EURASIP Journal on Applied Signal Processing 2004:13, 2053–2065 c ○ 2004 Hindawi Publishing Corporation Design of Ultraspherical Window Functions with Prescribed Spectral Characteristics

A method for the design of ultraspherical window functions that achieves prescribed spectral characteristics is proposed. The method comprises a collection of techniques that can be used to determine the three independent parameters of the ultraspherical window such that a specified ripple ratio and main-lobe width or null-to-null width along with a user-defined side-lobe pattern can be achieved. Other known two-parameter windows can achieve a specified ripple ratio and main-lobe width; however, their side-lobe pattern cannot be controlled as in the proposed method. A comparison with other windows has shown that a difference in performance exists between the ultraspherical and Kaiser windows, which depends critically on the required specifications. The paper also highlights some applications of the proposed method in the areas of digital beamforming and image processing.


INTRODUCTION
Windows are time-domain weighting functions that are used to reduce Gibbs' oscillations resulting from the truncation of a Fourier series. Their roots date back over one-hundred years to Fejer's averaging technique for a truncated Fourier series and they are employed in a variety of traditional signal processing applications including power spectral estimation, beamforming, and digital filter design. Despite their maturity, windows functions (or windows for short) continue to find new roles in the applications of today. Very recently, windows have been used to facilitate the detection of irregular and abnormal heartbeat patterns in patients in electrocardiograms [1,2]. Medical imaging systems, such as the ultrasound, have also shown enhanced performance when windows are used to improve the contrast resolution of the system [3]. Windows have also been employed to aid in the classification of cosmic data [4,5] and to improve the reliability of weather prediction models [6]. With such a large number of applications available for windows that span a variety of disciplines, general methods that can be used to design windows with arbitrary characteristics are especially useful.
Windows can be categorized as fixed or adjustable [7]. Fixed windows have only one independent parameter, namely, the window length which controls the main-lobe width. Adjustable windows have two or more independent parameters, namely, the window length, as in fixed windows, and one or more additional parameters that can control other window characteristics [8,9,10,11,12,13]. The Kaiser and Saramäki windows [8,9] have two parameters and achieve close approximations to discrete prolate functions that have maximum energy concentration in the main lobe. The Dolph-Chebyshev window [10] has two parameters and produces the minimum main-lobe width for a specified maximum side-lobe level. The Kaiser, Saramäki, and Dolph-Chebyshev windows can control the amplitude of the side lobes relative to that of the main lobe. The ultraspherical window has three parameters, and through the proper choice of these parameters, the amplitude of the side lobes relative to that of the main lobe can be controlled as in the Kaiser, Saramäki, and Dolph-Chebyshev windows; and in addition, arbitrary side-lobe patterns can be achieved. To facilitate the application of the ultraspherical window to the diverse range of applications alluded to earlier, a practical and efficient design method is required that can utilize its inherent flexibility.
In this paper, a method is proposed for designing ultraspherical windows that achieves prescribed spectral characteristics such as specified ripple ratio, main-lobe width, null-to-null width, and a user-defined side-lobe pattern. The paper is structured as follows. Section 2 presents some performance measures for windows. Section 3 introduces the ultraspherical window and some formulas for generating its coefficients from three independent parameters. Aspects of the window's frequency spectrum and its equivalence to other windows are also discussed. Section 4 proposes a method for designing ultraspherical windows that achieve prescribed spectral characteristics. The method entails a variety of short algorithms that calculate two of the three independent parameters based on the prescribed spectral characteristics. Section 5 proposes an empirical equation that can be used to accurately predict the window length required to achieve multiple prescribed spectral characteristics simultaneously. Section 6 compares the ultraspherical window's effectiveness in achieving prescribed spectral characteristics with respect to other conventional windows. Section 7 presents examples and demonstrates the accuracy of the proposed method. Section 8 describes two applications of the proposed method in the areas of beamforming and image processing. Section 9 provides concluding remarks.

