Precompensation for anticipated erasures in LTI interpolation systems

Abstract

This paper considers compensation of anticipated erasures in a discrete-time (DT) signal such that the desired interpolation can still be accomplished, with minimum error, through a linear time-invariant (LTI) filter. The algorithms presented may potentially be useful in the compensation of a fault in a digital-to-analog converter where samples are dropped at known locations prior to reconstruction. Four algorithms are developed. The first is a general solution that, in the presence of erasures, minimizes the squared error for arbitrary LTI interpolation filters. In certain cases, e.g., oversampling and a sinc-interpolating filter, this solution is specialized so it perfectly compensates for erasures. The second solution is an approximation to the general solution that computes the optimal, finite-length compensation for arbitrary LTI interpolation filters. The third is a finite-length windowed version of the oversampled, sinc-interpolating solution using discrete prolate spheroidal sequences. The last is an iterative algorithm in the class of projection onto convex sets. Analysis and results from numerical simulations are presented.

Full text

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. XX, XX 20051
Pre-compensation for Anticipated Erasures in LTI
Interpolation Systems
Sourav Dey, Member, IEEE, Andrew Russell, Member, IEEE, and Alan Oppenheim, Fellow, IEEE
Abstract—This paper considers compensation of anticipated
erasures in a discrete-time (DT) signal such that the desired inter-
polation can still be accomplished, with minimum error, through
a linear time-invariant (LTI) filter. The algorithms presented
may potentially be useful in the compensation of a fault in a
digital-to-analog converter where samples are dropped at known
locations prior to reconstruction. We develop four algorithms.
The first is a general solution that, in the presence of erasures,
minimizes the squared error for arbitrary LTI interpolation
filters. In certain cases, e.g. oversampling and a sinc-interpolating
filter, we specialize this solution so it perfectly compensates for
erasures. The second solution is an approximation to the general
solution that computes the optimal, finite-length compensation
for arbitrary LTI interpolation filters. The third is a finite-
length, windowed version of the oversampled, sinc-interpolating
solution using discrete prolate spheroidal sequences. The last is
an iterative algorithm in the class of projection onto convex sets.
Analysis and results from numerical simulations are presented.
Index Terms—erasures, interpolation, LTI reconstruction,
erasure compensation, discrete prolate sphroidal sequences,
projection-onto-convex-sets, broken pixels
EDICS: DSP-RECO Signal Reconstruction or
DSP-SAMP Sampling, Extrapolation, and Interpolation
I. INTRODUCTION
I
variety of techniques for signal processing. In one of the most
common forms of interpolation, the DT signal is represented
as impulses on a uniformly spaced grid and the interpolation is
accomplished by low-pass filtering this uniform impulse train.
In most situations, the lowpass filter is fixed. Thus, if erasures
occur in the DT signal, without additional compensation, the
desired interpolation will be distorted.
For example, consider a system designed for low-pass
reconstruction from uniform samples in which, perhaps be-
cause of a faulty digital-to-analog (D/A) converter, specific
samples are forced to zero. This problem might occur in a
flat-panel display with defective pixel LEDs. Such displays
inherently rely on a form of low-pass filtering accomplished
by viewing the display from an appropriate distance. As
illustrated in Figure 6(a), with defective LEDs and without
additional compensation, the perceived output is degraded. As
NTERPOLATING a continuous-time (CT) signal from a
discrete-time (DT) representation is an integral part of a
Manuscript received June 3, 2004; revised December 9, 2004. The associate
editor coordinating the review of this manuscript and approving it for
publication was Dr. Fredrik Gustafsson.
S. Dey, A. Russell, and A. Oppenheim are with the Digital Signal Process-
ing Group in the Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology, Room 36-615, 77 Massachusetts Av-
enue, Cambridge, MA 02139 USA (e-mail: sdey@mit.edu; air@alum.mit.edu;
avo@mit.edu)
we develop in this paper, under certain conditions, it is possible
to compensate for the erasures by adjusting the other sample
values, and still achieve a suitable reconstruction at the output
of a linear time-invariant (LTI) filter.
It is important to note that this problem is not one of data
recovery. It is assumed that the correct values of the DT signal
are known. It is the conversion process preceding the low-pass
filter that forces particular values to zero. In the example of
the display, the video card knows the value to transmit to the
defective pixel, but the pixel itself cannot display it because
it is broken. In anticipation of these erasures, we can change
the original DT signal so the distortion after interpolation is
reduced.
