204
12 Cryptocurrencies
more information than the channel capacity C across a communication channel. The
noise in the channel destroys transmitted information and we have to detect which
bits were flipped and then flip them back. This requires extra, redundant, information
that we need to send along with the real information, the payload. We only name
Hamming codes as an example of a class of error-correcting codes that stuff the
payload into larger data frames. This allows the receiver to detect and correct flipped
bits.
In the previous sections the transmitted symbols were selected from a discrete
set. Next, we briefly discuss extending the theory to continuous sets in the following
section.
12.4 Continuous Information Channels
An example where continuous signals appear is the transmission of the information—
the music—as a continuous stream of voltages u, from an old-fashioned record player
to the audio amplifier and on to the speakers. The voltages x = u/u 0 , normalized to
some reference voltage u 0 , have a continuous probability distribution p(x), which
can be approximated by a histogram of the voltages measured in small time intervals,
normalized by the total number of measurements. In Fig. 12.4 we show the amplitudeprobability distributions for the first movement of Beethoven’s fifth symphony on
the left and for a sine on the right. The amplitude of the sine was chosen to have
the same rms as that of the distribution on the left-hand side. The dashed red curves
show Gaussians with the rms of the distributions and is the same on both plots.
Having such probability distributions available, we can now generalize the concept
of entropy, introduced in (12.2), to continuous variables x
H [x] = −
p(x) log ( p(x)) dx ,
(12.20)
Fig. 12.4 The solid black lines show the amplitude-probability distribution of the first movement
of Beethoven’s fifth symphony (left) and that of a sine (right). The rms of the two Gaussians, shown
as dashed red curves, are equal and also equal to the corresponding black curves
12 Cryptocurrencies
more information than the channel capacity C across a communication channel. The
noise in the channel destroys transmitted information and we have to detect which
bits were flipped and then flip them back. This requires extra, redundant, information
that we need to send along with the real information, the payload. We only name
Hamming codes as an example of a class of error-correcting codes that stuff the
payload into larger data frames. This allows the receiver to detect and correct flipped
bits.
In the previous sections the transmitted symbols were selected from a discrete
set. Next, we briefly discuss extending the theory to continuous sets in the following
section.
12.4 Continuous Information Channels
An example where continuous signals appear is the transmission of the information—
the music—as a continuous stream of voltages u, from an old-fashioned record player
to the audio amplifier and on to the speakers. The voltages x = u/u 0 , normalized to
some reference voltage u 0 , have a continuous probability distribution p(x), which
can be approximated by a histogram of the voltages measured in small time intervals,
normalized by the total number of measurements. In Fig. 12.4 we show the amplitudeprobability distributions for the first movement of Beethoven’s fifth symphony on
the left and for a sine on the right. The amplitude of the sine was chosen to have
the same rms as that of the distribution on the left-hand side. The dashed red curves
show Gaussians with the rms of the distributions and is the same on both plots.
Having such probability distributions available, we can now generalize the concept
of entropy, introduced in (12.2), to continuous variables x
H [x] = −
p(x) log ( p(x)) dx ,
(12.20)
Fig. 12.4 The solid black lines show the amplitude-probability distribution of the first movement
of Beethoven’s fifth symphony (left) and that of a sine (right). The rms of the two Gaussians, shown
as dashed red curves, are equal and also equal to the corresponding black curves