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Analog and digital filters with α-splines
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Analog and digital filters with
α -splines ✩
77 78
Miguel Ángel Raposo-Sánchez a,∗ , José Sáez-Landete b , Fernando Cruz-Roldán b a b
Department of Física y Matemáticas, Escuela Politécnica Superior de la Universidad de Alcalá, 28805 Alcalá de Henares, Spain Department of Teoría de la señal y Comunicaciones, Escuela Politécnica Superior de la Universidad de Alcalá, 28805 Alcalá de Henares, Spain
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Article history: Available online xxxx Keywords: Filtering theory Spline functions and analog filters Finite impulse response (FIR) digital filters Least integral squared error Principally flat
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Analytical expressions of analog and digital α -spline filters with continuous transition bands are derived. Previous works based on β -spline functions consider the order of the spline function, which must be a natural number, as a parameter to sharpen the transition band. However, the inclusion of an α -spline model provides greater flexibility since its order can be a real number, it depends on three parameters, and it can be used to model the transition band in the frequency domain. This paper presents the formulation of analog α -spline filters and the corresponding discretization to obtain the frequency and the impulse responses of their digital counterparts. There is also shown the design of α -spline digital filters minimizing two different least integral squared errors. As these methods do not provide direct control over the stopband attenuation, a third method which leads to principally flat digital filters is introduced. Several numerical examples demonstrate the efficiency and flexibility of the proposed techniques. © 2017 Published by Elsevier Inc.
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1. Introduction
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Analog and digital filters are widely required in different areas of signal processing [1–9]. A common method of designing digital filters is based on transformation methods from their analog counterpart. In contrast, windowing is a classical method for directly obtaining the digital filter, and one of its essential concerns is the design of smoothing window functions with good characteristics [1,10–12]. Several authors have proposed the use of spline functions for filter design [13–17]. In order to eliminate the Gibbs phenomenon, a transition band is introduced to continuously connect the passband to the stopband, removing the discontinuity present in a conventional brick-wall ideal filter. This transition band is the result of the convolution of rectangular pulses with the ideal frequency response. The design can be extended to include a ρ th-order spline, leading to filters with several desirable properties. First, the use of spline functions in the filter design allows analytical leastsquared-error approximation finite impulse response (FIR) filters, along with a reduction of the approximation ripple. Second, the use of splines gives explicit control over the transition band, and it provides almost as good an approximation as more complicated
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✩
This work was partially supported by the Spanish Ministry of Economy and Competitiveness through project TEC2015-64835-C3-1-R. Corresponding author. E-mail address: [email protected] (M.Á. Raposo-Sánchez).
*
http://dx.doi.org/10.1016/j.dsp.2017.03.003 1051-2004/© 2017 Published by Elsevier Inc.
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• The formulation of α -spline analog filters is presented. The
59 60
numerical approaches. The resulting FIR filters are easy to calculate and to program because several formulas that relate the ρ th-order transition-band spline function with the desired stopband attenuation, and the filter order with the transition bandwidth, have been proposed [15–17]. In [15,16], the filter design is based on β -spline models, and recently the use of α -spline functions to shape the transition band of FIR filters has been proposed in [17]. α -Spline functions are defined in [18]. They are the result of convolving two types of rectangular pulses of different widths in the frequency domain. α -Spline functions result in transition functions with non-integer exponents. Based on a modified version in the frequency domain of the α -spline functions, a consistent mathematical formulation to design digital filters was presented in [17]. That paper introduced a new formulation to employ non-integer values for the spline order. Furthermore, an optimization method to obtain the parameters of the α -spline digital window was also provided, resulting in design improvements. This present paper focuses rather on α -spline analog filters, and two practical and simple methods to design digital filters are also presented. In more detail, the main contributions of this paper can be summarized as follows:
mathematical expressions of both the analog frequency and the impulse responses are derived. Then, their corresponding discretization shows that the discrete-time filter accurately matches the model proposed in [17].
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( p −1) times
1
[ p ,q]
2
1
3
−1
f (t ) = F
6 7 8
p + αq
{φ()} =
10
a (− jt )
13 15
Fig. 1. Rectangular convolving functions to obtain
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a least integral squared error approximation are introduced, following two different criteria: a) the minimization of the integral squared error over the entire frequency spectrum and b) the minimization of the integral squared error in both the passband and the stopband. The integral of the square of the error is an optimization criterion often used in filter design because it is a measure of the energy of the approximation error. • The design of principally flat α -spline filters is discussed. This alternative design method provides direct control over the stopband attenuation achieved by the digital filter. Computer simulations have also been run and the obtained results demonstrate good performance in terms of the transition bandwidth and stopband attenuation. The rest of this paper is organized as follows. In Section 2, analytical expressions for both the frequency and the impulse responses of the α -spline analog filter are derived. That section also describes the discretization of the analog filter to obtain its digital counterpart. The optimization procedures based on the least integral squared error are addressed in Section 3, whereas Section 4 introduces the design of principally flat α -spline filters. Both sections illustrate the performance of the filters, comparing several designs to other techniques; finally, some concluding remarks are given in Section 5.
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α a (− jt )
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e
(4)
.
2.1.
a t − 2(jαp + α q)
−e
j α a t 2( p +α q)
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76
,
(5)
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(6)
φ () =
2 π ( p +α q )
To calculate the Fourier transform of F 1 following functions of order :
86
(t ), let us consider the
a
0,
and
φα () =
otherwise,
(7)
α a
0,
, || <
α a , 2 ( p +α q )
otherwise,
G +, () = + =
94 95
, ≥ 0, 0,
(8)
< 0,
g (t ) =
where denotes the analog frequency variable, α (0 ≤ α ≤ 1) provides control over the width of φα () (see Fig. 1), and a is a parameter related to the bandwidth of the convolving function, which has a direct influence on the transition bandwidth of the designed filter. [ p ,q] The convolving functions 1 () are generated by performing, in the frequency domain, p + q − 1 convolution operations between p functions φ (), and q functions φα ():
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where ∈ N. Then, using the Euler gamma function in its integral form, we get
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! . 2π (− jt ) +1
(9)
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By introducing (9) in (6), and using the algebraic expansion of the powers of a binomial,
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1
=
(− jt ) p +q
2π g p +q−1 (t )
( p + q − 1)!
(10)
,
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(t ) =
2π ( p + α q) p +q g p +q−1 (t )
=κ ·
(1)
(2)
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where
αq ap+q ( p + q − 1)! q p
e
· e
− 2( pj+aαtq)
a t − 2(jαp + α q)
j a t
p
− e 2( p+αq) −q j α a t 2 ( p + α q ) −e
C kl e
g p +q−1 (t ),
(11)
k =0 l =0
κ=
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2π ( p + α q) p +q
αq ap+q ( p + q − 1)!
C kl = (−1)(k+l)
δα ,kl =
110
116
j δα ,kl a t
where 2 π ( p +α q )
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F
[ p ,q]
, || <
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g (t ) −→ G +, (),
F1
a , 2 ( p +α q )
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[ p ,q]
Let us consider the following two types of rectangular pulses, both having area 2π :
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α -Spline kernels
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[ p ,q] [ p ,q] F 1 (t ) = F−1 1 () = [ f (t )] p [ f α (t )]q p j a t − 2( pj+aαtq) 2( p +α q) e − e ( p + αq) p +q = −q . j α a t αq ap+q (− jt ) p+q − 2(jαp+αa qt ) e − e 2( p+αq)
2. Analog filter with α -splines
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we obtain
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α -splines.
• The design and the performance of α -spline digital filters by
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−e
j a t 2( p +α q)
and finally
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21
p + αq
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e
− 2( pj+aαtq)
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f α (t ) = F−1 {φα ()} =
11
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Similarly
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(3)
In the time domain, φ() can be written as
4
17
(q−1) times
() = φ () ∗ . . . ∗ φ () ∗ φα () ∗ . . . ∗ φα () .
k + αl p + αq
p k
−
122
,
123
1 2
q l
124
,
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.
Calculating the Fourier transform of (11) and employing the frequency shift property, the α -spline functions in the frequency domain can be written as
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1. For || > (c + a /2):
1 2
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H T () = 0.
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2. For −c −
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a
< < −c +
2
H T () = B
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q
p
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C kl
k =0 l =0
9
a 2
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:
+ c − δα ,kl a
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p +q
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,
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+
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where
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B=
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p +q
( p + α q) . ( p + q)!α q
3. For || < c −
14
a 2
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Fig. 2.
H T () = B
α -Spline convolving function.
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:
q p
C kl 1 −
k =0 l =0
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k + αl
p +q
p + αq +
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= 1.
