Probability of Error
The probability density function of having an error of a given size can be modelled by a Gaussian function; the mean value will be the relative sent value, and its variance will be given by:
where \Phi_N (f) is the spectral density of the noise within the band and Hr (f) is the continuous Fourier transform of the impulse response of the filter hr (f).
The probability of making an error is given by:
where, for example, P_{e|H_0} is the conditional probability of making an error given that a symbol v0 has been sent and P_{H_0} is the probability of sending a symbol v0.
If the probability of sending any symbol is the same, then:
If we represent all the probability density functions on the same plot against the possible value of the voltage to be transmitted, we get a picture like this (the particular case of L = 4 is shown):
The probability of making an error after a single symbol has been sent is the area of the Gaussian function falling under the functions for the other symbols. It is shown in cyan for just one of them. If we call P+ the area under one side of the Gaussian, the sum of all the areas will be: 2 L P^+ - 2 P^+. The total probability of making an error can be expressed in the form:
We have now to calculate the value of P+. In order to do that, we can move the origin of the reference wherever we want: the area below the function will not change. We are in a situation like the one shown in the following picture:
it does not matter which Gaussian function we are considering, the area we want to calculate will be the same. The value we are looking for will be given by the following integral:
where erfc is the complementary error function. Putting all these results together, the probability to make an error is:
from this formula we can easily understand that the probability to make an error decreases if the maximum amplitude of the transmitted signal or the amplification of the system becomes greater; on the other hand, it increases if the number of levels or the power of noise becomes greater.
This relationship is valid when there is no intersymbol interference, i.e. g(t) is a Nyquist function.
Read more about this topic: Amplitude-shift Keying
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