CHARACTERIZATION OF WINDOWS
Windows are frequently compared and classified in terms of their spectral characteristics. The frequency spectrum of a window is given by where W 0 (e jωT ) is called the amplitude function, N is the window length, and T is the interval between samples. The amplitude and phase spectrums of a window are given by A(ω) = |W 0 (e jωT )| and θ(ω) = −ω(N − 1)T/2, respectively, and |W 0 (e jωT )|/W 0 (e 0 ) is a normalized version of the amplitude spectrum. The normalized amplitude spectrum of a typical window is depicted in Figure 1. Two parameters of windows in general are the null-tonull width B n and the main-lobe width B r . These quantities are defined as B n = 2ω n and B r = 2ω r , where ω n and ω r are the half null-to-null and half main-lobe widths, respectively, as shown in Figure 1. An important window parameter is the ripple ratio r which is defined as (see Figure 1). The ripple ratio is a small quantity less than unity and, in consequence, it is convenient to work with the reciprocal of r in dB, that is, R can be interpreted as the minimum side-lobe attenuation relative to the main lobe and −R is the ripple ratio in dB. Another parameter that may be used to quantify a window's side-lobe pattern is the side-lobe roll-off ratio, s, which is defined as Figure 1: A typical window's normalized amplitude spectrum and some common spectral characteristics.
where a 1 and a 2 are the amplitudes of the side lobe nearest and furthest, respectively, from the main lobe (see Figure 1). If S is the side-lobe roll-off ratio in dB, then s is given by For the side-lobe roll-off ratio to have meaning, the envelope of the side-lobe pattern should be monotonically increasing or decreasing. These spectral characteristics are important performance measures for windows. When analyzing narrowband signals, such as sinusoids, weak signals can easily be obscured by nearby strong signals. The width characteristics provide a resolution measure between adjacent signals while the ripple ratio determines the worst-case scenario for detecting weak signals in the presence of strong narrowband signals. The side-lobe roll-off ratio provides a simple description of the distribution of energy throughout the side lobes, which can be of importance if prior knowledge of the location of an interfering signal is known. Further explanation of the usefulness of these spectral characteristics can be found in [11].

THE ULTRASPHERICAL WINDOW
The coefficients of a right-sided ultraspherical window of length N can be calculated explicitly as [12,14] In (6) µ, x µ , and N are independent parameters and w[(N − n − 1)T] = w(nT). A normalized window is obtained as The binomial coefficients can be calculated as (9) The independent parameter x µ can be expressed as where β ≥ 1 and x (µ) N−1,1 is the largest zero of the ultraspherical polynomial C µ N−1 (x). The new independent parameter β is the so-called shape parameter and can be used to set the null-to-null width of a window to 4βπ/N, that is, β times that of the rectangular window [9]. Throughout the paper, x (λ) n,l denotes the lth zero of the ultraspherical polynomial C λ n (x). Unfortunately, closed-form expressions for the zeros of this polynomial do not exist but the zeros can be found quickly using the following iterative algorithm which is valid for l = 1 and rnd(n/2) yielding the largest and smallest zeros, respectively. The rounding operator is defined as where int(y) is the integer part of y and is also known as the floor operator. Due to the symmetry relation C µ n(−x) = (−1) n C µ n(x), only the positive zeros need to be considered.
Step 3 If |y k+1 − y k | ≤ ε, then output x * = y k+1 and stop. Set k = k + 1, and repeat from Step 2. In this algorithm, ε is the termination tolerance. A good choice is ε = 10 −6 which would cause the algorithm to converge in 3 to 6 iterations. Equation (12) in Step 1 represents the lowest upper bound for the zeros of the ultraspherical polynomial [16]. In Step 2, the Newton-Raphson method is used to obtain the next estimate of the zero.
The amplitude function of the ultraspherical window is given by where C µ n(x) is the ultraspherical polynomial which can be calculated using the recurrence relationship given in (14).
The Dolph-Chebyshev window is a special case of the ultraspherical window and can be obtained by letting µ = 0 in (6), which results in where is the Chebyshev polynomial of the first kind. In the Dolph-Chebyshev window, the side-lobe pattern is fixed, that is, (1) all side lobes have the same amplitude and (2) a minimum main-lobe width is achieved for a specified side-lobe level. Hence this window is usually designed to yield a specified ripple ratio r. To design a Dolph-Chebyshev window, x µ is calculated using the relation [10] x µ = x 0 = cosh 1 Alternatively, the Dolph-Chebyshev window can be designed to yield a specified null-to-null width β times that of the rectangular window. This can be accomplished by using (10) where x (µ) N−1,1 = x (0) N−1,1 is the largest zero of the Chebyshev polynomial of the first kind T N−1 (x), which is given by .
The Saramäki window is a special case of the ultraspherical window and can be obtained by letting µ = 1 in (6), which results in where is the Chebyshev polynomial of the second kind. The Saramäki window, like the Kaiser window, is known for achieving close approximations to discrete prolate functions and is designed to yield a null-to-null width β times that of the rectangular window. This can be accomplished by using is the largest zero of the Chebyshev polynomial of the second kind U N−1 (x), which is given by Another special case of interest is the case where µ = 1/2 in (6), which results in where P n (x) is the Legendre polynomial which can be calculated using the recurrence relationship for r = 2, 3, . . . , n, where P 0 (x) = 1 and P 1 (x) = x.

