Lesson 9 Introduction Signal Spectral Analysis: Estimation of the - - PowerPoint PPT Presentation

Lesson 9

Introduction

 Signal Spectral Analysis: Estimation of the power spectral

density

 The problem of spectral estimation is very large and has

applications very different from each other

 Applications:

 To study the vibrations of a system  To study the stability of the frequency of a

• scillator

 To estimate the position and number of signal sources in an

acoustic field

 To estimate the parameters of the vocal tract of a speaker  Medical diagnosis  Control system design  In general To estimate and predict signals in time or in space

Study of radio frequency spectrum in a big city

 side by side there are the various radio and

television channels, the signals cell phone, radar signals, etc.

 The frequency ranges, if considered with

sufficient bandwidth, are occupied by signals totally independent of each other, with different amplitudes and different statistical characteristics

 To analyze the spectrum, it seems logical

to use a selective receiver that measures the energy content in each interval frequency.

 We will seek the most accurate possible

estimate of these energies in the time available without making any further assumptions, not looking for models of signal generation

Non Parametric Spectral Analysis

Study of radio frequency spectrum in a big city

 The non-parametric spectral analysis is

a conceptually simple matter if you use the concept of ensemble average.

 if you have enough realizations of the

signal, just calculate the discrete Fourier transform and averaging the powers, component by component.

 However, rarely you have numerous

replicas of the signal; often, you have available a single replica, for an interval

• f time allotted

 To determine the power spectrum, you

have to use additional assumptions such as stationarity and ergodicity

Analysis of the speech signal

 consider the spectrum of acoustic signal due

to vibration or a voice signal

 in this case, all of the signal as a whole has

unique origins and then there will be the relationship between the contents

• f the various spectral bands.

 it must first choose a model for the

generation of the signal and then determine the parameters of the model itself

 For example, it will seek the parameters of a

linear filter that, powered by a uniform spectrum signal (white noise) produces a power spectrum similar to the spectrum under analysis

Parametric Spectral Analysis

Analysis of the speech signal

 Obviously, the success of the technique

depends on the quality and parametric correctness of the model chosen.

 Valid models lead to a parsimonious signal

description , that is characterized by the minimum number of parameters necessary

 This will lead to a better estimate

• f these parameters and then to optimal

results

 the parametric spectral analysis leads to the

identification of the model and this opens a subsequent phase of study to understand and then possibly check the status and evolution the system under observation

Formal Problem Definition

 Let be y= {y(1), y(2), . . . , y(N)} a second order

stationary random process,

 GOAL: to determine an estimate of its power

spectral density for ω ∈ [−π, +π]

 Observation

 We want  The main limitation on the quality of most PSD

estimates is due to the quite small number of data samples N usually available

 Most commonly, N is limited by the fact that the signal

under study can be considered wide sense stationary

• nly over short observation intervals

) ( ˆ  

) (  | ) ( ) ( ˆ |      

Two possible way(1)

 There are two main approaches

Non

Parametric Spectral Analysis Parametric Spectral Analysis

explicitly estimate the covariance

• r the spectrum of the process

without assuming that the process has any particular structure assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters. In these approaches, the task is to estimate the parameters of the model that describes the stochastic process

Non Parametric Estimation: Priodogram

 The periodogram method was introduced by Schuster

in 1898.

 The periodogram method relies on the definition of

the PSD

 in practice the signal y(t) is only available for a finite

interval

Periodogram Power Specrtal Density Estimation

Correlogram

 Spectrum estimation can be interpreted as an

autocorrelation estimation problem.

Correlogram Power Specrtal Density Estimation the estimate of the covariance lag r(k),

• btained from the available

sample {y(1), y(2), . . . , y(N)}

Estimation of Autocorrelation Sequence(ACS)

 There are two standard way to obtain an estimate

 unbiased estimate  biased estimate  Both estimators respect the symmetry properties of the ACS  The biased estimate is usually preferred, for the following reasons:

 the ACS sequence decays rather rapidly so that r(k) is quite small for

large lags k

 the ACS sequence is guaranteed to be positive semidefinite. is not the

case for the unbised definition

 if the biased ACS estimator is used in the estimation the correlogram is

eqaul to the periodogramm

) ( ˆ k r

Statistical Performance

 Both Periodogram and Correlogram are

asymptotically unbiased:

 Both have large variance, even for large N

 can be large for big k  even if the errors

are small, there are ”so many” that when summed in the PSD error is large | ) ( ) ( ˆ | k r k r 

| ) ( ) ( ˆ | k r k r  | ) ( ) ( ˆ | k k   

Periodogram Bias

Bartlett window. Frequency Domain

Bartlett window

 Ideally, to have zero bias, we

want WB(ω) = Dirac impulse δ(ω)

 The main lobe width

decreases as 1/N.

