1. 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 [9-13], but window method is one of the most frequently used methods [2-4]. 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
, the expression
(1)
is called sinc sum function.
Some properties of the sinc sum function are proved as follows:
(i) Global symmetry: 
and
;
(ii) Stair shape:
;
(iii) Local symmetry: 
and
;
(iv) Local extrema certainty: Let
. Then (a) 
has a local maximum value if 
is an odd number; (b) 
has a local minimum value if k is an even number;
(v) Oscillation regularity: If 
increases from 0 to
, then 
oscillates with decaying magnitude above or below 
alternatively along with the increase of
;
(vi) Extreme value stability: Let 
be large enough. Then the extreme value and some sub-extreme values of 
are almost unchangeable along with the change of L.
2. New HT Expression
2.1. Truncated Ideal HT
HT formula in frequency domain is
(2)
Corresponding HT formula in time domain is
(3)
We truncate it as follows:
(4)
where 
and
(5)
The frequency response of 
is
(6)
In the third step Euler formula is used.
Let
. Then it becomes
(7)
This is a new form of frequency response of the truncated HT. Let
(8)
Theorem 1 
of (8) has following properties:
(i) 
and
;
(ii)
;
(iii)
;
(iv)
;
(v) 
has a local maximum value if 
is an odd number and has a local minimum value if 
is an even number on the interval
;
(vi) 
oscillates in the vicinity of 1 on the interval
.
Proof. According to (1), (8) becomes
(9)
where 
and 
are two sinc sum functions.
Substituting 
for 
in (9) we get
(10)
By property (ii) of the sinc sum function we get
(11)
By property (i) of the sinc sum function we get
(12)
From (2.8) and (2.11) we get
(13)
Substituting 
in both sides yields
.
Part (i) of the proof is complete.
Substituting 
for 
in (9) we get
(14)
By property (ii) of the sinc sum function and (2.8),
(15)
Part (ii) of the proof is complete.
Substituting 
for 
in (9) we get
(16)
By property (ii) of the sinc sum function we get
(17)
By property (i) of the sinc sum function and (2.8),
(18)
Part (iii) of the proof is complete.
Substituting 
for 
in (9) we immediately get
(19)
Part (iv) of the proof is complete.
By the property (iv) of the sinc sum function, 
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 
relates to the parity of L. For L even, 
and 
have extrema of the same kind. In this case, 
, 
and 
have extrema of the same kind at any extreme point of
. For 
odd, 
and 
have extrema of different kinds at each extreme point. But by property (v) of the sinc sum function we always have 
on the interval
. Then we know that 
and
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
. Therefore, 
has a local maximum value if 
is an odd number and has a local minimum value if 
is an even number on the interval
, no matter how 
is odd or even.
Part (v) of the proof is complete.
By property (v) of the sinc sum function, 
and 
are both oscillates in the vicinity of 
on the interval
. Then from (9) we easily know that 
oscillates in the vicinity of 1 on the interval.
The proof is complete.
From Theorem 2.1 we can see that 
is symmetrical about
, odd symmetrical about both 
and
, and periodic with period
.
In addition, 
oscillates with decaying amplitude in the vicinity of 1 both from 
to 
and from 
to
. Because the proof is too complexity, we do not prove it in the paper.
2.2. New HT Formula
For 
we construct an expression as follows:
(20)
where 
are undetermined weights and
(21)
Theorem 2 
of (21) has following properties:
(i) 
and
;
(ii)
;
(iii)
;
(iv)
;
(v) 
has a local maximum value if 
is an odd number and has a local minimum value if 
is an even number on the interval
;
(vi) 
oscillates in the vicinity of 1 on the interval 
and in the vicinity of 0 on both the interval 
and the interval 
Proof. According to (1), (21) becomes
(22)
where 
and 
are four sinc sum functions.
For convenience, let
(23)
and
(24)
Then (22) becomes
(25)
Comparing (23) and (24) with (9), respectively, we get
(26)
and
(27)
Then (25) becomes
(28)
Substituting 
for 
in (28) we get
By Theorem 2.1 (i) we get
Comparing it with (2.27), we get
(29)
Substituting 
in both sides yields
.
