Asymptotic Behavior of Cosine Windows

Raghavendra G. Kulkarni

Bharat Electronics, HMC Division

Bangalore, India

The asymptotic behavior of the even-cosine series windows presented here reveals that the series has an asymptotic decay 6 dB/octave higher than the odd-cosine series. The coefficients of the series are determined for the condition of maximum asymptotic decay, and the frequency response plots of the series are given.

It is well known that an ideal rectangular bandpass filter requires an impulse response of infinite duration, which is unrealizable in practice. Truncating this infinite impulse response after a certain time duration results in undesirable ripples (known as Gibb's ripples) in the passband and stopband. Windows (finite even time functions), when multiplied with the infinite impulse response of the ideal rectangular filter, tend to smooth the Gibb's ripples in the frequency response; however, this improvement is at the cost of widening the transition width. A window is characterized in the frequency domain in terms of its mainlobe width, peak sidelobe response and asymptotic decay. The widely used window functions are the Dolph-Chebyshev window, kaiser window, Gaussian window and cosine windows. An exhaustive discussion on windows has been provided by Harris^{1} and Nuttall.^{2 }

The cosine windows comprise a wide range of window functions, including the Hanning, Hamming and Blackman windows,^{1} the cosine temporal weightings,^{2} and the odd- and even-cosine series.^{3} Nuttal^{2} discusses the criteria for maximizing asymptotic decay and minimizing the sidelobe peaks, along with the trade-off in achieving one characteristic at the expense of the other. The odd-cosine series functions presented by Malocha and Bishop^{3} are the same as the cosine temporal weightings,^{2} with the coefficients of the series determined by minimizing the sidelobe peaks. The coefficients of the even- and odd-cosine series functions are further refined in Kulkarni and Lahiri^{4} to obtain improved sidelobes. The even- and odd-cosine series have an even and odd number of kernels in their respective frequency responses as described by Malocha and Bishop.^{3}

This article presents the asymptotic behavior of the even-cosine series and compares it with that of the odd-cosine series. The findings reveal that the even-cosine series has a 6 dB/octave higher asymptotic decay of the sidelobes than the odd-cosine series.

ASYMPTOTIC BEHAVIOR WHEN DECAY IS MAXIMIZED

The asymptotic behavior of the odd-cosine series (or the cosine temporal weightings) has been explained in past literature.^{2} The asymptotic behavior of the even-cosine series has not been discussed previously and is presented here. For this purpose, the even-cosine series is expressed in the time and frequency domains as ^{4 }

where the time domain function, h_{N2} , exists only in the interval from T/2 to T/2. The coefficients b_{n} are real and positive and are determined by the criteria applied on the sidelobes of the frequency function H_{N2} .

The function H_{N2} is rearranged in a convenient form for this discussion as

after recognizing that the term [1 {(2n + 1)^{2} /4T^{2} f^{2} }]^{1} has been expressed in a power series. Note from Equation 3 that when the first term in the series is not equal to zero such that

the function decays as 1/f^{2} , resulting in a 12 dB/octave decay. Note that the minimum asymptotic decay that can be obtained with the even-cosine series is 12 dB/octave; for cosine temporal weightings^{2} it is 6 dB/octave. However, if

and if the second term (numerator part) is non-zero such that

the function will decay as 1/f^{4} , giving a 24 dB/octave decay. In the same way, if the first two terms in the power series vanish such that

and the third term is not equal to zero, as in

then the function will decay as 1/f^{6} , resulting in a 36 dB/octave decay.

From the discussion on unique coefficients,^{4} note that b_{0} = 1 and only b_{1} can be determined, leading to a two-term even-cosine series. From Equation 5, only b_{1} and b_{2} can be determined, and a three-term cosine series results. The coefficients of the even-cosine series up to four terms are determined and listed in Table 1 along with mainlobe width, peak sidelobe level and asymptotic decay. The frequency domain plots of H_{12} to H_{42} are shown in Figure 1. Comparing the asymptotic decays of the even- and odd-cosine series from the table, note that for a given N, the even-cosine series has a 6 dB/octave higher decay of sidelobes than the odd-cosine series.

With the condition of maximizing asymptotic decay, the odd- and even-cosine series together can be represented by a single expression as

where k assumes integer values. When k is odd, the even-cosine series is obtained; when k is even, the odd-cosine series results. The window function given in Equation 6 is called the cosine powered window, or Hanning window.^{1}

ASYMPTOTIC BEHAVIOR OF SIDELOBES WHEN THE SIDELOBE PEAKS ARE MINIMIZED

When the condition of minimizing sidelobe peaks is used for determination of coefficients of the even-cosine series, then Equation 4 is not satisfied and the inequality expression governing the coefficients

prevails. The coefficients b_{n} are given in the literature.^{4} With these coefficients the asymptotic decay achievable is 12 dB/octave. However, for the criteria of minimizing sidelobe peaks, the odd-cosine series presents a 6 dB/octave decay of the sidelobes.^{2} Thus, the even-cosine series has a 6 dB/octave higher decay of the sidelobes when compared with the odd-cosine series. Table 2 lists various parameters along with the asymptotic decay of the sidelobes for an even- and odd-cosine series for the condition of minimizing sidelobe peaks. The values of peak sidelobes are obtained using the results in the literature.^{4}

CONCLUSION

The asymptotic behavior of an even-cosine series has been presented and compared with the odd-cosine series. It was found that the even-cosine series has a 6 dB/octave higher asymptotic decay than the odd-cosine series. The coefficients of the series were determined for the condition of maximum asymptotic decay. The frequency responses of the series up to the fourth order were also shown.

ACKNOWLEDGMENT

The author thanks the management of Bharat Electronics for supporting this work. *

References

1. F.J. Harris, "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform," Proceedings of the IEEE, Vol. 66, January 1978, pp. 5183.

2. A.H. Nuttal, "Some Windows with Very Good Sidelobe Behavior," IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. ASSP-29, No. 1, February 1981, pp. 8491.

3. D.C. Malocha and C.D. Bishop, "The Classical Truncated Cosine Series Functions with Applications to SAW Filters," IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Vol. UFFC-34, January 1987, pp. 7585.

4. R.G. Kulkarni and S.K. Lahiri, "Improved Sidelobe Performance of Cosine Series Functions," IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Vol. 46, March 1999, pp. 464466.

Raghavendra G. Kulkarni received his BE degree in electronics and communications from Karnatak University in 1979, his MTech degree from Indian Institute of Technology, Madras in 1982 and his PhD degree from Indian Institute of Technology, Kharagpur in 1998. He joined the HMC Division of Bharat Electronics, Bangalore in 1982 and since then has been engaged in the design and development of SAW devices. He won the company's R&D award in 1989 for his work on SAW filters. Kulkarni has authored several technical papers in the area of SAW devices and has applied for two patents. He is a member of IMAPS-India and IEEE. Currently, Kulkarni is working as a manager guiding SAW device development activities in Bharat Electronics' HMC Division.