ON THE EIGENSTRUCTURE OF UNDERSPREAD WSSUS CHANNELS
Werner Kozek1 and Andreas F. Molisch2
NUHAG, Dept. of Mathematics, University of Vienna
Strudlhofgasse 4, A-1090 Vienna, Austria, [email protected]
1
2
INTHFT, Vienna University of Technology, Gusshausstrasse 25/389
A-1040 Vienna, Austria, [email protected]
ABSTRACT
We consider the problem of nding pulses that suer from
minimum distortion when transmitted over an underspread
time{varying multipath propagation channel. Mathematically, this corresponds to the design of properly time/frequency localized functions that are approximate eigenfunctions of the channel. We consider the problem of determining this localization in terms of the channel's spreading function (deterministic case) and scattering function
(stochastic WSSUS case).
1. INTRODUCTION
In mobile radio systems, one is often faced with the task
of transmitting pulses with as little distortion as possible,
given incomplete a priori knowledge about the channel. The
usual rule of thumb says that pulses with duration smaller
than the coherence time and a bandwidth smaller than the
coherence bandwidth are approximate eigenfunctions of the
channel and thus suer from little distortion. However,
apart from these restrictions, there are no rules for nding the pulses that suer really the "least" distortion.
Besides these problems, there are also problems with
the exact denition of coherence time and coherence bandwidth. The terminology seems to originate from [1] where
mainly tropo{/ionospheric scattering is considered. The coherence bandwidth f is essentially the frequency so that
one has RHx(f ) = 0:5RHx (0), when RHx(f ) is the spectral correlation function of the received signal; the coherence time is dened analoguously. These denitions are
inherently based on the WSSUS (wide{sense stationary uncorrelated scattering) model. They are valuable for an intuitive understanding but suer from three problems: (i)
the denitions are not applicable to more general stochastic channels; (in modern mobile communication scenarios
the strict WSSUS model does not always result in a viable short{term channel characteristic particularly in situations where specular reections dominate over diuse scattering). However, the WSSUS assumption is often used
(without physical justication) for want of a better model.
(ii) The denitions are not directly applicable to deterministic channels which have gained importance in the mobile
This work was partly supported by the Austrian FWF Grant
S-7001 and the "METAMORP" project SMT4-CT96-2093 of the
European Union.
radio community. (iii) The autocorrelation functions can
be non{monotone, leading to an inevitable ambiguity in
the determination of coherence time/bandwidth.
These problems have led us to undertake a more precise mathematical analysis of the problem allowing a formal synthesis of "optimum" pulses both for deterministic
and WSSUS channels. This description is based on viewing
the channel as a linear operator. We will see that normal
operators play a central role, because only these have bases
of (approximate) eigenfunctions.
Outline. In Section 2, we will give a strict mathematical
description of eigenfunctions for linear time{invariant and
linear frequency{invariant systems. Section 3 discusses the
existence of eigenfunctions in time{varying systems, and
reviews the concepts of both the spreading function, and
the time{varying transfer function. Section 4 discusses underspread systems. Section 5 deals with WSSUS systems,
and gives a formal description for the synthesis of optimum
pulses in these environments. We consider both pulses that
are suitable for TDMA and for CDMA systems.
2. SINGLY{DISPERSIVE CHANNELS
LTI Channels|Mathematics. A time{delay T is the
basic linear operator for a time{invariant multipath propagation channel in the sense that
Z
Z
Hx(t) = h( )T x(t)d = h( )x(t )d; (1)
where Hx(t) is the output signal and x(t) the input signal.
Sinusoids are generalized eigenfunctions of linear time{
invariant (LTI) systems, i.e., for an LTI system H with
transfer function H (f ), one has formally Hx(t) = H ( )x (t);
j t
with x (t) := e 2 , and the transfer function dened as
transform of the impulse response H (f ) :=
Rthe Fourier
h(t)e j2t .