PRESCRIBED SPECTRAL CHARACTERISTICS
With the appropriate selection of the parameters µ, x µ , and N, ultraspherical windows can be designed to achieve prescribed specifications for the side-lobe roll-off ratio, the ripple ratio, and one of the two width characteristics simultaneously. Parameter µ alters the side-lobe roll-off ratio, x µ provides a trade-off between the ripple ratio and a width characteristic, and N allows different ripple ratios to be obtained for a fixed width characteristic and vice versa. In some applications the window length N may be fixed. Such a scenario limits the designer's choice in achieving prescribed specifications for the side-lobe roll-off ratio and either the ripple ratio or a width characteristic but not both. For the case where N is adjustable, a prediction of N is possible which allows one to achieve prescribed specifications for the side-lobe roll-off ratio, the ripple ratio, and a width characteristic simultaneously.
In the subsections to follow, algorithms are proposed that achieve each prescribed specification to a high degree of precision. Some important quantities to be used are identified in Figure 2 which depicts a plot of C µ N−1 (x) for the values µ = 2 and N = 7. The modified sign (msgn) and max functions are defined as

Side-lobe roll-off ratio
To generate an ultraspherical window for a fixed N and a prescribed side-lobe roll-off ratio s, one can select the parameter µ appropriately. This can be accomplished by solving the one-dimensional minimization problem where the values of C µ n(x) are given by (14), and x N−2,1 can be found using Algorithm 1 with l = 1, λ = µ + 1, n = N − 2, and ε = 10 −6 . The zero x (µ+1) Simple algorithms such as dichotomous, Fibonacci, or golden section line searches, as outlined in [17], can be used to perform the minimization in (27). The lower and upper bounds on µ in (27) can be set to If s = 1, then no minimization is necessary and µ = 0 yields the Dolph-Chebyshev window. The bound µ L = −0.9999 was chosen because C µ N−1 (x) has a singularity at µ = −1. Also, for values of µ ≤ −1.5, the zeros of the ultraspherical polynomial overlap rendering the resulting window useless for our purposes. The bound µ H = 10 was chosen because the improvements in the side-lobe roll-off ratio that can be achieved for values of µ > 10 are negligible. The ultraspherical window imposes limits on the sidelobe roll-off ratio that can be achieved for low values of N. For example, if N = 7, window designs with S = 20 log 10 s outside the range −10.19 < S < 12.78 dB are not possible for any value of µ. For this reason, the side-lobe roll-off ratio's design range must be limited for a given N to that produced using µ L = −0.9999 and µ H = 10. The limiting values are shown in Table 1 for window lengths in the range 5 ≤ N ≤ 20 which spans the practical design range −20 ≤ S ≤ 60 dB.

Null-to-null width
To generate an ultraspherical window with µ and N fixed and a prescribed null-to-null half width of ω n rad/s, one can select the parameter x µ appropriately. This can be accomplished by calculating x µ using the expression where the zero x (µ) N−1,1 can be found using Algorithm 1 with l = 1, λ = µ, n = N − 1, and ε = 10 −6 .