 For small values of N,

WB(ω) may differ quite a bit from δ(ω)

 If the unbised estimation

the window is rectangular

Summary Bias analysis

 Note that, unlike WB(ω), WR(ω) can assume negative

values for some values of ω, thus providing estimate of the PSD that can be negative for some frequencies.

 The bias manifests itself in different ways

 Main lobe width causes smearing (or smooting): details in

φ(ω) separated in f by less than 1/N are not resolvable.

 periodogram resolution limit = 1/N  Sidelobe level causes leakage  For small N, severe bias  As N → ∞, WB (ω) → δ(ω), so φ(ω) is asymptotically

unbiased

Periodogram Variance

 As N → ∞

 inconsistent estimate  erratic behavior

The Blackman-Tukey method

 Basic idea: weighted correlogram, with small weight

applied to the estimated covariances r(k) with large k

 The BT periodogram is a locally smoothed

 Variance decreases substantially (of the order of M/N)  Bias increases slightly (of the order 1/M)  The window is chosen so as to ensure that the spectral

density of estimated power is always positive

lag window Frequency Domain

 Let be  It is possible to prove that  This means that the more slowly the window decays to zero in one domain,

the more concentrated it is in the other domain

 The equivalent temporal width, N e is determined by the window length (Ne =

2M) for rectangular window, Ne = M for triangular window).

 Since N e βe = 1 also the bandwidth β e is determined by the window length

 As M increases, bias decreases and variance increases ⇒ choose M as a tradeoff

between variance and bias. As a rule of thumb, we should choose M ≤ N/10 in order to reduce the standard deviation of the estimated spectrum at least three times, compared with the periodogram

 Choose window shape to compromise between smearing (main lobe width) and

leakage (sidelobe level). The energy in the main lobe and in the sidelobes cannot be reduced simultaneously, once M is given

Choice of the BT window

The Bartlett method

 Basic idea: split up the available sample of N

• bservations into L = N/M subsamples of M
• bservations each, then average the periodograms
• btained from the subsamples for each value of ω.

Welch method

 Similar to Bartlett method, but allow overlap of

subsequences (gives more subsequences, thus better averaging) and use data window for each periodogram; gives mainlobe-sidelobe tradeoff capability

 if K = M, no overlap as in Bartlett method  if K = M/2, 50% overlap, S = 2M/N data segments  The Welch method is approximately equal to Blackman-

Tuckey with a non-rectangular lag window

• verlap

Daniell

 It can be proved that, for

are nearly uncorrelated random variables for

 The basic idea of the Daniel method is to perform

local averaging of 2J + 1 samples in the frequency domain to reduce the variance by about 2J + 1

 As J increases:

 bias increases (more smoothing)  variance decreases (more averaging)

Non parametric estimation summary

 The non-parametric spectral analysis is a conceptually

simple matter if you use the concept of ensemble average

 Goal is to estimate the covariance or the spectrum of

the process without assuming that the process has any particular structure

 Priodogram- Correlogram  Asymptotically unbiased, inconsistence  None of the methods we have seen solves all the

problems of the periodogram

 Parametic estimation…

Matlab Examples:

Periodogram

 Exercise 1.a

 Estimate the power spectral density of the signal

“flute2” by means of periodogram

 Hint on periodogram:

 the spectrum estimation using periodogram is given by the

following equation

Matlab Examples:

Periodogram

 Pseudocode:

 load the file flute2.wav  consider 50ms of the input signal (y)  estimate PSD using periodogram:

 N = length(y);  M = 2^ceil(log2(N)+1); %number of frequency bins  phip = (1/N)*abs(fft(y,M)).^2;

Matlab Examples:

Periodogram

 Exercise 1.b

 Quantify the bias and variance of the periodogram  Hint on periodogram:

 Periodogram is asymptotically unbiased and has large

variance, even for large N.

