Proceedings of the 17th World Congress
The International Federation of Automatic Control
Seoul, Korea, July 6-11, 2008

Stability of Signal Reconstruction Filters
via Cardinal Exponential Splines
Masaaki Nagahara ∗ Yutaka Yamamoto ∗∗Pramod P. Khargonekar ∗∗∗∗Department of Applied Analysis and Complex Dynamical Systems,
Graduate School of Informatics, Kyoto University, Kyoto 606-8501,
JAPAN (TEL: +81 75 753 5903; email: nagahara@i.kyoto-u.ac.jp).
∗∗
Department of Applied Analysis and Complex Dynamical Systems,
Graduate School of Informatics, Kyoto University, Kyoto 606-8501,
JAPAN (TEL: +81 75 753 5901; email: yy@i.kyoto-u.ac.jp).
∗∗∗
College of Engineering, University of Florida, 300 Weil Hall,
Gainesville, Florida, FL 32611-6550 U. S. A.
(TEL: +1 352 392 6000; email: ppk@ufl.edu).
Abstract: There is a new trend in digital signal processing. It is gradually recognized thatwhile the processing is done in the digital domain, its performance must be measured in theanalog domain. This framework was proposed by the present authors and co-workers, and also
recently proposed by Unser and his co-workers. While our approach relies on modern sampleddata control theory which minimizes an analog H ∞ performance criterion, Unser independently
proposed an oblique projection method. This paper examines their method, and shows that
their method often leads to instability of designed filters. A comparison with the sampled-data
method is made, along with some design examples, which shows the advantage of the sampleddata method.
1. INTRODUCTION
Expanding signals in terms of prespecified basis functions
is a fundamental problem in signal processing. Fourier andwavelet analysis are typical examples. In so doing, it isrequired that such an expansion gives a faithful result,i.e., the error is either zero or very small, and that it iseffective in the sense that one needs only a small numberof terms to attain sufficient accuracy.
Such an expansion is performed on an a priori given setof data on signals. In the digital context, such data arelikely to be given in the form of sampled values of a target
signal. Typically, one is given a set of uniformly sampled
signal values {f (nh)}∞n=−∞ , with sampling period h, andthen compute a series function expansion as desired. We
do not therefore know the intersample values of the signal
and hence we have only partial information of the original continuous-time signal f (t). This loss of information
should be in some way recovered to obtain a desirable
expansion.
A usual, and now quite standard, approach is to rely on
the sampling theorem (Shannon (1949); Zayed (1996)),
assuming that the original signal does not contain any
frequency components beyond the Nyquist frequency π/h,which is half of the sampling frequency 2π/h.
In examining the hypotheses underlying the sampling
theorem, we however notice that there can be certain
points that can be improved. For example, it is much more
⋆ This work was supported in part by the JSPS Grant-in-Aid for
Scientific Research (B) No. 18360203, and also by Grant-in-Aid for
Exploratory Research No. 17656138.

978-1-1234-7890-2/08/$20.00 © 2008 IFAC

natural to assume that the class of signals obeys a certain
frequency decay curve, extending even beyond the Nyquist
frequency. It is then desirable to design a digital (discretetime) filter that optimally reconstructs the original analog
signals. Due to the nature of analog characteristics, itis plausible that such an optimal filter gives a result
different from a more familiar and rather standard perfectreconstruction filters (Vaidyanathan (2000)) based on thediscrete-time performance measures.
Thus our desirable target is to design a digital (discretetime) filter that optimizes an analog performance index
for analog signals. This philosophy and framework were
proposed by the present authors in, e.g., Khargonekar
and Yamamoto (1996); Yamamoto et al. (2000). To thisend, we have shown that modern sampled-data control
theory can be very effectively used for signal processing.
This is more than natural since this new theory madeit possible to design a digital controller that optimallycontrols an analog performance of a continuous-time plant.
This new theory was developed in the 90’s and represents afundamental advance in the design philosophy of sampleddata control theory which was previously based on only
discrete-time performance measured at sampled points.
This new theory is adopted to digital signal processing in
Khargonekar and Yamamoto (1996). While there are sometechnical modifications necessary for this, this is basicallyin the scope of sampled-data theory, and now effectively
solvable. Due to the nature of analog characteristics,
this new theory can optimize the intersample responsesof the processed signals. This is in marked contrast to
the conventional philosophy of digital signal processing.

