white noise & properties

26
8/7/2019 White Noise & Properties http://slidepdf.com/reader/full/white-noise-properties 1/26 By ANANDA RAJ M C I MTECH, DEC NMAM, Nitte 10/15/2010 1

Upload: pikumarkrishna

Post on 08-Apr-2018

234 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 1/26

By

ANANDA RAJ M C

I MTECH, DEC

NMAM, Nitte

10/15/2010 1

Page 2: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 2/26

CONTENTS

� INTRODUCTION TO NOISE

� DIFFERENT TYPES OF NOISE

� WHITE NOISE

� GAUSSIAN PROCESS

� ADDITIVE WHITE GAUSSIAN NOISE

10/15/2010 2

Page 3: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 3/26

INTRODUCTION

� Noise is an inevitable part of our existence- disturbance , unwantedsignal

� In scientific study, noise arise in many

ways: natural process

� sensors and recording systems

Measuring of Noise:

� Frequency Spectrum

� Noise Power 

10/15/2010 3

Page 4: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 4/26

Different Types Of NOISES

� Johnson Noise

� Shot Noise

� Color Noise

� White Noise

� Additive White Gaussian Noise (AWGN)

10/15/2010 4

Page 5: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 5/26

Johnson Noise

� Electrons Motion in Electrical Resistors

� Squared Voltage Noise Density

vn2 = 4kTR (units of volts/sqrt(Hz))

K->Boltzman¶s Constant, T->Abs. Temperature,

R->Resistance, f->Bandwidth

V rms = Sqrt(integral (f 1 to f 2) vn 2 df)

= Sqrt ( 4kTR f)

10/15/2010 5

Page 6: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 6/26

Shot Noise

� Motion of Discrete Electrons

� Arises in Electronic Devices such as Diodes andTransistors

�Exhibits Gaussian Probability Density Function

10/15/2010 6

Page 7: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 7/26

Color Noises

� White Noise : Power Density is Constant Over

a Finite Frequency Range

Pink Noise : Power Density Proportional to 1/f � Blue Noise : Power Density Proportional to f 

� Purple Noise : P. D Proportional to f^2

�Brown Noise : P.D Proportional to 1/f^2

� Black Noise : P. D is Constant for a finite

frequency range above 20khz

10/15/2010 7

Page 8: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 8/26

White Noise

� Random Process

� Noise in which all frequency components present( zero to infinity)

� White is used in the sense that White Light Contains Equalamounts of all Frequencies within the Visible Band

� Flat Power Spectral Density

Mean is Zero

� Autocorrelation function

10/15/2010 8

Page 9: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 9/26

Mathematical Representation

A Random Vector w is a White random vector if 

and only if its Mean vector and Autocor

-relation Matrix are the following:

10/15/2010 9

Page 10: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 10/26

White Random Process

A continuous time random process w (t ) is a whitenoise process if and only if its mean functionand autocorrelation function satisfy the

following:

i.e. zero mean process for all time & has infinitepower at zero time shift since its AutoCorrelation is Dirac Delta Function

10/15/2010 10

Page 11: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 11/26

The Previously Shown Autocorrelation function

yields (By Fourier Transform) the followingPower Spectral Density of White Noise:

Where , k is Boltzmann's Constant

Te is Noise Equivalent Temperature

10/15/2010 11

Page 12: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 12/26

Characteristics of White Noise

(a) Power Spectral Density

(b) Autocorrelation Function

10/15/2010 12

Page 13: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 13/26

E.g.: Ideal Low pass Filtered Noise

Consider an White Noise w(t) with mean zeroand power spectral density is appliedto an ideal low pass filter of Bandwidth B and

magnitude response of one. Then, the PowerSpectral Density of the noise n(t) is given by,

10/15/2010 13

Page 14: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 14/26

The Autocorrelation Function of n(t) is Inverse

Fourier Transform of Power Spectral Density

We see has its maximum value

at the Origin and it passes through zero at

10/15/2010 14

Page 15: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 15/26

Characteristics Of Low Pass Filtered

White Noise

(a) Power Spectral Density

(b) Autocorrelation Function

10/15/2010 15

Page 16: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 16/26

Gaussian Process

Consider X(t) random process that starts at t=0

and lasts at t=T. Suppose also that we weight

the random process X(t) by some function g(t)

and then integrate the product g(t)*X(t) overthis observation interval, there by obtaining a

random variable Y defined by,

10/15/2010 16

Page 17: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 17/26

Page 18: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 18/26

Where is the Mean and is the Variance

of the Random Variable Y. A plot of this

probability density function is shown below,for the special case when Gaussian Random

Variable Y is Normalized to have a Mean

of zero and a Variance of one, as shown by

10/15/2010 18

Page 19: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 19/26

Gaussian Noise

Gaussian Noise is defined as Noise with

Gaussian Amplitude Distribution.

It is commonly used as Additive White Noise to

yield the Additive White Gaussian

Noise(AWGN).

10/15/2010 19

Page 20: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 20/26

White Gaussian Noise MATLAB

MATLAB Command:

Y= wgn (m, n, p);

The above Command Generates an m-by-n matrix of White Gaussian Noise and p is Power of Y in dBW.The default Load Impedance is One ohm.

Ex: To generate a column vector of 100 containing realwhite Gaussian noise of power 0dB W. Use thisCommand

Y1= wgn(100, 1, 0);

10/15/2010 20

Page 21: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 21/26

Additive White Gaussian Noise

10/15/2010 21

� Noise is Additive i.e. addition of White Noise

and Gaussian Distribution

� Noise is White i.e. the power spectral densityis flat in all frequencies

� Gaussian i.e. the samples (amplitude) have aGaussian distribution

Page 22: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 22/26

AWGN MATLAB

� MATLAB Command:

y = awgn (x, snr);

The above command adds the White Gaussian Noise to thevector signal x.

The snr is Scalar specifies the signal to noise ratio per sample,in dB. This syntax assumes the power of x is zero.

To add the signal power we use following command,

y= awgn (x, snr, sigpower);

10/15/2010 22

Page 23: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 23/26

MATLAB Example Program

t = 0:.1:10;

x = sin(2*pi*t); % Create sine wave signal.

y = awgn (x,10, 'measured'); % Add white%Gaussian noise and measures the signal

%power

plot(t, x, t, y) % Plot both signals.legend('Original signal', 'Signal with AWGN');

10/15/2010 23

Page 24: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 24/26

Application White Noise

� Emergency Vehicle Sirens

� Electronic Music, Audio Synthesis

� Generate Impulse Responses

� Frequency Response Testing

� Sound Masking - Tinnitus Masker

10/15/2010 24

Page 25: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 25/26

Page 26: White Noise & Properties

8/7/2019 White Noise & Properties

http://slidepdf.com/reader/full/white-noise-properties 26/26

THANK YOU

10/15/2010 26