II. PROBLEM STATEMENT
Mathematically, the problem can be viewed as one of
sampling grid conversion. Consider a discrete-time (DT) rep-
resentation, x[n], on a uniform grid I,
I = {0,±1,±2,±3,...}
such that conversion to a uniform impulse train with spacing
T and interpolation using a filter, h(t), returns our desired
continuous-time (CT) signal r(t),
?
It should be emphasized that x[n] is not restricted to
being samples of a function. Compensation of the erasure
pertains only to the interpolation of a CT signal from a DT
representation.
Because of the erasure, we are forced to represent the signal
on a non-uniform grid, I?, that is the grid I with one point
removed. For convenience, and without loss of generality, we
choose n = 0 as the erasure point so that
r(t) =
n∈I
x[n]h(t − nT)
(1)
I?= {±1,±2,±3,...}
Our goal is to find a DT representation, ˆ x[n], on the non-
uniform grid, I?, that minimizes the squared interpolation
error. Specifically, we desire an interpolation ˆ r(t),
?
that minimizes the energy of the error,
?∞
ˆ r(t) =
n∈I?
ˆ x[n]h(t − nT)
(2)
E2=
−∞
|ˆ r(t) − r(t)|2dt =
?∞
−∞
|e(t)|2dt (3)

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where e(t) = ˆ r(t) − r(t). We can equivalently express (2) as
∞
?
if we impose the additional constraint that ˆ x[0] = 0. For
convenience, we define ˆ x[n] − x[n] = c[n], or equivalently,
ˆ x[n] = x[n] + c[n]. Combining (1), (4), and our definition
of c[n], e(t) can be expressed as an expansion in the basis
{h(t − nT)},
∞
?
with the constraint c[0] = −x[0]. We can equivalently rewrite
equation (3),
?
n=−∞
ˆ r(t) =
n=−∞
ˆ x[n]h(t − nT)
(4)
e(t) =
n=−∞
c[n]h(t − nT)
(5)
E2=
?∞
−∞
∞
?
c[n]h(t − nT)
?2
dt(6)
The objective then is to determine c[n] for n = 0 that
minimizes E2. It is important to note that expressing ˆ x[n]
in this form of x[n] with additive compensation c[n] is
not restrictive, both in the general case and in our further
development where we impose certain additional restrictions
on ˆ x[n]. While in this paper we only consider full erasure, we
can generalize to the case where the affected value is fixed to
any constant value, ˆ x[0] = C. In this case, the constraint on
the compensation is c[0] = −x[0] + C. For convenience, and
without loss of generality, we focus exclusively on the case
where C = 0, corresponding to full erasure.
If {h(t − nT)} forms an orthogonal set, e.g the shift-
orthogonal sinc-interpolating kernel or B-splines, then by
Parseval’s relation,
?∞
The scaling on the right side of (7) assumes that ?h(t −
nT)?2 = T2, as is standard in sampling theory. Note that
in the orthogonal case from (7), the error is the energy
of c[n]. Since the value of c[0] is constrained, the optimal
compensation signal is
−∞
|e(t)|2dt = T
∞
?
n=−∞
|c[n]|2
(7)
c[n] = −x[0]δ[n]
(8)
This is a degenerate solution, equivalent to not compen-
sating the erasure. Consequently, non-trivial solutions are
possible only if {h(t − nT)} is not an orthogonal set.
In Section 3 of this paper, we discuss the general solution
to the problem, where h(t) is an arbitrary LTI filter and all
values of x[n] may be adjusted. In Section 4, we focus on
the case where h(t) is bandlimited and derive solutions that
compensates with zero error for the case of an oversampled,
sinc-interpolating filter (LPF). In Section 5, we consider the
case in which only a finite number of values may be adjusted,
and we derive the optimal solution for arbitrary LTI inter-
polation filters. Sections 6 and 7 present two alternatives for
finite-length compensation, a heuristic solution that is an ap-
proximation to the optimal solution and an iterative algorithm
which converges to the optimal solution for oversampled, sinc-
interpolating filters. Lastly, in Section 8, we discuss how the
algorithms developed may be use to compensate an image with
pixels that are permanently set to zero.