Main-lobe width
To generate an ultraspherical window with µ and N fixed and a prescribed main-lobe half width of ω r rad/s, one can select the parameter x µ appropriately. This can be accomplished by calculating x µ using the expression where x a is defined by C µ N−1 (x a ) = msgn(µ) · max(a, b) as identified in Figure 2. Parameter x a is found through a threestep process. First, the zero x (µ+1) N−2,1 is found using Algorithm 1 with l = 1, λ = µ + 1, n = N − 2, and ε = 10 −6 , and then the parameter a = |C as seen in Figure 2, parameter x a is found using a modified version of Algorithm 1 where (13) is replaced by and the starting point given in (12) is replaced by In the modified algorithm, l = 1, λ = µ, n = N − 1, and ε = 10 −6 .

Ripple ratio
To generate an ultraspherical window with µ and N fixed and a prescribed ripple ratio r, one can select the parameter x µ appropriately. The parameter x µ is found through a threestep process. First, the zero x (µ+1) N−2,1 is found using Algorithm 1 with l = 1, λ = µ + 1, n = N − 2, and ε = 10 −6 and then the parameter a = |C Third, the parameter x µ is found using a modified version of Algorithm 1 where (13) is replaced by and the starting point given in (12) is replaced by Instead of finding the largest zero of In the modified algorithm l = 1, λ = µ, n = N − 1, and ε = 10 −6 .

PREDICTION OF N
In some applications designers may be able to choose the window length N. In such applications, the extra degree of freedom allows for more flexible window designs to be obtained. Specifically, solutions that are required to meet both a prescribed ripple ratio and width characteristic are possible. In this section, an empirical equation is proposed that predicts the ultraspherical window length N required to achieve a prescribed side-lobe roll-off ratio, ripple ratio, and mainlobe width simultaneously. To obtain an equation for N, we employ the performance factor [18] which is used to give a normalized width that is approximately independent of N. Rearranging (34), an expression for N is obtained as where N is rounded up to the nearest integer. From (35), it becomes clear that N can be predicted by obtaining an accurate approximation of D.

Measurements and tendencies of D
To obtain realistic data for the approximation of D, windows of length N = 7, 9, 13, 19, 51, 127, and 255 were designed to cover the range 20 ≤ R ≤ 100 in dB for the parameter range −0.9999 ≤ µ ≤ 10. Figure 3 shows plots of D versus R in dB for the two cross-sections µ = 1 and 10. The plots tend to be quadratic and are representative for the range −0.9999 ≤ µ ≤ 10 considered in this paper. Note the approximately linear behavior for N = 255 indicating the independence of the performance factor D with respect to N for large N, which agrees with previous observations concerning the performance factor D [18].

Data-fitting procedure
Before approximating D, the allowable error in the datafitting procedure must be determined. From (35), we note that for N 1 a per-unit error in D gives approximately the  Table 3: Model coefficients a ijk in (37) (S < 0).
same per-unit error in N, that is, For example, if N = 127 and a relative error in D of 1.00% is assumed, that is, ∆D/D = 0.01, then an equivalent error of 1.26 samples in N occurs. Errors of this magnitude have been considered acceptable in the past [18] as N may be in error by at most 1 or 2 and only for high window lengths. Thus, the relative error ∆D/D ≤ 0.01 is sought throughout the approximation procedure. A general quadratic model was used for the approximation of D as a function of S in dB, R in dB, and the main-lobe half width ω r . Such a model takes the form where φ(i, j, k) = (S/20) i R j ω k r . The coefficients a i jk were found through a linear least-squares solution of the overdetermined system of sampled data points {S, R, ω r , D} where D is the dependent variable.
Two separate sets of 27 coefficients were found for the ranges 0 ≤ S ≤ 60 and −20 ≤ S ≤ 0 given in dB and are provided in Tables 2 and 3, respectively. Two sets were required to produce accurate solutions due the nature of D and its relation to positive and negative S values. Figure 4 shows plots of the relative error of the predicted D versus R for various  window lengths over the cross sections µ = 1 and −0.6. The mean of the absolute relative error for the approximations given by Tables 2 and 3 is 0.2874 and 0.2266%, respectively. Less error occurs for the coefficients in Table 3 because the approximation was performed over a smaller range of S than that used for Table 2. The absolute relative error exceeds 1.0% only for small values of R less than 20 and large values of R greater than 100.
In an attempt to reduce the number of approximation model coefficients, the quadratic model where was investigated which yields 10 coefficients as opposed to 27. Using the same data fitting technique as before, the mean of the absolute relative error for the entire approximation was found to be 1.0911%. In 70% of the predictions, the absolute error was less than 1.0%. On the basis of the above experiments, N can be accurately predicted using the formula where D aprx is given by the 27-term approximation model described in (37) using the coefficients provided in Tables 2  and 3. The same process can be used to predict N for other width characteristics such as the null-to-null or 3 dB widths.