Matlab Examples:

Bias and variance of the periodogram

 Goal: quantify the bias and variance of the periodogram  Pseudocode:

 compute R realizations of N samples white noise

 e = randn(N,R);  for each realization:

 filter white noise by means of a LTI filter Y(z) = H(z)E(z)  compute the periodogram spectral estimate  phip(i,:) = (1/N)*abs(fft(y,N)).^2;

 end  compute the ensemble mean: phip(RxN)

 phipmean = mean(phip);

 compute the ensemble variance

 phipvar = var(phip);

 Plot

Matlab Examples:

Bias and variance of the periodogram

Matlab Examples:

Bias and variance of the periodogram

Matlab Examples:

Correlogram

 Exercise 2

 Estimate the power spectral density of the signal “flute2”

by means of correlogram.

 Hint on correlogram:

 the spectrum estimation using correlogram is given by the

following equation:

Matlab Examples:

Correlogram

 Goal: Estimate the power spectral density using the correlogram  Pseudocode:

 load the file flute2.wav  consider 50ms of the input signal (y)  estimate ACS

 [r lags] = xcorr(y, 'biased');  r = circshift(r,N);

 estimate PSD using correlogram:

 N = length(y);  M = 2^ceil(log2(2*N-1)+1); %number of frequency bins

 phic = fft(r,M);

Matlab Examples:

Correlogram

 MATLAB Hint: Matlab provides the functions:  [r lag]=xcorr(x,’biased’) that produces a

biased estimate of the autocorrelation (2N-1 samples)

• f the stationary sequence “x”. “lag” is the vector of lag

indices [-N+1:1:N-1].

 r = circshift(r,N) that circularly shifts the

values in the array r by N elements. If N is positive, the values of r are shifted down (or to the right). If it is negative, the values of r are shifted up (or to the left).

Matlab Examples:

Correlogram

Matlab Examples:

Modified periodogram (Blackman-Tukey)

 Exercise 3.a

 Estimate the power spectral density of  the signal “flute2” by means of Blackman-Tukey

method.

 Hints on B-T method:

 The spectrum estimation using BT method is given by the

following equation

 Goal: Estimate the power spectral density using the B-T method  Pseudocode:

 load the file flute2.wav  consider 50ms of the input signal -->N = length(y);  estimate ACS

 [r lags] = xcorr(y, 'biased');

 window with a bartlett window of the same length

 rw = r.*bartlett(2*N-1);  r = circshift(r,N);

 estimate PSD using BT:

 Nfft = 2^ceil(log2(2*N-1)+1);  phiBT = real(fft(r,Nfft));

Matlab Examples:

Modified periodogram (Blackman-Tukey)

Matlab Examples:

Modified periodogram (Blackman-Tukey)

Matlab Examples:

Modified periodogram (Blackman-Tukey)

 Exercise 3.b

 Goal: quantify the bias and variance of the BT method  Pseudocode:

 compute R realizations of N samples white noise

 e = randn(N,R);  for each realization:  filter white noise by means of a LTI filter Y(z) = H(z)E(z)  compute the BT spectral estimate  end

 compute the ensemble mean: phip(RxN)

 phipmean = mean(phip);

 compute the ensemble variance

 phipvar = var(phip);

 Plot

Matlab Examples:

Modified periodogram (Blackman-Tukey)

Matlab Examples:

Modified periodogram (Bartlett Method)

 Exercise 4

 Estimate the power spectral density of the signal

“flute2” by means of Bartlett method.

 Hint on Bartlett method :

 split up the available sample of N observations into L = N/M

subsamples of M observations each, then average the periodograms obtained from the subsamples for each value of ω.

Matlab Examples:

Modified periodogram (Bartlett Method)

 Goal: Estimate the power spectral density using the Baralett Method  Pseudocode:

 load the file flute2.wav  consider 50ms of the input signal -->N = length(y);  define the number of subsequences L and the number of samples

for each of them M=ceil(N/L)

 for each subsequence:

 consider the right samples: yl = y(1+l*M : M+l*M);  estimate periodogram: (1/M)*abs(fft(yl)).^2  mean periodograms of the subsequences:

 phil = phil + (1/M)*abs(fft(yl)).^2;  phiB=phil/L;

 end

Matlab Examples:

Modified periodogram (Bartlett Method)

Matlab Examples:

Modified periodogram (Welch Method)

 Exercise 5

 Estimate the power spectral density of the signal

“flute2” by means of Welch method.