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17th IFAC World Congress (IFAC'08)
Seoul, Korea, July 6-11, 2008

x

H1(s)

S

c1

Q(z)

c2

H

H2(s)

y

Σ
Fig. 1. Sampled-data signal reconstruction.
In particular, this theory can optimize the analog H ∞performance index, which means the maximum of thefrequency response gain. Hence it can control the highfrequency intersample behavior even beyond the Nyquist
frequency.
A very similar philosophy is also presented and proposed
recently by Unser and co-workers (Unser (2005)). To comeup with a solution method, they introduced a new type of
splines called cardinal exponential B-splines (Unser andBlu (2005)). This consists of B-spline like functions thatare exponentials between sampled points. Assuming thatthe original signals are linear combinations of exponentials,
it is natural to conceive that such signals are expressible as combinations of such piecewise exponential basis
functions. The motto is “Think analog, act digital” in
Unser (2005) which is exactly the same as the one insampled-data signal processing theory since Khargonekar
and Yamamoto (1996).
Toward a solution of such an expansion, Unser (2005)
proposed a method based on oblique projections. The
performance is a square norm. They gave a system solution
to this problem.
This paper intends to closely examine the merit of Unser’s
method. In particular, we focus our attention to thestability of designed filters. In fact, Unser (2005) claims
that their method always gives rise to a stable filter.
We will examine this closely, and actually this claim
is indeed false, that is, the proposed method in Unser(2005) can, in general, lead to unstable filters. This is notcoincidental, but is closely connected to the limiting zerosof discretized continuous-time transfer functions, and it
shows that Unser (2005) neglects analog characteristics
described by plants with order more than 1.
The paper is organized as follows: We start with examining
Unser’s method, and then give a detailed stability analysis.
We will show the relation with Astrom’s fundamental
result, and show a counterexample in the stability analysis.
2. SIGNAL RECONSTRUCTION BY CARDINALEXPONENTIAL SPLINES
2.1 Perfect reconstruction by oblique projection
We state and review the signal reconstruction framework
by cardinal exponential B-splines (Unser (2005)). Consider
the system shown in Fig. 1. Hereafter we assume, without
loss of generality, that our sampling period is 1.
Let x(t) ∈ L2 denote an exogenous signal. Then it isfiltered through an analog low-pass filter H1 (s), becoming
semi-bandlimited following the characteristics of H1 (s). It
is further sampled by the ideal sampler
S : H1 L2 ∋ u → v ∈ ℓ2 , v[k] := u(k), k = 0, 1, 2, . . . ,and becomes a digital signal c1 [k]. We here note that if the
low-pass filter H1 (s) satisfies a decay estimate|H1 (jω)| ≤ M (1 + |ω|2 )−α/2
for some M > 0 and α > 1/2, then for every function u in
H1 L2 , the resulting sampled sequence v belongs to ℓ2 (see,e.g., Kannai and Weiss (1993)). This condition is satisfied,for example, for low-pass filters with transfer function ofrelative degree greater than or equal to 1. Then the digital
filter Q(z), to be designed below, produces another digital
signal c2 [k], which is converted back to an analog signalby the zero-order hold H defined by∞
β0 (t − k)v[k],H : ℓ2 ∋ v → u ∈ L2 , u(t) :=k=0

where β0 (t) is the hold function
1, t ∈ [0, 1),β0 (t) :=0, otherwise.
The output of H is then filtered through an analog filter
H2 (s) to obtain the final objective analog signal y(t). The
signal reconstruction problem is to find a digital filter
Q(z) which makes the output y as close to the input x
as possible.
Let h1 (t) and h2 (t) denote the impulse responses of H1 (s)and H2 (s), respectively. Then, the digital signal c1 [k] isrepresented as follows.
c1 [k] = S(h1 ∗ x)[k] ∞x(t)h1 (k − t)dt=
0 ∞=x(t)φ1 (t − k)dt
(φ1 (t) := h1 (−t))
0= x(·), φ1 (· − k).
On the other hand, the output analog signal y(t) isrepresented as follows.
∞y(t) =
c2 [k]φ2 (t − k),
k=0

φ2 (t) = [β0 (·) ∗ h2 (−·)](t).
Now define the following spaces:
V1 := span{φ1 (· − k)}∞k=0 ,V2 := span{φ2 (· − k)}∞
k=0 .