III. GENERAL SOLUTION
In this section we derive the optimal compensation for
arbitrary LTI interpolation filters h(t). We first focus on the
case of a single erasure. In that case, the solution can be
computed by minimizing the objective function (6) subject
to the constraint p = c[0] + x[0] = 0, using the method of
Lagrange multipliers. Defining the Lagrangian q = E2+ λp
and setting to zero the partial derivatives with respect to c[k]
for all k = 0,
∂
∂c[k]q =
?∞
−∞
2
?
∞
?
n=−∞
c[n]h(t − nT)
?
h(t − kT)
?
(9)
= 2
∞
?
n=−∞
c[n]
??∞
−∞
h(t − nT)h(t − kT)dt
= 0
(10)
which simplifies to,
∞
?
n=−∞
c[n]φhh((k − n)T) = 0
(11)
where φhh(τ) is the CT deterministic autocorrelation of the
filter h(t), defined as
?∞
For k = 0, the derivative has an extra term with λ?, the
Lagrange multiplier. Specifically,
φhh(τ) =
−∞
h(t)h(t − τ)dt(12)
∂
∂c[0]q = 2
∞
?
n=−∞
c[n]φhh(−nT) + λ?= 0
(13)
Eqs. (11) and (13) can be combined into a single condition
(14), where all the constant terms have been incorporated into
λ = −1
∞
?
The second derivative in both cases reduces to,
?∞
which is always positive, ensuring that the optimization
finds a minimum. Since n and k are integers, the values
φhh((n−k)T) are uniform samples of the CT autocorrelation
φhh(τ). If we define a DT sequence φhh[n] = φhh(nT) of
the uniform samples, equation (14) can then be expressed in
the frequency domain as (16), where Φhh(ejω) is the discrete-
time Fourier transform (DTFT) of φhh[n]. Note that since h(t)
is not restricted to be bandlimited, Φhh(ejω) in general has
aliased components. We should also note that when h(t) is
2λ?.
n=−∞
c[n]φhh((k − n)T) = λδ[k]
(14)
∂2
∂c[k]2q = 2
−∞
|h(t − kT)|2dt = 2φhh(0)
(15)
not bandlimited to π/T, φhh[n] =?
mh[m]h[m−n], the DT
autocorrelation of the sampled filter h[n] = h(nT).

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Incorporating the constraint c[0] = −x[0], the Lagrange op-
timization reduces to two equations in the frequency domain,
C(ejω)Φhh(ejω) = λ
?
(16)
1
2π
<2π>
C(ejω)dω = −x[0]
(17)
where λ and C(ejω) are the unknowns. In the case where,
Φhh(ejω) = 0,∀ω and the integral, κ =
?∞
−∞
1
Φhh(ejω)dω
converges, these two equations have a unique solution,
Copt(ejω) =
−x[0]/κ
Φhh(ejω)
(18)
In the case where ∃ω : Φhh(ejω) = 0, the solution is not
unique and cannot be represented by (18). In this important
case, where Φhh(ejω) has zeros on the unit circle, the optimal
solutions compensate perfectly with zero error. We can derive
the optimal solution by inspection, expressing the error (6) in
the frequency domain. To express (6) in the frequency domain,
we first expand the square and define a new signal k[n],
?∞
nm
??∞
?
?
K(ejω), the DTFT of k[n], can be expressed as,
E2=
−∞
??
?
c[n]c[m]h(t − nT)h(t − mT)
?
dt
(19)
=
?
n
?
m
c[n]c[m]
−∞
??
h(t − nT)h(t − mT)dt
?
(20)
=
n
c[n]
m
c[m]φhh[n − m]
??
?
?
k[n]
(21)
K(ejω) = C(ejω)Φhh(ejω)
(22)
Eq. (21) is the inner-product of c[n] and k[n]. Using
Plancherel’s theorem for DT sequences [1], we can express
(21) in frequency domain as,
?
=
<2π>
where Φ∗
symmetric. If c[n] has frequency components only where
Φhh(ejω) = 0, while meeting the constraint c[0] = −x[0], it
results in perfect compensation with zero error. A particularly
simple solution is a pair of impulses, i.e. if Φhh(ejωz) = 0
then,
E2=
<2π>
C(ejω)K∗(ejω)dω(23)
?
|C(ejω)|2Φhh(ejω)dω(24)
hh(ejω) = Φhh(ejω) because φhh[n] is even and
Copt(ejω) = −x[0]π (δ(ω − ωz) + δ(ω + ωz))
is a solution that compensates with zero error.