An LTI system as a linear operator acting on L2 (R)
generally does not have proper eigenfunctions. The eigenstructure of LTI systems is characterized by the fact that
these are noncompact and normal linear operators. Normality in a linear operator sense means:
H H = HH ;
(2)
where H denotes the adjoint operator. Normality is a fundamental prerequisite for the existence of a "diagonalizing"
transform in the abstract sense of Gelfand (which for LTI
operators is just the Fourier transform) [2].
In most engineering applications the distinction between
generalized and proper eigenvalues/eigenfunctions can be
circumvented without loss of rigor by considering the so{
called approximate eigenvalues (approximate point spectrum) [2, p.118]: A 2 C is called approximate eigenvalue if
and only if there exists2 a sequence fn g 2 L2 (R) such that
limn!1 kHn n k = 0; kn k = 1. In the LTI case
the approximate eigenfunctions associated to a certain
frequency and to an approximation level ":
kH H ( ) k2 < ";
are obtained by (i) taking a low{pass prototype signal and (ii) shifting it to the center frequency (t) = M (t) := (t)ej2t ;
(3)
where M denotes the modulation operator. These approximate eigenfunctions are highly ambiguous because by
the denition of an LTI system any time{shifted version of
M satises the same estimate. The maximum admissible bandwidth to achieve a given " depends on the atness
of the transfer function. For systems with nite{duration
impulse response the transfer function is bandlimited and
by Bernstein's inequality (see [3, p.144]) it has a bounded
derivative. This leads to the following characterization of
approximate eigenfunctions.
Theorem 1. Let H be an LTI system with (i) impulse
response h( ) = 0, for 2= [ 0 ; 0 ], and (ii) unit maximum
gain, then any time{frequency (TF) shifted version of an
L2"(R){normalized lowpass function with bandwidth B <
is an "{approximate eigenfunction of H:
20
k H Mf Tt H (f ) Mf Tt k2 ":
Discussion. Theorem 1 has dened a bandwidth B that a
signal must not exceed in order to be a "narrowband" signal
in the sense of mobile radio (i.e. in order to suer "negligible" distortions of the pulse during transmission). This
denition circumvents the problems that usually occur in
the denition of such a bandwidth: (i) it gives a precise
mathematical meaning to what is "negligible", by relating
the parameter " to the deviation from "zero" distortion.
(ii) The only knowledge it requires about the channel is the
maximum delay 20 in the channel. Note that Theorem 1
does not exclude the existence of {approximate eigenfunctions with larger bandwidth than B for one specic system
with maximum delay 20 (such a function is then, however,
not guaranteed to be an approximate eigenfunction of a
dierent system). An obvious disadvantage of our mathematical model is the fact that a nite{duration impulse
response is |in its strict sense| unphysical.
LFI Channels. Linear frequeny{invariant (LFI) systems
correspond to time{selective, frequency{nonselective fading
with the input{output relationZ
Hx(t) = q(t)x(t) = Q( )M x(t)d
where Q( ) is the Fourier transform of q(t) and M denotes
the modulation operator (see (3)). The maximum L2 (R){
gain is given by kHk = kqk1 :
The generalized eigenfunctions of an LFI system are
time{shifted versions of a Dirac delta distribution. Approximate L2 (R){eigenfunctions are obtained by short{duration
pulses, when we dene duration strictly via the support.
Theorem 2. Let H be an LFI system with (i) maximum narrowband Doppler{shift 0 and (ii) unit maximum
gain, then any TF shifted version of a temporally centered
function with duration D < 2" 0 ; is an "{approximate
eigenfunction:
k HMf Tt q(t) Mf Tt k2 ":
For the admissible pulse duration of the signal in an LFI{
system, analogous considerations as for the bandwidth in
an LTI{system are applicable. However, in practice one
major dierence occurs: the Doppler shift e.g. in a mobile
radio system is truly limited, namely to twice the maximum
speed of transmitter, receiver, or scatterers.
