EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS
Eur. Trans. Telecomms. (2008)
Published online in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/ett.1320
Signal Processing
Causal discrete-time system approximation of non-bandlimited continuous-time
systems by means of discrete prolate spheroidal wave functions
Roman Tzschoppe∗ and Johannes B. Huber
Institute of Information Transmission, University of Erlangen-Nuremberg, Germany
SUMMARY
In general, linear time-invariant (LTI) continuous-time (CT) systems can be implemented by means of LTI
discrete-time (DT) systems, at least for a certain frequency band. If a causal CT system is not bandlimited, the
equivalent DT system may has be to non-causal for perfectly implementing the CT system within a certain
frequency band. This paper studies the question to which degree a causal DT system can approximate the CT
system. By reducing the approximation frequency band, the approximation accuracy can be increased—at
the expense of a higher energy of the impulse response of the DT system. It turns out, that there exists a
strict trade-off between approximation accuracy, measured in the squared integral error, and energy of the
impulse response of the DT system. The theoretically optimal trade-off can be achieved by approximations
based on a weighted linear combination of the discrete prolate spheroidal wave functions (DPSWFs). The
results are not limited to the case of approximating a CT system by means of a causal DT system, but they
generally hold for the approximation of an arbitrary spectrum by means of a spectrum of an indexlimited
time sequence. Copyright © 2008 John Wiley & Sons, Ltd.
1. INTRODUCTION
If the transfer function of a continuous-time linear timeinvariant (CT-LTI) system is bandlimited, the CT system can
be perfectly implemented by a discrete-time (DT) system,
when the sampling frequency is chosen according to the
sampling theorem. The DT system is referred to as the
impulse-invariant DT version of the CT system, since the
impulse response of the DT version is obtained by sampling
the impulse response of the CT system [5].
In case a CT system is not bandlimited, an impulseinvariant version does not exist. Nevertheless, if the input
signal is bandlimited, there exists a DT system perfectly
approximating the CT system within the frequency band of
the bandlimited input signal. But if the CT system is causal,
it turns out that the DT system has to be non-causal, in
general.
Discrete prolate spheroidal sequences (DPSSs) and their
counterpart in frequency domain, the discrete prolate
spheroidal wave functions (DPSWFs), have properties,
which make them very interesting for various applications,
such as secure communications [1], filter design [2], channel
equalisation [3] and channel prediction [4]. One of the
most important properties is that the first DPSS of the
orthogonal set of indexlimited DPSSs comprise the highest
energy concentration within a limited frequency band.
All succeeding DPSSs have a decreasing concentration.
This property is utilised in this paper for causal DT
system approximation of causal non-bandlimited CT
systems under an energy constraint of the DT impulse
response.
The paper is organised as follows. After introducing
some basic notations, DT system implementations of CT
systems are treated in Section 3. Apart from the review
* Correspondence to: Roman Tzschoppe, Institute of Information Transmission, University of Erlangen-Nuremberg, Cauerstr. 7/LIT, D-91058 Erlangen,
Germany. E-mail: [email protected]
Copyright © 2008 John Wiley & Sons, Ltd.
Received 31 October 2007
Revised 17 April 2008
Accepted 26 June 2008
R. TZSCHOPPE AND J. B. HUBER
of the well-known case of bandlimited CT systems, it is
shown that in the case of non-bandlimited CT systems,
the DT system in general has to be non-causal, even
if the CT system is causal. Section 4 demonstrates,
that for identifying non-bandlimited CT systems, system
identification methods based on DT signals also have to
estimate a non-causal DT impulse response. The last but one
Section 5 analyses to which extent a causal DT system can
approximate a causal CT system. The tools for this analysis
are the DPSWFs. Two examples illustrate all obtained
insights, and Section 6 finally concludes the paper.
Figure 1. Continuous-time LTI system.
Figure 2. Equivalent discrete-time LTI system for bandlimited
input signals.
2. NOTATION
Throughout this paper, the following notation is used: CT
signals and their spectra have an index c, like xc (t) and
Xc (f ), DT signals and their spectra have no index, like
x[k] and X(F ). The relationship between DT index k and
CT is: t = kT . The Fourier transform and its inverse are
defined by
def
Hc (f ) =
def
∞
−∞
∞
hc (t) e−j2πft dt
(1)
Hc (f ) ej2πft df
(2)
hc (t) =
−∞
and the discrete Fourier transform and its inverse are
defined by
def
H(F ) =
∞
h[k] e
−j2πkF
def
h[k] =
1/2
−1/2
3.1. Bandlimited continuous-time systems
A well-known sufficient condition for the existence of an
equivalent DT-LTI system is that both CT input signal
xc (t) and CT impulse response of the LTI system hc (t)
are bandlimited [5]. Therefore, the sampling rate 1/T of
both ideal continuous-to-discrete-time (C/D) and discreteto-continuous-time (D/C) converter has to be chosen such
that the sampling theorem is fulfilled
Xc (f ) = 0,
Hc (f ) = 0,
|f | >
(3)
k=−∞
bandlimited input signal can be equivalently described
by a DT-LTI system (see Figure 2). First, the analysis is
restricted to the case of bandlimited CT-LTI systems. Then,
the analysis is extended to the more general case of nonbandlimited CT-LTI systems.
