Random Lecture 2 Flashcards
what is the best way to represent random signals
is in terms of frequency content
What is the fourier transform
X(w) = integral from -inf to inf of x(t) * e^(-iwt) * dt
What is the inverse fourier transform
x(t) 1/2pi * integral from - inf to inf of X(w)*e^(iwt) dw
What does the fourier transform do
convert between the time domain and frequency domain
How does a sine wave appear after fourier transofmr
single line at frequency of sine wave, magnitude equal to amplitude of sine wave
What is the kronecker delta function
delta ij equal to 1 if i = j and zero everywhere else
What is the projection properties due to result of kronecker delta definition
sum from i=-inf to inf ai*dela ij = aj
What can we use the delta function to do
project functins from x to y
For analogue functions how do we project them
integratal from -inf to inf of f(x)*delta(x,y) dx = f(y)
What is the formula for the delta function
delta (t,t’) = 1/2pi * integral -inf to inf of e^(iw(t-t’)) dw
How can the dirac deltafunction be rewritten delta(t,t’)
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
What does delta function delta (t - t’) look like
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
What is the fourier transform of a product of two time signals fourier transofmr (y1(t)*y2(t))
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
What is the convultion theorem
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
What is the inverse of the convultion theorem
Inverse fourier transform of Y1(w)Y2(w) = integral from - inf to inf of y1(tau)y2(t - tau) d tau
Why does FRF H(w) = Y(w)/X(w) not work for random signals
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
What is the autocorrelation function
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
what is the mathematicaly definition of an autocorrelation function and what does this assume
phi xx(tau) = E[x(t +tau)*x(t)] Assumes data is stationary otherwise it should depend on t as well as tau
What is the result of an autocorrelation function of x(t) = sin (2pit/tau’)
Will be regular peaks in phi xx(tau) when tau = n*tau’
What happens to an autocorrelation function if x(t) is zero mean
phi xx(tau) = E[x(t)^2] = sigmax^2 if x is not zero mean phi xx(0) = mean square of the process
If x(t) is stationary what can we do the origin of t in an autocorrelation function
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)
What does the crosscorrelation function
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
What is the mathematical definition of the crosscorrelation function
phi yx(tau) = E[y(t+tau)*x(t)] phi xy(tau) = E[x(t+tau)*y(t)]
How are the crosscorrelation phi yx(tau) and phi xy(tau) related
Because of stationarity phi yx(tau) = phi xy(-tau) and phi xy(tau) = phi yx(-tau)
What type of function is the autocorrelation function
An even function of tau
What is the fourier transform of the autocorrelation function
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
Why can we interpret Sxx(w) as a power density
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
If Sxx(w) is equal to 2piP what is the signal
Value is equal to a constant and weights all frequencies equally, thus it is white noise
Suppose Sxx(w) is equal to 2PiP what does phixx(tau) equal
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
What is the fourier transofrm of the cross correlation function equal to
Syx(w) = Y(w)X(w)*
What are Sxx and Syx(w) called
Sxx is sometimes called the autospectral density
Syx is the cross spectral density
How can the spectral densities Sxx(w) and Syx(w) be used to find the frequency response function of a random signal
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
What is duhamels integral
y(t) = integral from - inf to inf of h(w)*x(t-tau) dtau
What can we recover with you autocorrelation functions of x and y
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
Starting from Duhamels integral show that ybar = Hxbar
See presentation
What is Sxx
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