COMPARISON WITH OTHER WINDOWS
For a fixed window length, two-parameter windows such as the Kaiser, Saramäki, and Dolph-Chebyshev windows can control the ripple ratio. The three-parameter ultraspherical window can control the ripple ratio as well as the side-lobe roll-off ratio. For comparison's sake, ultraspherical windows of the same length were designed to achieve the side-lobe roll-off ratio and main-lobe width produced by the Kaiser window, for values of the Kaiser-window parameter α in the range [1,10], and the resulting ripple ratios for the two window families were measured and compared. The Dolph-Chebyshev and Saramäki windows were excluded from the comparison because these windows are special cases of the ultraspherical window that can be readily obtained by fixing parameter µ to 0 and 1, respectively. Figure 5a shows plots of the side-lobe roll-off ratio in dB obtained for Kaiser windows of varying length versus D = 2ω r (N −1) and Figure 5b shows a plot of ∆R which is defined as where R U and R K are the values of R for ultraspherical and Kaiser windows, respectively, in dB for the same length, side roll-off ratio, and main-lobe width. As can be seen, the ultraspherical window offers a reduced ripple ratio for low val- corresponding main-lobe half width, say ω rU , for which the ultraspherical window gives a better ripple ratio than the Kaiser window. For main-lobe half widths that are larger than ω rU , the Kaiser window gives a smaller ripple ratio. A plot of ω rU versus N is shown in Figure 6. From this plot, a formula can be obtained for ω rU as where the coefficients are presented in Table 4. In effect, if the point [N, ω r ] is located below the curve in Figure 6, the ultraspherical window is preferred, and if it is located above the curve, the Kaiser window is preferred.    A consequence of rounding N up to the nearest integer is that one prescribed spectral characteristic is oversatisfied. For the method presented in this paper, one will always achieve S and either ω r or R to a high degree of precision by using either (30) or the procedure described in Section 4.4 as appropriate to calculate parameter x µ . In this example, we oversatisfy R by using (30). Figure 9 shows the amplitude spectrums of the windows obtained. Both designs meet the prescribed characteristics and oversatisfied R by To examine the accuracy of the window length prediction formula, windows were designed to achieve the same prescribed characteristics with window lengths taken to be one less than predicted by (40), that is, for (a) N − 1 = 80 and (b) N − 1 = 82. Figure 10 shows the amplitude spectrums obtained for N and N − 1 in the critical area near the main-lobe edge. All windows were found to satisfy the S and ω r specifications; however, both windows  The required Kaiser-window parameter α for (a) and (b) can be predicted using the formula [19]

EXAMPLES
as α = 6.8514 and 9.4902 producing main-lobe half widths of ω r = 0.1462 and 0.1964 rad/s, respectively. Ultraspherical windows were designed to achieve the same side-lobe roll-off ratio and main-lobe widths as the Kaiser windows measured as

APPLICATIONS
The ultraspherical window function has been presented in terms of its spectral characteristics to facilitate its use for a diverse range of applications. The flexibility provided by our ability to control the side-lobe roll-off ratio has enabled us to develop a method for the design of FIR filters that satisfy prescribed specifications, which leads to improved filter specifications relative to the Kaiser window method [20,21]. In this section, two other window applications, beamforming and image processing, are presented to illustrate the benefits obtained by exercising the proposed methods flexibility.