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4. For c − 2a < < c + 2a :
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H T () = 1 − B
23
q
p
C kl
k =0 l =0
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− c − δα ,kl a
p +q
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.
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+
90
We can write the above equations in a compact form as follows:
26
H T () = 1 − B
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q
p
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C kl
k =0 l =0
30
|| − c − δα ,kl a
p +q +
38
100
Fig. 3. Frequency responses of an analog ideal low-pass filter and an tion.
[ p ,q]
1
p
() = κ ·
k =0 l =0
40
p
=κ ·
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=κ
45
h T (t ) = h i (t ) · F 1 C kl G +,( p +q−1) ( − δα ,kl a )
p +q−1 · a
·
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C kl
k =0 l =0
[ p ,q]
Fig. 2 shows an example of 1
<
a 2
− δα ,kl a
h i (t ) =
p +q−1 .
(12)
+
(), which is nonzero for
−a 2
<
53
.
2.2. The frequency response of the analog filter In order to obtain the frequency response of the analog filter, the following convolution is carried out:
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[ p ,q]
H T () = H i () ∗ 1
() =
57
1 2π
58
∞
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[ p ,q]
H i (ξ ) 1
2π
e
j t
−c
( − ξ )dξ ,
H i () =
(17)
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110
[ p ,q]
whereas F 1 (t ) is the inverse Fourier transform of 1 given by (6), or equivalently
⎛
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[ p ,q]
⎞p ⎛
(),
a t
⎞p ⎛
α a t
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(18)
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As a result, the continuous-time impulse response of the analog filter can be finally written as follows:
111 113
⎞q
at sin 2(αp +α q ) ⎠ ·⎝ ⎠ . α a t 2 ( p +α q )
⎞q
c sin(c t ) ⎝ sin 2( p +αq) ⎠ ⎝ sin 2( p +αq) ⎠ h T (t ) = · · . a t α a t π c t 2 ( p +α q ) 2 ( p +α q )
(13) where
102
107
c sin (c t ) d = , π c t
⎛
−∞
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1
c
sin 2( p+aαt q) [ p ,q] F 1 (t ) = ⎝ a t 2 ( p +α q )
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(16)
106
p +q−1
q p
(t ).
101
where h i (t ) is the inverse Fourier transform of the ideal filter, given by
C kl − δα ,kl a +
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The inverse Fourier transform of (13) can be expressed as [ p ,q]
q
k =0 l =0
43
α -spline func-
q
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47
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37
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2.3. Continuous-time impulse response
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(15)
,
where −∞ < < ∞, 0 ≤ α < 1, p ≥ 1 and q ≥ 1 with p , q ∈ N.
31
91
(19)
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1, 0,
2.4. Digitalization of the analog filter
|| < c ,
(14)
otherwise, [ p ,q]
c is the cutoff frequency and 1
() is the α -spline function
given by (12). In order to carry out the above convolution, let us consider different subintervals of (see Fig. 3):
127 128
Once the impulse response of the analog filter has been computed, the design of the discrete-time digital filter can be based on discretizing the corresponding continuous-time response. To this end, let us consider t = nT , or in the frequency domain ω = T
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Fig. 4. Integral squared approximation error versus the half-length of the
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α -spline filter.
90
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and d = a T , assuming without loss of generality that the sampling period is T = 1. In this case, the frequency response of the resulting digital filter can be expressed as
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H T (ω) = 1 − B
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⎛
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C kl
|ω| − ωc − δα ,kl d
h T [n] =
40
d n ωc sin(ωc n) ⎝ sin 2( p+αq) · d n π ωc n 2 ( p +α q )
⎞p ⎛
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+
⎞
q α n sin 2( p +dαq) ⎠ ·⎝ ⎠ , α d n 2 ( p +α q )
−∞ < n < ∞, and represents an infinite impulse response (IIR) filter. 3. FIR filter design by least integral squared error approximation The integral of the square of the error is an optimization criterion often used in filter design because it is a measure of the energy of the approximation error. In this way, the main goal of the design is to obtain a computationally realizable filter with an impulse response of finite length, N = 2M + 1, whose frequency response H (ω) approximates the theoretical frequency response H T (ω) minimizing the integral squared error, defined as
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(20)
(21)
41 42
,
L I S E =
1 2π
ωr 2
| H (ω) − H T (ω)| dω,
(22)
−ωr
where H (ω) is the frequency response of the finite impulse response h[n]. Depending on how the truncation is done, the resulting filter exhibits some properties which approximate, to a greater or lesser extent, the properties of the specified theoretical filter. Below we will show the design of α -spline filters following two different criteria: a) the minimization of the integral squared error over the entire frequency spectrum and b) the minimization of the integral squared error in both the passband and the stopband.
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This problem can be formulated as the minimization of
p +q
where ω denotes the digital frequency variable, −π ≤ ω < π , 0 ≤ α < 1, p ≥ 1 and q ≥ 1 with p , q ∈ N. Furthermore, the analog impulse response reduces to the form
36 38
q
k =0 l =0
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p
3.1. Minimization of the total integral squared error (MITISE)
T = =
π
1 2π
∞
| H (ω) − H T (ω)|2 dω =
95 96
|h[n] − h T [n]|2
97
n=−∞
−π
M
94
|h[n] − h T [n]|2 + 2
n=− M
∞
98 99
|h T (n)|2 .
(23)
101
n = M +1
This function is minimized if h[n] = h T [n], − M ≤ n ≤ M, i.e., the best approximation to the frequency response, in the sense of minimizing (23), is obtained using the coefficients of the theoretical impulse response. Burrus et al. addressed this design for modeling transition bands using β -spline functions [15]:
hb [n] =
sin(ωc n)
πn
sin(d n/(2ρb ))
d n/(2ρb )
ρb
d N 2π
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.
(24)
109 110
Generally, the effect of the order of the β -spline functions, ρb , on the frequency response is not obvious. In [15], as a result of numerical experimentation, it is stated that the optimum value of ρb that minimizes (23), for a filter of length N and a given width of the transition band, is
ρb = 0.62
100
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= 0.62 f N ,
(25)
where f is the width of the transition band, expressed in Hertz. Our proposal minimizes (23) using as impulse response the α -spline filter given by (21). In order to evaluate the effects of truncating the filter h T [n], the logarithm of the approximation error as a function of the filter length has been measured for different values of ρ , as in [15]. The results, shown in Fig. 4, have been obtained assuming that the cutoff frequency and the transition bandwidth are fixed. As can be seen, the error monotonically decreases as the filter length increases, for a given value of p + α q. Moreover, there appear different regions (that correspond to the regions near the zeros of the α -spline functions) with flat shape whose width increases for increasing p + α q. The infinitely long h T [n] should be truncated just as the flat region begins, because increasing the length of the truncated impulse response in those
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Fig. 5. Magnitude frequency responses of discrete low-pass filters with
α - and β -spline functions minimizing the total integral squared error.
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Table 1 Example design 1. Simulation results for the optimized total integral squared error (ωc = 0.5π , d = 0.2π , M = 36).
94 95
30
Filter design technique
εaisae
ε pbe
εsbe
εdev
A s (dB)
96
31
β -spline (ρ = 4.526) (Ref. [15]) α -spline Eq. (21) with ( p , q, α ) = (4, 1, 0.2)
6.506 · 10−3 6.643 · 10−3
4.215 · 10−11 6.970 · 10−12
4.215 · 10−11 6.970 · 10−12
1.890 · 10−5 1.626 · 10−5
94.5 95.8
97
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99
regions does not decrease the error significantly. Computer simulations and numerical experimentation have led us to an expression that relates the transition bandwidth, f = d /(2π ) in Hertz, the half-length of the filter, M = ( N − 1)/2, and the optimum order of the α -spline function for q = 1:
f M = 0.857( p + αq)|q=1 = 0.857( p + α ).
(26)
This order leads to an acceptable degree of accuracy. Hence, we can describe the filter design algorithm as follows.
M
εaisae =
MITISE design procedure
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Step 1. Select the length of the filter (N = 2M + 1), the transition bandwidth (d ) and the cutoff frequency ωc . Step 2. Calculate the order of the α -spline function from (26). Step 3. Compute the 2M + 1 coefficients of the filter using (21) (− M ≤ n ≤ M).
