A Flexible Window Function for Spectral Analysis

D igital Object Identifier 10.1109/MSP.2009.935422A Flexible Window Function for Spectral Analysis

R

egarding window functions used in spectral analysis, the most important performance measures are 3-dB bandwidth and sidelobe attenuation. For

many window functions, Hanning and Hamming for example, we have no con-trol over a window’s 3-dB bandwidth and sidelobe attenuation for a given w indow length. For other window functions—Kaiser, Gaussian, and Chebyshev—we can reduce those windows’ 3-dB band-width to get improved spectral resolu-tion. However, with these later window functions (what we refer to as “conven-tional windows”), spectral resolution improvement comes at the expense of sidelobe attenuation reduction that degrades our ability to avoid undesirable spectral leakage. Likewise we can increase those windows’ sidelobe attenu-ation, but only by sacrificing desirable spectral resolution. This article describes a novel window function that enables us to control both its 3-dB bandwidth (spec-tral resolution) and sidelobe attenuation (spectral leakage) independently.

The 3-dB bandwidth, sidelobe atten-uation, and roll-off rate are used to measure the performance of windows for power spectral density (PSD) esti-mation [1]–[3]. I mproved frequency

resolution of the estimated PSD can be obtained if we reduce a window’s 3-dB bandwidth. The sidelobe attenuation means the difference between magni-tude of the mainlobe and the maximum magnitude of the sidelobes. The side-lobe roll-off rate is the asymptotic decay rate of sidelobe peaks. Unde-sirable spectral leakage [4]–[6] can be reduced by increasing sidelobe attenu-ation and roll-off rate. Therefore, an ideal window for PSD estimation has zero bandwidth and infinite sidelobe attenuation such as an impulse func-tion in frequency domain.

The conventional windows are able to control 3-dB bandwidth or sidelobe attenuation by only one parameter in general [1], [7]–[12]. Thus, they cannot control these two characteristics inde-pendently. In other words, if we reduce a window function’s 3-dB bandwidth, the sidelobe attenuation is also reduced, and vice versa [5], [6]. This behavior is the cause of the tradeoff problem between good frequency resolution and acceptable spectral leakage in the esti-mated PSD. The Butterworth window does not have this problem because it allows control of the 3-dB bandwidth and sidelobe attenuation independently.Butterworth windows are used as antialiasing filters to reduce the noise in the reconstructed image in previous research [13]. They are also used to remove the edge effect of the matched filter output in pattern matching algo-rithm [14]. The transfer function of a Butterworth filter is adopted as a win-dow in those applications. However, in this article, a portion of the impulse response of a Butterworth filter is called the Butterworth window and its characteristics in PSD estimation are analyzed.

BUTTERWORTH WINDOW

The Butterworth window can be obtained by the standard Butterworth filter design procedure. I mportant to us is the fre-quency magnitude response H 1f 2 of a Butterworth filter, denoted by [2] and [6]

|H 1f 2|51/

? 11a f f c

b 2N

, (1)where f is frequency in hertz.

The Butterworth filter is character-ized by two independent parameters, 3-dB cutoff frequency f c and filter order N . The cutoff frequency and order of the Butterworth filter serve as parameters that control the bandwidth and sidelobe attenuation of the Butterworth window. The cutoff frequency of a filter has the identical meaning with the bandwidth of a window. However, the cutoff frequency is represented as a half of the bandwidth since the bandwidth of a window refers to two-sided frequency from negative to positive, while the cutoff frequency of a filter refers to only one-sided positive frequency. Our desired window spectrum is identical to the frequency response of the Butterworth filter. Thus, the inverse Fourier transform is applied to the Butterworth filter’s frequency response, in (1), to obtain the filter’s impulse response, and a portion of that response becomes the Butterworth window in the time domain.

SIMULATION AND

PERFORMANCE ANAL YSIS

I n our simulation, the frequency and impulse responses of Butterworth filters are investigated to design the Butterworth window by varying the cutoff frequency f c and filter order N . The sampling frequen-cy f s is set to 2,048 Hz. The magnitude

l evels of the impulse response of a “DS P Tips and Tricks” introduces

practical design and implementation signal processing algorithms that you may wish to incorporate into your designs. We welcome readers to submit their contributions. Contact Associate Editors Rick Lyons (R.Lyons@https://www.wendangku.net/doc/161397721.html,) or C. Britton Rorabaugh (dspboss@https://www.wendangku.net/doc/161397721.html,).