3. DETERMINISTIC LTV SYSTEMS
Spreading Representation. A general linear time{varying
(LTV) system H potentially causes both time and frequency
shifts. The (asymmetrical) spreading function of an LTV
system H establishes a weighted decomposition into these
fundamental eects:
Z Z
Hx(t) =
SH (; )M T x(t) d d:
(4)
In a mobile radio channel, the spreading function is proportional to the power of the contribution from a scatterer that
causes a time delay and a frequency shift ; it is thus the
natural physical interpretation of the channel. Mathematically, the spreading function is well{dened as a unitary
representation of Hilbert{Schmidt (HS) operators on the
TF plane. HS operators on L2 (R) are linear integral operators with square{integrable kernel [2]. LTI/LFI operators
are never HS, they can be simulated by a HS operator only
over a limited time/frequency period. However, the denition of SH (; ) can be extended to these systems with the
help of distributions, i.e., for an LTI/LFI system SH (; )
is ideally concentrated on the / {axis (see Fig.1).
HS operators need not be normal (see (2)) and, as a
consequence, need not have eigenfunctions. In other words,
there may be time{varying channels that make a distortionfree transmission impossible. However, we can anticipate
that for underspread channels, approximate eigenfunctions
do exist: the LTI-system, which has eigenfunctions, is a
limit case of underspread systems; thus the "exact" eigenfunctions of the LTI system will be the limit case of the
"approximate" eigenfunctions of the underspread system.
Hence, in anticipation of the approximate normality of underspread operators we concentrate on normal HS operators. It is noteworthy that while a normal HS operator has
exact eigenfunctions, these are not apt for practical applications, because they are not at all TF localized and the
eigenbasis in its entirety is totally unstructured (cannot be
generated in a TDMA{like manner).
It is important to note that SH (; ) induces a TF formulation of the quadratic form:
hHx; xi = hSH ; Axi ;
(5)
where Ax(; ) denotes the (asymmetrical) ambiguity function of an L2 (R){signal [3, p.284]
Z
Ax(; ) = hx; M T xi = x(t)x(t )e
j 2t
dt (6)
The quadratic form plays a fundamental role in detection
and eigenvalue estimation as it corresponds to a matched
lter [4, 5].
Time{Varying Transfer Function. The spreading function uniquely determines a linear time{varying system and
gives an intuitively clear relation to the physics of the channel. However, TF selective weighting (in the sense of generalizing the frequency{domain ltering in the LTI-case) is
more intuitively reected by the concept of a time{varying
transfer function [6]. Zadeh's time{varying transfer function can be dened via symplectic Fourier transform of the
asymmetrical spreading function
Z Z
ZH (t; f ) :=
SH (; )e j2(t f )d d:
By the terminilogy, one is tempted to interpret ZH (t; f )
as a "TF{parametrized eigenvalue distribution"; indeed for
LTI systems one has ZH (t; f ) = H (f ) and for LFI systems ZH (t; f ) = q(t). However, for a general LTV system a pointwise eigenvalue interpretation cannot be maintained, i.e., in general kHk 6= kZH k1 and ZGH (t; f ) 6=
ZG (t; f )ZH (t; f ) [7].
4. UNDERSPREAD SYSTEMS
Denition and Properties. Underspread systems are
characterized by a limited narrowband Doppler shift and
limited maximum time{delay (see Figure 1):
SH (; ) = 0;
for (; ) 2= [ 0 ; 0 ] [ 0 ; 0 ]: (7)
ν
ν
τ
(a)
τ
(b)
ν
ν
ν0
τ
(c)
τ0
τ
(d)
Figure 1: Support of SH (; ) for: (a) LTI system, (b) LFI
system, (c) identity, (d) underspread system (0 0 1)
LTI and LFI systems can be seen as limit cases of underspread systems. The total spread
H := 40 0 ;
is a fundamental condition number for the identication
and frequency domain representation of underspread systems [7]. Generally speaking, with decreasing H the theory of underspread systems approaches standard LTI system theory independent of the spreading ratio 0 =0 .
