BY:

RUPAK KUMAR GUPTA

(M.TECH 1ST YEAR STUDENT)

R.NO 15304020

PONDICHERRY UNIVERSITY DEPARTMENT OF ELECTRONICS AND

COMMUNICATION

To convert the analog signals into samples of digital signals is called the DSP.

In this Subject, we will study the power spectrum estimation of DSP.

In metrological phenomena Fluctuation in air due to

change the pressure and temperature, that is the good

characterized by Spectral estimation aims at extracting

information about the system, from its observed

output (in the absence of the input) and a signal ,

when it is associated with noise characterized by

statistics as Random process.

For attempting to estimation, based on a finite length

because of size of any image is finite (available of data).

Even with a large image, in order to make the assumption

of stationary but in contrast to parametric methods, non-

parametric methods do not make any assumptions on the

data-generating process or model (e.g. autoregressive

model).

The ideal power spectrum of a signal x (n) can be

computed by first finding the ideal autocorrelation r(x) � � k = lim→∞[ � + � � + � � � ]�=

Even with a large image, in order to make the assumption of

stationary, often only a small window of the image is

considered at a time. � � k = [� � � + � � � ]− −��=

Now length of data that record that we select is determined by

the rapidly of the time variation in the signal statistics.

The autocorrelation function of a random process (selected finite

window) is the appropriate statistics average, after that we will

apply Fourier Transform because random signal is in the Time

Domain. Hence that results yield the power Density Spectrum.

Also, the ideal power spectrum is “blurred” or “smoothened” by the low pass filter, and the amount of blurring is mainly determined by the width of the main lobe. Power spectrum at any particular frequency is “leaking” into the side lobes, possibly “masking” the power spectra at the nearby frequencies

By choosing a different types of window, the width

of the main lobe and the amplitudes of the side lobes

of the low pass filtered can be varied.

On one hand, the width of main lobe determines

the spectral resolution.

The larger the width of the main lobe, the more blurring

effect the low pass filter has on the power spectrum.

On the other hand, the amplitudes of the side lobes

determine the spectral masking effect.

The higher the amplitudes of the side lobes, the more

masking effect of the power spectrum at a given frequency

has on the nearby frequencies.

A best spectral estimator, provides an estimate

which has as high a resolution and as

low a bias and variance as possible, a given length of data.

These factors depend on the associated noise, data length,

window etc (that means signal scenario).

This variance reduction is due to smearing of the signal

spectrum with the broad main lobe of the spectrum of the

window and this results in a poor resolution

Parametric method we assume that signal is output of a system having white noise as an input. We model the system and get its parameters

i.e. coloring filter coefficients and predict the power spectrum.

Here we compare the Yule-Walker method & Burg method for Power spectrum estimation.

Application of Power Spectrum estimation WIENER FILTER FEATURE EXTRACTION

In this topic, analyzes five common non-parametric

PSE methods

They are:

Periodogram Method,

Modified Periodogram Method,

Bartlett’s Method,

Welch’s Method,

Blackman-Tukey Method.

All these above methods has no assumption (estimation )about how the data were generated so it is called nonparametric methods.

1. PERIODOGRAM A simple-minded method of estimating the

autocorrelation of a signal of finite length is called the

Periodogram Method.

The well known form of power density spectrum estimation is

called Periodogram

The classic method for estimation of the power spectral

density of an N sample record is the periodogram.

periodogram doesn’t converge to the True Power Density

That means it is not a

consistent estimation of true power density spectrum.

So the emphasis of classical Non-parametric method is

on obtaining a consistent estimate of power spectrum

through some averaging and smoothing operations

performed directly on the periodogram or directly on the

autocorrelation.

The variance of the basic periodogram is large, though it

has good resolution, low bias and good signal detectability

even at high noise levels

2. MODIFIED PERIODOGRAM Modified periodogram method is used for power Spectrum

density estimation purpose.

The main difference between periodogram method and

modified periodogram method is window technique.

For smoothing purpose of periodogram, the window technique

is used in Modified periodogram method.

3. Bartlett Method (Averaging Periodograms Method)

Both the Periodogram Method and Modified Periodogram

Method do not give zero variances as the data length

approaches infinity.

One way to enforce zero variance by Bartlett method we

divide the signal into blocks, find their period grams and

average them to get the Power spectrum.

Example The Bartlett’s Method divides the signal of length N into

K segments, with each segment having length L = N /K . The

Periodogram Method is then applied to each of the K segments.

The average of the resulting estimated power spectra is taken as

the estimated power spectrum of the Bartlett’s Method.