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To illustrate its performance, the design of lowpass linear-phase digital filters of length 73 (M = 36), with transition bandwidth d = 0.2π and cutoff frequency ωc = 0.5π , is considered. From (25), previously reported in [15] for β -spline digital filters, we obtain ρb = 4.526, whereas from (26), the value is p + α q = 4.2, leading to the following set of parameters ( p = 4, α = 0.2, q = 1). Fig. 5 shows the magnitude frequency responses of the designed linear-phase filters. Note that the transition bandwidths are virtually identical, whereas the α -spline filter exhibits better performance in both the passband and the stopband. Table 1 compares both filters for the average integral squared approximation error, defined as
∞
|hi [n] − h [n]|2 + 2
n=− M
100
|hi [n]|2 ,
n = M +1
where h i [n] is the ideal impulse response; the passband error, given by
ω p 1 | H (ω) − H i (ω)|2 dω, ε pbe =
π
103 104 105
108 109
where H i (ω) is the ideal frequency response; the stopband error
εsbe =
102
107
0
1
101
106
π
45 46
98
π
110 111 112
2
| H (ω)| dω;
113
ωs
114
the maximum passband deviation, obtained as
115
εdev = max | H (ω) − H i (ω)| ,
116
|ω|<ω p
117
and the minimum stopband attenuation A s . The best obtained results are indicated by the boldface numbers, and as can be observed, the filter designed with (26) outperforms that with (25), with the exception of the average integral squared approximation error.
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3.2. Minimization of integral squared error on the pass and stopband (MISEPS)
124
In this case, the integral squared error on the pass and stopband is minimized:
127
ω p π 1 1 2 | H (ω) − H T (ω)| dω + | H (ω) − H T (ω)|2 dω. ps =
π
π
0
ωs
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Fig. 6. Minimization of the integral squared error approximation.
26
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27
93
(27)
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In [15], this problem is addressed empirically, concluding that the optimum value, ρb that minimizes (27) for length N and a given width of the transition band f Hertz, can be calculated by
ρb = K N 1 , where
⎧ ⎨ 0.453 + 0.386/ N 1 , N 1 ≤ 1.25, K = 0.774 − 0.0251N 1 , 1.25 < N 1 < 5, ⎩ 0.648, 5 < N 1 ,
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=2
∞
2
|h T [n]| .
(30)
n = M +1
We have also observed that the spline order that minimizes (30) is approximately the same for any combination of M = ( N − 1)/2 and d such that M · d is fixed. The minimum value of decreases as M increases. Fig. 6 shows the logarithm of as a function of the α -spline order, parametrized with d · M = 6.384π and with different combinations of M and d . The α -spline order to minimize the function is p + α q|q=1 = 3.7. Different tests have been carried out, varying the parameter x = M · d , with the aim of finding a preliminary relation which is a fairly accurate approximation, between the x-parameter and the order, ρ = p + α q, considering q = 1, of the α -spline function which minimizes the approximation error in both bands. In this sense, several numerical experiments have been carried out for different intervals of the variable x = M · d (see Table 2), arriving at a set of pairs (x = M · d , ρ = p + α q). From the above, a second-degree polynomial model
65 66
ρ = a · x2 + b · x + c
94
ρ by means of (31).
95
x = M · d
a
b
c
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1.75π ≤ x < 4.00π 4.00π ≤ x < 5.75π 5.75π ≤ x < 7.50π 7.50π ≤ x < 9.25π 9.25π ≤ x < 10.75π 10.75π ≤ x < 12.50π 12.50π ≤ x < 14.25π
−0.0294 −0.0178 −0.0097 −0.0103 −0.0055 −0.0056 −0.0061
0.5394 0.5745 0.4256 0.5555 0.3542 0.4208 0.5195
−1.1575 −1.9071 −0.9189 −2.7449 −0.0147 −1.1893 −3.3449
97
(31)
98 99 100 101 102 103
(29)
1 = 1.12 · f , and d = 2π 1 [15]. We have also performed empirical tests to find the optimal value for the α -spline order, in terms of their length and transition bandwidth. We have observed that, in all cases studied, the optimum value that minimizes (27) matches the value that minimizes the function
45 46
(28)
Table 2 Coefficients to obtain
has been developed to determine the order of the α -spline. The coefficients (a, b, c ) obtained by polynomial regression are shown in Table 2 for each considered interval. In this case, the algorithm to design the filter is given below.
104 105 106 107 108 109
MISEPS design procedure Step 1. Select the length of the filter (N = 2M + 1), the transition bandwidth (d ), and the cutoff frequency (ωc ). Step 2. Calculate the order of the α -spline function from (31) using the coefficients (a, b, c ) given in Table 2. Step 3. Compute the 2M + 1 coefficients of the filter using (21) (− M ≤ n ≤ M).
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In order to compare the results obtained using α - and β -spline functions, low-pass 115-length digital filters (M = 57), with transition bandwidth d = 0.112π and cutoff frequency ωc = 0.45π , have been designed. For the above values, (28) and (29) provide the optimal order for the β -spline of ρb = 4.17312, whereas the optimal order of the α -spline functions, obtained from (31), is p + α q|q=1 = 3.7, leading to the set of values ( p = 3, α = 0.7, q = 1). Fig. 7 shows the magnitude frequency responses of the designed filters. It is observed again that the transition bandwidths are virtually identical, whereas the proposed α -spline filter exhibits improvements in both the passband and the stopband. Table 3 shows the aforementioned quality parameters obtained from the designed linear-phase filters. Also in this case, the pro-
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Fig. 7. Magnitude frequency responses of discrete low-pass filters with
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α - and β -spline minimizing both the passband and the stopband squared integral error.
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Table 3 Example design 2. Simulation results for the optimized integral squared error in the pass and stopband (ωc = 0.45π , d = 0.112π π , M = 57).
28 29 30 31 32 33
36 37
ε pbe
εsbe
εdev
A s (dB)
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3.803 · 10−3 3.936 · 10−3
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1.423 · 10−10 1.152 · 10−11
7.706 · 10−5 4.402 · 10−5
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posed approach outperforms the β -spline-based technique in almost all quality measures (except the average integral squared approximation error). 4. Principally Flat (PF) filter design
40 41 42 43 44 45 46 47 48 49 50 51 52 53
In this section, we introduce an alternative design method which provides control over the stopband attenuation achieved by the filter. According to Roark et al. [16], β -spline functions with non-integer orders can be used to model principally flat filters. Furthermore, a number of heuristic arguments have been presented [16, Section B], in order to obtain digital filters whose frequency response is principally flat (PF) in the passband. It is argued that the heuristic arguments are supported by empirical evidence observed during extensive testing of the filter designs. Following [16], the impulse response of β -spline based PF filters is given by
h P F ,β [n] =
54 55
sin(ωc n)
πn
sin(π n/( M + 1)
π n/( M + 1)
ρ .
(32)
Moreover, a relation between
56 57 58 59 60
• • • •
the β -spline order ρ , M = ( N − 1)/2, with N being the filter length, the transition bandwidth d , the stopband attenuation A s ,
61 62 63 64 65 66
96
εaisae
38 39
95
Filter design technique
34 35
94
has been empirically determined and is given in [16, Table I]. Analogous relations, using the α -spline functions, can be found in [17, Tables 1 and 2]. Since the arguments used in [16] and [17] for PF filter design are identical, the α -spline based PF impulse response can be written as
h P F ,α [n] =
sin(ωc n)
πn
·
·
sin(π n/( M + 1))
97 99 101
π n/( M + 1)
sin(απ n/( M + 1))
απ n/( M + 1)
102
q
103
.
(33)
In order to obtain a principally flat design, we propose the relation ρ = p + α q, in such a way that for a given ρ with different values of ( p , α , q), different filters are obtained. The resulting filters present similar or better properties than those obtained with β -spline functions when non-integer orders are used. Finally, the PF-based filter design algorithm is described as follows.
104 105 106 107 108 109 110 111 112 113
PF design procedure Step 1. Select the stopband attenuation ( A s ), the transition bandwidth (d ) and the cutoff frequency (ωc ). Step 2. Calculate the values of the half-order of the filter (M) and the order ρ = p + α q of the α -spline function from [17, Tables 1 and 2]. Step 3. Compute the 2M + 1 coefficients of the filter using (33) (− M ≤ n ≤ M).
114 115 116 117 118 119 120 121 122 123
In the example below, we design β - and α -spline PF lowpass linear-phase digital filters by determining, from [16, Table I] or from [17, Tables 1 and 2], the values of M and ρ = p + α q that must be used in (32) or (33), respectively, to achieve the desired filter characteristics. The desired specifications are the following:
124 125 126 127 128 129
• Minimum stopband attenuation A s = 80 dB. • Cutoff frequency ωc = 0.5π . • Transition bandwidth d = 0.25π .
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Table 4 Example design 3. Simulation results for the optimized principally flat filter design (ωc = 0.5π , d = 0.25π π , M = 25, A s = 80 dB).