(1)

Summarizing, the above setup says the following: The
exogenous signal x is filtered through an analog filter
H1 (s). This in general limits the bandwidth of the filteredsignal, usually in a low-pass characteristic, according to
the decay characteristic of H1 (s). The filtered signal isthen sampled. The shifted linear span of φ1 (·) is V1 .
Likewise V2 is precisely the space of outputs producedby the hold device H and H2 (s). Note, however, that V1
never represents the totality of signals obtained by filteringx ∈ L2 by H1 . The latter space is infinite-dimensional,
while V1 is spanned by linear (possibly infinitely many)
combinations of shifted φ1 (t) = h1 (−t).
Suppose, for the moment, that we want to reconstruct
given signals in V1 with those in V2 . That is, we consider
the following signal reconstruction problem:
Problem A: Given an arbitrary exogenous signal in x ∈L2 , find y ∈ V2 such that y optimally approximates x in
the sense of L2 .

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w

H

u

H2 (s)xΣy(s − γ1 )(s − γ2 ) · · · (s − γM ),(s − α1 )(s − α2 ) · · · (s − αN )α := (α1 , . . . , αN ; γ1 , . . . , γM ).

Hα (s) =e−

Then the B-spline function corresponding to Hα (s) is
given by Unser (2005)

Fig. 2. Error system Ew
In an analogy with the well known projection theorem,
Unser (2005) proposed that x − y be orthogonal to V1 .
Note that this gives a best approximant if φ1 belongs to
V2 . Since φ1 does not necessarily belong to V2 , one needs amodification. Namely, the optimal reconstruction solution
proposed in Unser (2005) is characterized byx(·) − y(·), φ1 (· − k) = 0, k = 0, 1, 2, . . . .
(2)This idea states that there remains no extra component
in the error x − y that can be expanded with elements in
V1 . The filter Q(z) corresponds to the oblique projection
Unser (2000) of x(t) onto V2 orthogonal to V1 .
Definition 1. A linear operator P : L2 → V2 is an obliqueprojection on V2 orthogonal to V1 if the following three
conditions hold:(1) P x = x, ∀x ∈ V2 .
(2) x − P x ∈ V1⊥ , ∀x ∈ L2 .
(3) P x = 0, ∀x ∈ V1⊥ .
We quote the following proposition from Unser (2005):
Proposition 1. The optimal filter, denoted by Qop , achieving (2) makes the operator Σ (see Fig. 1) to be the obliqueprojection on V2 orthogonal to V1 .
Now consider the error system shown in Fig. 2. In thisfigure, Σ is the operator from x to y shown in Fig. 1. By
Ew we denote the error system given by Fig. 2. We thenhave the following proposition.
Proposition 2. The optimal filter Qop achieving (2) is also
optimal in the sense that
Qop (z) = argmin Ew 2 = argmin Ew ∞ .
Q(z)

Q(z)

Proof. Assume that the input w is the discrete-time delta
function
1, k = 0,δ[k] =
0, otherwise.
Then x ∈ V2 . Since Σ is an oblique projection,y = Σx = x.
That is, the impulse response of the error system Ew iszero.
2

Next, assume that the input w is in ℓ . In this case, x is
also in V2 and e = y −x = 0. That is, Qop (z) is the optimalfilter to make Ew ∞ = 0.
2
In the next subsection, we will show a formula for the filter
Qop (z).