1
2T
(5)
Choosing the transfer function of the DT-LTI system
H(F ) ej2πkF dF
(4)
where the frequency variables are related by F = fT . The
correspondence between hc (t), and Hc (f ) is compactly
denoted by F{hc (t)} = Hc (f ) and the correspondence
between h[k] and H(F ) equivalently by F∗ {h[k]} = H(F ).
Complex conjugation is indicated by (·)∗ .
3. DISCRETE-TIME SYSTEM
IMPLEMENTATION OF CONTINUOUS-TIME
SYSTEMS
This section deals with the question under which
circumstances a CT-LTI system (see Figure 1) with
Copyright © 2008 John Wiley & Sons, Ltd.
F
H(F ) = Hc
,
T
1
2
(6)
1
2T
(7)
|F | <
yields an effective CT transfer function
Hc,eff (f ) ≡ Hc (f ),
|f | <
The DT system is denoted as an impulse-invariant version
of the CT system since both impulse responses are
related by
h[k] = Thc (kT )
(8)
that is, the DT impulse response is a scaled and sampled
version of the CT impulse response.
Eur. Trans. Telecomms. (2008)
DOI: 10.1002/ett
CAUSAL DISCRETE-TIME SYSTEM APPROXIMATION
3.2. Non-bandlimited continuous-time systems
If the CT system is not perfectly bandlimited, aliasing will
occur when choosing h[k] according to Equation (8). In
cases where |Hc (f )| tends to zero for |f | → ∞, one can
argue that choosing 1/T high enough will lead to a rather
negligible effect of aliasing. An example is the design of
a DT lowpass filter by sampling the CT impulse response
of an analogue lowpass filter. But there are cases, where
|Hc (f )| will not tend to zero in principle—at least within
the relevant frequency range. One example therefore is the
transfer function of near-end crosstalk (NEXT) in twisted
multipair cables. The NEXT transfer function between two
pairs is generally modelled being proportional to f 0.75 , for
example Reference [6]. Several measurements show a high
compliance with this model—at least for frequencies up
to 30 MHz [7, 8]. In this case, the effect of aliasing will
be dramatic, and a choice of the DT impulse response
according to Equation (8) will lead to an effective CT
transfer function Hc,eff (f ) which is totally different from
Hc (f ) for |f | < 1/(2T ).
But there are many applications where such CT systems
are excited by bandlimited signals and the DT system has
to represent the CT system within this limited frequency
band, only. In this case, it is possible to design a DTLTI system fulfilling Equation (6) simply by applying
the inverse discrete Fourier transform (Equation (4)) to
Hc (F/T ). This is equivalent to sampling the bandlimited
CT impulse response hc,bl (t)
hc,bl (t) =
∞
−∞
rect(fT )Hc (f ) ej2πft df
t
1
sinc
∗ hc (t)
T
T
τ
1 ∞
sinc
=
hc (t − τ) dτ
T −∞
T
h[k] = T hc,bl (t)
=
t=kT
(9)
(10)
where rect(x) is defined by
If the CT-LTI system is causal, that is hc (t) = 0 ∀t < 0,
and hc (t) = 0—at least for one t 0—the bandlimited
impulse response hc,bl (t) is non-causal. This follows
directly from the proof, that if a bandlimited CT signal is
zero on any time interval of non-zero length, it is identically
zero everywhere (cf. Appendix 5B, in Reference [9]).
Equations (9) and (10) show that the equivalent DT-LTI
system fulfilling Equation (6) is non-causal in general,
although H(F ) is not bandlimited. If all zeros of hc,bl (t) for
t < 0 are located
at −nT (n ∈ N), h[k] will be causal (e.g.
if hc (t) = ∞
c
l=0 l δ(t − lT ), cl ∈ R). But in general, not all
zeros of hc,bl (t) for t < 0 will follow this strict constraint,
and in this case h[k] will be non-causal.
In summary, a CT-LTI system with bandlimited input can
be equivalently described by a DT-LTI system, regardless
whether the CT-LTI system is bandlimited or not. But in the
non-bandlimited case, the DT-LTI system has to be noncausal in most cases.
4. DISCRETE-TIME SYSTEM
IDENTIFICATION OF NON-BANDLIMITED
CONTINUOUS-TIME SYSTEMS
Not only for DT-LTI system implementation of a nonbandlimited CT-LTI system, but also when identifying a
non-bandlimited CT-LTI system by means of DT signals,
the corresponding DT impulse response turns out to be noncausal, as shown in the sequel.