Beamforming
In radar, ocean acoustics, and ultrasonics it is necessary to design antenna or transducer systems with specific directivity properties, that is, for point-to-point communication systems, a high gain in one direction with low gain in all other directions is considered desirable. Known as beamforming, this activity shapes the radiation pattern (or beam) of a transmitted signal so that most of its energy propagates towards the intended receiver or target. Similarly, when receiving signals, the receiver sensitivity (or beam) can be directed towards the transmitter or source to receive the maximum signal strength possible. Directing and focusing signal energy in this fashion leads to the rejection of interference from other sources and to reduced power requirements for transmitter and receiver power, which in turn provides cost savings. One practical and common antenna/transducer configuration is the linear array, which is characterized by having all its radiating elements positioned in a straight line. Linear arrays can consist of one continuous radiating element or a number of individual discrete elements. Generally, discrete elements are favored because of their capability to dynamically change the directivity properties of the array. The array factor (AF) is used to describe an array's directivity properties. For a broadside array of length N with amplitude excitations for each isotropic element being symmetrical about the center of the array, the AF is given by [22] AF(θ) =            r n=1 a n cos (2n − 1)u for odd N, r n=1 a n cos 2(n − 1)u for even N, where u = the spatial frequency (degrees/m) = πd λ cos θ, θ = the bearing angle (degrees), d = the spacing between elements (m), λ = the wavelength of the signal (m), a n = the excitation coefficients or currents (A), a n =        a n , n = 1, 1 2 a n , n = 1, The relationship between AF(θ) and a n is analogous to the relationship between W(e jωT ) and w(nT). This similarity allows window design techniques to be applied directly to the design of antenna arrays. As in window designs, the trade-off between the main-lobe width and the side-lobe level of the AF is of primary importance. In the uniform array the excitation coefficients are all equal, as in the rectangular window, and hence the main-lobe width of the AF is narrow and sidelobe levels are large. At the other extreme, the binomial array's AF has no side lobes but has of a large main-lobe width. Practical difficulties also arise with the implementation of the binomial array because the difference between excitation coefficients can be considerable leading to disparate current requirements. The Dolph-Chebyshev array, which offers an adjustable trade-off between the main-lobe width and side-lobe level, overcomes the implementation difficulties associated with the binomial array and is generally accepted as being a practical compromise between the uniform and binomial arrays. The Dolph-Chebyshev array's AF suggests it is best used when no prior knowledge of the interference sources is available, that is, the likelihood of interference is equal at all locations. However, if the general location of interference sources can be identified, not much can be done to compensate with the Dolph-Chebyshev array. One solution could be to use the more flexible three-parameter ultraspherical weights instead of the twoparameter Dolph-Chebyshev weights, in which case the excitation coefficients are given by a n = w (r + n − 1)T for n = 1, 2, . . . , r, where w(nT) are the coefficients provided by (6) resulting in This is equivalent to the amplitude function of the ultraspherical window given in (16) with the substitution u = ωT/2. Similarly, all the techniques developed in this paper are easily transferable to customizing the directivity properties of linear arrays. Fair comparisons between the two AFs can be made by designing ultraspherical and Dolph-Chebyshev arrays of the same length and the same null-to-null width, and then measuring the ripple ratios. To accomplish this, we make cos(ω n /2) in (29) equal for both the Dolph-Chebyshev and ultraspherical arrays, which yields the relation where x 0 is given by (19). Substituting and rearranging yields the closed-form expression for the ripple ratio that the Dolph-Chebyshev array of the same length and null-to-null width would produce compared to an ultraspherical array. This expression can be used to judge how much ripple ratio is sacrificed to attain a given side-lobe pattern. Figure 11 shows enlarged plots around the first null of three ultraspherical arrays designed with N = 31, ω n = 0.5 rad/s (θ n = 28.6479 degrees), and S = −10, 0, and 10 dB. The first side-lobe peak is 4.38 dB less for the case S = −10 dB and 3.84 dB more for the case S = 10 relative to the peak for the case S = 0 (i.e., the Dolph-Chebyshev array). On the other hand, the furthest side-lobe peak (not shown) is 5.62 dB more for S = −10 and 6.16 dB less for S = 10 dB relative to the peak for S = 0. The ripple ratio for the Dolph-Chebyshev array is given by (49) as −58.35 dB. An important observation is that the positioning of the second null weighs heavily on the amplitude of the first side lobe, which, in turn, is very important in determining the amplitude of the remaining side lobes. To this extent, an alteration in the amplitude of the first side lobe greatly influences the amplitude of the remaining side lobes in an inverse fashion, that is, increasing the first side-lobe amplitude decreases most of the remaining side-lobe amplitudes. Experimental results indicate that the side lobe envelope of the ultraspherical array tends to cross that of the Dolph-Chebyshev array within the first three side lobes adjacent to the main lobe. In this respect, negative S values are preferred to the Dolph-Chebyshev array for narrowband interference sources that are confined to this region. Alternatively, positive S values are preferred for interference sources that fall past this region. Using the methods proposed in this paper, antenna array designers are provided with an easy-to-use visual design approach for deciding what amount of trade-off between sidelobe pattern and ripple ratio is best for their particular situation.