(as K ∞ variance approaches zero.)

only at the cost of resolution, the averaged PSE provides a

lower variance, depending upon the number of segments

averaged and the window used, for a given length of data.

Bartlett based method is used for reducing the variance and

frequency resolution values in the noise spectrum.

Therefore the value of K can be used to design the trade-off

between spectral resolution and variance. With a larger number

of segments, the variance is reduced but at the expense of

spectral resolution. Conversely, with a smaller number of

segments, the spectral resolution (i.e. the bandwidth of the

main lobe) is reduced at expense of larger variance.

Performance characteristics of BARTLETT Method for PSE

We observe that the Bartlett PSE is asymptotically unbiased &

if K is allowed to increase with an increase in N, the estimate

is also consistent.

Hence, asymptotically, this estimate is characterized by the

quality factor.

��= K= N/M The frequency resolution of the Bartlett estimation,

measured by taking the 3-dB width of the main loab

of the rectangular is

f = .

M= 0.9/Δf

Hence ��= 0.9/Δf = 1.1N Δf

4. Welch Method: Averaging the Modified periodogram

Welch method is the similar to Bartlett Method but only some

modifications data segments can be overlapping in welch

method.

And second thing is to window the data segments prior to

computing the periodogram

When data length is short Blackman tuckey method is better

than Welch method but as the data length increase Welch

Method gives excellent results.

Welch method got better precision but less frequency

resolution as compare to Bartlett method.

Therefore this method is used to calculate the PSD of a signal

with reducing the effect of noise in DSP .

The Welch’s Method eliminates the trade off between spectral resolution and variance in the Bartlett’s Method by allowing

the segments to overlap.

However, overlapping introduces correlations between

segments. In practice, the amount of overlapping is typically

between 50% to 75%.

Performance characteristics of WELCH Method for PSE

Under the two conditions given by the

quality factor is:

L = N/M for no overlap

8L/9 16N/9M for 50 % overlap.

The spectral width of triangular window

at the 3-db points is

��=

f = .

Q factor in term of N & Δf

0.78 NΔf

1.39N NΔf

5. BLACKMAN-TUKEY METHOD:

Black man turkey method is also known as

smoothing periodograms.

In this method we windowed the auto-correlation sequence

and take Fourier transform to get power spectrum estimate

(Periodogram) in effect we smooth out the Periodogram.

For values of data points of m approaching N, the variance of

these estimation is very high.

Performance characteristics of BLACKMAN & TUKEY Method for PSE Following equation shows that the value of

estimate is asymptotically unbiased

MATHEMETICAL RELATED TO THESE

THREE METHODS.. S.NO

ESTIMATE

QUALITY FACTOR

FREQ RESOLUTION

(Δf)

QUALITY FACTOR

1.

BARLETT

Q= K= N/M

0.9/M

1.11NΔf

2.

WELCH METHOD

L=N/M (for no overlap)

8L/9=16N/9M (for 50% overlap and Triangular window)

1.28/M

1.39NΔf

3.

BLACKMAN & TUKEY METHOD

1.5N/M

0.64/M

2.34NΔf

APPLICATION OF POWER SPECTRUM For Feature Extraction, we calculate the ‘mean square error’ in order to calculate the performance of above five methods.

Another purpose is to analyze how power spectrum can be used for feature extraction

to detect cancer in an ultrasound image of a prostate. Feature extraction does not

depend much on the the variance of the power spectrum and spectral resolution.

Because of the Periodogram Method is best for feature extraction

The feature extraction is depends mainly on the overall shape, described

by the slope and the y-intercept, of the linearized power spectrum. Since

it does not closely depend on the local characteristics of the power

spectrum, such as spectral resolution and variance, the complexity

becomes the critical factor in choosing the best estimation method for this

feature extraction. It was found that the the Periodogram Method is the

simplest and one of the most accurate estimation method for this feature

extraction

CONCLUSION

Frequency resolution is low in Bartlett method because of

windowing(that means leakage frequency due to side lobes), instead of

Frequency resolution of Bartlett method is more than Welch Method and

Blackman-Tukey Method.

Welch Method has got better precision but less frequency resolution

than Bartlett method

Blackman-tukey methods better precision than Bartlett & Welch Method

methods

Blackman-Tukey Method has better variance (even at large lags).

Quality factor of Blackman-Tukey more than Bartlett & Welch Method

When data length is short Blackman tuckey method is better

than Welch method but as the data length increase Welch

Method gives excellent results.

Analysis of signals in the frequency domain with non uniform

sampling times is efficient for estimation of random signals.

In future it can be extended to estimation of communication

signals like audio signals, speech signals and video signals.