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Filter design technique
εaisae
ε pbe
εsbe
εdev
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96
31
β -spline (ρ = 3.537) (Ref. [16]) α -spline Eq. (33) with ( p , q, α ) = (3, 1, 0.537) α -spline Eq. (33) with ( p , q, α ) = (2, 2, 0.768) α -spline Eq. (33) with ( p , q, α ) = (1, 3, 0.846)
1.009 · 10−2 9.732 · 10−3 9.549 · 10−3 9.486 · 10−3
1.457 · 10−10 1.142 · 10−10 2.519 · 10−11 1.522 · 10−11
1.457 · 10−10 1.142 · 10−10 2.519 · 10−11 1.522 · 10−11
1.009 · 10−4 8.546 · 10−5 3.503 · 10−5 3.485 · 10−5
79.9 81.4 89.1 89.2
97
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
5. Conclusions
52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
99 100 101
For the β -spline-based digital filter, we arrive at M = 25 and ρ = 3.537. On the other hand, different combinations of ( p , α , q) can be considered to get p + α q = ρ = 3.537. In particular, we tested the combinations (3, 0.537, 1), (2, 0.768, 2) and (1, 0.846, 3) for the proposed α -spline method. Fig. 8 shows the frequency responses of the obtained β and α -spline PF lowpass linear-phase digital filters, and Table 4 shows their errors. According to their magnitude responses and the quality parameters of Table 4, the proposed approach outperfoms the β -spline design in terms of minimum stopband attenuation and transition bandwidth. The proposed approach provides the lower least average integral squared, passband and stopband errors. As can also be seen in Table 4, the α -spline filter designed with the set ( p , q, α ) = (1, 3, 0.846) improves all quality parameters.
50 51
98
In this paper, the formulation of α -spline analog filters and its connection to their digital counterparts is presented. The α -spline approach is a natural extension of β -spline models, thoroughly studied in [15,16]. We derive mathematical expressions for the frequency and the impulse responses of the analog filters, and we present the corresponding discretization which accurately matches the model shown in [17]. Furthermore, two new criteria for the design of the α -spline digital filters, based on a least integral squared error approximation, are also presented. As these methods do not provide direct control over the stopband attenuation of the resulting filter, principally flat α -spline digital filters have also been introduced. Some representative examples illustrate that the proposed design approaches outperform those obtained through β -spline functions.
Acknowledgments
102 103
The authors would like to thank the anonymous Reviewers and the Handling Editor for their constructive suggestions which have helped in improving the paper.
104 105 106 107
Appendix A. Supplementary material
108 109
Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.dsp.2017.03.003. References [1] A. Antoniou, Digital Filter: Analysis, Design, and Applications, 2nd edition, McGraw-Hill, New York, 2000. [2] R. Schaumann, H. Xiao, M.E. Van Valkenburg, Design of Analog Filters, 2nd edition, Oxford University Press, 2009. [3] Y.-D. Jou, F.-K. Chen, WLS design of FIR Nyquist filter based on neural networks, Digit. Signal Process. 21 (1) (2011) 17–24. [4] L. Biagiotti, C. Melchiorri, FIR filters for online trajectory planning with time- and frequency-domain specifications, Control Eng. Pract. 20 (12) (2012) 1385–1399. [5] N. Karaboga, F. Latifoglu, Elimination of noise on transcranial Doppler signal using IIR filters designed with artificial bee colony ABC-algorithm, Digit. Signal Process. 23 (3) (2013) 1051–1058. [6] B. Boashash, G. Azemi, A review of time–frequency matched filter design with application to seizure detection in multichannel newborn EEG, Digit. Signal Process. 28 (2014) 28–38. [7] P. Upadhyay, R. Kar, D. Mandal, S. Ghoshal, V. Mukherjee, A novel design method for optimal IIR system identification using opposition based harmony search algorithm, J. Franklin Inst. 351 (5) (2014) 2454–2488. [8] L.-W. Chen, Y.-D. Jou, F.-K. Chen, S.-S. Hao, Eigenfilter design of linear-phase FIR digital filters using neural minor component analysis, Digit. Signal Process. 32 (2014) 146–155.
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[9] D. Jeon, S. Kim, B. Kwon, H. Lee, S. Lee, Prototype filter design for QAM-based filter bank multicarrier system, Digit. Signal Process. 57 (2016) 66–78. [10] T.W. Parks, C.S. Burrus, Digital Filter Design, Wiley-Interscience, New York, 1987. [11] T. Saramäki, Adjustable windows for the design of FIR filters – a tutorial, in: Proceedings of the 6th Mediterranean Electrotechnical Conference, Ljubljana, Yugoslavia, 1991, pp. 28–33. [12] P. Martín, F. Cruz-Roldán, T. Saramäki, A new window for the design of cosinemodulated multirate systems, in: 2004 IEEE International Symposium on Circuits and Systems, vol. 3, 2004, pp. 529–532. [13] M. Unser, A. Aldroubi, M. Eden, B-spline signal processing. Part I – theory, IEEE Trans. Signal Process. 41 (2) (1993) 821–833. [14] M. Unser, Splines: a perfect fit for signal and image processing, IEEE Signal Process. Mag. 16 (6) (1999) 22–38. [15] C.S. Burrus, A.W. Soewito, R.A. Gopinath, Least squared error FIR filter design with transition bands, IEEE Trans. Signal Process. 40 (6) (1992) 1327–1340. [16] R.M. Roark, M.A. Escabí, B-spline design of maximally flat and prolate spheroidal-type FIR filters, IEEE Trans. Signal Process. 47 (3) (1999) 701–716. [17] M.Á. Raposo-Sánchez, J. Sáez-Landete, F. Cruz-Roldán, α -Spline design of finite impulse response digital filters, Signal Process. 122 (2016) 204–212. [18] J. Ibánez, I. Santamaría, C. Pantaleón, L. Vielva, Parametric smoothing of spline interpolation, in: 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing, (ICASSP’04), Montreal, Canada, vol. 2, 2004, pp. 597–600.
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Miguel Á. Raposo-Sánchez was born in Buenos Aires, Argentina, in 1955. He received the degree in Physics from the Universidad de Valladolid (UVA), Spain, in 1981, and the degree in Economics from the Universidad de Alcalá (UAH), Spain, in 1995. Mathematics teacher at the secondary school from 1982 to 1989. Since 1990, he has been a Professor with the Department of Física y Matemáticas of the Universidad de Alcalá, Madrid, Spain. His teaching and research
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interests are in digital signal processing, filter design, acoustical engineering and bioelectromagnetism.
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José Sáez-Landete was born in Valdeganga (Albacete), Spain, in 1977. He received the M.S. degree in physics from the Universidad de Zaragoza, Spain, in 2000 and the Ph.D. degree in physics from the Universidad Complutense de Madrid, Spain, in 2006. Since 2006, he has been with the Signal Theory and Communications of the Universidad de Alcalá, Madrid, Spain, where he is an Associate Professor. His research interests include digital signal processing, image processing, filter design, digital communications and optimization. Fernando Cruz-Roldán was born in Baena, Spain, in 1968. He received his Technical Telecommunication Engineer degree from the Universidad de Alcalá (UAH), Spain, in 1990, his Telecommunication Engineer degree from the Universidad Politécnica de Madrid (UPM), Spain, in 1996, and Ph. D. in Electrical Engineering from the UAH, in 2000. Dr. Cruz-Roldan received the Universidad de Alcalá Prize for the most outstanding doctoral dissertation in the engineering discipline. He joined the Department of Ingeniería de Circuitos y Sistemas (UPM), in 1990, where from 1993 to 2003, he was an Assistant Professor. From 1998 to February 2003, he was a Visiting Lecturer at Universidad de Alcalá. In March 2003, he joined the Universidad de Alcalá, Spain, as an Associate Professor, and since November 2009, he has been a Professor with the Department of Teoría de la Señal y Comunicaciones, Universidad de Alcalá. His teaching and research interests are in digital signal processing, filter design, and multirate systems applied to subband coding and digital communications.
