Here are some more points for your curve. These are for BPSK with the coding used with several deep space (interplanetary) spacecraft:

    r=1/2 k=7 convolutional: Eb/No 4.5 dB, eff 0.5 bps/Hz

    Voyager (RS+r=1/2,k=7): Eb/No 2.4 dB, eff 0.437 bps/Hz

    Cassini (RS+r=1/6 k=15): Eb/No 0.6 dB, eff 0.146 bps/Hz

    CCSDS r=1/6 turbo large block: Eb/No 0.0 dB 0.167 bps/Hz

    As you can see, there’s not a whole lot of room for improvement after turbo codes.

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