The system identification task is to estimate the CT
transfer function Hc (f ) for |f | < 1/(2T ), if input and
output signals are merely accessible in DT, as depicted in
Figure 3. All signals are assumed to be zero-mean, weak
stationary random signals. W.l.o.g., the input signal xc (t)
is assumed to be bandlimited to the Nyquist frequency
1/(2T ).†
The CT cross-correlation function (CCF) is denoted by
def
def
φyc xc (τ) = E{yc (t + τ)xc∗ (t)}. Equivalently, φyx [κ] =
E{y[k + κ]x∗ [k]} indicates the DT CCF. The
corresponding cross-power spectral densities (CPSD)
def
are identified by φyc xc (τ) = E{yc (t + τ)xc∗ (t)} and by
def
(11)
φyx [κ] = E{y[k + κ]x∗ [k]}. CT and DT autocorrelation
functions (ACF) and power spectral densities (PSD) are
denoted in the same manner.
sinc(x) denotes the sinc function sin(πx)
πx , and ∗ denotes linear
convolution.
† Higher frequency components are not necessary for identifying H (f )
c
within |f | < 1/(2T ), and are completely suppressed by the ideal C/D
converters.

|x| < 1/2

 1,
def
rect(x) = 1/2, |x| = 1/2


0,
|x| > 1/2
Copyright © 2008 John Wiley & Sons, Ltd.
Eur. Trans. Telecomms. (2008)
DOI: 10.1002/ett
R. TZSCHOPPE AND J. B. HUBER
Figure 3. Discrete-time system identification of a continuoustime LTI system.
Due to ideal sampling (x[k] = xc (kT ), y[k] = yc (kT )),
the following relationships between the DT and CT
ACF/CCF holds
φxx [κ] = φxc xc (κT )
φyx [κ] = φyc xc (κT )
(12)
and for the corresponding PSD/CPSD:
F
,
T
1
F
yx (F ) = yc xc
,
T
T
xx (F ) =
1
x x
T c c
|F | <
1
2
|F | <
1
2
(13)
For the CPSD between output and input signal of the
effective DT system holds
yx (F ) = Heff (F ) · xx (F ),
|F | <
1
2
|f | <
1
2T
Obviously, a time sequence and its spectrum cannot
coexistently be indexlimited and bandlimited. Slepian [10]
studied the questions to which extent the spectrum of an
indexlimited time sequence can be bandlimited, to which
extent the time sequence of a bandlimited spectrum can be
indexlimited and to which extend a time sequence can be
concentrated both in time and in frequency. Therefore, he
introduced certain time sequences, the DPSSs, and certain
frequency functions, the DPSWFs. Only the definitions
and the most important properties of the DPSSs and the
DPSWFs are reviewed in the following, further details
are provided in Reference [10].‡ Computation of the
DPSSs from its definition (cf. Equation (20)) is numerically
problematic, especially for large sequence lengths. A
numerically stable method can be found in Reference [11].
The CT counterpart of DPSSs and DPSWFs are extensively
studied in References [12–15].
5.1.1. Discrete prolate spheroidal wave functions
The DPSWFs are defined in Reference [10] as the
eigenfunctions Um (F ) of the following integral equation
(15)
It follows directly from Equations (13)–(15), that Heff (F )
is given by Equation (6). The corresponding effective DT
impulse response is consequently given by Equations (9)
and (10), and thus is non-causal in general.
5. CAUSAL DISCRETE-TIME SYSTEM
APPROXIMATION OF NON-BANDLIMITED
CONTINUOUS-TIME SYSTEMS
Even though in some cases, it is not feasible to introduce a
sufficient signal delay for a causal implementation of a noncausal DT-LTI system, it may be highly desirable to identify
Copyright © 2008 John Wiley & Sons, Ltd.
5.1. Introduction to discrete prolate spheroidal wave
functions and discrete prolate spheroidal sequences
(14)
and equivalently for the CT system
yc xc (f ) = Hc (f ) · xc xc (f ),
the non-causal DT impulse response. An important example
thereof is the estimation of the CT transfer function Hc (f )
for |f | < 1/(2T ), if input and output signals are merely
accessible in discrete-time, as considered in Section 4. This
estimation of the CT transfer function may then be helpful to
design a causal DT-LTI system approximating the CT-LTI
system. When limiting the desired frequency range used for
approximating Hc (f ), this method provides better results
than attempting to identify directly a causal DT-LTI system.
W
−W
sin(Nπ(F − F ))
Um (F ) dF = λm Um (F )
sin(π(F − F ))
0 W 1/2,
−∞ < F < ∞
m = 0, 1, . . . , N − 1
(16)
Equation (16) has only N eigenvalues λ0 , λ1 , . . . , λN−1 .
They are distinct, real, positive and are ordered such that
λ0 > λ1 > · · · > λN−1 > 0
(17)
‡ Note that in this paper slightly different definitions for the discrete Fourier
transform and its inverse are used.
