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Chapter 1: The Shanon-Hartley Theorem

The Shannon-Hartley theorem specifies the maximum amount of information that can be encoded over a specified bandwidth in the presence of noise. It serves as an upper ceiling for radio transmission technologies. Understanding how this relates to signal-to-noise ratio can help you understand how much bandwidth it is possible to obtain over a wireless link.

The Shannon-Hartley theorem specifies the maximum amount of information that can be encoded over a specified bandwidth in the presence of noise. It serves as an upper ceiling for radio transmission technologies.

The Nyquist Limit

Shannon-Hartley derives from work by Nyquist in 1927 (working on telegraph systems). Nyquist determined that the number of independent pulses that could be put through a telegraph channel per time unit is limited to twice the bandwidth of the channel. This is expressed in math as:

fp ≤ 2B
The Nyquist Limit Function

In other words, the number of pulses at a given frequency is less-than or equal to twice the bandwidth. For example, a channel with a 100 Hertz bandwidth can encode no more than 200 symbols per second. A 10 Mhz bandwidth channel can encode no more than 20 million symbols per second.

Hartley's Law

In 1928, Hartley wanted to formalize the amount of information per second that could be encoded into a given bandwidth. There are multiple ways of representing Hartley’s law, but the most common is:

R ≤ 2B log2(M)
Hartley's Law Function

R is the data-rate, in bits-per-second. M is derived as follows:

M = 1 + A / ∆V

A is the amplitude of the signal, in volts (for example, -5V to 5V would be an amplitude of 10), while ∆V represents the precision of the receiver.

This assumes an error-free environment, which to all practical intents and purposes does not exist. Therefore, Hartley’s law is commonly used only as a building-block for the Shannon-Hartley law.

Shannon-Hartley

Shannon built upon Hartley’s law by adding the concept of signal-to-noise ratio:

C = B log2 1 + S / N
C is Capacity, in bits-per-second. S and N represent signal and noise respectively, while B represents channel bandwidth.
The Shannon-Hartley Function

For example, given a 16 Mhz channel and a signal-to-noise ratio of 7:

C = 16 * log2(1+7) = 16 * log2(8) = 48 mbit/s.

Combining Shannon-Hartley with Polarity and MIMO

Utilizing modern antennas that simultaneously transmit data across multiple channels, one needs to multiply the Shannon-Hartley number for a given bandwidth by the number of channels. For example, 16 Mhz channels on a 2x2 antenna array could provide a theoretical upper-limit of 192 mbit/s with a signal-to-noise ratio of 7.

Practical Uses for Shannon-Hartley

The above is very theoretical, and is likely to be off-putting to the first-time reader. Therefore, it is necessary to show some practical uses for this knowledge.

The first example shows the relative increase in theoretical data capacity by bandwidth:

Figure 4: Shannon Limit Capacity by Bandwidth

This shows that given a constant signal-to-noise ratio of 7, data capacity increases linearly as bandwidth is increased. Remember that this is for a single 1x1 antenna on a fixed channel size; actual bandwidth numbers are multiplied by the number of data-streams available.

A second example shows the effects of noise on a single 10 Mhz channel:

Figure 4=5: Shannon Limit Capacity by SNR

This emphasizes the importance of signal-to-noise ratio on radio performance. For a single 10 mhz channel, capacity degrades considerably at lower signal-to-noise ratio levels. This underlines a point made throughout this book: improve your signal before you add bandwidth!

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