We mention two asymptotic properties of underspread
systems (modO(H ) means one has an approximate equality where the approximation gets better with decreasing
H ): (i) Underspread systems are asymptotically normal:
HH = H H modO(H ):
(Recall that LTI and LFI systems are normal and normality
of a HS operator is a prerequisite for the existence of complete bases of eigenfunctions.) (ii) One has asymptotical
validity of the multiplicative calculus of transfer functions:
ZGH (t; f ) = ZG (t; f )ZH (t; f ) modO(H ):
Approximate Eigenfunctions. In section 2, we analyzed
LTI and LFI systems assuming incomplete a priori knowledge, namely in form of a maximum delay. The following
theorem generalizes these results in that we assume no more
knowledge of SH (; ) than 0 ; 0 :
Theorem 3. Let H be an underspread system with
normalized maximum gain:
kHk = kZH k1 = 1 modO(H )
and (t) be L2 (R){normalized (A (0; 0) = 1) and adapted
to H by proper localization of A (; ):
jA (; ) 1j " ; for (; ) 2 [ 0 ; 0 ] [ 0 ; 0 ]: (8)
Then, Mf Tt and ZH (t; f ) are approximate eigenpairs:
k H Mf Tt ZH (t; f ) Mf Tt k2 " modO(H )
In an LTI system, we have seen that the "optimum"
pulse (in the sense of minimum distortions) has innite duration; it really has zero distortions. In an underspread
LTV system, on the other hand, there exists an optimum
pulse with a nite duration, and also nite distortions. This
is clear from an intuitive point of view, because an innite{
duration pulse gets distorted simply because of the time{
variance of the transfer function; if, on the other hand, the
pulse is made too short, it gets a broad spectrum, and is
distorted by the frequency selectivity of the channel. The
existence of an optimum pulse can also be proven mathematically by considering the impact of Heisenberg's uncertainty principle on the shape of ambiguity functions: One
has
T F 1;
where T and F are the duration/bandwidth of formulated via the second{order moments which are equal to
second{order derivatives of A (; ): (see [3, p.291]):
1 @ 2 A (0; 0) = 4 Z t2 j (t)j2 dt = T 2 ;
(9)
2
@
Z
2
1 @ A
2
2
2
@ 2 (0; 0) = 4 f j (f )j df = F : (10)
5. WSSUS SYSTEMS
Denition. A wide{sense stationary uncorrelated scattering (WSSUS) channel is characterized by i) being zero{
mean, ii) showing uncorrelated amplitudes of the TF shifts:
E fSH (; )g = 0;
(11)
E fSH (; )SH ( 0 ; 0 )g = CH (; )( 0 )( 0 );(12)
here, the function CH (; ) is the so{called scattering function [6, 1]. The scattering function establishes a fairly detailed, yet |in a deterministic sense| incomplete, a priori
knowledge about a time{varying channel.
Pulse Optimization. For a WSSUS channel, we can
give an explicit optimization rule for the pulseshape (t)
in terms of the scattering function, the optimization aims
at minimum expected orthogonal distortion, where the expectation is over the WSSUS ensemble:
Theorem 4. Given a WSSUS channel with scattering
function CH (; ) then the unit{energy pulse with minimum
expected orthogonal distortion is obtained by maximizing
an L2 (R2 ){inner product of CH (; ) and jA (; )j2 :
opt := arg min
EH kH k2 jhH; ij2
= arg max
CH ; jA j2 ; k k = 1:
(13)
Closed form solutions to (13) are hard to obtain in general (in the classical signal synthesis problem A (; ) does
not appear magnitude{squared [8]). It turns out, however, that under additional assumptions about the channel,
closed{form solutions are possible.
Elliptical Symmetry. For elliptical shape of the scattering function, i.e.,
CH (; ) = C0
2
0
2
+ 0
!
one can show [9] that normalized and properly scaled Hermite functions establish local optima. Hermite functions
form an orthonormal basis of L2 (R), they are dened as
[10]
r
k 2 k
hk (t) = 4 20 p1 2p1 e 00 t dtd k e
0
k!
