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As the number of signal samples N increases, we have

lim N

E [P XX ( f )] =

m=

r xx (m)e j2 fm = P XX ( f ) (10.27)

For a Gaussian random sequence, the variance of the periodogram can be obtained as

Var [P XX ( f )] = P 2 XX ( f ) 1 + sin 2 fN N sin 2 f

2

(10.28)

As the length of a signal record N increases, the expectation of the periodogram converges to the powerspectrum P XX ( f ) and the variance of P XX ( f ) converges to P 2 XX ( f ) . Hence the periodogram is an unbiasedbut not a consistent estimate.

The periodograms can be calculated from a DFTof thesignal x (m) , or from a DFTof the autocorrelationestimates r xx (m) . In addition, the signal from which the periodogram, or the autocorrelation samples, areobtained can be segmented into overlapping blocks to result in a larger number of periodograms, whichcan then be averaged. These methods and their effects on the variance of periodograms are consideredin the following.

10.4.2 Averaging Periodograms (Bartlett Method)

In this method, several periodograms, from different segments of a signal, are averaged in order to reducethevariance of theperiodogram. TheBartlett periodogramis obtained as theaverage of K periodogramsas

P B XX ( f ) = 1K

K

i= 1

P ( i) XX ( f ) (10.29)

where P (i) XX ( f ) is the periodogram of the ith segment of the signal. The expectation of the Bartlett

periodogram P B XX ( f ) is given byE [P B XX ( f )] = E [P

( i) XX ( f )]

= N 1

m= ( N 1)

1 |m| N

r xx (m)e j2 fm (10.30)

= 1 N

1/ 2

1/ 2

P XX (v)sin ( f v) N sin ( f v)

2

dv

where (sin fN / sin f )2/ N is the frequency response of the triangular window 1 | m| / N . From Equa-tion (10.30), the Bartlett periodogram is asymptotically unbiased. The variance of P B XX ( f ) is 1/ K of thevariance of the periodogram, and is given by

Var P B XX ( f ) = 1K

P 2 XX ( f ) 1 + sin 2 fN N sin 2 f

2

(10.31)

10.4.3 Welch Method: Averaging Periodograms from Overlapped and Windowed Segments

In this method, a signal x (m) , of length M samples, is divided into K overlapping segments of length N ,

and each segment is windowed prior to computing the periodogram. The ith

segment is dened as x i(m) = x (m + iD ) , m = 0, . . . , N 1, i = 0, . . . , K 1 (10.32)

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where D is the overlap. For half-overlap D = N / 2, while D = N corresponds to no overlap. For the i th

windowed segment, the periodogram is given by

P (i) XX ( f ) = 1

NU

N 1

m= 0

w(m) x i(m)e j2 fm

2

(10.33)

where w(m) is the window function and U is the power in the window function, given by

U = 1 N

N 1

m= 0

w2(m) (10.34)

The spectrum of a nite-length signal typically exhibits side-lobes due to discontinuities at the endpoints.The window function w(m) alleviates the discontinuities and reduces the spread of the spectral energyinto the side-lobes of the spectrum. The Welch power spectrum is the average of K periodograms obtainedfrom overlapped and windowed segments of a signal:

P W XX ( f ) = 1K

K 1

i= 0

P ( i) XX ( f ) (10.35)

Using Equations (10.33) and (10.35), the expectation of P W XX ( f ) can be obtained as

E [P W XX ( f )] = E [P(i) XX ( f )]

= 1 NU

N 1

n= 0

N 1

m= 0

w(n)w(m)E [ x i(m) x i(n)]e j2 f (n m)

= 1 NU

N 1

n= 0

N 1

m= 0

w(n)w(m)r xx (n m)e j2 f (n m)

=

1/ 2

1/ 2

P XX (v)W (v f )dv (10.36)

where

W ( f ) = 1 NU

N 1

m= 0

w(m)e j2 fm2

(10.37)

and the variance of the Welch estimate is given by

Var [P W XX ( f )] = 1K 2

K 1

i= 0

K 1

j= 0

E P (i) XX ( f ) P( j) XX ( f ) E P

W XX ( f )

2

(10.38)

Welch has shown that for the case when there is no overlap, D = N ,

Var [P W XX ( f )] =Var [P ( i) XX ( f )]

K 1

P 2 XX ( f )K 1

(10.39)

and for half-overlap, D = N / 2,

Var [P W XX ( f )] = 98K 2

P 2 XX ( f ) (10.40)

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10.4.4 BlackmanTukey Method

In this method, an estimate of a signal power spectrum is obtained from the Fourier transform of thewindowed estimate of the autocorrelation function as

P BT XX ( f ) = N 1

m= ( N 1)

w(m) r xx (m)e j2 fm (10.41)

For a signal of N samples, the number of samples available for estimation of the autocorrelation value atthe lag m, r xx (m) , decrease as m approaches N . Therefore, for large m, the variance of the autocorrelationestimate increases, and the estimate becomes less reliable. The window w(m) has the effect of down-weighting the high variance coefcients at and around theend points. Themean of the BlackmanTukeypower spectrum estimate is

E [P BT XX ( f )] = N 1

m= ( N 1)

E [r xx (m)]w(m)e j2 fm (10.42)

Now E [r xx (m)] = r xx (m)w B(m) , where w B(m) is the Bartlett, or triangular, window. Equation (10.42) maybe written as

E [P BT XX ( f )] = N 1

m= ( N 1)

r xx (m)wc(m)e j2 fm (10.43)

where wc(m) = w B(m)w(m) . The right-hand side of Equation (10.43) can be written in terms of theFourier transform of the autocorrelation and the window functions as

E [P BT XX ( f )] =

1/ 2

1/ 2

P XX ( )W c( f )d (10.44)

where W c( f ) is the Fourier transform of wc(m) . The variance of the BlackmanTukey estimate isgiven by

Var [P BT XX ( f )] U N

P 2 XX ( f ) (10.45)

where U is the energy of the window wc(m) .

10.4.5 Power Spectrum Estimation from Autocorrelation of Overlapped Segments

In the BlackmanTukey method, in calculating a correlation sequence of length N from a signal recordof length N , progressively fewer samples are admitted in estimation of r xx (m) as the lag m approaches

the signal length N . Hence, the variance of r xx (m) , increases with the lag, m. This problem can be solvedby using a signal of length 2 N samples for calculation of N correlation values. In a generalisation of this method, the signal record x (m) , of length M samples, is divided into a number K of overlappingsegments of length 2 N . The ith segment is dened as

x i(m) = x (m + iD ) , m = 0, 1, . . . , 2 N 1 (10.46)

i = 0, 1, . . . , K 1

where D is the overlap. For each segment of length 2 N , the correlation function in the range of 0 m N is given by

r xx (m) = 1

N

N 1

k = 0

x i(k ) x i(k + m) , m = 0, 1, . . . , N 1 (10.47)

In Equation (10.47), the estimate of each correlation value is obtained as the averaged sum of N products.