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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 3, MARCH 1997
Similarly to Fig. 6, the result shown in Fig. 7 was obtained by
using the COBA and the NFOBA algorithms. From the figure, we can
observe that the NFOBA algorithm may become unstable initially due
to the normalization procedure. Thus, to overcome the divergence, the
NFOBA algorithm was not normalized during the initial two blocks.
The figure shows that the COBA algorithm has better convergence
rate than the NFOBA algorithm, and their adaptation accuracies are
almost the same.
To illustrate the effect of nonstationary environments on the
proposed algorithms, the results shown in Fig. 8 were drawn by
using the time-varying unknown system. From the results, we can
see that the TOBA and COBA algorithms outperform the SOBAF
and NFOBA algorithms in tracking property as well as convergence
rate. In addition, the tracking property of the TOBA algorithm is
superior to that of the COBA algorithm.
A Multiple Window Method for
Estimation of Peaked Spectra
Maria Hansson and Göran Salomonsson
Abstract— A multiple window method for estimation of a peaked
power density spectrum is designed. The method optimizes a filter
function utilizing the Karhunen–Loève basis functions of a known peaked
spectrum as windows to reduce variance and bias in the locality of the
frequency peak. For improving performance, a penalty function is used to
suppress the sidelobes outside a given bandwidth. The improved windows
are obtained as the solution of a generalized eigenvalue problem.
The bias at the frequency peak is reduced due to matching windows,
while the variance is decreased by averaging uncorrelated periodograms.
The method is compared with the Thomson multiple window estimator
as well as to a single Hanning window in a simulation.
I. INTRODUCTION
V. CONCLUSIONS
The TOBA, SOBA, and COBA algorithms have been developed
based on the preconditioning technique. The TOBA algorithm produces fast convergence speed, and the SOBA and COBA algorithms
yield computational efficiency. Through computer simulations, it has
been shown that the proposed algorithms are very fast, as compared
with the OBA, SOBAF, and NFOBA algorithms, and their tracking
properties are superior to those of the OBA, SOBAF, and NFOBA
algorithms. Furthermore, the algorithms have no instability problem
existing in the SOBAF and NFOBA algorithms.
REFERENCES
[1] G. A. Clark, S. K. Mitra, and S. R. Parker, “Block implementation of
adaptive digital filters,” IEEE Trans. Acoust., Speech, Signal Processing
vol. ASSP-29, pp. 744–752, June 1981.
[2] W. B. Mikhael and F. H. Wu, “Fast algorithms for block FIR adaptive digital filtering,” IEEE Trans. Circuits Syst., vol. CAS-34, pp.
1152–1160, Oct. 1987.
, “A fast block FIR adaptive digital filtering algorithm with
[3]
individual adaptation of parameters,” IEEE Trans. Circuits Syst., vol.
36, pp. 1–10, Jan. 1989.
[4] T. Wang and C. L. Wang, “On the optimum design of the block
adaptive FIR digital filter,” IEEE Trans. Signal Processing, vol. 41, pp.
2131–2140, Jan. 1993.
[5] D. Mansour and A. H. Gray, Jr., “Unconstrained frequency-domain
adaptive filter,” IEEE Trans. Acoust., Speech, Signal Processing, vol.
ASSP-30, pp. 726–734, Oct. 1982.
[6] G. Panda, B. Mulgrew, C. F. N. Cowan, and P. M. Grant, “A self
orthogonalizing efficient block adaptive filter,” IEEE Trans. Acoust.,
Speech, Signal Processing, vol. ASSP-34, pp. 1573–1582, Dec. 1986.
[7] C. H. Yon and C. K. Un, “Fast multidelay block transform-domain
adaptive filters based on a two-dimensional optimum block algorithm,”
IEEE Trans. Circuits Syst., vol. 41, May 1994.
[8] P. Delsarte and Y. V. Genin, “The split Levinson algorithm,” IEEE
Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 470–478,
June 1986.
[9] G. S. Ammar and W. B. Gragg, “Superfast solution of real positive
definite Toeplitz systems,” SIAM J. Matrix Anal. Appl., vol. 29, pp.
61–67, 1988.
[10] L. A. Hageman and D. M. Young, Applied Iterative Methods. New
York: Academic, 1981.
[11] G. H. Golub and C. F. V. Loan, Matrix Computations. Baltimore, MD:
Johns Hopkins University Press, 1983.
[12] T. K. Ku and C. J. Kuo, “Design and analysis of Toeplitz preconditioners,” IEEE Trans. Signal Processing, vol. 40, pp. 129–141, Jan.
