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Design of State Digital Filters B. Pˇseniˇcka, F. Garc´ıa-Ugalde, J. Savage, S. Herrera-Garc´ıa, and V. Dav´ıdek Abstract—In this correspondence, we develope a very simple algorithm for the design of a second-order state-space digital filter. The design equations presented here were derived on the condition that the sensitivity of the coefficients in the state-space digital filter (SDF) are spread uniformly to all coefficients of the SDF. The design algorithm has been shown to provide SDF realizations having a minimum output round-off noise while preserving low coefficient sensitivity and freedom from limit cycles. Fig. 1. Signal flow graph of a state digital filter of the second order.
Index Terms—Digital filter design, quantization effects.
I. STATE DIGITAL FILTER STRUCTURES Modeled as a linear discrete time-invariant N th-order singleinput/single-output system with constant coefficients, a digital filter (DF) can be described by the state equations
u(
n
Au( ) + B ( ) T ) = C u( ) + D ( )
+ 1) =
n
y (n
x n
n
(1)
x n
where
u(
n)
state vector; single input; y (n) single output; and ; ; , and are the (generally complex) state matrices with the sizes of N 2 N; N 2 1; N 2 1, and 1 2 1, respectively. Using the transform, the transfer function and impulse response of this digital filter can be obtained in the form x(n)
ABC z
D
H (z )
=
Y
(z )
X (z )
=
D + [CT( I 0 A)01B]
(2)
z
I is the identity matrix. D n01 ( )= CA B
where
h n
for for
n
Fig. 2. Signal flow graph of the second canonical structure (direct realization form II) for a second-order IIR DF.
=0 0
(3)
n >
Let the transfer function of a second-order DF be given by H (z )
or equivalently H (z )
=
A0
+
z
=
01 + A2 z 02 1 + B1 z 01 + B2 z 02
A0
+ A1 z
(4)
01 (A1 0 A0 B1 ) + z 02 (A1 0 A0 B2 ) : 1 + B1 z 01 + B2 z 02
(5)
The state matrices of a second-order SDF acquire the form
A=
a11
a12
a21
a22
B=
b1 b2
CT = [ 1 c
c2
]
D=
d:
(6)
An example of a second-order SDF structure is depicted in Fig. 1. Manuscript received April 5, 1996; revised December 10, 1997. The associate editor coordinating the review of this paper and approving it for publication was Dr. Sawasd Tantaratana. B. Pˇseniˇcka, F. Garc´ıa-Ugalde, and J. Savage are with the Universidad Nacional Aut´onoma de M´exico, Facultad de Ingenier´ıa, M´exico (e-mail:
[email protected]). S. Herrera-Garc´ıa is with the Cinvestav Avenida Inst. Polyt. Nacional, M´exico City, Mexico (e-mail:
[email protected]). V. Dav´ıdek is with the the Department of Circuit Theory, Czech Technical University, Prague, Czech Republic. Publisher Item Identifier S 1053-587X(98)05950-9.
Fig. 3. State digital filter of the second order as an equivalent realization to the second canonical structure (direct realization form II) of a second-order IIR DF.
The DF analysis method described in [1] and [2] can be applied to obtain the state matrices of any DF. For the second canonical structure of the second order in Fig. 2 (direct realization form II), we obtain, e.g., (depending on the numbering of the state nodes)
A = 00
B1 B2
1 0
B=
A1 A2
0 0
A0 B1 A0 B2
CT = [1
0]
D=
A0 :
(7) By combining the state matrices (7) with the SDF structure in Fig. 1, we can obtain an equivalent filter for this particular canonical structure depicted in Fig. 3.
1053–587X/98$10.00 1998 IEEE
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 9, SEPTEMBER 1998
Fig. 4. Frequency response of the designed fourth-order DF’s for six-bit coefficients length and the Cauer approximation: Original DF frequency response (solid line), frequency response of the SDF realization (dotted line), and frequency response of the CDF realization (dashed line).