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Digital Signal Processing
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Analog and digital filters with
α -splines ✩
77 78
Miguel Ángel Raposo-Sánchez a,∗ , José Sáez-Landete b , Fernando Cruz-Roldán b a b
Department of Física y Matemáticas, Escuela Politécnica Superior de la Universidad de Alcalá, 28805 Alcalá de Henares, Spain Department of Teoría de la señal y Comunicaciones, Escuela Politécnica Superior de la Universidad de Alcalá, 28805 Alcalá de Henares, Spain
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a r t i c l e
i n f o
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a b s t r a c t
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Article history: Available online xxxx Keywords: Filtering theory Spline functions and analog filters Finite impulse response (FIR) digital filters Least integral squared error Principally flat
29 30 31
Analytical expressions of analog and digital α -spline filters with continuous transition bands are derived. Previous works based on β -spline functions consider the order of the spline function, which must be a natural number, as a parameter to sharpen the transition band. However, the inclusion of an α -spline model provides greater flexibility since its order can be a real number, it depends on three parameters, and it can be used to model the transition band in the frequency domain. This paper presents the formulation of analog α -spline filters and the corresponding discretization to obtain the frequency and the impulse responses of their digital counterparts. There is also shown the design of α -spline digital filters minimizing two different least integral squared errors. As these methods do not provide direct control over the stopband attenuation, a third method which leads to principally flat digital filters is introduced. Several numerical examples demonstrate the efficiency and flexibility of the proposed techniques. © 2017 Published by Elsevier Inc.
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33
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35 36
101
1. Introduction
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
Analog and digital filters are widely required in different areas of signal processing [1–9]. A common method of designing digital filters is based on transformation methods from their analog counterpart. In contrast, windowing is a classical method for directly obtaining the digital filter, and one of its essential concerns is the design of smoothing window functions with good characteristics [1,10–12]. Several authors have proposed the use of spline functions for filter design [13–17]. In order to eliminate the Gibbs phenomenon, a transition band is introduced to continuously connect the passband to the stopband, removing the discontinuity present in a conventional brick-wall ideal filter. This transition band is the result of the convolution of rectangular pulses with the ideal frequency response. The design can be extended to include a ρ th-order spline, leading to filters with several desirable properties. First, the use of spline functions in the filter design allows analytical leastsquared-error approximation finite impulse response (FIR) filters, along with a reduction of the approximation ripple. Second, the use of splines gives explicit control over the transition band, and it provides almost as good an approximation as more complicated
61 62 63 64 65 66
✩
This work was partially supported by the Spanish Ministry of Economy and Competitiveness through project TEC2015-64835-C3-1-R. Corresponding author. E-mail address: [email protected] (M.Á. Raposo-Sánchez).
*
http://dx.doi.org/10.1016/j.dsp.2017.03.003 1051-2004/© 2017 Published by Elsevier Inc.
102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124
• The formulation of α -spline analog filters is presented. The
59 60
numerical approaches. The resulting FIR filters are easy to calculate and to program because several formulas that relate the ρ th-order transition-band spline function with the desired stopband attenuation, and the filter order with the transition bandwidth, have been proposed [15–17]. In [15,16], the filter design is based on β -spline models, and recently the use of α -spline functions to shape the transition band of FIR filters has been proposed in [17]. α -Spline functions are defined in [18]. They are the result of convolving two types of rectangular pulses of different widths in the frequency domain. α -Spline functions result in transition functions with non-integer exponents. Based on a modified version in the frequency domain of the α -spline functions, a consistent mathematical formulation to design digital filters was presented in [17]. That paper introduced a new formulation to employ non-integer values for the spline order. Furthermore, an optimization method to obtain the parameters of the α -spline digital window was also provided, resulting in design improvements. This present paper focuses rather on α -spline analog filters, and two practical and simple methods to design digital filters are also presented. In more detail, the main contributions of this paper can be summarized as follows:
mathematical expressions of both the analog frequency and the impulse responses are derived. Then, their corresponding discretization shows that the discrete-time filter accurately matches the model proposed in [17].
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( p −1) times
1
[ p ,q]
2
1
3
−1
f (t ) = F
6 7 8
p + αq
{φ()} =
10
a (− jt )
13 15
Fig. 1. Rectangular convolving functions to obtain
16
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a least integral squared error approximation are introduced, following two different criteria: a) the minimization of the integral squared error over the entire frequency spectrum and b) the minimization of the integral squared error in both the passband and the stopband. The integral of the square of the error is an optimization criterion often used in filter design because it is a measure of the energy of the approximation error. • The design of principally flat α -spline filters is discussed. This alternative design method provides direct control over the stopband attenuation achieved by the digital filter. Computer simulations have also been run and the obtained results demonstrate good performance in terms of the transition bandwidth and stopband attenuation. The rest of this paper is organized as follows. In Section 2, analytical expressions for both the frequency and the impulse responses of the α -spline analog filter are derived. That section also describes the discretization of the analog filter to obtain its digital counterpart. The optimization procedures based on the least integral squared error are addressed in Section 3, whereas Section 4 introduces the design of principally flat α -spline filters. Both sections illustrate the performance of the filters, comparing several designs to other techniques; finally, some concluding remarks are given in Section 5.
55 56 57
α a (− jt )
60 61 62 63 64 65 66
e
(4)
.
2.1.
a t − 2(jαp + α q)
−e
j α a t 2( p +α q)
75
76
,
(5)
79 80 81 82 83 84
(6)
φ () =
2 π ( p +α q )
To calculate the Fourier transform of F 1 following functions of order :
86
(t ), let us consider the
a
0,
and
φα () =
otherwise,
(7)
α a
0,
, || <
α a , 2 ( p +α q )
otherwise,
G +, () = + =
94 95
, ≥ 0, 0,
(8)
< 0,
g (t ) =
where denotes the analog frequency variable, α (0 ≤ α ≤ 1) provides control over the width of φα () (see Fig. 1), and a is a parameter related to the bandwidth of the convolving function, which has a direct influence on the transition bandwidth of the designed filter. [ p ,q] The convolving functions 1 () are generated by performing, in the frequency domain, p + q − 1 convolution operations between p functions φ (), and q functions φα ():
96 97
where ∈ N. Then, using the Euler gamma function in its integral form, we get
98 99 100 101
! . 2π (− jt ) +1
(9)
102 103
By introducing (9) in (6), and using the algebraic expansion of the powers of a binomial,
104 105 106
1
=
(− jt ) p +q
2π g p +q−1 (t )
( p + q − 1)!
(10)
,
107 108 109
(t ) =
2π ( p + α q) p +q g p +q−1 (t )
=κ ·
(1)
(2)
92 93
where
αq ap+q ( p + q − 1)! q p
e
· e
− 2( pj+aαtq)
a t − 2(jαp + α q)
j a t
p
− e 2( p+αq) −q j α a t 2 ( p + α q ) −e
C kl e
g p +q−1 (t ),
(11)
k =0 l =0
κ=
111 112 113 114 115 117 118 119 120 121
2π ( p + α q) p +q
αq ap+q ( p + q − 1)!
C kl = (−1)(k+l)
δα ,kl =
110
116
j δα ,kl a t
where 2 π ( p +α q )
89 91
F
[ p ,q]
, || <
88 90
g (t ) −→ G +, (),
F1
a , 2 ( p +α q )
85 87
[ p ,q]
Let us consider the following two types of rectangular pulses, both having area 2π :
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α -Spline kernels
72 73
[ p ,q] [ p ,q] F 1 (t ) = F−1 1 () = [ f (t )] p [ f α (t )]q p j a t − 2( pj+aαtq) 2( p +α q) e − e ( p + αq) p +q = −q . j α a t αq ap+q (− jt ) p+q − 2(jαp+αa qt ) e − e 2( p+αq)
2. Analog filter with α -splines
58 59
71
we obtain
53 54
α -splines.
• The design and the performance of α -spline digital filters by
51 52
−e
j a t 2( p +α q)
and finally
14
21
p + αq
12
20
e
− 2( pj+aαtq)
70
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f α (t ) = F−1 {φα ()} =
11
19
Similarly
9
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18
67
(3)
In the time domain, φ() can be written as
4
17
(q−1) times
() = φ () ∗ . . . ∗ φ () ∗ φα () ∗ . . . ∗ φα () .
k + αl p + αq
p k
−
122
,
123
1 2
q l
124
,
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.
Calculating the Fourier transform of (11) and employing the frequency shift property, the α -spline functions in the frequency domain can be written as
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1. For || > (c + a /2):
1 2
67 68
H T () = 0.
3 4
2. For −c −
5
69
a
< < −c +
2
H T () = B
7
8
q
p
6
C kl
k =0 l =0
9
a 2
70
:
+ c − δα ,kl a
71
p +q
72
,
73
+
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where
10 11
B=
12 13
76
p +q
( p + α q) . ( p + q)!α q
3. For || < c −
14
a 2
15 16 17
Fig. 2.
H T () = B
α -Spline convolving function.
77 78 79 80
:
q p
C kl 1 −
k =0 l =0
18 19
k + αl
p +q
p + αq +
81 82
= 1.
83 84 85
4. For c − 2a < < c + 2a :
20 21 22
H T () = 1 − B
23
q
p
C kl
k =0 l =0
24 25
86
− c − δα ,kl a
p +q
87 88
.