Generalization to multiple erasures requires constrained
minimization with multiple constraints. For example, assume
that there are two erasures at indices n1and n2. The erasures
specify two constraints,
(25)
p1= c[n1] + x[n1] = 0
p2= c[n2] + x[n2] = 0
(26)
As before, we minimize E2subject to these constraints.
Defining the Lagrangian q = E2+λ?
zero the partial derivatives with respect to c[k] for all k results
in a condition of the form,
1p1+λ?
2p2and setting to
∞
?
n=−∞
where λ1= −1
(27) can be expressed as,
c[n]φhh((k−n)T) = λ1δ[k−m1]+λ2δ[k−m2] (27)
2λ?
1and λ2= −1
2λ?
2. In the frequency domain
C(ejω)Φhh(ejω) = λ1e−jωm1+ λ2e−jωm2
(28)
As in the single-sample case, if Φhh(ejω) = 0,∀ω then
a unique solution to (28) exists. If ∃ω1,ω2 : Φhh(ejω1) =
0,Φhh(ejω2) = 0,|ω1| = |ω2|, then solutions that compensate
with zero error exist. The exact form of the multiple-erasure
perfect compensation solutions are explored in the following
section. Both of these solutions, the unique one and the perfect
one, generalize to N erasures in a straightforward manner.
IV. BANDLIMITED SOLUTIONS
In this section, we focus on the special case where e(t) is
bandlimited and h(t) is a band-limiting filter with H(jω) = 0
for ω > π/T. In this case, by Parseval’s relation, there is
perfect norm equivalence between CT and DT and the error
can be determined directly in the discrete-time domain,
?∞
Where e[n] = e(nT) is samples of e(t), expressed as c[n]
filtered by h[n] = h(nT), a sampled representation of the
interpolation filter h(t),
−∞
|e(t)|2dt = T
∞
?
n=−∞
|e[n]|2
(29)
e[n] = e(nT) =
∞
?
k=−∞
c[k]h(T(n −k)) =
∞
?
k=−∞
c[k]h[n −k]
(30)
In this case, the entire problem can be posed in DT.
Combining (30), (29), and (3), our error metric reduces in
the frequency domain to
?
This expression in analogous to (24), except since h(t)
is band-limited, Φhh(ejω) = |H(ejω)|2, where H(ejω) is
the DTFT of the sampled CT filter h[n] = h(nT). If
E2=T
2π
<2π>
|H(ejω)C(ejω)|2dω(31)
∀ω,H(ejω) = 0 the optimal solution for the band-limited case
is,
Copt(ejω) =
−x[0]/κ
|H(ejω)|2
(32)
where κ =?∞
x[n] are samples of a band-limited function and h(t) is an
ideal low-pass, sinc-interpolating filter with a guard-band.
Specifically, let x(t) be a band-limited, continuous-time
signal that is at least slightly oversampled. In addition, we
assume that 1/T = RΩc/π, where x(t) is band-limited to
Ωcand R > 1 is the oversampling ratio. We denote the ratio
−∞
1
|H(ejω)|2dω. The solution (32) precludes an
important special case: the classical sampling model where

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π/R by γ. x(t) is represented by its samples x[n] = x(nT)
on the uniform grid I. Low-pass filtering the scaled impulse-
train p(t) =?∞
response of an ideal, sinc-interpolating filter with cutoff Ωc.
Note that because of oversampling, Ωc< π/T so {h(t−nT)}
is not an orthogonal set. Since h(t) is bandlimited, we can pose
the problem in DT. In this case, h[n] = h(nT), an ideal DT
sinc-interpolating filter with cutoff γ = ΩcT = π/R and gain
1/T. Without loss of generality, we normalize the time and
frequency axes such that T = 1 and Ωc= π/R.