1992.
Power density spectrum estimation methods can be divided into
two main groups—nonparametric and parametric—where the periodogram and window methods belong to the former. Windows
may be fixed, e.g., the Hanning window; others have parameters
for adjusting the sidelobe attenuation, e.g., the Kaiser window. The
window methods decrease the variance of the spectrum estimate by
smoothing [1].
Another nonparametric spectrum estimator is based on multiple
windows. The data windows are chosen to give uncorrelated periodograms, and the variance of the estimate is decreased by an
averaging procedure [2]–[4]. The frequency resolution is predetermined, and reported results show that the method gives small bias
and low variance as long as the power density spectrum is flat.
This correspondence addresses a multiple window method that
is matched to a peaked power density spectrum in order to obtain
both low bias at the peak and low variance of the estimate. The
windows are derived by optimizing a filter function. The optimization
procedure can be constrained, resulting in control of the sidelobes to
prevent leakage from frequencies outside the resolution bandwidth,
cf. [5]. The windows are obtained as the solution of a generalized
eigenvalue problem.
The correspondence is organized with the problem formulation and
solution in Section II. Numerical examples are given in Section III.
Section IV concludes the paper.
II. PEAK MATCHED MULTIPLE WINDOWS
The power density spectrum Sx (f ) of the real valued stationary
random process x(n) is given. The spectrum is assumed to have a
peak located at f = 0: We would like to estimate the spectrum from
N samples x = [x(0) 1 1 1 x(N 0 1)]T of the process by using the
estimator
K
S^x (f ) =
i S^i (f )
(1)
i=1
where
2
N 01
S^i (f ) =
x(n)hi (n)e0j 2fn :
(2)
n=0
Equation (2) is a windowed periodogram obtained by using the data
window hi = [hi (0) 1 1 1 hi (N 0 1)]T : A weighted sum of K
Manuscript received July 5, 1995; revised October 22, 1996. The associate
editor coordinating the review of this paper and approving it for publication
was Dr. Gregory H. Allen.
The authors are with the Department of Applied Electronics, Signal
Processing Group, Lund University, Lund, Sweden.
Publisher Item Identifier S 1053-587X(97)01871-0.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 3, MARCH 1997
periodograms provides the estimate in (1). We will design windows
1 1 1 K , giving a small bias as well as a low variance
estimate of Sx (f ) in the neighborhood of the peak frequency. Low
bias is obtained by matching the windows to the peak of Sx (f ),
whereas reduction of the variance is established with uncorrelated
periodograms S^i (f ) at the peak. Thus, a frequency local estimate
is desired. To prevent leakage from regions outside a predetermined
interval of width B , the Fourier transforms Hi (f ) of hi ; i = 1 1 1 1 K
have to be bandlimited to the interval (0B=2; B=2): The mainlobes
of the windows should be inside this band, and the sidelobes of each
window should be as low as possible.
The multiple window estimation method can be considered to be
a filtering procedure in a filter bank. The impulse responses of the
subfilters are hi ; i = 1 1 1 1 K: Given the input signal x(n), the power
of the output signal within the frequency interval (0B=2; B=2) is
K
B=2
PB =
i
jHi (f )j2Sx(f ) df
0B=2
i=1
K
= ihTi RB hi :
(3)
i=1
hi ; i = 1
The (N 2 N ) Toeplitz covariance matrix R B has the elements
rB (l) = rx (l) 3 B sinc(Bl); 0 jlj N 0 1, where rx (l) is the
covariance function of x(n); sinc(x) = sin(x)=x, and 3 denotes
the convolution operator. In (3), PB is the power of x(n) within the
mainlobe of the windows. We want to find the K window functions
hi that maximize PB : The optimization is performed subject to the
constraint
PZ =
=
K
i=1
K
i=1
i
1=2
01=2
2
(4)
where Sz (f ) with the corresponding Toeplitz covariance matrix R Z
is chosen for suppression of the sidelobes of the windows.
The solution with respect to hi is the set of eigenvectors of the
generalized eigenvalue problem
RB q i = iRZ q i ;
2 1 1 1 N :
i=1
111 N
(5)
1 The eigenvectors corresponding
K largest eigenvalues are used as windows hi = q i ; i =
1 1 1 1 K; and are known as peak matched multiple windows (PM
MW’s). The windows are orthogonal to the covariance matrix R B ,
implying uncorrelated periodograms S^i (f ) at the peak frequency. The
covariance matrix R Z is chosen in order to grant certain properties
to the estimate S^x (f ):
where
to the
Two examples have been investigated:
• RZ = I
• RZ = RG
where R G corresponds to a penalty frequency function. In the first
example, the Karhunen-Loève basis functions are used as windows.