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Fig. 6. Frequency response of the state digital filter. TABLE I COMPARISON BETWEEN SDF AND CDF SENSITIVITIES
Fig. 5. Signal-to-round-off-noise ratios at the outputs of the analyzed DF’s for the Cauer approximation and different word lengths for both the canonical and the state-space realizations of the second-order DF sections. L1 norm-based scaling.
However, it is worth noting that this equivalent filter has not been designed with respect to any optimization criteria and, consequently, that the DF’s in Figs. 2 and 3 are equivalent only as far as the DF transfer function is concerned. Recently, a new interest for the research work in the field of the design of SDF’s has emerged due to the paper published by Smith et al. [3]. In this article, a thorough analysis of floating-point round-off noise in second-order state-space DF’s has been presented. More precisely, second-order state-space DF structures implemented using a floating-point arithmetic have been found to have output round-off error variance independent of any scaling of the state variables. Moreover, low fixed-point round-off noise realizations have been proved to show low floating-point round-off noise, and vice versa. Consequently, low round-off noise second-order state-space DF structures can be implemented in both a scaled and unscaled form using floating-point arithmetic without any degradation of the low-output round-off noise property. Since the article by Mullis and Roberts [4] that appeared in 1976, many contributions addressing the problem of the design of low noise
fixed-point second-order DF structures [5]–[16] have been published. If we substitute (6) in (2), then we can obtain the transfer function of the state digital filter in the form
01
02
H (z) = d + 1 + 1 z1 z0+1 + 2 z2 z02 where the constants
and
(8)
are expressed by equations
d=D 1 = b1 c1 + b2 c2 2 = c2 b1 a21 + c1 b2 a12 0 c1 b1 a22 0 c2 b2 a11 1 = 0tr A 2 = det A:
(9)
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A
A
and det and tr denote the determinant and the trace of the matrix , respectively. Based on (9), four equivalent realizations that differ only in coefficient signs are possible; they are characterized by the sets of SDF coefficients [11]
A
f f f f
g 01020102 g 22 1 0 2 1 0 2 g 22 0 1 2 0 1 2 g
a11 ; a11 ;
0 0
a12 ; a12 ;
0 0
b ;
a21 ; a
b ;
;b ;
a21 ; a
;
c ;
c ;d
b ;c ;
c ;d
b ;b ;
II. SENSITIVITIES OF SECOND-ORDER STATE DIGITAL FILTERS Let us consider the transfer function of a second-order DF in the form of (16)
a11 ; a12 ; a21 ; a22 ; b1 ; b2 ; c1 ; c2 ; d a11 ; a12 ; a21 ; a22 ;
In the following part, a new algorithm for the design of unscaled second-order SDF’s based on sensitivity analysis of the zeros and poles of the transfer function of second-order SDF’s will be derived.
2 0i Ai z i=0 2 0i = 1+ Bi z i=1
(10) H (z )
=
c ;c ;d :
01 11 2 =
z
01
p
= 1;
i
= 1; 2
=
and F2 (z )
=
(12)
01 + (a21 b1 0 a11 b2 )z 02 b2 z 1 0 tr Az 01 + det Az 02
(13)
z
c2
=0 =0
=
a11
z
z
04
2B2 2 A1 2A2
04
B2
0 16 A
(17)
:
A0 A2
z
B1
z
+ 2B2 z 01
=z
z
A1 z
02 + 2A2 z 03
A1 z
01 + 2A2 z 02
z
=z
z
=z
(18)
1
1
=
SA
=z
z
1
=
SA
01 + 2B2 z 02 1
=
SA
A1
+ 2A2 z 01
=z
z
where, for example, SA is the sensitivity of zeros z01;2 of the transfer function (4) with respect to the coefficient A1 =
SA
@ z01;2 @ A1
=
0
(14)
A0
=
A1
= b1 c1 + b2 c2
0 1 tr A
A2
=
c2 b1 a21
d
1 det A + 1 = 0tr A d
B
B2
= det
@ H (z )
@ H (z )
@ A1
@z
:
(19)
z =z
d
+ c1 b2 a12
0
c1 b1 a22
0
c2 b2 a11
(20)
A
:
By combining (5) and (20) and by applying the implicit function theory, we can obtain the sensitivities of the poles of the transfer function of a second-order SDF, taking into account the SDF coefficients, in the forms of z
Sa
z
Sa
z
Sa
has been demonstrated and gives the result of the design of low-noise, fixed-point, second-order state-space structures. Straightforward design equations for low-roundoff-noise statespace structures have been published in [11].