Part (i) of the proof is complete.
Substituting 
for 
in (28) we get
By Theorem 2.1 (ii) we get
Comparing it with (2.27), we get
(30)
Part (ii) of the proof is complete.
Substituting 
for 
in (28) we get
By Theorem 2.1 (iii) we get
Comparing it with (2.27), we get
(31)
Part (iii) of the proof is complete.
Substituting 
for 
in (28) we get
By Theorem 2.1 (iv) we get
Comparing it with (2.27), we get
(32)
Part (iv) of the proof is complete.
By Theorem 2.1 we know from (26) that 
has a local maximum value if 
is an odd number and has a local minimum value if 
is an even number on the interval
, and we know from (27) that 
has a local maximum value if 
is an odd number and has a local minimum value if 
is an even number on the interval
. Therefore, 
and 
have local extrema of the same kind at each extreme point on the interval
. Then from (25) we know that any common extreme point of 
and 
is the extreme point of 
on the interval. Thus 
has a local maximum value if 
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 
oscillates in the vicinity of 1 on the interval 
and in the vicinity of −1 on the interval 
and that
; we know from (27) that 
oscillates in the vicinity of 1 on the interval 
and in the vicinity of −1 on the interval 
and that
.
Then from (2.24) we know that 
oscillates in the vicinity of 1 on the interval 
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
.
The proof is complete.
From Theorem 2.2 we can see that 
is symmetrical about
, odd symmetrical both about 
and about
, and periodic with period
.
Comparing (2.7) with (2.20), we know that 
is a special case of 
with
. We call each 
sub-amplitude response of HT.
Replacing 
by 
in (2.21), we get
(33)
By property (iv) of the sinc sum function, we easily know that 
and 
always have extrema of different kinds at each integer point 
except 
and 
on the interval
.
A plot of 
and 
for 
are illustrated in Figure 1. We can see that the above analysis is identical with the figure.
By
, (21) becomes
(34)
Using triangular identity, we have
Using Euler formula, we get
(35)
By
, (2.19) becomes
(36)
Substituting (2.34) in (2.35), we get
(37)
Corresponding impulse response is
(38)
or
(39)
From the above discussions about 
we can see that if the value of each 
is proper the superposition of all 
can make the maximum deviation of 
decreased because their changes in opposite directions can counteract mostly. Then we can obtain HT formulas with good performances. According to the property (vi) of the sinc sum function, 
is almost unchangeable along with the change of 
if it is large enough. Thus 
can be arbitrary under this condition. In general, 
can be considered large enough.
Equation (38) contains a window function as follows:
(40)
Now the weight 
is the widow constant.
For convenience, let
(41)
It is the amplitude of (2.19).
3. Examples of Obtaining HT Formulas
We choose 
(correspondingly, 
through 4). Then (41) becomes
(42)
We can select 5 points of
. In general, it is better to select the local extreme points of 
denote the 5 points, respectively. Then from (42) we get 5 equations as follows:
(43)
(44)
(45)
(46)
(47)
Solving the simultaneous equations, we can obtain 
through
.
Now we choose
, and 
as above five points and let
. Substituting them in (8) and (21), respectively, we get
, 
, 
,
, and
, in turn. We do not list theses values here.
, and 
are selected with 3 cases listed in 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.
Solving (3.2) we get 
through 
listed in table 3.1, too. Then from (39) we obtain HT formula as follows:
(48)
Corresponding window is
(49)
For
, we get corresponding maximum passband ripple of each HT. In Table 1, Ap is the maximum passband ripple and 
is from the following relationship:
(50)
where 
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).
4. 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
, 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
(a)
(b)
(c)
Figure 2. Comparison between new windows with other windows in terms of HT performances for L = 63, respectively. (a) Magnitude response obtained by using new window (No. 1), Kaiser window (β = 3:83) and Hanning window; (b) Magnitude response obtained by using new window (No. 2), Kaiser window (β = 4:98) and Hamming window; (c) Magnitude response obtained by using new window (No. 3), Kaiser window (β = 7:5) and Blackman window.
in the vicinity of 0 dB from some local extrema to the point with the normalized frequency being 0.5 in Figure 2 and 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.
5. 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.