In the infamous Shannon-Hartley theorem the bandwidth of the channel is measured in Hz (Hertz) but the channel capacity is measured in bps (bits per second). So, either I'm missing something obvious or there's a dimensional mismatch in the equation.. Can somebody please help me understand?
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3 Answers
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In the equation:
$$C = B \cdot log_2 (1 + \frac{S}{N})$$
The B represents the bandwidth in Hz, and the log2(1 + S/N) represents the "information density" that you can achieve as a result of the signal to noise ratio. This expression has units of "bits/cycle", but this is rarely stated explicitly, since it's technically a dimensionless quantity. It's basically a measure of how many distinct signalling states (e.g., voltage levels) you can reliably distinguish at the receiver, given the noise level in the channel.
So, if bandwidth has units of Hz, or cycles/second, and the rest has units of bits/cycle, you end up with bits/second.
answered Jan 25, 2014 at 12:59
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\$\begingroup\$ I'm very confused now, is the Shannon-Hartley theorem technically wrong? \$\endgroup\$– gpoCommented Jan 25, 2014 at 13:07
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\$\begingroup\$ No, it's perfectly fine to use "symbolic" units such as cycles and bits, even if they're dimensionless in the scientific sense. In the scientific sense, both sides of the equation have units of 1/seconds. \$\endgroup\$ Commented Jan 25, 2014 at 13:19
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\$\begingroup\$ I'm still confused :$ Is there a definition for "dimensionful/less" units and why is a bit (which is the basic unit of information) a "dimensionless" one? \$\endgroup\$– gpoCommented Jan 25, 2014 at 13:24
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\$\begingroup\$ All dimensionless quantities are scientifically equivalent, because they have no physical manifestation. All scientific "dimensionful" units can ultimately be expressed in terms of the fundamental physical units of length, mass, charge and/or time. Dimensionless units are just theoretical abstractions like "a bit of information". \$\endgroup\$ Commented Jan 25, 2014 at 13:39
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Both units have dimensions of inverse time. Hertz, formerly known as cycles per second, has the same dimensions as angular frequency, radians per second.
In other words, bits, cycles, and radians are not dimensionful quantities.
answered Jan 25, 2014 at 12:59
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\$\begingroup\$ I think your answer answers my question but I don't understand it actually. A bit is the basic unit of information which does not seem dimensionless to me.. How one decides what is a "dimensionful quantity" and what is not? i.e. How do you define dimensionful? \$\endgroup\$– gpoCommented Jan 25, 2014 at 13:16
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\$\begingroup\$ @gpo, a dimensionful quantity has dimensions of some combination of the 7 primary dimensions: mass, length, time, temperature, electric current, amount of light, and amount of matter. Note that only dimensionless quantities can be arguments of, for example, sine, log, etc. since, otherwise, we would be adding together quantities with differing dimensions. See: mne.psu.edu/cimbala/Learning/General/units.htm \$\endgroup\$ Commented Jan 25, 2014 at 13:50
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\$\begingroup\$ I can now upvote you, sweet :) Thank you for your answer. \$\endgroup\$– gpoCommented Jan 25, 2014 at 16:15
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B.T = Cycles/Time x Time = Cycles = A count of events - a unitless number
S/N = Energy/Energy = Unitless
Information = B.Tlog(1 + kS/N) = unitless - bits - cycles
Capacity = Blog(1 + kS/N) = bit/s - or cycles/second - 1/time