89
+
90
We can write the above equations in a compact form as follows:
26
H T () = 1 − B
28
29
q
p
27
C kl
k =0 l =0
30
|| − c − δα ,kl a
p +q +
38
100
Fig. 3. Frequency responses of an analog ideal low-pass filter and an tion.
[ p ,q]
1
p
() = κ ·
k =0 l =0
40
p
=κ ·
41 42 44
=κ
45
h T (t ) = h i (t ) · F 1 C kl G +,( p +q−1) ( − δα ,kl a )
p +q−1 · a
·
49 50
C kl
k =0 l =0
[ p ,q]
Fig. 2 shows an example of 1
<
a 2
− δα ,kl a
h i (t ) =
p +q−1 .
(12)
+
(), which is nonzero for
−a 2
<
53
.
2.2. The frequency response of the analog filter In order to obtain the frequency response of the analog filter, the following convolution is carried out:
54 55 56
[ p ,q]
H T () = H i () ∗ 1
() =
57
1 2π
58
∞
61 62 63 64 65 66
[ p ,q]
H i (ξ ) 1
2π
e
j t
−c
( − ξ )dξ ,
H i () =
(17)
103 104 105
110
[ p ,q]
whereas F 1 (t ) is the inverse Fourier transform of 1 given by (6), or equivalently
⎛
108 109
[ p ,q]
⎞p ⎛
(),
a t
⎞p ⎛
α a t
112 114
(18)
115 116 117
As a result, the continuous-time impulse response of the analog filter can be finally written as follows:
111 113
⎞q
at sin 2(αp +α q ) ⎠ ·⎝ ⎠ . α a t 2 ( p +α q )
⎞q
c sin(c t ) ⎝ sin 2( p +αq) ⎠ ⎝ sin 2( p +αq) ⎠ h T (t ) = · · . a t α a t π c t 2 ( p +α q ) 2 ( p +α q )
(13) where
102
107
c sin (c t ) d = , π c t
⎛
−∞
59 60
1
c
sin 2( p+aαt q) [ p ,q] F 1 (t ) = ⎝ a t 2 ( p +α q )
51 52
(16)
106
p +q−1
q p
(t ).
101
where h i (t ) is the inverse Fourier transform of the ideal filter, given by
C kl − δα ,kl a +
46 48
The inverse Fourier transform of (13) can be expressed as [ p ,q]
q
k =0 l =0
43
α -spline func-
q
39
47
97 99
34
37
96 98
2.3. Continuous-time impulse response
33
36
94 95
32
35
92 93
(15)
,
where −∞ < < ∞, 0 ≤ α < 1, p ≥ 1 and q ≥ 1 with p , q ∈ N.
31
91
(19)
118 119 120 121 122 123 124 125 126
1, 0,
2.4. Digitalization of the analog filter
|| < c ,
(14)
otherwise, [ p ,q]
c is the cutoff frequency and 1
() is the α -spline function
given by (12). In order to carry out the above convolution, let us consider different subintervals of (see Fig. 3):
127 128
Once the impulse response of the analog filter has been computed, the design of the discrete-time digital filter can be based on discretizing the corresponding continuous-time response. To this end, let us consider t = nT , or in the frequency domain ω = T
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Fig. 4. Integral squared approximation error versus the half-length of the
24
α -spline filter.
90
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91
and d = a T , assuming without loss of generality that the sampling period is T = 1. In this case, the frequency response of the resulting digital filter can be expressed as
29 30 31
H T (ω) = 1 − B
34 35
⎛
37 39
C kl
|ω| − ωc − δα ,kl d
h T [n] =
40
d n ωc sin(ωc n) ⎝ sin 2( p+αq) · d n π ωc n 2 ( p +α q )
⎞p ⎛
43 44 45 46 47 48 49 50 51 52 53 54
57 58 59 60 61 62 63 64 65 66
+
⎞
q α n sin 2( p +dαq) ⎠ ·⎝ ⎠ , α d n 2 ( p +α q )
−∞ < n < ∞, and represents an infinite impulse response (IIR) filter. 3. FIR filter design by least integral squared error approximation The integral of the square of the error is an optimization criterion often used in filter design because it is a measure of the energy of the approximation error. In this way, the main goal of the design is to obtain a computationally realizable filter with an impulse response of finite length, N = 2M + 1, whose frequency response H (ω) approximates the theoretical frequency response H T (ω) minimizing the integral squared error, defined as
55 56
(20)
(21)
41 42
,
L I S E =
1 2π
ωr 2
| H (ω) − H T (ω)| dω,
(22)
−ωr
where H (ω) is the frequency response of the finite impulse response h[n]. Depending on how the truncation is done, the resulting filter exhibits some properties which approximate, to a greater or lesser extent, the properties of the specified theoretical filter. Below we will show the design of α -spline filters following two different criteria: a) the minimization of the integral squared error over the entire frequency spectrum and b) the minimization of the integral squared error in both the passband and the stopband.
92 93
This problem can be formulated as the minimization of
p +q
where ω denotes the digital frequency variable, −π ≤ ω < π , 0 ≤ α < 1, p ≥ 1 and q ≥ 1 with p , q ∈ N. Furthermore, the analog impulse response reduces to the form
36 38
q
k =0 l =0
32 33
p
3.1. Minimization of the total integral squared error (MITISE)
T = =
π
1 2π
∞
| H (ω) − H T (ω)|2 dω =
95 96
|h[n] − h T [n]|2
97
n=−∞
−π
M
94
|h[n] − h T [n]|2 + 2
n=− M
∞
98 99
|h T (n)|2 .
(23)
101
n = M +1
This function is minimized if h[n] = h T [n], − M ≤ n ≤ M, i.e., the best approximation to the frequency response, in the sense of minimizing (23), is obtained using the coefficients of the theoretical impulse response. Burrus et al. addressed this design for modeling transition bands using β -spline functions [15]:
hb [n] =
sin(ωc n)
πn
sin(d n/(2ρb ))
d n/(2ρb )
ρb
d N 2π
102 103 104 105 106 107 108
.
(24)
109 110
Generally, the effect of the order of the β -spline functions, ρb , on the frequency response is not obvious. In [15], as a result of numerical experimentation, it is stated that the optimum value of ρb that minimizes (23), for a filter of length N and a given width of the transition band, is
ρb = 0.62
100
111 112 113 114 115 116
= 0.62 f N ,
(25)
where f is the width of the transition band, expressed in Hertz. Our proposal minimizes (23) using as impulse response the α -spline filter given by (21). In order to evaluate the effects of truncating the filter h T [n], the logarithm of the approximation error as a function of the filter length has been measured for different values of ρ , as in [15]. The results, shown in Fig. 4, have been obtained assuming that the cutoff frequency and the transition bandwidth are fixed. As can be seen, the error monotonically decreases as the filter length increases, for a given value of p + α q. Moreover, there appear different regions (that correspond to the regions near the zeros of the α -spline functions) with flat shape whose width increases for increasing p + α q. The infinitely long h T [n] should be truncated just as the flat region begins, because increasing the length of the truncated impulse response in those
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Fig. 5. Magnitude frequency responses of discrete low-pass filters with
α - and β -spline functions minimizing the total integral squared error.
27 28 29
91 92 93
Table 1 Example design 1. Simulation results for the optimized total integral squared error (ωc = 0.5π , d = 0.2π , M = 36).
94 95
30
Filter design technique
εaisae
ε pbe
εsbe
εdev
A s (dB)
96
31
β -spline (ρ = 4.526) (Ref. [15]) α -spline Eq. (21) with ( p , q, α ) = (4, 1, 0.2)
6.506 · 10−3 6.643 · 10−3
4.215 · 10−11 6.970 · 10−12
4.215 · 10−11 6.970 · 10−12
1.890 · 10−5 1.626 · 10−5
94.5 95.8
97
32 33 34 35 36 37 38 39 40 41 42 43 44
99
regions does not decrease the error significantly. Computer simulations and numerical experimentation have led us to an expression that relates the transition bandwidth, f = d /(2π ) in Hertz, the half-length of the filter, M = ( N − 1)/2, and the optimum order of the α -spline function for q = 1:
f M = 0.857( p + αq)|q=1 = 0.857( p + α ).
(26)
This order leads to an acceptable degree of accuracy. Hence, we can describe the filter design algorithm as follows.
M
εaisae =
MITISE design procedure
47 48 49 50 51 52
Step 1. Select the length of the filter (N = 2M + 1), the transition bandwidth (d ) and the cutoff frequency ωc . Step 2. Calculate the order of the α -spline function from (26). Step 3. Compute the 2M + 1 coefficients of the filter using (21) (− M ≤ n ≤ M).