z Sa
= b2 c2
2
B1
B
B1 z
=
SB
(15)
b1 c1
0 16
Because (2) and (5) are expected to be equal, the following equations have to hold for the transfer function of the second order
lead to the design of the canonical (though scaled) structure in Fig. 3. Another example, which has characterized by [9, Eq. (18)] a22
(16)
1
=
SB
z
for the first and second state variable node, respectively. In fact, the value of p = 2 is usually applied to the design of broadband DF’s, whereas the value of p = 1 has proved to be appropriate when narrowband DF’s are synthesized. It can be shown that the scaling constraints (11), when appropriately handled, place two additional constraints on the SDF coefficients, thus leaving two degrees of freedom for further (scaled) design. For example, the constraining requirements a22
:
iz
z
01 + (a12 b2 0 a22 b1 )z 02 1 0 tr Az 01 + det Az 02
b1 z
=
(11)
where the transfer function Fi (z ) from the filter input to the ith state variable node takes the value of [13]. F1 (z )
z iz
z
By applying the implicit function theorem [17], the sensitivities of poles and zeros of the transfer function (4) with respect to the canonical filter coefficients can be derived as
Balanced Lp scaling has commonly been applied for fixed-point SDF implementations, requiring that all the Lp norms of the transfer functions from the filter input to each state variable node have the value of unity. That is, for second-order SDF’s
k
;
z01;2
—the maximization of the output signal-to-round-off-noise ratio; —the minimization of coefficient sensitivities; —freedom from limit cycles.
Fi (z )
0 0 01 ) 0 1 01 )
Then, the poles and zeros are given by the formulae
Clearly, four more coefficient constraints, which completely determine all of the second-order SDF coefficients up to the sign permutations of (10), must be defined. These allow four degrees of freedom in choosing the DF transfer function (4) realization. Generally, the four additional design constraints can be elicited from one or from a combination of a few of the following desirable goals: • a scaling of structure variables that aims to preserve an acceptable dynamic range for (usually fixed-point) variable representations; • a reduction of finite word-length effects, which frequently produces
k
2 (1 i=1 2 (1 i=1
A0
S
z
b
=
=
=
= =
a22 z B1 z
01 0 1
01 + 2B2 z 02
0
B1
a21 + 2B2 z 01
B1
a12 + 2B2 z 01
0
a11 z B1 z S
z
b
01 0 1
=z
z
(21) =z
z
01 + 2B2 z 02 =
z Sc
=z
z
z
=
=z z Sc
=0
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 9, SEPTEMBER 1998
1
z
where, for example, Sa is the sensitivity of poles z 1;2 of the transfer function of a second-order SDF taking into account the SDF coefficient a11
Saz
=
@z11;2 @a11
=
@ H1(z )
0 @a
11
@ H1(z) @z
:
= = = = =
When required, a scaled realization can always be obtained from an unscaled one by applying the transformations [13] 1
(23)
and then, the set of (24) and (25) can be derived according to the (5), (20), (21), and (23)1. One has been derived in [2] for a11 = a22 ; a12 = 0a21 ; c2 = 0b2 ; and b1 = c1 having the form
A B C T
D
where 0 ; 0 ; 0 , and 0 are the state matrices of the new (scaled) realization, and is a nonsingular diagonal (scaling transformation) matrix of the form
T=
c2 = 0b2 =
p
(
2 + 4 0 )=2
c1 = b1 = 0 + c22 : Another for the form
a11 = a22 ; a12 = 0a21 ; b1 = c2 ; and b2 = c1
takes
d = A0 a11 = a22 = 0B1 =2
a12 = 0a21 = B2 0 B12 = A1 0 A0 B1 = A2 0 A0 B2
= a11 + + p c1 = b2 = c2 = b1 = =2c1 :
4
2
=2a12
The set of (24) and (25) represent algorithms for the synthesis of unscaled second-order real coefficient SDF’s. 1 Here, there are nine SDF coefficients to compute and ten potential design constraints to satisfy.