Eur. Trans. Telecomms. (2008)
DOI: 10.1002/ett
CAUSAL DISCRETE-TIME SYSTEM APPROXIMATION
Their associated eigenfunctions Um (F ) form a set of
linearly independent real functions. They are normalised
such that
1/2
dUm (F ) 2
Um (F ) df = 1 Um (0) 0,
0
dF −1/2
As the DPSWFs, the DPSSs are doubly orthogonal (cf.
Reference [10])
N−1
k=0
F =0
m = 0, 1, . . . , N − 1
W
1/2
Um (F ) Un (F ) dF = λm
−W
m, n = 0, 1, . . . , N − 1
As one might expect, the DPSWFs and the DPSSs are
strongly related (for fixed parameters W and N). As shown
in Reference [10], the DPSWFs can be calculated from the
DPSSs by
Um (F ) Un (F ) dF
m = 0, 1, . . . , N − 1
5.1.2. Discrete prolate spheroidal sequences
m =
The DPSSs are defined in Reference [10] as the real solution
vm [k] to the following system of equations
k=0
N−1
k=0
m even
m odd
1,
j,
(24)
It is further pointed out in Reference [10], that the
indexlimited DPSSs can be derived from the DPSWFs by
sin(2πW(k − l))
vm [l] = λm vm [k]
π(k − l)
(20)
1
vm [k] =
m
1/2
−1/2
Um (F ) ejπ(N−1−2k)F dF
m = 0, 1, . . . , N − 1,
normalised such that
N−1
(23)
where
where δmn denotes the Kronecker delta [10].
k = 0, ±1, ±2, . . .
vm [k] e−jπ(N−1−2k)F
k=0
m, n = 0, 1, . . . , N − 1 (19)
0 W 1/2,
N−1
Um (F ) = m
= λm δmn
l=0
(22)
5.1.3. Connections between the DPSWFs and the
DPSSs
−1/2
N−1
vm [k]vn [k] = δmn
k=−∞
(18)
Note that both eigenvalues and eigenfunctions are functions
of N and W. But for brevity, the notations λm and Um (F ) are
mostly used instead of writing λm (N, W) and Um (N, W; F ).
The most important property of the DPSWFs is the
remarkable double orthogonality:
∞
vm [k]vn [k] = λm
k = 0, 1, . . . , N − 1
(25)
and the infinite DPSSs by
v2m [k] = 1
vm [k] 0
vm [k] =
N−1
(N − 1 − 2k)vm [k] 0
W
−W
Um (F ) ejπ(N−1−2k)F dF
k = 0, ±1, ±2, . . .
(26)
(21)
It is noteworthy to say, that the N different DPSS vm [k] are
associated with the λm , which are the eigenvalues of the
integral equation (16). The non-zero λm are again ordered
according to Equation (17). Both λm and vm [k] are of course
functions of N and W, but for brevity an explicit notation
is omitted again.
Copyright © 2008 John Wiley & Sons, Ltd.
m = 0, 1, . . . , N − 1,
k=0
m = 0, 1, . . . , N − 1
1
m λm
5.2. Spectral extension of a finite sequence
Slepian [10] showed that the following problem can be
solved by means of the DPSWFs. Let H(F ) be given for
|F | W. For W < |F | 1/2, H(F ) is spectrally extended.
Under all possible spectral extensions, only extensions that
correspond to time sequences that are indexlimited to the
Eur. Trans. Telecomms. (2008)
DOI: 10.1002/ett
R. TZSCHOPPE AND J. B. HUBER
index set {N0 , . . . , N0 + N − 1} are considered. Which
spectral extension has least energy?
The extension is only possible if for |F | 1/2
H(F ) =
N0
+N−1
ᾱm =
h[k] e−j2πkF
(27)
k=N0
or equivalently, if for |F | W, H(F ) lies in the span of the
N basis functions, the DPSWFs§
H(F ) = e−j2π(N0 +(N−1)/2)F
N−1
αm Um (F )
(28)
m=0
Due to the DPSWF orthogonality Equation (19), the
DPSWF weights αm can be easily determined by
αm =
1
λm
W
−W
H(F ) ej2π(N0 +(N−1)/2)F Um (F ) dF (29)
In Reference [10], it is stated that the extension of H(F )
according to Equation (28) for |F | 1/2 is the extension
of minimum energy, which is given by
1/2
−1/2
not lie in the span of the DPSWFs. But, by inserting Ht (F )
instead of H(F ) into Equation (29) yields coefficients
|H(F )|2 dF =
N−1
|αm |2
(30)
m=0
It is furthermore shown in Appendix A, that if Equation (27)
is valid for |F | W, then the spectral extension of H(F )
to the interval W < |F | 1/2, such that the corresponding
time sequence is indexlimited to {N0 , . . . , N0 + N − 1},
is unique. Other possibilities of extending H(F ) to the
interval W < |F | 1/2, such that the corresponding time
sequence is indexlimited to {N0 , . . . , N0 + N − 1}, do not
exist.