 Hint on Welch method :

 similar to Bartlett method but: allow overlap of subsequences

use data window for each periodogram

Matlab Examples:

Modified periodogram (Welch Method)

 Goal: Estimate the power spectral density using the Baralett Method  Pseudocode:

 load the file flute2.wav  consider 50ms of the input signal -->N = length(y);

 define:

 the number of samples for each subsequence: M  the number of new samples for each subsequence: K=M/4  the number of subsequences: S= N/K - (M-K)/K;  the window: v = hamming(M) ;

 P = (1/M)*sum(v.^2);

 for each subsequence

 consider the right samples: xs = x(1+s*K : M+s*K) ;  window the subsequence: v.*xs

 estimate periodogram: (1/(M*P))*abs(fft(v.*xs)).^2

 mean periodograms of the subsequences:  phis = phis+ (1/(M*P))*abs(fft(v.*xs)).^2 ;  phiW = phis/S;



end

Matlab Examples:

Modified periodogram (Welch Method)

Parametric Spectral Estimation

 Consider a sequence of independent samples {xn} and

with it feeding a filter H(ω), if the transform of the sequence {xn} is indicated X(ω), the output will be:

 The power spectral density of the process white {xn} is

constant because for sequences of independent samples of length N, the components of the discrete Fourier transform are all of equal value root mean square (assuming T = 1):

Parametric Spectral Estimation

 The parametric spectral analysis consists in determining the

parameters of the filter H (ω) in such a way that the spectrum in the filter output resembles as much as possible to the spectrum of the signal {yn} analyzed

 We will have spectral analysis Moving Average (MA) if the filter has a z-

transform characterized by all zeros

 We have the case Auto Regressive (AR) when The filter is all-pole

autoregressive

 We have the mixed case ARMA (Auto Regressive Moving Average) in

the more general case of poles and zeros.

 We will see that the parametric spectral analysis all zeros practically

coincides with the modified non parametric techniques.

 The spectral techniques AR instead are of very different nature and

innovative

 The spectral analysis ARMA then, is less frequently used also because

as is known, any filter can be represented with only zeros or only poles and a mixed representation serves only for a description of the same (or a similar) transfer function with a lower number of parameters.

All Zeros Analysis: Moving Average(MA)

 Consider a FIR filter with impulse response {bh} whose

z-transform is characterized by all zeros

 Let be {xn} the sequence white at the filter and the

sequence {yn} colored output

 The autocorrelation function of the

sequence {yn} is:

 Our problem is to determine a filter {bh} or its transformed

B(z) (the solution is not unique) from a estimate of the autocorrelation data

 For now let's assume that the estimate available is very

good, so we can pretty much assume that we know the autocorrelation function

 Switching to z-transform can be seen that R(z) as:  R(z) is a polynomial in the variable z-1 that for each root,

has also the root and reciprocal conjugated.

 a way of determining a filter B(z), assigned the

autocorrelation function R (z), is to find the roots of R(z) and for example assign to B (z) all the H roots outside of the circle unit and to B*(1/z) all other (inside).

All Zeros Analysis: Moving Average(MA)

All Zeros Analysis: Moving Average(MA)

 following the strategy of the previous slide, B*(1/z) is

minimum phase, otherwise we can choose B*(1/z) at maximum phase or mixed phase, in different ways 2H

 Note that it is not enough that R (z) is a any

polynomial for identifying a filter B(z).

 In fact, there would be reciprocal pairs of zeros but it

could also have simple zeros on unit circle, in which case it would not be possible to find the B(z) because you can not associate a mutual positioned root

 But in this case, the symmetrical polynomial R(z) would

not represent an autocorrelation function

 the values ​of the R(z) on the unit circle,

contain changes sign when passing through the zeros and then negative values

 while instead a power spectrum, Transform

Fourier autocorrelation function, is always positive

Minimum Phase Maximum Phase Mixed Phase

Truncation of the autocorrelation

 We have an autocorrelation function equal to zero for m>H  It is always possible to find 2H filters of length H + 1 that

powered by white sequences return in the output the autocorrelation

 However, there are only estimates of the autocorrelation

function: if the estimate is made with the correlation of the sequence padded with zeros, then its Fourier transform, (the periodogram) is always positive.

 in this case, the length of the filter is excessive because, due

to the dispersion of the estimate, the samples of the autocorrelation estimated will never be zero

 If we want to limit the length of the filter to H, it should be

squash to O, the autocorrelation function in H samples, windowing it so that the spectrum remains positive, and then multiplying it by a window which in practice is always the triangular

Truncation of the autocorrelation

 In conclusion, to make a all zeros parametric

estimation, it is necessary to window the autocorrelation function with a triangular window of length 2H;

 Incising H greater will be the resolution of the spectral

parameter estimation

 In essence, it is seen that this technique of spectral

estimation coincides with that of the smoothing periodogram

Analysis All Poles(AR)

 Preliminary Observations

 This technique of spectral estimation is very important

for many reasons.

 it should be noted that the IIR filters having unlimited

impulse response, they can produce spectra of large spectral resolution with a limited number of parameters

 lend themselves to the description of phenomena that have a

long coherence time, i.e. where the process uncurreled very slowly.

