Chapter 6. Design of Variable Transition Bandwidth FIR Filters Using
6.3 Design of Variable Transition Bandwidth FIR Filters
6.3.1 Generation of Variable Kaiser Window
The frequency response of the variable window for FIR filter design is characterized as
( )
( )where the parameter p is setting to be the variable passband edge ωp in the range
1, 2
p p
⎡ω ω ⎤
⎣ ⎦ , and the coefficients w n are assumed to be symmetric at p( ) n=0 and expressed as the polynomials of the parameter p as
( )
( )
Now, based on Kaiser window function (6.5), we first make Kaiser window be variable in FIR filter design. Defining variable parameter ωp=p in (6.11) and after some manipulations, the minimum attenuation A can be formulated in the p function as s
( )
28.72(
c)
7.95s
A p ω p N π
= − + (6.15)
and therefore the parameter α is controlled over p in (6.7), then Kaiser window coefficients (6.5) can be expressed in the function of p as
( )
Thus, the problem reduces to find the optimal window coefficients w n m such that
(
,)
w n p( ) could approximate w n as well as possible. Here, the objective errors function between k( )p( )
Then, the optimal window coefficient vector An can be obtain by
1 -1 .
An=−2 Q Pn n (6.21)
6.3.2 Design of Variable Transition Bandwidth FIR Filters with Simplest Finite Impulse Response
After variable Kaiser window is developed in previous section, we would use (6.13) to multiply by the simplest finite lowpass impulse response (6.4) , then the windowed variable transition bandwidth finite impulse response coefficients can be achieved as
( ) ( ) ( )
and its corresponding zero-phase frequency response is
( ) ( )
( )
Fig. 6-3 The magnitude response of the designed lowpass variable transition bandwidth filter with the simplest finite impulse response when N=13, 4M= , 0.45ωc= π, ωp1=0.35π and ωp2=0.44π .
6.3.3 Design of Variable Transition Bandwidth FIR Filters with WLS Impulse Response
To further improve the performance of the designed filter, the WLS technique [17], [24] is used to find the optimal impulse response coefficients. The desired variable transition bandwidth frequency response is given by
(
,)
1, 0 expressed as the polynomial of the parameter p as( )
( )
Then the transfer function can be rewritten as
( ) ( )
and its frequency response
( ) ( )
where
is designed to approach the desired response (6.24). Due to the symmetric assumption of
p( )
w n and h n , the coefficients p( ) h n m have the following symmetry
(
,)
(
,) (
,)
, for 1 .h n m =h −n m ≤ ≤n N (6.30)
Then, the WLS method is used to find the optimal coefficients h n m . Thus, the objective
(
,)
error function between D
(
ω,p)
and H e(
jω,p)
can be expressed asTo find the optimal B , (6.31) is differentiated with respective to B and set the result to zero. Finally, B can be achieved by the formula
1 1 . 2 B B
B=− Q P− (6.33)
Without loss of generality, the elements of P and B Q are detail derived in the closed- B forms as
Now, multiplying h n m by
(
,)
w n m , the transfer function of the designed variable(
,)
windowed FIR filter can be expressed as
( ) ( ) ( )
According to chapter 1, (6.35) can also be implemented in the Farrow structure as shown in Fig. 1-1 .To demonstrate the effectiveness of the proposed method, the magnitude response of (6.35) is shown in Fig. 6-4 with N=13, 4M= , 0.45ωc= π, ωp1=0.35π and ωp2=0.44π .
Fig. 6-4 The magnitude response of the designed lowpass variable transition bandwidth filter with the WLS impulse response when
13
6.4 Conclusions
This chapter presents the technique for generating the variable window. By using this window, the lowpass variable transition bandwidth FIR filters can be designed easily. Then, to improve the performance of the designed frequency response, the WLS method is used to find the optimal impulse response coefficients. Moreover, in the procedures of deriving error functions, all elements of the relative vectors and matrices can be calculated in the closed- forms by truncating the series expansion of the Bessel function. In the future work, this method would be researched in 2-D window design, such as McClellan transformation.
Chapter 7
Conclusions and Future Works
In this thesis, the design of variable digital filter has been investigated in various fields and applications. In terms of variable fractional-delay, first, we have proposed a new coefficient relationship method to design VFD FIR filters, such can achieve high performance of frequency response in the whole band and less number of the designed coefficients then latest one. Second, contrast to FIR filters, the design of VFD allpass filters is also introduced.
Although the design of allpass filter is difficult to analyze, the latest noniterative WLS design is presented to avoid iterative integrals and simplify the procedure of finding the optimal solution. Comparing FIR filters with allpass filters, FIR cases have the simpler structure to implement designed filters than allpass cases, but in the viewpoint of design accuracy, allpass cases are more accurate. So, how to choose the suitable filter between these two is based on the design requirement.
On the other hand, the design of variable magnitude response digital filters is researched as well. First, two methods (binomial series expansions and Taylor series expansions) have been proposed to design variable fractional-order differintegrators, which can derive the objective WLS error function into closed-form and the other advantage of these two methods is that more accurate frequency response can be achieved at high frequency band than the conventional methods. Second, in terms of variable 2-D filters, the variable 2-D subfilters and variable 1-D prototype filter were designed with the same variable parameter to construct variable 2-D FIR filters by using McClellan transformation. In the transformation, the variable structure was proposed to implement the designed filters. Some design examples were detail derived to illustrate the effectiveness of the proposed method, such as variable 2-D fan-type filters, variable circularly symmetric filters and variable elliptically symmetric filters with arbitrary orientation.
Finally, except the variable delay and magnitude filters, the variable window was also generated which can be applied in designing variable transition bandwidth FIR filters. This window is based on the adjustable Kaiser window, and can be implemented in the Farrow structure. By applying the WLS approach, the optimal impulse response can be obtained to improve the performance of the frequency response.
In the future, the research of variable digital filter will keep up and it will be classified into three aspects. First, some new structures and modified Farrow structures could be investigated to achieve high performance of the designed filters. Second, the new approach techniques and error criterions are worth to study due to the different optimal coefficients would be obtained in the distinct objective error function. Third, the variable impulse response of the designed filters can be expressed in other basis forms except the polynomials of parameter. The new variable basis may achieve more approximate response of the designed filter than polynomial one. All these aspect will be researched and developed into 2-D filters.
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