Proceeding analogously to the previous section, we can find
a solution that compensates with zero error. According to (31),
minimizing the energy of e[n] is equivalent to minimizing the
energy of C(ejω) in the pass-band [−γ,γ] of the interpolation
filter H(ejω). If c[n] has no frequency components for |ω| <
γ, while meeting the constraint c[0] = −x[0], it results in
perfect compensation with zero error. There are an unlimited
number of signals that meet this criteria. A particularly simple
solution is,
cideal[n] = −x[0](−1)n
in which case Cideal(ejω) is an impulse at ω = π. In theory,
this solution only requires that R = 1+?, where ? is non-zero
but otherwise arbitrarily small. More broadly, it can be shown
in a straightforward manner that any high-pass signal c[n],
that meets the constraint
2π
frequency components for |ω| < γ is a solution.
For a fixed amount of oversampling, perfect compensation
can be extended to multiple erasures. The problem is nomi-
nally more difficult because there are multiple constraints to
satisfy. For example, with two erasures at indices i and j,
perfect compensation can be of the form,
n=−∞x[n]δ(t − tn) through h(t) gives our
desired interpolation r(t), where h(t) =sin(Ωct)
πt
is the impulse
(33)
1
?π
−πC(ejω)dω = −x[0], with no
c[n] = α1c1[n − i] + α2c2[n − j]
Where c1[n] and c2[n] are perfect compensation sequences for
the erasures at i and j, respectively. Imposing the constraints
c[i] = −x[i] and c[j] = −x[j], we define a 2 × 2 system of
linear equations that can be solved for the scale factors, α1
and α2,
(34)
α1c1[0] + α2c2[i − j] = −x[i]
α1c1[j − i] + α2c2[0] = −x[j]
For a guard-band of size ? > 0, we can always choose c1[n]
and c2[n] such that these linear equations are non-singular. In
fact, we can always choose sinusoids of different frequencies
such that the resulting linear equations have a unique solution.
For example, in the 2×2 case, we can choose c1[n] = cos[πn]
and c2[n] = cos[(π −?
with a guardband, compensating N erasures requires solving
an N ×N system of linear equations for the scale factors. The
error remains zero because the sum of band-limited signals is
also band-limited, i.e. the space of finite-energy band-limited
signals is closed. This reasoning can also be used to extend
the perfect solutions for the general case, where h(t) is not
bandlimited, to compensate for multiple erasures.
(35)
(36)
2)n]. Thus, in the case of an ideal LPF
V. OPTIMAL FINITE APPROXIMATION
While the solutions of Section 3 and 4 are optimal and some
even result in zero error, they typically have infinite length. In
addition, computation of the filter inverse generally requires
additional complex calculations in the form of spectral factor-
ization. Consequently, optimal compensation is impractical to
implement.
In this section, to mitigate these problems, we design c[n]
with the constraint of finite length. The solution we develop
is general and applicable when h(t) is an arbitrary LTI filter.
We impose the constraint that
ˆ x[n] = x[n], n = N
(37)
where N is the finite set of points to be adjusted. This is
equivalent to restricting the compensation signal, c[n], to be
non-zero only for n ∈ N. In general, the set N may be
non-sequential, but for simplicity, we focus on symmetric
sequential compensation where N= [−N−1
the derivation below is valid for any set N with |N| = N.
Following a derivation analogous to that in Section 2, we
use Lagrange optimization with the additional finite length
constraint. The minimization produces N +1 linear equations,
?
?
c[0] + x[0] = 0
2,N−1
2]. However,
n∈N
c[n]φhh((k − n)T) = 0, k = 0
(38)
n∈N
c[n]φhh(−nT) +λ
2= 0, k = 0
(39)
(40)
As shown in Section 3, the second-derivatives are all
positive, thus equations (38), (39), (40) have a unique solution
corresponding to the optimal compensation signal for the given
value of N. We express these equations in block matrix form,
?
where λ is the Lagrange multiplier, δ is a vector with all zero
entries except for a 1 as the center element, and cnis a vector
representation of c[n]. φhh is a Toeplitz, symmetric matrix
containing N samples of φhh(τ), the autocorrelation of h(t).
φhh
δT
1
2δ
0
??
cn
λ
?
=
?
0
−x[0]
?
(41)
φhh(i,j) = φhh((i − j)T), |i − j| ∈ N
For certain degenerate interpolation filters, φhh may be
singular. We do not consider such cases. Assuming φhh
is invertible, the block matrix (41) can be solved for the
Lagrange multiplier and optimal finite-length compensation
signal, cofax[n].