Analysis All Poles(AR)

 Let {xn} the output sequence of an IIR filter of order N,

1/AN(z) powered by a white sequence {wn}.

 The autocorrelation of the output sequence is

Frequency Domain IIR filter Yule–Walker equations

Yule–Walker equations

 Yule Walker equations can be obtained in the matrix form

indicating the following vectors with the symbols:

 multiplying on the left by ξN and considering the expected

value, whereas E[xN wn] = O we obtain

Yule–Walker equations

 The matrix of the coefficients of the YW equations is a Toeplitz matrix;

 It is symmetric (or Hermitian, for complex sequences) and

all the elements belonging to the same diagonal or subdiagonale are equal to each other.

 The matrix is ​characterized by N numbers.

 Rewriting in matrix form, the equations of Yule Walker is obtained

Yule–Walker equations

 For completeness, we add the complete formulation,

easily verified, in which also appears the first equation that contains the variance the sequence of input white σ2w

Autoregressive Spectral Estimation

 Once you find the ah,N starting form the N values ​of the

autocorrelation function for m = 1,. . . , N is immediate determine the components of the continuous spectrum of the signal {xi};

 the function of autocorrelation, is determined for all

values ​of temporal index m;

 It is eqaul to the assigned got m= 1:N  It computed by the Yule Walker equation for all the other  the truncation of the autocorrelation function does not

involve problems; simply, the values ​of the autocorrelation predicted by using the equations YW, does not coincide with the actual measurements, when the spectral estimation is done with an order (the value of N) too low

Exercises:

AR and MA spectral Estimation

 Exercise 1:

 Given a process

 Compute the first 3 samples [r0, r1, r2] of the autocorrelation of the

process xn

 Parametric spectral estimation of the MA process (order 1)  Parametric spectral estimation of the AR process (order 1)

White noise E[wn]=0 and

 The signal xn is a MA process of order 2:

Exercise 1: Solution :

Autocorrelation

 The generic FIR filter of an MA (1) estimate has

transformed:

 The autocorrelation of the process MA(1)is:  The estimate MA (1) of the process requires the use of

the first two samples of the autocorrelation of the process suitably truncated with a window with positive transformed

Exercise 1: Solution :

MA(1) parametric spectral Estimation

Triangular Window Autocorrelation

 We choose the minimum phase solution  zero associated with the process MA (1) is in Nyquist

Exercise 1: Solution :

MA(1) parametric spectral Estimation

 The generic AR(1) all-pole filter has transformed  We use the Yule–Walker equations

Exercise 1: Solution :

AR(1) parametric spectral Estimation

Y-W Autocorrelation

 Exercise 2:

 Spectral estimation of complex sinusoidal waveform

 Compute the autocorrelation sequence  parametric spectral estimation of the AR process (order 1)  What happens adding white noise to the signal ?  What happens (qualitatively) increasing the autoregressive filter

• rder ?

Exercises:

AR and MA spectral Estimation

 The autocorrelation of the process is:

Exercise 2: Solution :

Autocorrelation

 We use the Yule–Walker equations

Exercise 2: Solution :

AR(1) parametric spectral Estimation

Y-W the pole of the AR(1) is in wo it lies on the unit circles Power Spectrum

 Autocorrelation  Using the Y-W equation

Exercise 2: Solution

What happens adding white noise to the signal ?

In the presence of a white noise the frequency of the pole model of the AR (1) remains unchanged The radial position is changed. The pole is closer to the origin of a quantity proportional to the signal to noise ratio

Exercise 2: Solution

What happens adding white noise to the signal ?

 In the case without noise, the estimate provided is the

• ptimal one. Therefore, all the additional poles of

higher order AR processes will be positioned at the

• rigin of the unit circl

 Instead, in the case with noise, increasing the order N

• f the AR model, the poles are arranged inside the unit

circle,

 all at the same radial position relative to the center and

at frequencies distant from

Exercise 2: Solution

What happens incising the AR order?