1
λm
W
−W
Ht (F ) ej2π(N0 +(N−1)/2)F Um (F ) dF (31)
which, inserted into Equation (28) instead of αm , provide
an approximation for Ht (F ) in the interval (−W, W). The
costs of doing this kind of approximation is an energy
increase outside the interval (−W, W). For small values of
W, the approximation accuracy will be high at the costs of
a high energy increase, and vice versa. The questions to be
answered are the following. To which extent can an arbitrary
target function be approximated in the interval (−W, W) by
means of a spectrum of a sequence limited to the index set
{N0 , . . . , N0 + N − 1}, under the constraint of a limited
total energy? Furthermore, how can the approximation be
accomplished?
W.l.o.g. the function H(F ) approximating Ht (F ) can be
represented uniquely by
H(F ) = e−j2π(N0 +(N−1)/2)F
N−1
βm Um (F )
(32)
m=0
according to Equation (28) using some weights βm , which
have to be specified later. The accuracy of approximation
in (−W, W) is measured by the squared integral
error:
ε=
W
−W
|Ht (F ) − H(F )|2 dF
(33)
and the total energy according to Equation (30) by
E=
1/2
−1/2
|H(F )|2 dF =
N−1
|βm |2
(34)
m=0
5.3. Minimum energy spectral approximation
5.3.1. Introduction
In general, for an arbitrary target function Ht (F ) Equations
(27) and (28) do not hold. The reason is that Ht (F ) does
§ The corresponding equation in Reference [10] is erroneous. The pair of
parenthesis around N − 1 is missing.
The corresponding equation in Reference [10] is erroneous. From the
symmetry of the integrand of Equation (29) follows that (even for a realvalued time sequence) for even m the weight αm is real, and for odd m
imaginary.
Copyright © 2008 John Wiley & Sons, Ltd.
5.3.2. Approximation with limited number of DPSWFs
Since the indexlimited DPSSs have the highest energy
concentration in (−W, W) of all indexlimited sequences,
a straightforward approximation method would be to use a
limited number N N of DPSSs and their corresponding
DPSWFs, respectively. Since the energy concentration of
the spectrum of vm [k] is given by λm (cf. Reference [10]),
and because the λm are sorted according to Equation
(17), it is best to use the DPSSs/DPSWFs with smallest
Eur. Trans. Telecomms. (2008)
DOI: 10.1002/ett
CAUSAL DISCRETE-TIME SYSTEM APPROXIMATION
function of E, respectively
indices first
βm =
ᾱm ,
0,
0 m N − 1
N − 1 < m < N − 1
(35)
ε=
N ,
5.3.3. Approximation with optimally weighted
DPSWFs
The previously described method provides a fairly good, but
not the optimum trade-off between squared integral error
and total energy. Apart from this, the trade-off is merely
possible in discrete steps (N reasonable possibilities) and
not continuously. In Appendix B, it is shown that it is
best, not to limit the number of used DPSWFs/DPSSs,
but to utilise a weighted combination of all N DPSWFs/
DPSSs simultaneously. The optimal weighting factors
βm , which minimise ε for a given energy E, are
given by
βm =
λm ᾱm
λm + µ(E)
(36)
where the real-valued, positive energy-dependent constant
µ(E) has to be chosen, such that Equation (34) is
fulfilled
N−1
m=0
λ2m |ᾱm |2
!
=E
(λm + µ(E))2
(37)
The corresponding time sequence with best spectral fit
(in the squared integral error sense) in (−W, W) and of
limited energy can be expressed by means of the DPSSs
(cf. Equation (25))
−W
+
Increasing
yields a better spectral fit in (−W, W) at the
cost of increasing the energy outside (−W, W).
W
N−1
m=0
h[k] =
εmin =
m=N0
λm
−2
λm + µ(E)
(39)
W
−W
|Ht (F )|2 dF −
N−1
λm |ᾱm |2
(40)
m=0
lim λm =
N→∞
1,
0,
m = 2NW(1 − δ)
m = 2NW(1 + δ)
where x denotes the largest integer
the smallest integer value x, and δ
number between 0 and 1 (cf. Reference
optimum DPSWFs/DPSSs weights βm
approximations are given by
λm ᾱm
1+µ(E) ,
λm ᾱm
µ(E) ,
(41)
value x, x
is an arbitrary
[10]). Thus, the
for bandlimited
m = 2NW(1 − δ)
m = 2NW(1 + δ)
(42)
(38)
By varying µ, a continuous trade-off between energy E of
the sequence and squared integral error in (−W, W) can be
achieved. Inserting the optimal DPSWF weights according
to Equation (36) into Equation (33) unveils, after some
straightforward steps, ε as a function of µ, and as an implicit
Copyright © 2008 John Wiley & Sons, Ltd.
This equation also provides the squared integral error for
the method of limiting the number of DPSWFs according
to Equation (35), when limiting the sum to the first N terms.
