A Bit About Shannon's Equation


Background.  If we know the bandwidth of a channel we should be able to figure out its capacity.  But the formula for doing this depends on how well the signal is transmitted.  The relationship that Claude Shannon developed in 1938 follows and is called Shannon's Equation.



SNR stands for Signal to Noise Ratio.  Obviously, the ability to discern a signal in a particular range of frequencies influences the range's capacity.  Notice that the capacity increases as either the bandwidth and/or the SNR increases as you'd expect.

Now we work some examples.

Example.  Phone Line Capacity. A phone line normally has a bandwidth, in the more precise usage of the word, of 3000Hz.  This is measured from 300 Hz to 3300 Hz.  The standard SNR for a phone line is 3162.  Based on this we get the following calculation.

C = 3000log2(1 + 3162) = 3000log2(3163) = 3000(11.62) 34,860 bps ~ 34.86 Kbps

Example.  Wireless Channel Capacity.  Since I don't really know the SNR of the wireless frequencies let's assume it is around the SNR of a phone line.  Depending on the wireless standard, the channels have differing widths.  Unfortunately, these channels usually overlap in some specifications such as the 802.11b.


Standard Overlapping Bandwidth Capacity
802.11b Yes 22 MHz C = 22,000,000log2(1 + 3162) = 22,000,000(11.62) = 255,640,000 bps ~ 256 Mbps
802.11a No 20 MHz C = 20,000,000log2(1 + 3162) = 20,000,000(11.62) = 232,400,000 bps ~ 232 Mbps
802.11g No    


So I know the SNR for wireless isn't this good!!! 

Let's assume the numbers for SNR in Forouzan's book are actually in dB.  Then a number of sources state that we want the following SNR for wireless channels.  These sources also state that these SNRs need to be determined and assessed!


Quality SNR
minimum 20 dB
excellent 40 dB


Thus the calculations change to the following.


Standard Overlapping Bandwidth Capacity
802.11b Yes 22 MHz C = 22,000,000log2(1 + 40) = 22,000,000( ) =  bps ~  Mbps
802.11a No 20 MHz C = 20,000,000log2(1 + 40) = 20,000,000( ) =  bps ~  Mbps
802.11g No    


These numbers still are not believable for a number of reasons.  So I need to do more checking!