Exercise 2: Solution

What happens incising the AR order?

Exercise 2: Solution

What happens incising the AR order?

Exercise 2: Solution

What happens incising the AR order?

 Autoregressive Model Hint:

 The parametric of model-based methods of spectral

estimation assume that the signal satisfies a generating model with known functional form, and then proceed in estimating the parameters in the assumed model Power Spectral Density

Matlab Examples:

Autoregressive Model

Autoregressive Model

 Goal: Estimate the power spectral density of the signal y by means of

AR model

 Consider the signal y defined by the differential equation:  y(n)=a1 y(n-1) + a2 y(n-2) + a3 y(n-3) + z(n)  Estimate {ap} and σz with an AR model (order p)  Plot estimated PSD and compare with the true PSD

 MATALB Hint: Matlab provides the functions:  [r lag]=xcorr(x,’biased’) that produces a biased estimate of

the autocorrelation (2N-1 samples) of the stationary sequence “x”. “lag” is the vector of lag indices [-N+1:1:N-1].

 R=toeplitz(C,R) that produces a non-symmetric Toeplitz matrix

having C as its first column and R as its first row.

 R=toeplitz(R) is a symmetric (or Hermitian) Toeplitz matrix.

Matlab Examples:

Autoregressive Model

 Pseudocode:

 Consider the signal y defined by the differential equation:

y(n)=a1 y(n-1) + a2 y(n-2) + a3 y(n-3) + z(n)

 sigmae = 10;  a = poly([0.99 0.99*exp(j*pi/4) 0.99*exp(-

j*pi/4)])

 b = 1 ;  z = sigmae*randn(N,1);  y = filter(b, a, z);

 Estimate {ap} and σz with an AR model (order p)



n=3;



r = xcorr(y , 'biased');

 Rx = toeplitz(r(N:N+n-1), r (N:-1:N-n+1));  rz = r(N+1:N+n ) ;  theta = -Rx^(-1)*rz;  varz = r(N) +sum(theta.*r(N-1:-1:N-n));

 Plot estimated PSD and compare with the true PSD

Matlab Examples:

Autoregressive Model

Matlab Examples:

Autoregressive Model

Spectral Estimation Application:

Linear Prediction Coding (Just a Hint)

 The object of linear prediction is to estimate the

• utput sequence from a linear combination of input

samples, past output samples or both :

 The factors a(i) and b(j) are called predictor coefficients.

 

 

   

p i q j

i n y i a j n x j b n y

1

) ( ) ( ) ( ) ( ) ( ˆ

Linear Prediction Coding

Introduction

 Many systems of interest to us are describable by a

linear, constant-coefficient difference equation :

 If Y(z)/X(z)=H(z), where H(z) is a ratio of

polynomials N(z)/D(z), then

 Thus the predicator coefficient given us immediate access to the

poles and zeros of H(z).

 MA, AR and ARMA

 

 

  

q j p i

j n x j b i n y i a ) ( ) ( ) ( ) (

 

   

 

p i i q j j

z i a z D z j b z N ) ( ) ( and ) ( ) (

 Given a zero-mean signal y(n), in the AR model :

 The error is :  To derive the predicator we use the orthogonality principle, the

principle states that the desired coefficients are those which make the error orthogonal to the samples y(n-1), y(n-2),…, y(n-p).





  

p i

i n y i a n y

1

) ( ) ( ) ( ˆ





   

p i

i n y i a n y n y n e ) ( ) ( ) ( ˆ ) ( ) (

Linear Prediction Coding

Orthogonality principle

 Thus we require that

 Or,  Interchanging the operation of averaging and summing, and

representing < > by summing over n, we have

 The required predicators are found by solving these equations.

p ..., 2, 1, j for ) ( ) (     n e j n y

p 1,..., j , ) ( ) ( ) (    

 

 n p i

j n y i n y i a

Linear Prediction Coding

Orthogonality principle

 The orthogonality principle also states that resulting

minimum error is given by

 Or

 We can minimize the error over all time :

, ...,p , j r i a

j i p i

2 1 , ) (  

 



E r i a

p i i 



0

) ( , ...,p , j r i a

j i p i

2 1 , ) (  

 



E r i a

p i i 



0

) (

Linear Prediction Coding

Orthogonality principle