For large N, the set of sequences of bandwidth W that
are confined to an index set of length N has dimension
approximately 2NW, and for the eigenvalues of the DPSSs
follows
N→∞
∗m βm vm [k − N0 ]
λ2m |ᾱm |2
λm + µ(E)
In the limit µ → 0, the optimal DPSWF weights
βm tend to ᾱm , E attains its maximum value
2
Emax = N−1
m=0 |ᾱm | , and the minimal achievable
ε yields
lim βm =
N0
+N−1
|Ht (F )|2 dF
where λm ᾱm is given by Equation (31). For an arbitrary
target function Ht (F ), the integral in Equation (31) will
not vanish in general for N → ∞. Consequently, the
optimum weights βm are in general non-zero for m =
2NW(1 + δ). Thus, also in the asymptotic case, more
than 2NW DPSWFs/DPSSs contribute to the bandlimited
approximation, in general.
Eur. Trans. Telecomms. (2008)
DOI: 10.1002/ett
R. TZSCHOPPE AND J. B. HUBER
can achieve a squared integral error ε that is 20 dB lower
than the energy of Ht (F ) in (−W, W) at no energy increase,
just by limiting the approximation band to (−0.2, 0.2). By
further reducing W or by releasing the total energy, much
better approximations are attainable.
5.3.5. Example 2
In the second example, the target function is the transfer
function of a causal CT system with impulse response
Figure 4. Non-causal discrete-time impulse response of the target
transfer function.
hc,t (t) =
0,
t<0
1 −t/τ
,
τe
t0
(43)
and transfer function
5.3.4. Example 1
The first example treats the causal approximation of a noncausal DT system. The impulse response ht [k] of length 7,
whose transfer function Ht (F ) shall be approximated in
(−W, W) for W = 0.25, has one non-causal tap (see
Figure 4). First, the method of limiting the number of
utilised DPSSs/DPSWFs is considered (cf. Equations (32)
and (35)). Figure 5 shows the approximation results,
obtained for DPSS length N = 6 (N0 = 0) and all possible
choices of N .
The optimal trade-off between ε and E (for W = 0.25)
is depicted in Figure 6, together with the suboptimal
approximation method of limited number of DPSWFs and a
least-squares fit method. The energy E is normalised to the
def 1/2
energy of the target function Et = −1/2 |Ht (F )|2 dF , and
the squared integral error ε is normalised to the energy of the
(W) def W
target function in (−W, W) Et = −W |Ht (F )|2 dF , in
order to provide a fair measure of the squared integral error
irrespective of the chosen bandwidth parameter W. The
1/2
least-squares fit method minimises −1/2 w(F ) · |Ht (F ) −
H(F )|2 dF , where w(F ) is a weighting function, chosen
w(F ) = 1, for |F | W, and w(F ) = const., for |F | > W.
Varying the constant yields a suboptimal trade-off between
ε and E, which gets asymptotically optimal for E →
Emax (const. → 0, respectively). For the choice const. =
1, the circled point is reached. This point represents
the performance of a least-squares optimal approximation
(without weighting) of a non-causal DT impulse response
by a causal one. Figure 7 shows the best achievable tradeoff for different values of W (cf. Equation (39)). Thus, a
causal approximation for the given target function Ht (F )
Copyright © 2008 John Wiley & Sons, Ltd.
Hc,t (f ) =
1
1 + j2πfτ
(44)
By choosing Ht (F ) = Hc,t (F/T ), according to Equation
(6), the corresponding DT impulse response becomes noncausal (cf. Equations (9), and (10)). A cut-out of the infinite
impulse response is depicted in Figure 8 for T = τ/5. The
transfer function Ht (F ) of this non-causal DT impulse
response serves as a target transfer function in the interval
(−W, W), which shall be approximated by a causal DT
impulse response of length N (N0 = 0). As approximation
method, the DPSS/DPSWF method with optimum weights
(cf. Equations (36) and (37)) is chosen. Figure 9 shows
the normalised squared integral error over the sequence
length N, when the energy E is restricted to the energy of
the target transfer function Et . For the investigated values
of W, a saturation effect occurs when N is chosen larger
than ≈ 50. From that point on, significant improvements in
approximation quality can be achieved only by reducing the
approximation bandwidth W.
Figure 10 depicts the magnitude of the weights βm
for different energy restrictions 10 log10 (E/Et ) (N = 100,
W = 0.25). The weights βm for 0 dB and for 1 dB differ
significantly for m 2NW. For m 2NW, the difference
is negligible. The situation is different for a tighter energy
restriction, for example for 10 log10 (E/Et ) = −1 dB.
Although the magnitude of the weights are attenuated for
m 2NW, the weights are still non-zero for m > 2NW.
Even in this case, more than 2NW DPSWFs/DPSSs play a
role in achieving the best possible trade-off between total
energy and approximation quality in the squared integral
error sense.
Eur. Trans. Telecomms. (2008)
DOI: 10.1002/ett
CAUSAL DISCRETE-TIME SYSTEM APPROXIMATION
Figure 5. Approximation with limited number of DPSSs/DPSWFs. Left column: impulse responses h[k], middle column: magnitude
of transfer functions (dashed: |Ht (F )|), right column: phase of transfer functions (dashed: arg{Ht (F )}). Parameters: N = 6, N0 = 0,
W = 0.25.
