discrete signal processing
DESCRIPTION
TRANSCRIPT
DISCRETE RANDOM SIGNAL PROCESSING
ADSPUNIT-I
Contents are…Definition -DTPBernoulli’s ProcessMoments-Ensemble AveragesStationary Process-WSSMatrix FormsParseval’s TheoremWeiner-Khinchine relationPSDFiltering of Random ProcessSpectral factorizationBias-ConsistencySpecial types of RPYule-walker Equation
Discrete Time Random Process:A random variable may be thought of as
mapping from sample space of an experiment into a set of real or complex values.
A Discrete time random process may be thought of mapping from sample space Ω into a set of discrete time signals.
It is nothing but an Indexed sequence of random variables.
Example: Tossing a coin, Rolling a die
Bernoulli’s Process:The outcome of an event does not affect the
outcome of the other event at any time then the process is called as Bernoulli’s Process.
The Moments are,Mean :The average of outcomes
µ=1/n ∑ x(i), i=1 to nVariance: How far the random values is away
from the central mean. σ2 =1/n ∑ (x-µ)2, i=1 to n
Skewness: It deals symmetry with the mean values
S= ∑ (x-µ)3/σ3
Kurtosis: Flatness or stability of the systemK= ∑ (x-µ)4/σ4
ERGODICITY:When the time average of the process is equal to
the ensemble average. It is said to be “ergodic”. ie, E(X)= Complement of X
ENSEMBLE AVERAGES:
Mean: Mx(n)= E[x(n)]Variance: σ2x(n)=E[|x(n)-Mx(n)|2]Auto Correlation :Finding the relationship
between the random variables in the same process.
rx(k,l)=E[x(k) x*(l)]Auto Covariance: Cx(k,l)=E[|x(k)-Mx(k)|,|x(l)-
Mx*(l)|] Cross Correlation: rxy(k,l)=E[x(k) y*(l)]Cross Covariance: Cxy(k,l)=E[|x(k)-Mx(k)|,|y(l)-
My*(l)|]
RELATIONSRelation between rx &Cx:
Cx(k,l)=rx(k,l)-Mx(k) Mx*(l)Mean=0,
Cx(k,l)=rx(k,l)Relation between rxy &Cxy:
Cxy(k,l) =rxy(k,l)-Mx(k) My*(l)Mean=0,
Cxy(k,l)=rxy(k,l)•If the random process is uncorrelated means Cxy(k,l)=0. •If the two random process x(n) & y(n) are said to be orthogonal means rxy(k,l)=0
STATIONARY PROCESSA process is said to be stationary when all
the statistical averages (Mean, Variance etc.) are independent of time
i.e, For first order, Mx(n)=Mx
σ2x(n)=σ2xFor second order, rx(k,l)= rx(k-l,0)
rx(k,l)= rx(k-l)
Example: Quantization Error
WIDE SENSE STATIONARY PROCESS:Case:1The mean of the process is constant Mx.The autocorrelation of the process depends
on the difference on k,l.(k-l)The variance of the process is finite.Case:2x(n),y(n) a said to be jointly WSS if they are
independently WSS.rxy(k-l)=E[x(k) ,y*(l)]
PROPERTIES OF WSS & AUTO
CORRELATION:1. Symmetry rx(k)=rx*(-k)2. Mean square value rx(0)=E[|
x(n)|2]≥0 3. Maximum Value rx(k) ≤ rx(0)
4. Periodicity E[|x(n)-x(n-ko)|2]For the auto correlation Rxx…
MATRIX AND ITS PROPERTIESThe auto correlation & auto covariance can
be expressed in the form of matrix.PROPERTIES:The autocorrelation of a WSS process x(n)
is a Hermitian Toeplitz matrix.Non negative & definite.The eigen value λk are real value and non
negative.
IMPORTANT MATRIX FORMSOrthogonal Matrix A T =A-1 Hermitian Matrix [A*]T=[AT]*Skew Hermitian Matrix A=-AH
Toeplitz Matrix => All the diagonal elements are same.
Henkal Matrix M+N-1
PARSEVAL’S THEOREM (OR) RAYLEIGH ENERGY FORMULA
The sum or integral of the square of the function is equal to the sum or integral of square of the transform.
That is E<x,x>
WEINER KHINCHINE RELATIONFor a well behaved stationary random
process the power spectrum is equal to the Fourier transform of the autocorrelation function.
POWER SPECTRAL DENSITYThe PSD of the process is written by,
Px(ejw)=∑rx(k) e(-jwk) , k=-∞ to ∞
Power spectrum of x(n),Px (z)=∑ rx(k) z-k , k=-∞ to ∞
FILTERING OF RANDOM PROCESS
A linear shift-invariant (LSI) system (or filter) with a unit sample response h(n), applied to the case of a deterministic signal. The input is x(n) and the output is y(n).
Py ( z) = Px ( z)H ( z)H * (1/ z* )
SPECTRAL FACTORIZATION
Px ( z) = σ 02 H ( z)H * (1/ z* ) .
Wold Decomposition Theorem:
A general random process can be written as a sum of a regular random process xr (n)and a predictable process x
p (n) ,
x(n) = xr (n) + x p (n) ,
Bias-ConsistencyThe difference between the expected value of the
estimate and the actual value is called the ‘Bias’ B.B=ϴ-E[ϴ^
N]
ϴ - Actual Valueϴ^
N- Estimate ValueIf an estimate is biased ,Asymptotically Biased, Lt E[ϴ^
N]=0 N->∞
If an estimate is consistent, Mean Square Convergence
Lt |ϴ-E[ϴ^N]|2=0
N->∞
SPECIAL TYPES OF RPTypes are,ARMA Process –ARMA(p,q) AR Process (Auto Regressive)-ARMA (p,0)MA Process (Moving average)-ARMA (0,q)
YULE-WALKER EQUATION rx(k)+∑ap(l)rx(k-l) = σ2
vcq(k),0≤k≤q
0, k>q
l=-∞ to ∞