An Application of Sinc Sum Function in Hilbert Transformer

An application of the sinc sum function in Hilbert transformer (HT) is studied. The expression of the frequency response of HT is expressed with sinc sum functions. Some properties of sub-amplitude response of HT are proved by using the properties of the sinc sum function. A general HT formula is obtained theoretically and it contains a general window function. As an example three new window functions are obtained. Different from the existing window functions obtained from lowpass filters, these window functions are obtained directly from HT. Comparisons show that new windows are better than the Hanning, Hamming, Blackman and Kaiser windows in terms of HT performances.


Introduction
Hilbert transformer (HT) is widely used in engineering, such as damage diagnosis of rotors [1], electroencephalography analysis [2], detection in speech [3], extraction of modal characteristics [4], upmixing stereo signals [5], and vibration analysis [6].Hilbert-Huang transform is a technique developed in recent years.One of its main parts is HT [4,7].Reference [8] gives more application examples of HT.About the design method of HT, we can find some methods [9][10][11][12][13], but window method is one of the most frequently used methods [2][3][4].The reason is that window method is the simplest one of them and several windows have good performances.The well-known fixed windows are the Hanning, Hamming, Blackman windows and the most frequently used adjustable window is the Kaiser window.These windows are obtained according to the performances of lowpass filters.Consequently, windows with good performances can not easily be obtained because for finding satisfied windows three performances of passband, stopband and transition width of lowpass filters must be given attention to simultaneously.In the study of FIR filter design, a new function is defined as sinc sum function [14] by the author.The function has been used in the design of FIR filters, such as lowpass filters [14] and differentiators [15,16].Further study shows that it can be used in the design of FIR HT.
The definition of the sinc sum function is as follows [14].
For the positive finite integer and the real independent variable L x , the expression   is called sinc sum function.Some properties of the sinc sum function are proved as follows: (i) Global symmetry:    S x S x    and   0 0 S  ; (ii) Stair shape:     oscillates with decaying magnitude above or below alternatively along with the increase of x ; (vi) Extreme value stability: Let be large enough.Then the extreme value and some sub-extreme values of Y. L. WANG 2

Truncated Ideal HT
HT formula in frequency domain is Corresponding HT formula in time domain is We truncate it as follows: where and 2 The frequency response of is In the third step Euler formula is used.
This is a new form of frequency response of the truncated HT.Let and ;     0 Z v has a local maximum value if v is an odd number and has a local minimum value if is an even number on the interval ; oscillates in the vicinity of 1 on the interval L   .Proof.According to (1), (8) becomes where  are two sinc sum functions.
Substituting v  for in (9) we get v By property (ii) of the sinc sum function we get By property (i) of the sinc sum function we get From (2.8) and (2.11) we get By property (ii) of the sinc sum function and (2.8), Part (ii) of the proof is complete.
By property (ii) of the sinc sum function we get By property (i) of the sinc sum function and (2.8), Part (iii) of the proof is complete.Substituting for in (9) we immediately get Part (iv) of the proof is complete.
By the property (iv) of the sinc sum function,   π S v has a local maximum value if is an odd number and has a local minimum value if is an even number.The local extreme points of and have have extrema of the same kind at any extreme point of .For odd,

 π
and have extrema of different kinds at each extreme point.But by property (v) of the sinc sum function we always have  π have extrema of the same kind at any extreme point of on the interval.According to (iv) of this theorem the conclusion is still true on the interval oscillates in the vicinity of 1 on the interval.The proof is complete.
From Theorem 2.1 we can see that , odd symmetrical about both and v , and periodic with period .
oscillates with decaying ampli-tude in the vicinity of 1 both from to 0.5 and from to 0. .Because the proof is too complexity, we do not prove it in the paper.

New HT Formula
For 2 1 M L    e we construct an expression as follows: are undetermined weights and and in the vicinity of 0 on both the interval K   and the interval . where and are four sinc sum functions. and Comparing ( 23) and ( 24) with ( 9), respectively, we get and Then (25) becomes Comparing it with (2.27), we get Comparing it with (2.27), we get Part (ii) of the proof is complete.
Comparing it with (2.27), we get Comparing it with (2.27), we get Part (iv) of the proof is complete.By Theorem 2.1 we know from (26) that   Z v have local extrema of the same kind at each extreme point on the interval K v L K    .Then from (25) we know that any common extreme point of is an odd number and has a local minimum value if is an even number on the interval.
Part (v) of the proof is complete.By Theorem 2.1 (i) and (v), we know from (26) that and in the vicinity of −1 on the interval Then from (2.24) we know that and in the vicinity of 0 on the interval .By Theorem 2 (iv) we know that oscillates in the vicinity of 0 on the interval .L  The proof is complete.From Theorem 2.2 we can see that , odd symmetrical both about and about , and periodic with period .
Replacing by in (2.21), we get By property (iv) of the sinc sum function, we easily know that are illustrated in Figure 1.We can see that the above analysis is identical with the figure.
Using triangular identity, we have Using Euler formula, we get Substituting (2.34) in (2.35), we get Corresponding impulse response is Now the weight K W is the widow constant.For convenience, let It is the amplitude of (2.19).

Examples of Obtaining HT Formulas
We choose 4 M    (correspondingly, through 4).Then (41) becomes We can select 5 points of .In general, it is better to select the local extreme points of v v  denote the 5 points, respectively.Then from (42) we get 5 equations as follows: Solving the simultaneous equations, we can obtain through W . , and 5 as above five points and let .Substituting them in ( 8) and ( 21), respectively, we get For 127 L  , we get corresponding maximum passband ripple of each HT.In Table 1, Ap is the maximum passband ripple and  is from the following relationship: where l  is the lower cutoff frequency of magnitude responses.
It is obvious that for finding good window constants we only need to take into account two performances of maximum passband ripple and transition width in the magnitude response of HT.
If maximum passband ripple and lower cutoff frequency as two specifications are known, we can easily design HTs.The first step is to select a window from table 3.1.Then the corresponding  is obtained from the table.The next step is to compute according to (50) and further .The last step is to compute HT coefficients according to (48).

Compared with Other Windows
Now we compare passband ripples of frequency responses of HTs obtained by using three new windows with those obtained by using the Hanning, Hamming, Blackman and Kaiser windows [17] for 63 L  , respectively.About the Kaiser window, we select its parameter  in this way that the maximum passband ripple obtained by using each new window is the same as that obtained by using the Kaiser window.For seeing clearly and reducing paper length we only plot part of each curve in Figure 2. It is necessary to say that for each curve passband ripples oscillate with decaying magnitude

Conclusion
A general HT formula is deduced by using the sinc sum function and it contains a general window function.Three new windows are obtained directly from the magnitude responses of HT.These new windows belong to fixed window.They are two or three cosine terms more than the Hanning, Hamming and Blackman windows but much simpler than the Kaiser window.Comparisons show that these new windows are better than the Hanning, Hamming, Blackman and Kaiser windows in terms of HT performances. (a)
0.461382 0.493228 0.460071e−1 −0.395644e−3 −0.221528e−3 2.97e−3 1.48in the vicinity of 0 dB from some local extrema to the point with the normalized frequency being 0.5 in Figure2and the whole curve is symmetrical about the frequency.From the figure we can see that these new windows are better than the Hanning, Hamming, Blackman and Kaiser windows in terms of HT performances, respectively.

Table 1 .
They are selected based on the consideration that the resulting windows can easily be compared with the Hanning, Hamming and Blackman windows in terms of HT performances, respectively.