β̂α (s) = ∆α (s)Hα (s),where ∆α (s) is the localization operator (Unser and Blu(2005); Unser (2005)) defined as∆α (s) = (1 − eα1 e−s )(1 − eα2 e−s ) · · · (1 − eαN e−s ). (3)Note that each 1 − eαi e−s is a truncation operator (Mirkin(2003)), and the support of βα (t) (the inverse Laplacetransform of β̂α (s)) is contained in [0, N ].
By using this B-spline function, the optimal filter Qop (z)is given as follows. Let the poles and zeros of H1 (s) andH2 (s) be
α1 = (α11 , . . . , α1N1 ; γ11 , . . . , γ1M1 ),α2 = (α21 , . . . , α2N2 ; γ21 , . . . , γ2M2 ).
For brevity, we assume that αij ’s are distinct, but this is
not at all necessary. Then the optimal Qop (z) is obtainedby the following equation (Unser (2005)):∆α (z)∆α2 (z)Qop (z) = N +N +1 1
,
(4)
1
2
−k
β(α1 :0:α2 ) (k)zk=0where β(α1 :0:α2 ) := βα1 ∗ β0 ∗ βα2 , and ∆αi (i = 1, 2) isthe discretization of the localization operator of ∆αi (s),that is,∆αi (z) =First, let us consider the rational transfer function(1 − eαin z −1 ),i = 1, 2.

(5)

n=1

The filter (4) is based on cardinal exponential B-splines,
and the realization of this filter is easily executed by
the spline calculus (see Unser and Blu (2005); Unser(2005)) 1 . On the other hand, this filter is realized via
system inversion.
Theorem 1. The optimal filter Qop (z) in (4) can be equivalently realized as
1,(6)
Qop (z) =H12d (z)where H12d (z) is the step-invariant equivalent discretization of H1 (s)H2 (s), that is, if a state-space realization of
H1 (s)H2 (s) is given by {A, B, C, 0}, then


 1
A

H12d (z) = SH1 (s)H2 (s)H =  e
C0eAτ Bdτ  .
0Proof. First, we consider the denominator of Qop (z). The
Laplace transform of β(α1 :0:α2 ) (t) is given byβ̂(α1 :0:α2 ) (s) = L β(α1 :0:α2 ) (s)= L β(α1 :α2 ) ∗ β0 (s)1 − e−s.
= β̂(α1 :α2 ) (s)s

2.2 Characterization by system inversion
The formula of the optimal Qop (z) is obtained by the
cardinal exponential B-spline method (Unser and Blu(2005); Unser (2005)).

Ni


1

However, this filter is not guaranteed to be stable, as we see in thenext section. Of course, if x belongs to V2 and if there are no errors
in computation, the formula still works for unstable filters, but inpractice this does not work.

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17th IFAC World Congress (IFAC'08)
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w

H12d

Qop

−

H

H2

e

1

0.8

0.6

Fig. 3. Block diagram equivalent to Fig. 2

0.4

The coefficients β(α1 :0:α2 ) (k), k = 0, 1, 2, . . . are obtainedby sampling the inverse Laplace transform of β̂(α1 :0:α2 ) (s).
Since (1 − e−s )/s is the laplace transform of the holdfunction β0 (t) of H, the denominator of Qop (z) is the ztransform of the step-invariant transformation of β(α1 :α2 ) ,that is,

0.2

0

−0.2

−0.4

−0.6

−0.8

S β̂(α1 :α2 ) (s)H = S∆α1 (s)∆α2 (s)H1 (s)H2 (s)H.

−1

By using the relation Se−s = z −1 S, we have (see (3) and(5))
S∆α1 (s)∆α2 (s) = ∆α1 (z)∆α2 (z)S.
It follows thatS β̂(α1 :α2 ) (s)H = ∆α1 (z)∆α2 (z)SH1 (s)H2 (s)H= ∆α1 (z)∆α2 (z)H12d (z).
Then, since the numerator of Qop (z) is ∆α1 (z)∆α2 (z), we
conclude that∆α1 (z)∆α2 (z)1
Qop (z) =.
=H
12d (z)S β̂(α1 :α2 ) H
2

0

10

20

30

40

50

Fig. 4. Time response of Unser’s reconstruction system
w

W

x

Σ

y

e
−

e−Ls

3. STABILITY OF CARDINAL EXPONENTIAL
B-SPLINE RECONSTRUCTION

Fig. 5. Error system Ew for sampled-data delayed signal
reconstruction
1
Qop (z) =H12d (z)z 3 − 0.7263z 2 + 0.1621z − 0.01111=
0.05725z 3 + 0.07827z 2 + 0.006011z
is unstable. This Qop agrees exactly with the one obtained
via oblique projections; see, e.g.,
http://bigwww.epfl.ch/demo/Esplines/.
Fig. 4 shows a time response of the reconstruction system.
The output clearly diverges.