The windows are bandpass filters and have their main power in
the range 0B=2 f B=2: However, the finite length of the
windows gives sidelobes that cause leakage from frequencies outside
the resolution interval. In the second example, a penalty frequency
function
jf j > B=2
SG (f ) = G
1 jf j B=2
is used to decrease the leakage from the sidelobes. The corresponding
Toeplitz covariance matrix is R G : The ideal window functions fulfill
the relationship
K
i=1
i jHi (f )j2 = SG01 (f );
01=2 f 1=2
(7)
G is set to a large value, the sidelobes of jHi (f )j2 outside
jf j > B=2 will be suppressed by this factor. The suppression factor
is indicated in parenthesis, e.g., PM MW (30 dB) for G = 30 dB.
The weighting factor i is a parameter that can be chosen arbi2
trarily. We study a matched spectrum approach 6K
i=1 i jHi (f )j =
Sx (f ) in the local interval 0B=2 f B=2: The total filter
and if
function should have the same appearance as the peak to minimize
bias as well as give a low variance of the power spectrum estimate
in the neighborhood of the peak. The matched spectrum approach is
fulfilled with i = i =6K
i=1 i ; which is used in this paper.
III. RESULTS
The properties of the proposed method are investigated and compared with those of the Thomson spectrum estimator [2] and the
single Hanning window. The windows with N = 128 are calculated
from (5) with the known power density spectrum defined by
02C jf j=10B log
Sx (f ) = e
0
(e)
jf j B=2
jf j > B=2:
(8)
In (8), Sx (f ) is a peaked spectrum with Sx (0) = 1 and Sx (B=2) =
dB, where C = 20 dB throughout this correspondence. The
Toeplitz covariance matrix R B is calculated from this spectrum. The
resolution parameter is B = 0:08, and the number of windows is
K = 8: The windows are calculated for different values of the
sidelobe suppression parameter G: The Thomson multiple windows
are calculated from (5) with rB (l) = B sinc(Bl) and R Z = I : The
bias, variance, and mean square error are compared for an ARMA
spectrum and for white noise spectrum.
0C
jHi (f )j Sz (f ) df
ihiT RZ hi = 1
779
(6)
A. Numerical Example
In order to illustrate the proposed methods, a number of sequences
x(n); n = 0 1 1 1 N 0 1 of a second-order ARMA process, with poles
p1;2 = 0:97e6j 20:1 and zeros z1;2 = 0:97e6j 20:3 are simulated.
The number of sequences is 25, each with 128 data samples. The
true spectrum has a peak at f = 0:1 and a notch at f = 0:3: The 25
estimated spectra are plotted as dotted lines in Fig. 1, where the true
spectrum is depicted as the solid line. The suppression parameter G is
put to G = 30 dB (PM MW (30 dB)), and the result is compared with
the use of windows with R Z = I (PM MW) and with the spectrum
estimates from a single Hanning window and from Thomson multiple
windows with resolution width B = 0:08 and K = 8 windows
(Thomson8 MW). The variation of the spectra gives a clue about the
variance; the closeness of the curves to the true spectrum indicates
bias.
B. Calculation of Bias and Variance
The bias and variance can be calculated since the true ARMA
spectrum is known. The calculation is made for a number of discrete
frequencies. Both the bias and variance vary with the magnitude of
the spectrum, and to obtain a comparable measure for different frequencies, they are divided by E [S^x (f )] and E [S^x (f )]2 , respectively.
1053–587X/97$10.00  1997 IEEE
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 3, MARCH 1997
Fig. 1. Result of spectrum estimation of an ARMA process with different
methods. The estimation is carried out with 25 different realizations (dotted
lines). The true power density spectrum is depicted as the solid line.