53 54 55 56 57 58 59 60 61 62 63 64 65 66
To illustrate its performance, the design of lowpass linear-phase digital filters of length 73 (M = 36), with transition bandwidth d = 0.2π and cutoff frequency ωc = 0.5π , is considered. From (25), previously reported in [15] for β -spline digital filters, we obtain ρb = 4.526, whereas from (26), the value is p + α q = 4.2, leading to the following set of parameters ( p = 4, α = 0.2, q = 1). Fig. 5 shows the magnitude frequency responses of the designed linear-phase filters. Note that the transition bandwidths are virtually identical, whereas the α -spline filter exhibits better performance in both the passband and the stopband. Table 1 compares both filters for the average integral squared approximation error, defined as
∞
|hi [n] − h [n]|2 + 2
n=− M
100
|hi [n]|2 ,
n = M +1
where h i [n] is the ideal impulse response; the passband error, given by
ω p 1 | H (ω) − H i (ω)|2 dω, ε pbe =
π
103 104 105
108 109
where H i (ω) is the ideal frequency response; the stopband error
εsbe =
102
107
0
1
101
106
π
45 46
98
π
110 111 112
2
| H (ω)| dω;
113
ωs
114
the maximum passband deviation, obtained as
115
εdev = max | H (ω) − H i (ω)| ,
116
|ω|<ω p
117
and the minimum stopband attenuation A s . The best obtained results are indicated by the boldface numbers, and as can be observed, the filter designed with (26) outperforms that with (25), with the exception of the average integral squared approximation error.
118 119 120 121 122 123
3.2. Minimization of integral squared error on the pass and stopband (MISEPS)
124
In this case, the integral squared error on the pass and stopband is minimized:
127
ω p π 1 1 2 | H (ω) − H T (ω)| dω + | H (ω) − H T (ω)|2 dω. ps =
π
π
0
ωs
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Fig. 6. Minimization of the integral squared error approximation.
26
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27
93
(27)
28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
In [15], this problem is addressed empirically, concluding that the optimum value, ρb that minimizes (27) for length N and a given width of the transition band f Hertz, can be calculated by
ρb = K N 1 , where
⎧ ⎨ 0.453 + 0.386/ N 1 , N 1 ≤ 1.25, K = 0.774 − 0.0251N 1 , 1.25 < N 1 < 5, ⎩ 0.648, 5 < N 1 ,
47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
=2
∞
2
|h T [n]| .
(30)
n = M +1
We have also observed that the spline order that minimizes (30) is approximately the same for any combination of M = ( N − 1)/2 and d such that M · d is fixed. The minimum value of decreases as M increases. Fig. 6 shows the logarithm of as a function of the α -spline order, parametrized with d · M = 6.384π and with different combinations of M and d . The α -spline order to minimize the function is p + α q|q=1 = 3.7. Different tests have been carried out, varying the parameter x = M · d , with the aim of finding a preliminary relation which is a fairly accurate approximation, between the x-parameter and the order, ρ = p + α q, considering q = 1, of the α -spline function which minimizes the approximation error in both bands. In this sense, several numerical experiments have been carried out for different intervals of the variable x = M · d (see Table 2), arriving at a set of pairs (x = M · d , ρ = p + α q). From the above, a second-degree polynomial model
65 66
ρ = a · x2 + b · x + c
94
ρ by means of (31).
95
x = M · d
a
b
c
96
1.75π ≤ x < 4.00π 4.00π ≤ x < 5.75π 5.75π ≤ x < 7.50π 7.50π ≤ x < 9.25π 9.25π ≤ x < 10.75π 10.75π ≤ x < 12.50π 12.50π ≤ x < 14.25π
−0.0294 −0.0178 −0.0097 −0.0103 −0.0055 −0.0056 −0.0061
0.5394 0.5745 0.4256 0.5555 0.3542 0.4208 0.5195
−1.1575 −1.9071 −0.9189 −2.7449 −0.0147 −1.1893 −3.3449
97
(31)
98 99 100 101 102 103
(29)
1 = 1.12 · f , and d = 2π 1 [15]. We have also performed empirical tests to find the optimal value for the α -spline order, in terms of their length and transition bandwidth. We have observed that, in all cases studied, the optimum value that minimizes (27) matches the value that minimizes the function
45 46
(28)
Table 2 Coefficients to obtain
has been developed to determine the order of the α -spline. The coefficients (a, b, c ) obtained by polynomial regression are shown in Table 2 for each considered interval. In this case, the algorithm to design the filter is given below.
104 105 106 107 108 109
MISEPS design procedure Step 1. Select the length of the filter (N = 2M + 1), the transition bandwidth (d ), and the cutoff frequency (ωc ). Step 2. Calculate the order of the α -spline function from (31) using the coefficients (a, b, c ) given in Table 2. Step 3. Compute the 2M + 1 coefficients of the filter using (21) (− M ≤ n ≤ M).
110 111 112 113 114 115 116 117 118
In order to compare the results obtained using α - and β -spline functions, low-pass 115-length digital filters (M = 57), with transition bandwidth d = 0.112π and cutoff frequency ωc = 0.45π , have been designed. For the above values, (28) and (29) provide the optimal order for the β -spline of ρb = 4.17312, whereas the optimal order of the α -spline functions, obtained from (31), is p + α q|q=1 = 3.7, leading to the set of values ( p = 3, α = 0.7, q = 1). Fig. 7 shows the magnitude frequency responses of the designed filters. It is observed again that the transition bandwidths are virtually identical, whereas the proposed α -spline filter exhibits improvements in both the passband and the stopband. Table 3 shows the aforementioned quality parameters obtained from the designed linear-phase filters. Also in this case, the pro-
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α - and β -spline minimizing both the passband and the stopband squared integral error.
92
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93
Table 3 Example design 2. Simulation results for the optimized integral squared error in the pass and stopband (ωc = 0.45π , d = 0.112π π , M = 57).
28 29 30 31 32 33
36 37
ε pbe
εsbe
εdev
A s (dB)
β -spline (ρ = 4.173) (Ref. [15]) α -spline Eq. (21) with ( p , q, α ) = (3, 1, 0.7)
3.803 · 10−3 3.936 · 10−3
3.353 · 10−10 2.880 · 10−11
1.423 · 10−10 1.152 · 10−11
7.706 · 10−5 4.402 · 10−5
83.8 88.0
98
p
100
posed approach outperforms the β -spline-based technique in almost all quality measures (except the average integral squared approximation error). 4. Principally Flat (PF) filter design
40 41 42 43 44 45 46 47 48 49 50 51 52 53
In this section, we introduce an alternative design method which provides control over the stopband attenuation achieved by the filter. According to Roark et al. [16], β -spline functions with non-integer orders can be used to model principally flat filters. Furthermore, a number of heuristic arguments have been presented [16, Section B], in order to obtain digital filters whose frequency response is principally flat (PF) in the passband. It is argued that the heuristic arguments are supported by empirical evidence observed during extensive testing of the filter designs. Following [16], the impulse response of β -spline based PF filters is given by
h P F ,β [n] =
54 55
sin(ωc n)
πn
sin(π n/( M + 1)
π n/( M + 1)
ρ .
(32)
Moreover, a relation between
56 57 58 59 60
• • • •
the β -spline order ρ , M = ( N − 1)/2, with N being the filter length, the transition bandwidth d , the stopband attenuation A s ,
61 62 63 64 65 66
96
εaisae
38 39
95
Filter design technique
34 35
94
has been empirically determined and is given in [16, Table I]. Analogous relations, using the α -spline functions, can be found in [17, Tables 1 and 2]. Since the arguments used in [16] and [17] for PF filter design are identical, the α -spline based PF impulse response can be written as
h P F ,α [n] =
sin(ωc n)
πn
·
·
sin(π n/( M + 1))
97 99 101
π n/( M + 1)
sin(απ n/( M + 1))
απ n/( M + 1)
102
q
103
.
(33)
In order to obtain a principally flat design, we propose the relation ρ = p + α q, in such a way that for a given ρ with different values of ( p , α , q), different filters are obtained. The resulting filters present similar or better properties than those obtained with β -spline functions when non-integer orders are used. Finally, the PF-based filter design algorithm is described as follows.
104 105 106 107 108 109 110 111 112 113
PF design procedure Step 1. Select the stopband attenuation ( A s ), the transition bandwidth (d ) and the cutoff frequency (ωc ). Step 2. Calculate the values of the half-order of the filter (M) and the order ρ = p + α q of the α -spline function from [17, Tables 1 and 2]. Step 3. Compute the 2M + 1 coefficients of the filter using (33) (− M ≤ n ≤ M).