0 t22 :
(27)
(28)
hold for this unscaled case. It is easy to see, from (26) and (27), that
a011 = a11 a022 = a22 b10 = b1 =t11 b20 = b2 =t22 c10 = c1 t11 c20 = c2 t22 :
(29)
By the way of substituting (29) into (28)
a011 = a022 b10 c10 = b20 c20 :
(30)
Therefore, the algorithm (25), which has been developed on the basis of a SDF coefficient sensitivity analysis, yields (after scaling) the design equations for the synthesis of minimum roundoff noise second-order scaled SDF’s. However, keeping the recent aforementioned developments by Smith et al. [3] in view, the scaling (26) is not essential when floating-point arithmetic is going to be applied to the SDF implementation. Then, an unscaled realization obtained through (25) can be directly used without any degradation of the minimum output round-off noise property. Moreover, consider the root stability condition for the second-order SDF’s
kz1 k < 1:
(25)
(a11 + ) + a212 2
0
a11 = a22 b1 c1 = b2 c2
4
(24)
t11
The diagonal elements t11 and t22 are given by the Lp norms of the transfer functions (12) and (13), respectively. Analytical expressions for calculating these Lp norms have been published in [14] for the cases of p = 2 and p = 1. Among the algorithms (24) and (25), the design equations in (25) deserve special attention due to the following. Consider the design constraints (15) yielding the synthesis of minimum round-off noise-scaled SDF’s. From (25)
d = A0 a11 = a22 = 0B1 =2
a12 = 0a21 = B2 0 B12 = A1 0 A0 B1 = A2 0 A0 B2 2 = ( + a22 )2 4a12
(26)
T
z =z
Saz Saz Scz Scz Saz
A0 = T0 AT B0 = T0 B C0 = T C D0 = D 1
(22)
Since the locations of poles and zeros are supposed to be the same for both the canonical digital filter CDF and SDF realization, by comparing (18) and (21), we can find the relationships between the sensitivities of the SDF’s and CDF’s shown in Table I. Jackson et al. [18] have shown that for the state-space structure described by (6), minimum round-off error under scaling L2 will be obtained if (15) is valid. Suppose the same transfer function (4) is realized by a CDF and SDF. If the following conditions are satisfied, (the primary goal is to spread the overall sensitivity uniformly to all SDF coefficients)
Saz Saz Sbz Sbz Saz
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(31)
1;2
In [5], a condition for a second-order SDF realization without limit cycles has been derived taking the form of the inequality
ka 0 a k + kz1 k 1: 11
22
1;2
2
(32)
Because of (31) and (28) or (30), this inequality is always satisfied by the coefficients of a stable SDF structure computed using (25). The possibility of realization of bandpass IIR DF’s by a cascade structure of second-order SDF’s (25), giving a lower sensitivity of the DF transfer function to coefficient quantization errors and a lower output round-off noise compared with the cascade structure of second-order CDF’s, will now be demonstrated.
