EE422G
Signals and Systems Laboratory
Noise Analysis
Kevin D. Donohue
Electrical and Computer Engineering
University of Kentucky
Noise as a Random Variable
Noise is modeled as a random variable (RV), which is defined
as a function that maps an event into a real number.
There are 2 main characterizations of noise that are important
for designing signal processing and communication systems:
1. Distribution of amplitudes is characterized by the probability density
function (pdf), or its integral, the cumulative distribution function (cdf)
2. The correlation or influence between neighboring noise samples is
characterized by the autocorrelation (AC), or its Fourier transform, the
power spectral density (PSD).
The PDF
Given random signal x[n] with pdf pX(x), its probability of
occurring between values a and b is given by:
b
Pr {a X ≤b }=∫ p X  x dx
a
The cdf is the probability the RV X is less than x denoted by:
x
Pr { X ≤ x }=P x  x= ∫ p X  d 
−∞
PDF and CDF Example
Consider an exponential
noise distribution with
pdf:
Compute cdf and find
values of x such that the
probability of being less
than x1 is 0.5132 and less
than x2 is 0.9544. Find
the probability of x being
between x1 and x2
0 .8
pdf
0 .6
0 .4
0 .2
0
0
0 .5
1
1 .5
2
2 .5
x
3
3 .5
4
4 .5
5
4 .5
5
1
0 .8
X : 3 .4 3
Y : 0 .9 5 4 4
X : 0 .8
Y : 0 .5 1 3 2
0 .6
cdf
p X ( x)=0.9exp(−0.9 x)
for x≥0
and 0 Elsewhere
1
0 .4
0 .2
0
0
0 .5
1
1 .5
2
2 .5
x
3
3 .5
4
Parametric PDF Estimate
If the form of the distribution is known, then only the parameters
of the distribution need to be estimated from the data. For
example consider a Gaussian distribution with pdf given by:
(
(x−μ)
1
p X ( x)=
exp −
2
2
2σ
√2 π σ
with mean  and standard deviation 2.
2
)
So if N data samples (xn) are collected, the sample mean and
standard deviation is estimated by:
1
 =
N
N −1
∑ xn
n=0

N −1
1

 x n − 2
=
∑
N −1 n=0
Threshold Design
Assume that a noise process is exponential with
unknown mean. Given 10 noise samples, estimate a
threshold to detect a signal of greater power, such that
the probability of false alarm is 1 out of 10k tests.
Sample Data:
s=[ 0.20, 1.17, 0.69, 3.70, 1.3, 5.55, 0.46, 0.70, 0.36, 0.34 ]
Model:
 
1
x
p X  x= exp −
b
b
for x≥0
Threshold Design
Compute sample mean as an estimate of the b
10
parameter:
 1
b=
s =1.45
∑
10 i=1
i
Use in cdf to show a relationship between a false alarm
probability and the threshold value:
 