114 115 116 117 118 119 120 121 122 123
In the example below, we design β - and α -spline PF lowpass linear-phase digital filters by determining, from [16, Table I] or from [17, Tables 1 and 2], the values of M and ρ = p + α q that must be used in (32) or (33), respectively, to achieve the desired filter characteristics. The desired specifications are the following:
124 125 126 127 128 129
• Minimum stopband attenuation A s = 80 dB. • Cutoff frequency ωc = 0.5π . • Transition bandwidth d = 0.25π .
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93
Table 4 Example design 3. Simulation results for the optimized principally flat filter design (ωc = 0.5π , d = 0.25π π , M = 25, A s = 80 dB).
28 29
94 95
30
Filter design technique
εaisae
ε pbe
εsbe
εdev
A s (dB)
96
31
β -spline (ρ = 3.537) (Ref. [16]) α -spline Eq. (33) with ( p , q, α ) = (3, 1, 0.537) α -spline Eq. (33) with ( p , q, α ) = (2, 2, 0.768) α -spline Eq. (33) with ( p , q, α ) = (1, 3, 0.846)
1.009 · 10−2 9.732 · 10−3 9.549 · 10−3 9.486 · 10−3
1.457 · 10−10 1.142 · 10−10 2.519 · 10−11 1.522 · 10−11
1.457 · 10−10 1.142 · 10−10 2.519 · 10−11 1.522 · 10−11
1.009 · 10−4 8.546 · 10−5 3.503 · 10−5 3.485 · 10−5
79.9 81.4 89.1 89.2
97
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
5. Conclusions
52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
99 100 101
For the β -spline-based digital filter, we arrive at M = 25 and ρ = 3.537. On the other hand, different combinations of ( p , α , q) can be considered to get p + α q = ρ = 3.537. In particular, we tested the combinations (3, 0.537, 1), (2, 0.768, 2) and (1, 0.846, 3) for the proposed α -spline method. Fig. 8 shows the frequency responses of the obtained β and α -spline PF lowpass linear-phase digital filters, and Table 4 shows their errors. According to their magnitude responses and the quality parameters of Table 4, the proposed approach outperfoms the β -spline design in terms of minimum stopband attenuation and transition bandwidth. The proposed approach provides the lower least average integral squared, passband and stopband errors. As can also be seen in Table 4, the α -spline filter designed with the set ( p , q, α ) = (1, 3, 0.846) improves all quality parameters.
50 51
98
In this paper, the formulation of α -spline analog filters and its connection to their digital counterparts is presented. The α -spline approach is a natural extension of β -spline models, thoroughly studied in [15,16]. We derive mathematical expressions for the frequency and the impulse responses of the analog filters, and we present the corresponding discretization which accurately matches the model shown in [17]. Furthermore, two new criteria for the design of the α -spline digital filters, based on a least integral squared error approximation, are also presented. As these methods do not provide direct control over the stopband attenuation of the resulting filter, principally flat α -spline digital filters have also been introduced. Some representative examples illustrate that the proposed design approaches outperform those obtained through β -spline functions.
Acknowledgments
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The authors would like to thank the anonymous Reviewers and the Handling Editor for their constructive suggestions which have helped in improving the paper.
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Appendix A. Supplementary material
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Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.dsp.2017.03.003. References [1] A. Antoniou, Digital Filter: Analysis, Design, and Applications, 2nd edition, McGraw-Hill, New York, 2000. [2] R. Schaumann, H. Xiao, M.E. Van Valkenburg, Design of Analog Filters, 2nd edition, Oxford University Press, 2009. [3] Y.-D. Jou, F.-K. Chen, WLS design of FIR Nyquist filter based on neural networks, Digit. Signal Process. 21 (1) (2011) 17–24. [4] L. Biagiotti, C. Melchiorri, FIR filters for online trajectory planning with time- and frequency-domain specifications, Control Eng. Pract. 20 (12) (2012) 1385–1399. [5] N. Karaboga, F. Latifoglu, Elimination of noise on transcranial Doppler signal using IIR filters designed with artificial bee colony ABC-algorithm, Digit. Signal Process. 23 (3) (2013) 1051–1058. [6] B. Boashash, G. Azemi, A review of time–frequency matched filter design with application to seizure detection in multichannel newborn EEG, Digit. Signal Process. 28 (2014) 28–38. [7] P. Upadhyay, R. Kar, D. Mandal, S. Ghoshal, V. Mukherjee, A novel design method for optimal IIR system identification using opposition based harmony search algorithm, J. Franklin Inst. 351 (5) (2014) 2454–2488. [8] L.-W. Chen, Y.-D. Jou, F.-K. Chen, S.-S. Hao, Eigenfilter design of linear-phase FIR digital filters using neural minor component analysis, Digit. Signal Process. 32 (2014) 146–155.
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[9] D. Jeon, S. Kim, B. Kwon, H. Lee, S. Lee, Prototype filter design for QAM-based filter bank multicarrier system, Digit. Signal Process. 57 (2016) 66–78. [10] T.W. Parks, C.S. Burrus, Digital Filter Design, Wiley-Interscience, New York, 1987. [11] T. Saramäki, Adjustable windows for the design of FIR filters – a tutorial, in: Proceedings of the 6th Mediterranean Electrotechnical Conference, Ljubljana, Yugoslavia, 1991, pp. 28–33. [12] P. Martín, F. Cruz-Roldán, T. Saramäki, A new window for the design of cosinemodulated multirate systems, in: 2004 IEEE International Symposium on Circuits and Systems, vol. 3, 2004, pp. 529–532. [13] M. Unser, A. Aldroubi, M. Eden, B-spline signal processing. Part I – theory, IEEE Trans. Signal Process. 41 (2) (1993) 821–833. [14] M. Unser, Splines: a perfect fit for signal and image processing, IEEE Signal Process. Mag. 16 (6) (1999) 22–38. [15] C.S. Burrus, A.W. Soewito, R.A. Gopinath, Least squared error FIR filter design with transition bands, IEEE Trans. Signal Process. 40 (6) (1992) 1327–1340. [16] R.M. Roark, M.A. Escabí, B-spline design of maximally flat and prolate spheroidal-type FIR filters, IEEE Trans. Signal Process. 47 (3) (1999) 701–716. [17] M.Á. Raposo-Sánchez, J. Sáez-Landete, F. Cruz-Roldán, α -Spline design of finite impulse response digital filters, Signal Process. 122 (2016) 204–212. [18] J. Ibánez, I. Santamaría, C. Pantaleón, L. Vielva, Parametric smoothing of spline interpolation, in: 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing, (ICASSP’04), Montreal, Canada, vol. 2, 2004, pp. 597–600.
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Miguel Á. Raposo-Sánchez was born in Buenos Aires, Argentina, in 1955. He received the degree in Physics from the Universidad de Valladolid (UVA), Spain, in 1981, and the degree in Economics from the Universidad de Alcalá (UAH), Spain, in 1995. Mathematics teacher at the secondary school from 1982 to 1989. Since 1990, he has been a Professor with the Department of Física y Matemáticas of the Universidad de Alcalá, Madrid, Spain. His teaching and research
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interests are in digital signal processing, filter design, acoustical engineering and bioelectromagnetism.
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José Sáez-Landete was born in Valdeganga (Albacete), Spain, in 1977. He received the M.S. degree in physics from the Universidad de Zaragoza, Spain, in 2000 and the Ph.D. degree in physics from the Universidad Complutense de Madrid, Spain, in 2006. Since 2006, he has been with the Signal Theory and Communications of the Universidad de Alcalá, Madrid, Spain, where he is an Associate Professor. His research interests include digital signal processing, image processing, filter design, digital communications and optimization. Fernando Cruz-Roldán was born in Baena, Spain, in 1968. He received his Technical Telecommunication Engineer degree from the Universidad de Alcalá (UAH), Spain, in 1990, his Telecommunication Engineer degree from the Universidad Politécnica de Madrid (UPM), Spain, in 1996, and Ph. D. in Electrical Engineering from the UAH, in 2000. Dr. Cruz-Roldan received the Universidad de Alcalá Prize for the most outstanding doctoral dissertation in the engineering discipline. He joined the Department of Ingeniería de Circuitos y Sistemas (UPM), in 1990, where from 1993 to 2003, he was an Assistant Professor. From 1998 to February 2003, he was a Visiting Lecturer at Universidad de Alcalá. In March 2003, he joined the Universidad de Alcalá, Spain, as an Associate Professor, and since November 2009, he has been a Professor with the Department of Teoría de la Señal y Comunicaciones, Universidad de Alcalá. His teaching and research interests are in digital signal processing, filter design, and multirate systems applied to subband coding and digital communications.
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