Random Lecture 2

Question 1

what is the best way to represent random signals

Answer 1

is in terms of frequency content

Question 2

What is the fourier transform

Answer 2

X(w) = integral from -inf to inf of x(t) * e^(-iwt) * dt

Question 3

What is the inverse fourier transform

Answer 3

x(t) 1/2pi * integral from - inf to inf of X(w)*e^(iwt) dw

Question 4

What does the fourier transform do

Answer 4

convert between the time domain and frequency domain

Question 5

How does a sine wave appear after fourier transofmr

Answer 5

single line at frequency of sine wave, magnitude equal to amplitude of sine wave

Question 6

What is the kronecker delta function

Answer 6

delta ij equal to 1 if i = j and zero everywhere else

Question 7

What is the projection properties due to result of kronecker delta definition

Answer 7

sum from i=-inf to inf ai*dela ij = aj

Question 8

What can we use the delta function to do

Answer 8

project functins from x to y

Question 9

For analogue functions how do we project them

Answer 9

integratal from -inf to inf of f(x)*delta(x,y) dx = f(y)

Question 10

What is the formula for the delta function

Answer 10

delta (t,t’) = 1/2pi * integral -inf to inf of e^(iw(t-t’)) dw

Question 11

How can the dirac deltafunction be rewritten delta(t,t’)

Answer 11

As integral only depends on t-t’ can write
delta (t,t’) = delta (t - t’)
delta (t,t’) = 1/2pi * integral -inf to inf of e^(iw(t-t’)) dw

Question 12

What does delta function delta (t - t’) look like

Answer 12

at t =/ t’ e^(iw(t-t’)) osciallates infintely as often as t’ varies from - inf to inf, negative as often as positve and integral averages to zero
at t = t’ then delta (0) = infnite

Question 13

What is the fourier transform of a product of two time signals fourier transofmr (y1(t)*y2(t))

Answer 13

See tutorial for full steps but basic steps are
write yi(t) as inverse fourier transform of corresponding spectra and sub into definition of fourier transform
Simplify and interchage order of integrals as limits all same
Extract delta function
Project out value

Question 14

What is the convultion theorem

Answer 14

product of two random signals
written as (Y1 * Y2)*(w) = fourier transform of y1(t)*y2(t) = 1/2pi *integral from -inf to inf of Y1(w - omega)*Y2(omega) d omega

Question 15

What is the inverse of the convultion theorem

Answer 15

Inverse fourier transform of Y1(w)Y2(w) = integral from - inf to inf of y1(tau)y2(t - tau) d tau

Question 16

Why does FRF H(w) = Y(w)/X(w) not work for random signals

Answer 16

As cant take fourier transform of y(t) or x(t) mathemtically if random signals
Need to take random signals, convert to deterministic objects and can then fourier transform
Simplest way to do this is take mean

Question 17

What is the autocorrelation function

Answer 17

phi xx(tau) measure of how much a signal looks like itself when shifted by an amount tau
It is used to find regularities in data
Roughly speaking detects periodicity in data

Question 18

what is the mathematicaly definition of an autocorrelation function and what does this assume

Answer 18

phi xx(tau) = E[x(t +tau)*x(t)]
Assumes data is stationary otherwise it should depend on t as well as tau

Question 19

What is the result of an autocorrelation function of x(t) = sin (2pit/tau’)

Answer 19

Will be regular peaks in phi xx(tau) when tau = n*tau’

Question 20

What happens to an autocorrelation function if x(t) is zero mean

Answer 20

phi xx(tau) = E[x(t)^2] = sigmax^2
if x is not zero mean phi xx(0) = mean square of the process

Question 21

If x(t) is stationary what can we do the origin of t in an autocorrelation function

Answer 21

change origin of t to t-tau without changing the autocorrelation
phi xx(tau) = E[x(t +tau)*x(t)] = E[x(t)*x(t - tau)] = E[x(t - tau)*x(t)] = phi xx(-tau)

Question 22

What does the crosscorrelation function

Answer 22

Detects correlation between two functions x(t) and y(t), if put big X in and get a change in Y autocorrelation function will be able to detect
Detects casual relationships between singals - input and output

Question 23

What is the mathematical definition of the crosscorrelation function

Answer 23

phi yx(tau) = E[y(t+tau)*x(t)]
phi xy(tau) = E[x(t+tau)*y(t)]

Question 24

How are the crosscorrelation phi yx(tau) and phi xy(tau) related

Answer 24

Because of stationarity phi yx(tau) = phi xy(-tau) and phi xy(tau) = phi yx(-tau)

Question 25

What type of function is the autocorrelation function

Answer 25

An even function of tau

Question 26

What is the fourier transform of the autocorrelation function

Answer 26

Write out mathematically and looks like convolution function
integral from - inf to inf of x(t+tau)x(t) dt
From convolution theorem this is the product of two spectra i.e.
X(w)X(w) = magnitude of X(w) ^2 = Sxx(w)
* indicates complex conjugation

Question 27

Why can we interpret Sxx(w) as a power density

Answer 27

phixx(tau) = 1/2pi * integral from -inf to inf of Sxx(w)e^(iwtau) dw then
We know phixx(0) is equal to the variance sigmax^2 thus
phixx(0) = 1/2pi * integral from -inf to inf of Sxx(w) dw
simgax^2

Question 28

If Sxx(w) is equal to 2piP what is the signal

Answer 28

Value is equal to a constant and weights all frequencies equally, thus it is white noise

Question 29

Suppose Sxx(w) is equal to 2PiP what does phixx(tau) equal

Answer 29

phixx(tau) = integral from - inf to inf of Pe^(iwt) dw = 2piPdelta(tau)
Pulling out constant and just delta function without 2pi, get additional 2pidelta function constant P
as delta is only non zero at tau = 0, for white noise excitation x is only correlated with itself at tau= 0 if shift at all doesnt look like itself

Question 30

What is the fourier transofrm of the cross correlation function equal to

Answer 30

Syx(w) = Y(w)X(w)*

Question 31

What are Sxx and Syx(w) called

Answer 31

Sxx is sometimes called the autospectral density

Syx is the cross spectral density

Question 32

How can the spectral densities Sxx(w) and Syx(w) be used to find the frequency response function of a random signal

Answer 32

Syx(w) = Y(w)X(w)*
Sxx(w) = X(w)X(w)*
Syx(w) / Sxx(w) = Y(w)/X(w) = H(w)
if x(t) and y(t) are infinite samples

Question 33

What is duhamels integral

Answer 33

y(t) = integral from - inf to inf of h(w)*x(t-tau) dtau

Question 34

What can we recover with you autocorrelation functions of x and y

Answer 34

Syy(w) = Y(w) Y(w)*
Sxx(w) = X(w) X(w)*
Syy(w)/Sxx(w) = Y(w)Y(w)/X(w)X(w) = H(w) H(w)*
thus Syy(w) = magnitude of H(w)^2 *Sxx(w)
Can only recover magnitude of FRF lose phase

Question 35

Starting from Duhamels integral show that ybar = Hxbar

Answer 35

See presentation

Question 36

What is Sxx

Answer 36

power spectrum of the signal x
it is the fourier transform of the autocorrelation function phixx
Called power spectral density
Tells us how much power is concentrated around individual frequencies