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III. SOME PRACTICAL RESULTS To show some potential advantages of the proposed IIR DF realization scheme, four bandpass digital filters have been designed and analyzed having the following specifications. • Lowpass and highpass band edges are 250 Hz and 400 Hz, respectively. • Maximum attenuation in the passband is 3 dB. • The order of the filter is 4 (two sections of the second order). • The sampling frequency is 8000 Hz. The four DF’s have been designed for the following standard approximations of the ideal design specifications: BUT Butterworth approximation; CHE Chebyshev approximation; ICH inverse Chebyshev approximation; ELL elliptic (Cauer) approximation. Transfer functions of the digital filters have been computed by a program based on algorithms published in [15]. Filters have been designed in the cascade form of two sections of the second order. Both second-order sections have been realized by CDF and SDF. Coefficients of the second-order sections of the SDF’s have been obtained using the algorithm in (25). In order to evaluate effects of coefficient quantization2 (rounding), the transfer functions of the designed DF’s have been computed for both the canonical (direct realization form II) and the state-variable realizations of the second-order sections for different coefficient lengths. Fig. 4 illustrates the magnitudes of the transfer functions (frequency responses) of the designed DF’s for the Cauer approximation, for the canonical and state-space realization, and the 6-bit coefficients lengths. While the 16-bit quantization of the coefficients has not influenced the shapes of the magnitudes of the DF transfer functions, it is obvious from Fig. 4 that for Cauer approximation and 6-bit quantization, the frequency response of the SDF (dotted line) is much better in comparison with the frequency response of the CDF (dashed line). The designed DF’s have then been simulated for some particular coefficient lengths and a finite-length fixed-point arithmetic3 using a computer simulation program. As a further measure of the finite word-length effects [4], [16], [19], the signal-to-round-off noise ratio (SNR) in decibels has been evaluated at the outputs of the tested DF’s according to the formula SNR = 10 log
M y 2 (i) i=0
M [y(i) 0 y^(i)]2 i=0
;
M
2
N
(33)
where y (i) is the output sequence of the analyzed DF with the particular finite coefficient length (from 6 to 25 bits) simulated with full4 arithmetic, and y^(i) is the output sequence of the analyzed DF with the same finite coefficient length simulated with the fixed-point finite-length two’s complement saturation arithmetic. 2 Finite-precision numerical effects for a fixed-point arithmetic are divided into the four categories: 1) effects of coefficient quantization; 2) quantization noise and overflow errors in representing signals as fixedpoint numbers; 3) small-scale limit cycles due to the nonlinear quantization characteristics of fixed-point implementations; 4) large-scale limit cycles due to the nonlinear overflow characteristics of fixed point implementations. 3 Saturation 4A
two’s complement arithmetic with rounding has been simulated.
PC computer 32-bit floating-point arithmetic has been used to compute the reference output signal for the SNR evaluation.
The input signal for the experiments was a random sequence of = 1000 samples with the uniform probability density function nonzero in the open interval (01:0; 1:0). Both DF’s have had the same input signal and have been scaled as given below. Fig. 5 depicts the values of the SNR for Cauer approximations and different word lengths for both the canonical and the statevariable realization of the second-order DF sections. Each secondorder section has been individually scaled by applying a scaling constant located at the input to the particular second-order section. The value of the scaling constant G for the particular second-order section has been selected to be
M
M
= 1=max
G
k
i=0
j
k (i)j
(34)
h
where hk (i) denotes the impulse response from the DF input node to the kth DF node in which an overflow could occur. This kind of fixed-point implementation scaling is based on the L1 norm of hk (i)
k
k (i)kL
1
=
h
i=0
j
k (i)j
(35)
h
and guarantees that the signal value in each node of the network has been constrained to have a magnitude equal or less than 1 to avoid overflow, provided the number M is sufficiently large. IV. EXAMPLE As an example, we shall realize the state space digital notch filter and the canonic notch filter of sixth order having the transfer function H (z )
=
1:
0 0 125 581 039 01 + 1 02 :
z
:z
0 0 064 723 164 01 + 0 981 570 85 02 01 + 1 02 0 125 581 039 2 1 0 0 11210 635 794 01 + 0 938 155 107 02 01 + 1 02 1 0 0 125 581 039 2 1 0 0 184 000 618 01 + 0 981 570 85 02 1
:
z
:
:
:
:
z
z
:
:
:
z
:z
:
z
z
:
:z
z
:
:
z
(36)
Using the set of equations in (24), the values of the state-space digital filters in the cascade form are b21
=
b11
=
00 2489 00 0333 :
b22
=
:
b12
=
00 1841 00 1730 :
