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12 Cryptocurrencies
The distribution of voltages only defines the statistics of the voltages that may
appear, but we also care about every possible voltage being able to immediately
follow any other voltage. This means that we require that very sharp transitions
from one voltage level to any other are faithfully transported. But sharp transitions
are described by high Fourier harmonics. The highest useful frequency, called the
bandwidth W , will therefore limit how many different voltages we can transmit
across the channel per unit time T . According to Nyquist’s theorem [11], we need at
least two measurements per period to characterize a frequency. Therefore, a total of
n = 2T W distinct voltages can be transmitted during the time T through a channel
with bandwidth W . The total entropy transported during time T is then given by
n H g = 2T W H g and the corresponding rate ˙
H = n H g /T = 2W H g is given by
˙
H = W log
2π eσ
2
.
(12.25)
For a source, characterized by a rms signal level σ
2
= P S , we find the rate ˙
H [x] =
W log (2π eP S ) and for the noise on the channel with σ
2
= P N , it is ˙
H [N ] =
W log (2π eP N ). Moreover, since the source and the noise are uncorrelated the total
power P t arriving at the decoder is given by P t = P S + P N and the corresponding
entropy rate is ˙
H [y] = W log (2π e(P S + P N )).
The capacity C of the continuous channel is defined analogously to (12.19)
as the maximum of the mutual information rate ˙
I [x; y] = ˙
H [x] + ˙
H [y] − ˙
H [x, y]
over all possible inputs. By inserting ˙
H [x, y] = ˙
H [x] + ˙
H [y|x] from (12.16), we
obtain ˙
I [x; y] = ˙
H [y] − ˙
H [y|x]. Immediately following (12.16) we reasoned that
˙
H [y|x] is the entropy added to the signal by the noise, which is, as we know from
the previous paragraph, equals ˙
H [N ]. Inserting all quantities into the expression for
C = ˙
I [x; y] = ˙
H [y] − ˙
H [N ], we find
C = W log (2π e(P S + P N )) − W log (2π eP N ) = W log
1 +
P S
P N
. (12.26)
The entropy of both source and noise were assumed to be maximum-entropy Gaussian, such that we can identify the mutual information rate ˙
I [x; y] as the channel
capacity C.
We observe that the channel capacity C is determined by the bandwidth W and
the signal-to-noise ratio P S /P N . Higher bandwidth—the ability to transmit higher
frequencies—will allow us to transmit more information. Moreover, a small noise
power P N allows us to distinguish smaller differences in the payload signal, which
gives us more signal levels to encode information. As an example, consider the
communication on a wireless WLAN network, which benefits from increasing the
bandwidth by channel bonding, where two or more of the, by default, W = 20 MHz
wide channels, are used simultaneously in order to increase the download speed.
Moreover, the transmission amplifiers automatically increase the transmission power
P S , up to the legally allowed maximum, to maximize transmission capacity. Normally
the power level is chosen to match the rate at which the information is created at the
source—listening to an Internet radio station requires less bandwidth than watching
12 Cryptocurrencies
The distribution of voltages only defines the statistics of the voltages that may
appear, but we also care about every possible voltage being able to immediately
follow any other voltage. This means that we require that very sharp transitions
from one voltage level to any other are faithfully transported. But sharp transitions
are described by high Fourier harmonics. The highest useful frequency, called the
bandwidth W , will therefore limit how many different voltages we can transmit
across the channel per unit time T . According to Nyquist’s theorem [11], we need at
least two measurements per period to characterize a frequency. Therefore, a total of
n = 2T W distinct voltages can be transmitted during the time T through a channel
with bandwidth W . The total entropy transported during time T is then given by
n H g = 2T W H g and the corresponding rate ˙
H = n H g /T = 2W H g is given by
˙
H = W log
2π eσ
2
.
(12.25)
For a source, characterized by a rms signal level σ
2
= P S , we find the rate ˙
H [x] =
W log (2π eP S ) and for the noise on the channel with σ
2
= P N , it is ˙
H [N ] =
W log (2π eP N ). Moreover, since the source and the noise are uncorrelated the total
power P t arriving at the decoder is given by P t = P S + P N and the corresponding
entropy rate is ˙
H [y] = W log (2π e(P S + P N )).
The capacity C of the continuous channel is defined analogously to (12.19)
as the maximum of the mutual information rate ˙
I [x; y] = ˙
H [x] + ˙
H [y] − ˙
H [x, y]
over all possible inputs. By inserting ˙
H [x, y] = ˙
H [x] + ˙
H [y|x] from (12.16), we
obtain ˙
I [x; y] = ˙
H [y] − ˙
H [y|x]. Immediately following (12.16) we reasoned that
˙
H [y|x] is the entropy added to the signal by the noise, which is, as we know from
the previous paragraph, equals ˙
H [N ]. Inserting all quantities into the expression for
C = ˙
I [x; y] = ˙
H [y] − ˙
H [N ], we find
C = W log (2π e(P S + P N )) − W log (2π eP N ) = W log
1 +
P S
P N
. (12.26)
The entropy of both source and noise were assumed to be maximum-entropy Gaussian, such that we can identify the mutual information rate ˙
I [x; y] as the channel
capacity C.
We observe that the channel capacity C is determined by the bandwidth W and
the signal-to-noise ratio P S /P N . Higher bandwidth—the ability to transmit higher
frequencies—will allow us to transmit more information. Moreover, a small noise
power P N allows us to distinguish smaller differences in the payload signal, which
gives us more signal levels to encode information. As an example, consider the
communication on a wireless WLAN network, which benefits from increasing the
bandwidth by channel bonding, where two or more of the, by default, W = 20 MHz
wide channels, are used simultaneously in order to increase the download speed.
Moreover, the transmission amplifiers automatically increase the transmission power
P S , up to the legally allowed maximum, to maximize transmission capacity. Normally
the power level is chosen to match the rate at which the information is created at the
source—listening to an Internet radio station requires less bandwidth than watching