Copyright © 2008 John Wiley & Sons, Ltd.
Eur. Trans. Telecomms. (2008)
DOI: 10.1002/ett
R. TZSCHOPPE AND J. B. HUBER
Figure 6. Achievable trade-offs between normalised energy E and
normalised squared integral error ε by method of limited number
of DPSWFs, optimum DPSWF weights and a least-squares fit
method. Parameters: N = 6, N0 = 0, W = 0.25.
Figure 8. Cut-out of the infinite non-causal discrete-time impulse
response fulfilling F∗ {ht [k]} = Hc,t (F/T ), for |F | < 1/2, (T =
τ/5).
Figure 9. Normalised squared integral error over sequence length
N with total energy restriction E Et for W = 0.25, 0.3, 0.35,
0.4, 0.45, 0.5. Parameter: N0 = 0.
Figure 7. Best achievable trade-off between energy E and squared
integral error ε for W = 0.1, 0.2, 0.3, 0.4, 0.5 by optimum DPSWF
weights. Parameters: N = 6, N0 = 0.
6. CONCLUSIONS
If a causal CT system is not bandlimited, an impulseinvariant DT version does not exist. In general, a DT
system implementation (within a certain frequency band)
can only be achieved by a non-causal DT system, even
if the input signal of the CT system is bandlimited.
If the introduction of a sufficient signal delay for a
Copyright © 2008 John Wiley & Sons, Ltd.
causal implementation of the non-causal impulse response
is not allowed, a causal DT system approximation is
necessary. By limiting the approximation frequency band,
a higher approximation accuracy can be achieved than by
an unweighted least-squares optimal fit, but at the cost
of an increased energy of the impulse response of the
approximating system. It is shown that the theoretically
optimal trade-off between squared integral error and energy
of the impulse response of the approximating system can
be achieved by using a weighted linear combination of the
DPSWFs.
Eur. Trans. Telecomms. (2008)
DOI: 10.1002/ett
CAUSAL DISCRETE-TIME SYSTEM APPROXIMATION
APPENDIX A
A.1. Proof of the uniqueness of the spectral extension
of a finite sequence
If H(F ) for |F | W is given by
H(F ) =
N0
+N−1
h[k] e−j2πkF
(45)
k=N0
the spectral extension of H(F ) to W < |F | 1/2, such
that the corresponding time sequence is limited to
{N0 , . . . , N0 + N − 1}, is unique. Or stated in other words,
there exists no second time sequence h̃[k] limited to
{N0 , . . . , N0 + N − 1}, such that
H(F ) =
N0
+N−1
h̃[k] e−j2πkF
(46)
k=N0
is fulfilled for |F | W.
Intuitively, the uniqueness follows directly from the
double orthogonality of the DPSWFs (cf. Equation (19)).
Since the weights αm are uniquely determined by H(F )
in (−W, W) (cf. Equation (29)), the orthogonality over
(−1/2, 1/2) also guarantees the uniqueness over the entire
frequency range.
A strict proof follows by contradiction. The spectrum of
h̃[k] is assumed to be different from H(F ) for W < |F | 1/2:
H̃(F ) =
H(F ),
|F | W
H(F ) + H(F ), W < |F | 1/2
(47)
with H(F ) = 0, for |F | in (W, 1/2]. According to
Equation (4), h̃[k] is given by
h̃[k] = h[k] +
1/2
H̃(F ) ej2πkF dF
−1/2
(48)
def
= h̃[k]
where H̃(F ) is given by
Figure 10. Magnitude of optimal DPSS/DPSWF weights βm
for 10 log10 (E/Et ) = −1, 0, and 1 dB (from top to bottom).
Parameters: N = 100, N0 = 0, W = 0.25. Dashed line indicates
2NW = 50.
Copyright © 2008 John Wiley & Sons, Ltd.
H̃(F ) =
0,
H(F ),
|F | W
W < |F | 1/2
(49)
Eur. Trans. Telecomms. (2008)
DOI: 10.1002/ett
R. TZSCHOPPE AND J. B. HUBER
It is trivial to prove that, if H̃(F ) is bandlimited, h̃[k]
cannot be indexlimited, in contradiction to the assumption,
that h̃[k] is indexlimited.