2
2
0 t
0
: (14)
The rst order Hermite function is the Gaussian which will
typically achieve the global optimum, also it is the only admissible TMDA{like pulse.
High{order Hermite functions (k 1) are CDMA{like
pulses [11] in the sense that (i) they are spread over a large
region of the TF plane (T F = k), (ii) one has a desirable
sharp concentration of the ambiguity function [10, p.27] :
jAhk (; )j (r(4k +1 2))1=4 for 4k 1+ 2 r 2k + 1;
q
with r := ( 0 )2 + ( 0 )2 .
Low Cost Matching Rule. A low{cost approximate
pulse design both in the sense of (13) and the minimization of " in (8) can be formulated via the second{order
moments as dened in (9), (10):
T 0
F = 0 :
(15)
This matching rule is consistent both with the above analytic solution and with the naive combination of the LTI
and LFI matching rules (Theorems 1, 2).
6. SUMMARY AND CONCLUSION
In this paper, we have discussed pulses that suer minimum distortion when transmitted over LTV (mobile radio)
channels. For deterministic channels, we have assumed limited knowledge about the channel, namely only the maximum duration of the impulse response and the maximum
Doppler shift. With this knowledge, it is possible to give
precise upper limits for the admissible bandwidth and duration of a pulse if a certain transmission distortion is not
to be exceeded. These denitions bear some relation to the
well-known concept of "coherence bandwidth/time"; however, they do not rely on WSSUS assumptions, and also do
not suer from ambiguities due to non{monotone correlation functions. We have furthermore shown that for underspread systems, approximate eigenfunctions exist, and
that Zadeh's transfer function gives approximate eigenvalues. Zadeh's transfer function is thus useful in this context;
however, for a pulse design, the spreading function is the
appropriate representation. For WSSUS channels that have
elliptical symmetry, we have found that Hermite functions
are optimum pulses. The lowest-order Hermite function
(Gaussian bell) is the optimum TDMA{like pulse, while
higher-order Hermite functions have CDMA{like properties.
7.
REFERENCES
[1] R.S. Kennedy. Fading Dispersive Communication Channels. Wiley, New York, 1969.
[2] W. Rudin. Functional Analysis. McGraw Hill, New York,
1973.
[3] A. Papoulis. Signal Analysis. McGraw Hill, New York,
1984.
[4] H.L. Van Trees. Detection, Estimation and Modulation
Theory. Wiley, New York, 1971.
[5] G. Jourdain and G. Tziritas. Communication over fading
dispersive channels. Signal Processing, 27:672{679, 1980.
[6] P.A. Bello. Characterization of randomly time{variant linear channels. IEEE Trans. Comm. Syst., 11:360{393, 1963.
[7] W. Kozek. On the transfer function calculus for underspread
LTV channels. IEEE Trans. Signal Processing, 45(1), January 1997.
[8] C.H. Wilcox. The synthesis problem for radar ambiguity
functions. Technical Report 157, Mathematics Research
Center, United States Army, University Wisconsin, Madison (Wisconsin), 1960.
[9] W. Kozek. Matched Weyl{Heisenberg expansions of nonstationary environments. PhD thesis, Vienna University of
Technology, 1997.
[10] S. Thangavelu. Lectures on Hermite and Laguerre Expansions. Princeton University Press, Princeton (NJ), 1993.
[11] G. Wornell. Spread{signature CDMA: Ecient multiuser
communication in the presence of fading. IEEE Trans. Info.
Theory, 41(5):1418{1438, 1995.
[12] V. Filimon, W. Kozek, W. Kreuzer, and G. Kubin. LMS
and RLS tracking analysis for WSSUS channels. In Proc.
IEEE ICASSP{93, pages III/348{351, Minneapolis (MN),
1993.
[13] J.D. Parsons. The Mobile Radio Propagation Channel. Pentech Press, London, 1992.