Transcript Neuronal Computation Using High Order Statistics

DCSP-5: Noise
Jianfeng Feng
Department of Computer Science Warwick
Univ., UK
[email protected]
http://www.dcs.warwick.ac.uk/~feng/dsp.html
Assignment
Question 8
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load handel
for i=1:73113
noise(i)=cos(100*i*0.1);
end
x=y+noise';
sound(x)
Noise in communication systems:
probability and random signals
I = imread('peppers.png');
imshow(I);
noise = 1*randn(size(I));
Noisy = imadd(I,im2uint8(noise));
imshow(Noisy);
Noise in communication systems:
probability and random signals
Noise is a random signal.
By this we mean that we cannot predict its value.
We can only make statements about the
probability of it taking a particular value, or
range of values.
The Probability density function (pdf) p(x) of a
random signal, or random variable x is defined
to be the probability that the random variable x
takes a value between x0 and x0 +dx.
We write this as follows:
p(x0 )dx =P(x0 <x< x0 +dx)
The probability that the random variable will take
a value lying between x1 and x2 is then the
integral of the pdf over the interval [x1 x2]:
From the rules of integration:
P(x_1<x<x_2)=P(x_2)-P(x_1)
Continuous distribution.
An example of a continuous distribution is the
Normal, or Gaussian distribution:
where m=m, s is the mean and standard
variation value of p(x).
The constant term ensures that the distribution is
normalized.
Continuous distribution.
This expression is important as many actually
occurring noise source can be described by it,
i.e. white
noise or coloured noise.
Generating f(x) from matlab
How would this be used?
If we want to know the probability of, say, the
noise signal, n(t), having the value [-v_1, v_1],
we would evaluate: P(v_1)-P(-v_1)
The distribution function P(x) is usually written in
terms of as a function of the error function erf(x).
The complementary error function erfc is defined
by
erfc(x)=1- erf(x)
Discrete distribution.
Probability density functions need not be
continuous.
If a random variable can only take discrete
value, its PDF takes the forms of lines.
An example of a discrete distribution is the
Poisson distribution
P ( n) 

n
n!
exp( )
We cannot predicate value a random variable
may take on a particular occasion.
We can introduce measures that summarise what
we expect to happen on average.
The two most important measures are the mean
(or expectation )and the standard deviation.
The mean of a random variable x is defined to
be
Expectation EX =
In the examples above we have assumed that the
mean of the Gaussian distribution to be 0, the
mean of the Poisson distribution is found to be
.
The mean of a distribution is, in common sense,
the average value.
The variance s is defined to be
The square root of the variance is called standard deviation.
The standard deviation is a measure of the
spread of the probability distribution around the
mean.
A small standard deviation means the distribution
are close to the mean.
A large value indicates a wide range of possible
outcomes.
The Gaussian distribution contains the standard
deviation within its definition.
In many cases the noise present in
communication signals can be modelled as a
zero-mean, Gaussian random variable.
This means that its amplitude at a particular time
has a PDF given by Eq. above.
The statement that noise is zero mean says that,
on average, the noise signal takes the values
zero.
We have already seen that the signal to
noise ratio is an important quantity in
determining the performance of a
communication channel.
The noise power referred to in the definition
is the mean noise power.
It can therefore be rewritten as
SNR= 10 log10 ( S / s2)
If only thermal noise is considered, we have
2
s =kT
mB
where T is the Boltzman's constant
(k=1.38 x 10-23 J/K)
Tm is the temperature
and B is the receiver bandwidth.
Correlation or covariance
• Cov(X,Y) = E(X-EX)(Y-EY)
correlation is normalized covariance
• Positive correlation
• Negative correlation
• No correlation
Stochastic process
• A stochastic process is a collection of
random variables x(t), for each fixed t, it is
a random variable
• For example, when x(t) is a Gaussian
variable, it is called a Gaussian process
• A special Gaussian process is called
white noise n(t)
• White noise is a random process with a flat
power spectrum.
• In other words, the signal contains equal power
within a fixed bandwidth at any center
frequency.
• White noise draws its name from white light in which the power
spectral density of the light is distributed over the visible band in
such a way that the eye's three color receptors are
approximately equally stimulated.
White noise vs. colour noise
• The most ‘noisy’ noise is a white noise
since its autocorrelation is zero, i.e.
corr(n(t), n(s))=0 when t > s
Otherwise, we called it colour noise since
we can predict some outcome of n(t),
given n(s)
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load handel
plot(y)
size(y)
x=randn(73113,1);
plot(abs(fft(z)))
z=y+.1*x;
plot(abs(fft(z)))
hold on
plot(abs(fft(y)),'r')
plot(abs(fft(.1*x)),'g')
Why do we love Gaussian?
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X+Y (two Gaussian) = another Gaussian