λ =2x[0]
(42)
κ
(43)
cofax[n] = −x[0]
κ
φ−1
hhδ(44)
where κ = δTφ−1
by (46) as Optimal Finite Approximation (OFAX). There
are efficient numerical techniques for computing cofax[n] that
exploit the Toeplitz, symmetric structure of φhh. In any case,
the computation only needs to be done once, since once c[n] is
determined it can be stored and retrieved when the algorithm
hhδ. We refer to the algorithm represented

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needs to be applied. This is the case with all of the finite-length
approximations explored, OFAX, DPAX, and IA.
The development above is general for any form of the
interpolation filter h(t). In the remaining part of this section,
we focus our analysis on the case that h(t) is an oversampled,
sinc-interpolating filter and the time-axis is normalized such
that T = 1. As shown in Section 4, the problem can be posed
directly in DT when h(t) is band-limited. In this case, h[n]
is a DT sinc-interpolating filter with cutoff γ. We denote its
autocorrelation matrix using Θγ, reserving φhhfor the general
case. Using frequency domain arguments, it is straightforward
to show that,
Θγ(i,j) = sinc(γ(i − j))
(45)
From [2] we know Θγis invertible. In this case, the OFAX
solution is,
cofax[n] = −x[0]
where κ = δTΘ−1
The OFAX algorithm was implemented for oversampled,
sinc-interpolating filters in MATLAB and examples of cofax[n]
were computed in which x[0] = −1. Figure 1(a) illustrates
cofax[n] and |Cofax(ejω)|, the magnitude of a 2048-point zero-
padded discrete Fourier transform (DFT), for various values
of N and γ = 0.9π. Figure 1(b) illustrates the same for
γ = 0.7π. As expected, cofax[n] is high-pass with the main-
lobe of the DFT centered at ω = π, and smaller energy
side-lobes at lower frequencies. As N increases, the energy
in the pass-band [−γ,γ] decreases, thus decreasing the error,
E2. Furthermore, for the same N, the solution for γ = 0.7π
performs better than that for γ = 0.9π because there is a larger
guard-band. Intuitively, the system is more oversampled, and
there is greater redundancy, so a better solution can be found
using fewer points.
Figure 2(a) illustrates E2as a function of N. The graph
shows that E2decreases approximately exponentially in N.
Since the OFAX algorithm generates the optimal solution,
the error curves shown in Figure 2(a) serve as a baseline for
performance of other finite-length choices for c[n].
In this case, where h(t) is an ideal, sinc-interpolating filter,
the solution becomes numerically unstable to the precision of
MATLAB, beyond E2= 10−9. In practical systems, though
the interpolation filter is never ideal. In general, non-ideal
h(t) impose looser restrictions, making the OFAX solution
better conditioned and computable to lower error values. Also,
in either case, the error can be made arbitrarily low by
performing the computation on a computer with arbitrarily
high precision. E2= 10−9is thus a worst-case bound on
minimum achievable error. Even then, in most contexts, an
error of 10−9= −180dB, compared to the signal, is more
than sufficient.
As in Section 3, generalization to multiple erasures requires
constrained minimization with multiple constraints. The devel-
opment is analogous to that above, except there are multiple
Lagrange multiplier terms, one for each erasure. In addition,
N will likely be non-sequential, with a group of points around
each erasure. Though rigorous, globally minimizing the OFAX
solution using every erasure is unnecessary in most cases. If
κ
Θ−1
γδ(46)
γδ.
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Fig. 1.
sampled, sinc-interpolating filters computed using OFAX. The plots for two
different cutoffs, γ = 0.7π and 0.9π, and N =7, 11, 21 are on the left. The
2048-point zero-padded Fourier Transform of each of these sequences is on
the right. The transforms are linearly interpolated for display.
Optimal finite-length compensation sequences, cofax[n], for over-
erasures are widely spaced, a reasonable result can be achieved
by superimposing single-erasure OFAX solutions.
VI. DISCRETE PROLATE APPROXIMATION
In this section, we develop another finite-length solution that
is an approximation to the optimal, finite-length solution for
oversampled, sinc-interpolating filters. With OFAX, we con-
struct a finite-length compensation signal directly from the im-
posed constraints. Alternatively, for the case of oversampled,
sinc-interpolating filters, we can start with the unconstrained
solution, cideal[n] = −x[0](−1)n, and truncate it through
appropriate windowing. In this case, we apply a finite-length