This implies that small-valued estimates are related in the same way
as large valued estimates. Bias is defined as
Bias S^x (f ) =
0 Sx (f )
E [S^x (f )]
E [S^x (f )]
(9)
where the expected value of the spectrum estimate is calculated to be
K
E [S^x (f )] = E
ihTi H (f )xxT (f )hi
i=1
K
=
i hiT H (f )Rx (f )hi :
(10)
i=1
8
8
In (10),
8i = diag[1
e0j 2f
8
8
1 1 1 e0j N 0
2 (
1)f
]
is the Fourier transform matrix. The variance of the spectrum estimate
is given by all combinations of the different periodogram covariances
K K
i cov(S^i (f )S^j (f ))
j =1 i=1
Variance S^x (f ) =
:
(11)
E [S^x (f )]2
Denoting hiT H (f )x = Ai and assuming x to be Gaussian gives
the covariance as
8
cov(S^i (f )S^j (f ))
H
cov(Ai AH
i Aj Aj )
H
H
H
H
= E [Ai Ai Aj Aj ] 0 E [Ai Ai ]E [Aj Aj ]
H
H
H H
= E [Ai Aj ]E [Ai A j ] + E [Ai Aj ]E [Ai Aj ]
T
H
2
T
2
= jh i
(f )R x (f )hj j + jh i (f )R x (f )hj j
=
8
8
8
(a)
(b)
Fig. 2. (a) Calculated bias. (b) Calculated variance for the ARMA process
(solid line PM MW (30 dB), dashed line PM MW, dotted line Hanning
window, dashed-dotted line
Thomson8 MW).
=
=
=
=
leakage causes a large bias at the notch. With the use of a penalty
function with G = 30 dB (solid line), the leakage is reduced, giving
a smaller bias.
The variance is depicted in Fig. 2(b). The multiple window methods show their superiority in variance reduction. On average, they all
have about one fourth of the Hanning window variance (dotted line),
which is one for all frequencies. The PM MW (dashed line) has a
large variance in the region of the notch f = 0:3 caused by sidelobe
leakage of the windows. The PM MW (30 dB) (solid line) shows, on
average, comparable results with the Thomson8 MW (dashed-dotted
line). At the peak of the spectrum f = 0:1, the Thomson8 MW
has the smallest variance as the bias of the peak is not considered.
However, a closer study of the peak as well as the surroundings shows
that the average variance of the peaked matched methods is smaller.
The disadvantage of suppressing the sidelobes in the PM MW is that
it gives a slightly worse variance in the area of the peak f = 0:1: The
variance at the notch f = 0:3 are, however, small for the PM MW
(30 dB) compared with the others and show the gain in the sidelobe
suppression. For the flat spectrum segment f > 0:3, the Thomson8
MW is close to the theoretical value for white noise, 1=8 = 0:125:
C. Comparison of Mean Square Error
Both low variance and small bias can be evaluated with the use
of the mean squared error, as this measure includes both variance
and squared bias. The normalized mean squared error MSE S^x (f )
is defined as
MSE S^x (f ) = E [S^x (f ) 0 Sx (f )]2 =E [S^x (f )]2
^x (f ) + Bias2 S
^x (f )
= Variance S
8
(12)
according to Walden et al. [6].
The calculated normalized bias of the ARMA process in Fig. 1 is
depicted in Fig. 2(a) for the different methods. The single Hanning
window (dotted line) shows small bias. The Thomson8 MW (dasheddotted line) has a fairly large bias at the peak f = 0:1 and the notch
f = 0:3: The behavior of the peak matched multiple windows PM
MW (dashed line) shows a reduced bias at f = 0:1, but the sidelobe
(13)
and an average value is calculated as
N 01
1
MSEav =
MSE S^x (k=N )
(14)
N k=0
where N = 128 is the number of frequencies in the DFT.
The windows tested are the PM MW (30 dB) and the PM MW
without sidelobe suppression. Two more cases with different G values
are included: G = 10 dB and G = 50 dB. The suppression of 50 dB
reduces the number of windows to K = 6 since the weighting factor
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 3, MARCH 1997
TABLE I
AVERAGE MEAN SQUARE ERROR COMPARED FOR THE ARMA
PROCESS AND FOR THE WHITE NOISE SPECTRUM. THE PM MW FOR
DIFFERENT G VALUES IS COMPARED WITH THE THOMSON MW WITH
K = 2; 4; 8 AS WELL AS TO THE SINGLE HANNING WINDOW
781
method suitable for estimation of peaked spectra as well as for spectra
with notches, e.g., ARMA processes. The result is also satisfactory
for white noise spectrum.
REFERENCES
[1] F. J. Harris, “On the use of windows for harmonic analysis with the
discrete Fourier transform,” Proc. IEEE, vol. 66, pp. 51–83, Jan. 1978.
[2] D. J. Thomson, “Spectrum estimation and harmonic analysis,” Proc.
IEEE, vol. 70, pp. 1055–1096, Sept. 1982.