It is claimed in Unser (2005) that the filter Qop (z) is stable
if cos(θ12 ) = 0, where θ12 is the angle between V1 and V2 .

4. ROBUST RECONSTRUCTION VIASAMPLED-DATA H ∞ OPTIMIZATION

We have shown in the previous subsection that the optimalfilter Qop (z) makes the transfer function of the error
system Ew (Fig. 2) zero. On the other hand, the block
diagram in Fig. 2 is equivalently transformed to Fig. 3.
From this block diagram, we easily see that the filter (6)makes the transfer function Ew zero.

However, this is not necessarily true according to thefollowing well-known result of limiting zeros by Åstöm etal. (1984):
Fact 3. For every continuous-time system with relative
degree strictly greater than 2, its step-invariant-discretizedsystem always possess an unstable zero provided that thesamping time is sufficiently small.
Even if the sampling time (= 1 in our present normalization) is not small compared to the time-constants of
H1 and H2 , the discretized system H12d (z) may still have
unstable zeros.
For example, consider
1,
H1 (s) =s+1

H2 (s) =1.
(s + 1.5)(s + 2)

An easy calculation via MATLAB shows cos(θ12 ) =0.4863 = 0. The zeros of the discretized system H12d (z)are{0, −1.28549, −0.0816767},
and hence the optimal filter

In the previous section, the reconstruction system proposed in Unser (2005) is not necessarily stable. Moreover,
Theorem 1 shows that this system does not take theintersample behavior into account.
In contrast, we have dealt in Khargonekar and Yamamoto(1996); Yamamoto et al. (2000) with the same problem via
sampled-data H ∞ control theory.
Consider the block diagram in Fig. 5 (cf. the error system
in Fig. 2). In this diagram, the input w is in L2 , and W
is a model of the input analog signal 2 . We assume that
W is a continuous-time linear time-invariant system withrational transfer function W (s). In the upper portion of
the diagram, input x is discretized, processed in discretetime, and then converted back to a continuous-time signaly (see Fig. 1). In the lower portion, the input analog
signal is delayed and forwarded to the adding point. The
2

In the case of the cardinal exponential B-spline reconstruction,
the input model W is H2 (s)H. The sampling theorem by Shannon
assumes that W is an ideal lowpass filter with cutoff frequency
ωc < π (Nyquist frequency).

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Seoul, Korea, July 6-11, 2008

25

−8

20

−9

15

−10

10
5 −2
10

−1

10
Normalized Frequency (×π rad/sample)

−11
Gain
(dB) −12

0

10

−13

Phase (degrees)

60

−14
40
−15
20
0 −2
10

−16
−1

10
Normalized Frequency (×π rad/sample)−17 −310

0

10

Fig. 6. Bode plot of reconstruction filters Qop (z): cardinal
exponential B-spline method (dot) sampled-data H ∞optimization (solid)

−1

0

10
Frequency (red/s)

10

Fig. 7. Frequency response (gain) of error system Ew :exponential B-spline method (dot) sampled-data H ∞optimization (solid)

objective is to attenuate the reconstruction error e withrespect to the H ∞ norm of the error system Ew . The
optimal filter is then obtained by modern sampled-data
control theory, and has the following advantages:2
1.5(1) the optimal filter is always stable;(2) the design takes the inter-sample behavior into account;(3) the system is robust against the uncertainty of W ;(4) the optimal FIR filter is also obtainable; via Linear
Matrix Inequalities (LMI).