Tae Hyun Yoon and Eon Kyeong Joo

[dsp TIPS&TRICKS ]

Butterworth filter are nearly zero after a certain point—almost all information that determines the filter’s frequency response is in the portion before that zero-magni-tude point of the impulse response. Therefore, it is expected that the suitable length of a Butterworth window can be determined by only a part of the infinite-time duration impulse response of a Butterworth filter.

Figure 1(a) shows the time-domain impulse response h1k2 of a unity-gain low-pass Butterworth filter when f c50.75 Hz and N53. I n that figure,

we show the initial positive-only portion

of the impulse response that becomes

our desired Butterworth window. The

2,139th sample of h1k2 is the point that

the magnitude of the impulse response

of the Butterworth filter becomes zero

for the first time.

The solid curve in Figure 1(b) is the

frequency spectrum of the 2,139-sample

Butterworth window. The 3-dB band-

width and sidelobe attenuation of this

window are 1.3 Hz and 24.3 dB, respec-

tively. In Figure 1(b), for comparison, we

show the frequency magnitude response

of the Butterworth filter as the dashed

curve. We see that there is no significant

difference between the magnitude

firm the performance. The signal x1t2

used for our simulation is

x1t25 0.84cos12p#52#t2

1 0.8cos12p#65.5#t2

1 0.3cos12p#85#t2

1 1.1cos12p#105#t2

10.35cos12p#140#t2

10.98cos12p#159#t2

1 0.6cos12p#174#t2

1 0.8cos12p#190#t2

1 cos12p#205#t2. (2)

The solid lines in Figure 2(a) show

the ideal PSD of x1t2. The dotted curve

in Figure 2(a) shows the estimated PSD

of a 2,139-sample rectangular windowed

x1t2, using Welch’s method [15], where

that window’s insufficient sidelobe

attenuation (spectral leakage) produces

[FIG1] Butterworth filter and window: (a) filter impulse response and window function and (b) the filter magnitude response and window spectrum. [TABLE 1] SUITABLE LENGTHS

OF BUTTERWORTH WINDOWS

(F S = SAMPLING FREQUENCY,

F C = CUTOFF FREQUENCY).

FILTER ORDER, N WINDOW LENGTH (SAMPLES)

1

j0.660#f s f c k

2

j0.705#f s f c k

3

j0.784#f s f c k

4

j0.890#f s f c k

5

j1.005#f s

c

k

[TABLE 2] FREQUENCY CHARACTERISTICS OF THE BUTTERWORTH WINDOWS.

FILTER

ORDER

3-DB

BANDWIDTH

SIDELOBE

ATTENUATION ROLL-OFF RATE

1 1.5 Hz10.0 dB

212 dB/OCT.

2 1.4 Hz18.2 dB

3 1.3 Hz24.3 dB

4 1.3 Hz28.8 dB

5 1.3 Hz30.4 dB

functions. Reference [16] provides spec-tral plots comparing Butterworth win-dows to the conventional window functions in Table 3. IMPLEMENTATION ISSUES

The computational time of PSD ■

estimation is not related to window type, but rather the window length and estimation method. So Butter-

worth windows have the same compu-

tational workload as the conventional window functions.

To use the computationally effi-■

cient radix-2 fast Fourier transform algorithm, we suggest that the time-

domain samples of Butterworth win-

dow should be zero padded to make the window length an integer power of two. As an alternative to zero pad-

ding, we can restrict the Butterworth window’s f c cutoff frequency to be

f c5

Kf s

2M11

, (3)

which leads to Butterworth windows that are 2M in length, where K is one of the scaling constants from Table 1, and M is an integer.

Because Butterworth windows are ■

not symmetrical, any specialized spec-

tral analysis scheme that requires the imaginary part of a window function’s

spectrum to be all zero

will not work with the

Butterworth windows.

CONCLUSIONS

We’ve shown that the

Butterworth window can be

obtained by the convention-

al Butterworth filter design

procedure. This window is

able to control the 3-dB

bandwidth and sidelobe

attenuation independently

by two parameters, the cut-

off frequency and the order

of the filter. As such, the

sidelobe attenuation can be

varied even if the 3-dB

bandwidth is fixed, and vice

versa. Therefore the tradeoff

problem between the fre-

quency resolution and spectral leakage in

the estimated PSD, unavoidable with the

conventional windows, can be solved by

the Butterworth window.

ACKNOWLEDGMENT

The authors thank Associate Editor

Richard (Rick) Lyons for his kind help in

preparation of this article.

AUTHORS

Tae Hyun Yoon (thyoon@ee.knu.ac.kr)

is a Ph.D. candidate at the School of

Electrical Engineering and Computer

Science, Kyungpook National University,

Daegu, Korea.

