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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:

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

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:

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!