b23
=
:
b13
=
: :
d1
= 1:0000
d2
a111
= 0:0323
a112
= 0:0608
a113
= 0:0920
a121
= 0:9902
a122
= 0:9666
a123
= 0:9864
00 9901
= 1:0000
00 0489 00 2466
a211
=
a221
= 0:0323
a222
c21
= 0:2489
c22
:
a212
=
d3
= 1:0000
00 9666 :
= 0:0608
= 0:1841
a213
a223 c23
=
00 9864 :
= 0:0920
= 0:0489:
State digital filters and canonic digital filters have been implemented with the signal processors TMS320C25. By means of the simulator, we have obtained the samples of the impulse response, and from the 40 samples via FFT, we have get the result that is presented in Fig. 6. V. CONCLUSION The design equations of the SDF presented in this correspondence are very simple, suitable for a calculator. The digital filter of the canonic form and the state space digital notch filter were implemented on a Texas Instruments TMS320C25 system. The set of equations in (24) were used for the design of the state space digital filter. The data word length for the TMS320C25 is 16 bits, and multiplier coefficients
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 9, SEPTEMBER 1998
accuracy was 13 bits, with the result of each multiplication truncated to 16 bits. The analog-to-digital and digital-to-analog conversions on the system have 12-bit accuracy. However, the state-variable cascade realization behaves considerably better for all tested approximations compared with the canonical cascade realization when a suitable scaling has been applied. Clearly, the state-variable 16-bit realization with the fixedpoint two’s-complement saturation arithmetic (e.g., on the DSP chip TMS320C25) offers a possibility to increase the signal-to-round-offnoise ratio at the output of the cascade realization of IIR DF’s in comparison with the tested canonical cascade realization.
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On Time-Dependent Wavelet Denoising Brani Vidakovic and Concha Bielza Lozoya
Abstract—In this correspondence, we address the shrinkage of wavelet coefficients and induced denoising in the time domain by taking into consideration the “time” behavior of a noisy signal. We illustrate our time adaptation paradigm in a thresholding procedure utilizing Bayesian hypothesis tests. Both one-dimensional (1-D) and two-dimensional (2D) signals are considered in examples to motivate and implement our method. Index Terms—Denoising, image processing, wavelet shrinkage.
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Wavelet shrinkage is a simple yet powerful tool in statistical modeling of data sets and signals. It can be described as a three-step procedure. Step 1) A discrete signal is transformed into a set of wavelet coefficients. Step 2) A shrinkage of the coefficients is performed. Step 3) The shrunken wavelet coefficients are transformed back to the domain of the original signal. Wavelet shrinkage is usually done by thresholding wavelet coefficients. Thresholding is a rule by which the coefficients with absolute values smaller than a fixed threshold are replaced by zeroes. Some other thresholding policies and choices of thresholds are reviewed in [12]. Many researchers have observed that scale-dependent denoising methods in the wavelet domain can improve visual and other characteristics of signals and images. Our aim is to describe and apply one possible approach to the adaptivity of shrinkage rules. “Time” dependent shrinkage has the threshold that depends on the relative position of the coefficient within the scale level of the decomposition. This shrinkage rule is fully determined via an empirical Bayes argument in testing precise hypotheses in the wavelet domain. We emphasize that shrinkage and denoising are two related but not identical actions. The former is performed in the wavelet domain with wavelet coefficients as arguments. The latter is a consequence in the time domain of the shrinkage action. The two are connected via Meyer’s result that states that the magnitudes of the wavelet coefficients determine the smoothness space to which the signal under consideration belongs. For a formal statement of this connection, refer to [7], [8], and [10]. A large body of research in the statistical and engineering communities focused recently on performing shrinkage/denoising by modeling in the wavelet domain. The models are supported by the data to produce statistically optimal shrinking methods. An early reference is the work of Mallat [9], who proposed the exponential power distribution as a realistic statistical model for wavelet coefficients.
Manuscript received January 27, 1997; revised March 5, 1998. This work was supported by NSF Grant DMS-9626159. The associate editor coordinating the review of this paper and approving it for publication was Dr. Xiang-Gen Xia. B. Vidakovic is with the Institute of Statistics and Decision Theory, Duke University, Durham, NC 27708 USA (e-mail:
[email protected]). C. Bielza Lozoya is with the Universidad Politecnica de Madrid, Madrid, Spain (e-mail:
[email protected]). Publisher Item Identifier S 1053-587X(98)05963-7.
1053–587X/98$10.00 1998 IEEE
