fundamentals of statistical signal processing(1) · introduction signal spectral analysis:...

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 oscillator

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 of 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 of 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 of 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 only over short observation intervals

)(ˆ

)(

0|)()(ˆ|

Two possible way(1) There are two main approaches

Non Parametric

Spectral Analysis

Parametric Spectral Analysis

explicitly estimate the covariance or 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), obtained 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 kr

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| krkr |)()(ˆ| krkr

|)()(ˆ| kk

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

observations into L = N/M subsamples of M observations each, then average the periodograms obtained 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

overlap

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


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;

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: 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:


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 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 = 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: 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:


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) 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 (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:



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:



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 order ?

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 optimal one. Therefore, all the additional poles of higher order AR processes will be positioned at the origin of the unit circl

Instead, in the case with noise, increasing the order N of 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?

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.


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


Spectral Estimation Application: Linear Prediction Coding (Just a Hint)

The object of linear prediction is to estimate the output 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

inyiajnxjbny10

)()()()()(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

jnxjbinyia00

)()()()(

p

i

iq

j

j ziazDzjbzN00

)()( 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

inyiany1

)()()(ˆ

p

i

inyia

nynyne

0

)()(

)(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,jfor 0)()( nejny

p1,...,j ,0)()()(0

n

p

i

jnyinyia


The orthogonality principle also states that resulting minimum error is given by

Or

We can minimize the error over all time :

, ...,p,jria ji

p

i

21 ,0)(0

Eriap

i

i 0

)(

, ...,p,jria ji

p

i

21 ,0)(0

Eriap

i

i 0

)(


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