where wk = xk + jyk , with real-valued numbers xk and yk ,
implying that
∂wk
= 1,
∂wk
∂ε(β0 , . . . , βN−1 , µ)
∂βn
W
= −
Ht∗ (F ) e−j2π(N0 +(N−1)/2)F Un (F ) dF
B.1. Derivation of optimal DPSWF weighting factors
To minimise Equation (33) under the constraint Equation
(34), the method of Lagrange multipliers is applied. The
solution is a stationary point of the function
def
ε(β0 , . . . , βN−1 , µ) =
W
−W
|Ht (F ) − H(F )|2 dF
+µ ·
N−1
|βm | − E
2
(50)
m=0
For ease of use, the integral is expanded before the partial
differentials are calculated
W
−W
|Ht (F ) − H(F )|2 dF
=
W
−W
−
−W
−
+
|Ht (F )|2 dF
W
W
−W
N−1
(53)
Considering Equation (53), one gets
APPENDIX B
∂w∗k
=0
∂wk
+ µβn∗
(54)
Setting the partial differential to zero yields the optimum
DPSWF weighting coefficients
W
1
βn =
Ht (F ) ej2π(N0 +(N−1)/2)F Un (F ) dF
λn + µ −W
=
λn ᾱn
λn + µ
(55)
The partial derivative of ε w.r.t. µ yields the energy
constraint Equation (37). The Langrangian multiplier µ
remains an implicitly defined function of the energy E.
LIST OF SYMBOLS
Ht (F ) ej2π(N0 +(N−1)/2)F
N−1
m=0
Ht∗ (F ) e−j2π(N0 +(N−1)/2)F
∗
βm
Um (F ) dF
N−1
βm Um (F ) dF
m=0
λm |βm |2
(51)
m=0
where H(F ) according to Equation (32) was inserted,
and the orthogonality of the DPSWFs according to
Equation (19) was utilised.
Since the DPSWF weighting factors can be imaginary,
Schwartz’ definition of a complex derivative is used
∂
1
=
∂wk
2
−W
+ λn βn∗
∂
∂
−j
∂xk
∂yk
Copyright © 2008 John Wiley & Sons, Ltd.
(52)
Variable Meaning
E
Et
Et(W)
F
f
k
H(F )
Ht (F )
Hc (f )
Hc,t (f )
h[k]
hc (t)
hc,t (t)
N
N
N0
T
t
Um (F )
vm [k]
(total) energy
(total) energy of a target function
energy of a target function in (−W, W)
(normalised) frequency variable w.r.t. DT
frequency variable w.r.t. CT
discrete-time (DT) variable
transfer function of a DT system
target transfer function of a DT system
transfer function of a CT system
target transfer function of a CT system
impulse response of a DT system
impulse response of a CT system
impulse response of a target transfer function w.r.t.
a CT system
DPSS length
number of utilised DPSSs (N N)
first DT index of an indexlimited sequence of
length N
sampling period
continuous-time (CT) variable
mth DPSWF
mth DPSS
Eur. Trans. Telecomms. (2008)
DOI: 10.1002/ett
CAUSAL DISCRETE-TIME SYSTEM APPROXIMATION
W
w(F )
αm
ᾱm
βm
δmn
δ
ε
m
λm
µ
xc xc (f )
yc xc (f )
xx (F )
yx (F )
φxx (τ)
φyx (τ)
φxx [κ]
φyx [κ]
τ
(normalised) cut-off frequency w.r.t. DT
weighting function
weight of the mth DPSS/DPSWF
weight of the mth DPSS/DPSWF
(optimal) weight of the mth DPSS/DPSWF
Kronecker delta
real number between 0 and 1
squared integral error
constant relating vm [k] and Um (F )
mth eigenvalue
Lagrangian multiplier (energy dependent)
CT power spectral density function
CT cross-power spectral density function
DT power spectral density function
DT cross-power spectral density function
CT autocorrelation function
CT cross-correlation function
DT autocorrelation function
DT cross-correlation function
constant w.r.t. Example 2
ACKNOWLEDGEMENT
The authors would like to thank Professor W. Kellermann for
pointing us to the DPSSs and DPSWFs.
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AUTHORS’ BIOGRAPHIES
Roman Tzschoppe received his Dipl.-Ing. degree in electrical engineering in 2001 from the University of Erlangen-Nuremberg,
Germany. Since 2001, he works as a Ph.D. student at the Institute for Information Transmission, University of Erlangen-Nuremberg,
Germany. His research activities include crosstalk cancellation for DSL, digital watermarking and steganography.
Johannes B. Huber received his Dipl.-Ing. degree in electrical engineering from the Technische Universität, München. From the
Universität der Bundeswehr, München, he received the Dr-Ing. degree with a thesis on coding for channels with memory and the Dr
Habil. degree with a thesis on trellis coded modulation. Since 1991, he is Professor at the Universität Erlangen-Nürnberg, Germany.
His research interests are information and coding theory, modulation schemes, high rate baseband transmission, algorithms for signal
detection and adaptive equalisation for channels with severe intersymbol interference, signalling, detection and equalisation for multipleinput multiple-output (MIMO) channels and concatenated coding together with iterative decoding. Professor Huber received the
Research Award of the German Society of Information Technology (ITG) in 1988 and 2000 and the Vodafone award for innovations
in mobile communications 2004. In 2005, he was elected as IEEE Fellow.
Copyright © 2008 John Wiley & Sons, Ltd.
Eur. Trans. Telecomms. (2008)
DOI: 10.1002/ett