, “An overview of multiple-window and quadratic-inverse spec[3]
trum estimation methods,” in Proc. ICASSP, 1994, pp. VI-185–VI-194.
[4] C. T. Mullis and L. L. Scharf, “Quadratic estimators of the power
spectrum,” in Advances in Spectrum Analysis and Array Processing, S.
Haykin, Ed. Englewood Clliffs, NJ: Prentice-Hall, 1991, pp. 1–57, ch.
1.
[5] J. L. Roux and J. Ménez, “A cost minimization approach for optimal
window design in spectral analysis of sampled signals,” IEEE Trans.
Signal Processing, vol. 40, pp. 996–999, Apr. 1992.
[6] A. T. Walden, E. McCoy, and D. B. Percival, “The variance of multitaper
spectrum estimates for real gaussian processes,” IEEE Trans. Signal
Processing, vol. 42, pp. 479–482, Feb. 1994.
is approximately zero for h7 and h8 : A comparison is made with
the Thomson multiple windows for different numbers of windows
and resolutions, i.e., Thomson8 MW, with resolution B = 0:08
(eight windows); Thomson4 MW (B = 0:04 and four windows)
and Thomson2 MW (B = 0:02 and two windows). The results are
described in Table I.
For the ARMA spectrum, the PM MW (30 dB) has the smallest
MSEav = 0:358, which is expected as these windows have both
small bias and low variance. The PM MW (10 dB) and PM MW have
larger MSEav as there is more sidelobe leakage of 0.587 and 1.079,
respectively. For the PM MW (50 dB), there is a strong suppression
of the sidelobes (50 dB), giving a MSEav = 0:399: The higher value
is caused by a reduction in the number of windows since the two last
weighting factors are close to zero. A comparison with the Thomson
multiple windows for different resolutions shows that these MSEav
values are larger due to a large bias and covariance of the different
periodograms, which are correlated for the nonsmooth spectrum.
For the white noise spectrum, the average mean squared error is
depicted in the second column of Table I. The lowest value among the
peaked matched methods is given by PM MW without any sidelobe
suppression. For the Thomson multiple windows, the result is close
to the theoretical value of zero bias and variance equal to 1/K. The
deviation is caused by the MSE at f = 0 and f = 0:5, where the
second term of (12) influences the result.
IV. CONCLUSIONS
A peak matched multiple window method for peaked spectra is
proposed. The resulting spectrum estimate has low variance and
bias in the neighborhood of the peak. The proposed method shows,
however, a large bias at a notch. This is due to large sidelobes of the
windows that cause leakage from frequencies outside the resolution
width. This leakage is suppressed with the use of a penalty function.
A local spectrum estimate with leakage control is achieved as the
sidelobe suppression is chosen with a parameter G: This makes the
Recursive versus Nonrecursive Correlation
for Real-Time Peak Detection and Tracking
Carl During
Abstract— This correspondence compares the recursive versus the
nonrecursive algorithms of correlation for a fast varying peak position by
simulations. The recursive algorithms give better response and accuracy
of time delay estimation, even though the sampling period decreases. The
average magnitude difference function (AMDF) is used as an application
example.
I. INTRODUCTION
This correspondence concerns real-time correlation peak detection
using the well-known average magnitude difference function (AMDF)
(see for example, [1]–[4]) as an application example. We compare the
recursive versus the nonrecursive algorithms of AMDF correlation for
a fast varying peak position by simulations. Discrete correlation in
the frequency or time domain are old techniques (see, for example,
[5]–[7]) used for comparing information (see, for example, [8]–[12])
but still useful in some applications (e.g., [13]–[16]). Even though the
following discussion will concentrate on measurements of mechanical
variables, a translation may be done to other case studies. The AMDF
is, for example, used by velocity sensors like the one illustrated in
Fig. 1 [17] or by the scene-matching technique implemented in the
cruise-missile guidance system (e.g., [18] and [19]). The required
information is here given by detecting, in real time, the extreme value
(peak) of a 1-D or 2-D discrete AMDF, respectively. The location of
the extreme value within the AMDF expresses how much a sequence
is shifted in time or space with respect to another sequence. This gives
Manuscript received June 14, 1995; revised June 12, 1996. The associate
editor coordinating the review of this paper and approving it for publication
was Prof. Allen Steinhardt.
The author is with the Department of Machine Design, Division of
Mechatronics, Royal Institute of Technology, Stockholm, Sweden.
Publisher Item Identifier S 1053-587X(97)01877-1.
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