−T
P fa =1−cdf T =exp
b
T =−b ln  P fa ≈−b ln P fa 
T ≈−1.45ln 1/10k=13.35
PDF Non-Parametric Estimation
If the form of the distribution is unknown, it can be
estimated with few assumptions using a normalized
histogram operation, which estimates the PDF over short
intervals (bins) based on percentage of sample data
occurring in the bin:
b
Pr [ a x≤b ]=∫ p  x  dx≈
a
b
∫ p  x dx=b−a  p  x ab 
samples∈[ a , b ]
total samples collected
for some x ab ∈[ a b ]
a
0 .8
0 .8
0 .7
0 .7
0 .6
0 .6
0 .5
0 .5
0 .4
0 .4
0 .3
0 .3
0 .2
0 .2
0 .1
0 .1
0
0
0 .5
1
1 .5
2
x
2 .5
3
3 .5
0
1
2
x
3
4
Signal Power and Moments
The RMS value of a random signal is equivalent to its
second moment.
√
√
1
1
2
S rms =
s
(t
)
dt
≈
∫
T T
N
N
∑ si2
i=1
If the signal is zero mean, the standard deviation is
equivalent to the RMS value.
Find the RMS value of:
s(t )=3+5sin (2 π 100 t )V
Find RMS value of a Gaussian random noise process with
mean 0.5 and variance of 4.
Signal to Noise Ratio
Signal to noise ratio in Decibels (dB) is defined as the
following power ratio:
SNR dB=10 log10
( )
σ
2
σ
2
s
=20 log 10
n
σs
σn
( )
where s is the RMS value of the signal and n
is the RMS value of the noise.
Assume a zero mean signal has an RMS value of 2 and
zero mean Gaussian noise has an RMS value of 1. The
noise and signal will be added together to simulate a
signal in noise. What must the noise signal be multiplied
by so that the resulting SNRdB is -2 dB?
Correlation
The correlation indicates how similar one signal is to
another. Related to this is the covariance, which removes
the signal mean:
 xy =E[ xy]=∫ ∫  x− x  y− y  p XY  x , y dx dy
x
y
Correlation is the same as above without the mean
subtraction. For zero mean signals, the covariance and
correlation are identical.
To estimate correlation from stationary (i.e. statistics do
not change over time) random signal segments of length
N, the sample correlation is used:
N −1
1
∑ x [n−k ] y [n]
Rxy [k ]=
N −k n=k
Autocorrelation
A signal can be correlated with a delayed version of itself
to determine influence (statistically) as samples get further
apart. This is the autocorrelation (AC) function:
1
Rxy [ k ]=
N
N −1
∑ y [n−k ] y [n]
n=k
This is referred to as the biased AC and k is often called
the lag, which represents the relative delay or shift
between the signals. The Matlab function xcorr() can be
used to perform these operations.
Note for k=0 the energy or second moment is computed.
This will always be the largest value for all AC lags. The
AC is often normalized by this value so the zero lag
becomes 1.
Convolution and Correlation
Recall the convolution operation:
N −1
w [k ]= ∑ x [ k −n] y [n]
n=k
The main difference between convolution and correlation
is time reversal of one of the signals before the multiply
and sum operation. The conv() operation in Matlab can be
used to implement correlation (with some minor
modifications to the input arguments)
R [k ]= 1
N
N −1
∑ x [n−k ] y [n]
n=k
Autocorrelation Example
x (n )
4
2
0
-2
-4
-2 0
-1 5
-1 0
-5
0
n
5
10
15
20
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Autocorrelation Example
Power Spectral Density
The power spectral density (PSD) of a random signal is its average
magnitude spectrum (phase is considered irrelevant and is removed).
For stationary random signals the PSD can be estimated from the
average of DFT magnitudes.
G −1
1
S [m]=
G
∑ ∣V g [m]∣
g=0
From long data segments the PSD is estimated from shorter
overlapping segments, referred to as Welch's method. For a single
segment g, FFT is computed from:
1
V g [m]=
N
N −1
∑
n=0

v [ Lgn] w [n]exp − j 2
nm
N FFT

for 0≤m N FFT
where w[n] is a window function used to taper the data down at the
edge of each extracted segment, and L is an increment (usually less
than N) to obtain an overlap between segments. NFFT is the number of
FFT points usually obtained from padding with zeros. The Matlab
function pwelch() can be used to implement the PSD computation.
PSD estimation process with hopping window method
FFT Magnitudes
Average
Windowed Segments
Power Spectral Density Estimation
The power spectral density (PSD) estimation performance is affected
by window length, window taper, and zero padding.
1. Window length is proportional to frequency resolution (i.e. the ability
to distinguish between 2 closely spaced frequencies).
1
Δf∝
T
2. Window tapering lowers resolution but reduces sidelobe artifacts
called spectral leakage.
3. Zero padding increases the number of grid points on the PSD
frequency axis. It does this through interpolations, so resolution is
not improved but can improve the location of a peak or a null by
using a finer grid spacing.
MATLAB Tips for Axes
If plotting a time signal (vector) the X-axis labels must
correspond to the same number of point as in the signal
vector. Given the sampling frequency is stored in variable
fs in Hz and the signal vector is stored in variable sig, a
time axis vector can be generated by:
>> tax = [0:length(sig)-1]/fs; % time axis in seconds
>> plot(tax,sig)
If plotting the frequency spectrum vector pd from an
NFFT point DFT, the frequency axis vector can be created
from:
>> fax = fs*[0:length(pd)-1]/length(pd); % Frequency axis in Hz
>> plot(fax,abs(pd))