1
0.5
0
−0.5

5. NUMERICAL EXAMPLES

−1
−1.5

Here we design the reconstruction filter Qop (z) by• cardinal exponential B-spline reconstruction (Unser(2005)), and• sampled-data H ∞ optimization (Khargonekar andYamamoto (1996); Yamamoto et al. (2000)).
Example 1. The analog filters are the same as those given
in Unser (2005)
1
1
H1 (s) =, H2 (s) =
.
s+1
s+2
The reconstruction delay is set to be m = 1 to beconsistent with Unser (2005). While the setup of Unser(2005) appears quite similar to the sampled-data setup(Fig. 5), we do need yet another transfer function W (s) tomake the sampled-data design work 3 . We here take W (s)as1
W (s) =.
s + 0.05
Fig. 6 shows the optimal filter Qop (z) designed by the
cardinal exponential B-spline method and sampled-data
H ∞ optimization. Fig. 7 shows the frequency response of
the error system Ew shown in Fig. 5. Sampled-data H ∞optimized system shows almost a flat characteristic. On
the other hand, the exponential B-spline reconstruction
3

−2

10

This W cannot be absorbed to H1 (s) in Unser’s framework, since
what is to be reconstructed is the weighted w, namely x, while Unser(2005) attempts to track unweighted input w, but with the expenseof stability as we already pointed out.

−2
0

5

10

15

20

25

30

35

40

Fig. 8. Time response (exponential B-spline reconstruction)system shows much larger error in high frequencies. Figures 8 and 9 show the time responses against the squarewave by the exponential B-spline reconstruction system,
and by the sampled-data designed filter. The former showsa large amount of ripples around the edges.
In the above example, the sampling time 1 is close to thetime constants of H1 and H2 . This makes the advantageof the sampled-data design a little ambiguous. In the next
example, where the time constant of H2 is large comparedto the sampling period, there is a more obvious distinction.
Example 2. We take H2 (s) = 1/(s + 0.05). H1 and W
are the same as before. The zeros of the step-invariantdiscretization H12d of H1 (s)H2 (s) are {0, −0.7063}, andhence the filter based on exponential B-splines is stable. Indeed, we have Qop (z) = (z 2 −1.319z +0.3499)/(0.3614z 2 +0.2552z). This filter leads to one step delay (m = 1). On
the other hand, our design is free from the limit of thedelay, and hence we design the optimal filter with m = 1
and m = 4.

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and can take the intersample behavior into accout. The
numerical examples show the advantages of the sampleddata H ∞ design. The sampled-data design has also beenstudied extensively for sample-rate conversion (Ishii etal. (1999); Nagahara and Yamamoto (2000); Yamamoto(2006b)), fractional delay filters (Nagahara and Yamamoto(2003)), digital image processing (Kakemizu et al. (2005))
etc. For details, the readers are referred to these articles
and references therein.

2
1.5
1
0.5
0
−0.5

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−1.5
−2
0

5

10

15

20

25

30

35

40

Fig. 9. Time response (sampled-data H ∞ optimization)
−26

−28

−30

−32

Gain (dB)

m=1

−34

−36

−38

−40
m=4

−42

−44 −3
10

−2

10

−1

10

0

10

Frequency (rad/s)

Fig. 10. Frequency response (gain) of error system Ew : exponential B-spline method (dash) sampled-data H ∞optimization with m = 1 (dash-dot) and m = 4 (solid)
Fig. 10 shows the frequency response of the error system Ew shown in Fig. 5. Sampled-data H ∞ optimized
system (m = 1) shows almost a flat characteristic. On
the other hand, the exponential B-spline reconstruction
system shows much larger error in high frequencies. Notealso that as we increase the delay step m, the error issubstantially reduced in the sampled-data design. While
we omit their time ersponses due to the lack of space,the design via cardinal exponential B-splines shows muchlarger ripples around the edges for square-wave responses.
6. CONCLUDING REMARKS
We have shown that the reconstruction filter proposed in
Unser (2005) is the inverse of the step-invariant discretization of the pertinent analog filters. Hence this reconstruction system does not take the intersample behavior ofcontinuous-time signals into account. We have also shownthat the obtained filter is not necessarily stable in spite of
the claim in Unser (2005). In contrast, the sampled-data
H ∞ optimal reconstruction proposed in Khargonekar andYamamoto (1996); Yamamoto et al. (2000) is always stable1419