Eon Kyeong Joo (ekjoo@ee.knu.ac.kr)

is a professor at the School of Electrical

Engineering and Computer Science,

[TABLE 3] COMPARISON OF FREQUENCY

CHARACTERISTICS.

WINDOW

3-DB

BANDWIDTH

SIDELOBE

ATTENUATION

RECTANGULAR0.879 Hz13.3-dB

TRIANGULAR 1.270 Hz26.5 dB

HANNING 1.367 Hz31.3-dB

KAISER a 5 20.980 Hz18.5 dB

a 5 4 1.172 Hz30.4 dB

CHEBYSHEV b 5 10.890 Hz20.1 dB

b5 2 1.172 Hz40.5 dB

BUTTERWORTH

(f c 5 0.439 Hz)

N 5 20.793 Hz18.2 dB

N 5 30.740 Hz24.3-dB

N 5 40.731 Hz28.8 dB

BUTTERWORTH

1N542

f c50.439 Hz0.731 Hz28.8 dB

f c51.500 Hz 2.815 Hz28.8 dB

f c52.500 Hz 4.720 Hz28.8 dB

[FIG2] Ideal and windowed PSD: (a) rectangular windowed and ideal PSDs and (b) PSDs using various Butterworth windows.

[dsp TIPS&TRICKS]c ontinued

Kyungpook National University, Daegu, Korea.

REFERENCES

[1] F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE, vol. 66, no. 1, pp. 51–83, Jan. 1978.

[2] S. Kay and S. Marple, “Spectrum analysis-a modern perspective,” Proc. IEEE, vol. 69, no. 11, pp. 1380–1419, Nov. 1981.

[3] M. H. Hayes, Statistical Signal Processing and Modeling. New York: Wiley, 1996.

[4] B. P. Lathi, Mod ern Digital and Analog Communication Systems, 3rd ed., New York: Oxford Univ. Press, 1998.

[5] N. Geckinli and D. Yavuz, “Some novel windows and a concise tutorial comparison of window fami-lies,” IEEE Trans. Acous. Speech Signal Processing, vol. ASSP-26, no. 6, pp. 501–507, Dec. 1978.[6] S. Orfanidis, Introduction to Signal Processing.

Englewood Cliffs, NJ: Prentice-Hall, 1996.

[7] S. Mitra and J. Kaiser, Hand book for Digital

Signal Processing. New York: Wiley, 1993.

[8] S. Bergen and A. Antoniou, “Design of ul-

traspherical window functions with prescribed

spectral characteristics,” EURASIP J. Appl. Signal

Process., vol. 2004, no. 13, pp. 2053–2065, Oct.

2004.

[9] A. Nuttall, “Some windows with very good side-

lobe behavior,” IEEE Trans. Acous. Speech Signal

Processing, vol. 29, no. 1, pp. 84–91, Feb. 1981.

[10] G. Andria, M. Savino, and A. Trotta, “Windows

and interpolation algorithms to improve electri-

cal measurement accuracy,” IEEE Trans. Instrum.

Meas., vol. 38, no. 4, pp. 856–863, Aug. 1989.

[11] G. Andria, M. Savino, and A. Trotta, “FFT-based

algorithms oriented to measurements on multi-

frequency signals,” Measurement, vol. 12, no. 1, pp.

25–42, Dec. 1993.

[12] J. Gautam, A. Kumar, and R. Saxena, “On the

modified Bartlett–Hanning window (family),” IEEE

Trans. Signal Processing, vol. 44, no. 8, pp. 2098–

2102, Aug. 1996.

[13] H. Baghaei, W. Wong, H. Li, J. Uribe, Y. Wang,

M. Aykac, Y. Liu, and T. Xing, “Evaluation of the ef-

fect of filter apodization for volume PET imaging us-

ing the 3-D RP algorithm,” IEEE Trans. Nucl. Sci.,

vol. 50, no. 1, pp. 3–8, Feb. 2003.

[14] S. Su, C. Yeh, and C. Kuo, “Structural analysis of ge-

nomic sequences with matched filtering,” in Proc. 25th

Ann. Int. Conf. IEEE Engineering in Medicine and Bi-

ology Society, Sept. 2003, vol. 3, pp. 2893–2896.

[15] P. Welch, “The use of fast Fourier transform for

the estimation of power spectra : A method based on

time averaging over short, modified periodograms,”

IEEE Trans. Audio Electroacous., vol. AU-15, no. 2,

pp. 70–73, June 1967.

[16] T. Yoon and E. Joo, “Butterworth window for

power spectral density estimation,” ETRI J., vol. 31,

no. 3, pp. 292–297, June 2009. [S P]