Image processing
With the ever-expanding gamut of computer monitors, hand-held devices such as digital cameras and video recorders, and high-end medical imaging systems, consumers can often base purchasing decisions on a few key image quality measures. On such measure is an image's contrast ratio (CR) which, simply put, defines the difference in light intensity between the darkest black and brightest white shades within an image. A high CR allows one to discern detailed differences between colors producing a crisp and sharp image. On the other hand, a low CR results in a blurring or smearing effect producing an image with little clarity. A direct consequence of the CR measure is its effect on an imaging system's capability to detect low-contrast objects residing near high-contrast objects, which can be of the utmost importance in some medical imaging applications, for example, detecting cancerous tumors. Also, interpretation of an image's quality has been shown, through human trials, to be directly related to the CR measure [23].
A number of imaging systems such as synthetic aperture radar (SAR) [24], computerized tomography (CAT scans) [24], and charge-coupled device (CCD)-based X-rays [25] construct images by using two-dimensional windowed inverse DFTs on spatial frequency-domain data. For these systems CR tolerance is usually specified in terms of the worstcase spectral leakage of the window function used, which is directly related to the window's main-lobe to side-lobe energy ratio (MSR). Strictly speaking, the CR is defined as [26] CR = E s + E m E s = 1 + MSR, where the side-lobe and main-lobe energies are given by respectively, and MSR = E m /E s . By referring to the window's spectral representation as the inner product of the Fourier kernel v = 1 e − jωT e − j2ωT · · · e − j(N−1)ωT with the window coefficient vector w, that is, W(e jωT ) = w T v, the side-lobe energy E s can be expressed in the form where sinc ω r (m − n) for m = n, where Q is a real, symmetric, positive-definite Toeplitz matrix. Using Parseval's theorem, the total energy is found as where a simple rearrangement yields the main-lobe energy E m . Thus a window's CR can be calculated as Using the flexible three-parameter ultraspherical window for the windowing operation, the side-lobe patterns can be easily adjusted to alter the energy contained in the side lobes and, consequently, the value of the CR measure. Figure 12 shows plots of the normalized CR versus the side-lobe rolloff ratio S in dB for various main-lobe half-width quantities. The curves are convex with easily discernible global maximum values. As such, the ultraspherical window that possesses the maximum CR for a given window length N and main-lobe width ω r can be found through the appropriate selection of S. This can be accomplished by solving the onedimensional optimization problem where vector w is calculated using (6) and the techniques described in Sections 4.1 and 4.3, the Q matrix is calculated using (55), S L = 0 dB, and S H = 30 dB. For the example with N = 31 and ω r = 0.4 rad/s, the solution of (58) yields a maximum CR value of 41.01 dB occurring at S = 17.75 dB. The corresponding parameters for the ultraspherical window are µ = 1.0810 and x µ = 1.0166.

CONCLUSIONS
A method for the design of ultraspherical windows that achieves prescribed spectral characteristics has been proposed. The method comprises a collection of techniques that can be used to determine the independent parameters of the ultraspherical window such that a specified ripple ratio, main-lobe width, or null-to-null width along with a specified side-lobe roll-off ratio can be achieved. The Kaiser, Saramäki, and Dolph-Chebyshev two-parameter windows can achieve a specified ripple ratio and main-lobe width; however their side-lobe patterns cannot be controlled as in the proposed method. Experimental results have shown that the desired characteristics can be achieved with a high degree of precision. The ultraspherical window includes both the Dolph-Chebyshev and Saramäki windows as particular cases and a difference in the performance of the ultraspherical and Kaiser windows has been identified, which depends critically on the required specifications. The paper has also shown that the proposed design method can be used to achieve improved performance in beamforming and image processing systems.