Space-Time-Frequency Degrees of Freedom: Fundamental Limits for Spatial Information

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by  Leif Hanlen Thushara Abhayapala

Space-Time-Frequency Degrees of Freedom:
Fundamental Limits for Spatial Information
Leif W. Hanlen† and Thushara D. Abhayapala†
National ICT Australia and Australian National University
Leif.Hanlen,Thushara.Abhayapala @nicta.com.au
{ }
Abstract—We bound the number of electromagnetic signals magnetic signals in space in Section II. This gives our main
which may beobserved over a frequencyrange [F−W,F+W] result in Theorem 1. Section III gives plots of the degrees of
7 a time interval [0,T] within a sphere of radius R. We show freedom, while Section IV provides a simple application of
0 that the such constrained signals may be represented by a
the dimensionality result to MIMO mutual information. We
0 series expansion whose terms are boundedexponentially to zero
2 beyond a threshold. Our result implies there is a finite amount draw conclusions in Section V. We formally define spatially
of information which may be extracted from a region of space constrainingsignalsasacompactoperator,anddevelopproofs
n
via electromagnetic radiation. in the Appendix.
a
J I. INTRODUCTION II. DIMENSIONALITY
9
Wireless communication is fundamentally limited by the
Existing dimensionality results for signals 3D space, with
physicsofthemedium.Electromagneticwavepropagationhas
] non-trivial bandwidth are limited by
T been given [1,2] as a motivation for developing such limits:
I information is ultimately carried on electromagnetic waves. =2WT +1 (1)
s. Narrowbanddegreesof freedom (dimensionality)results have DWT 2
c eπRF
[ been given for dense multipath [3–5] and subsequently ex- space = +1 (2)
D c
tended to sparse systems. Here, the signal bandwidth is neg- (cid:18)(cid:24) (cid:25) (cid:19)
1
ligible: dimensionality results are defined in wavelengths. where(1)isfrom[10],formalisedin[11]and(2)isfrom[12].
v
Narrow-band wavefields were shown to have limited con- Fundamentally, we seek to develop a result which combines
6
5 centrations [6]. The limit was based upon the free-space both (1) and (2): broadband, spatially diverse signals.
0 Helmholtz (wave) equation – a time independent variation of Source-free (propagating) electric fields Ψ(r,t), are solu-
1 theelectromagneticwave[7].Suchwavesmaybe represented tions of the free-space Maxwell wave equation [7]:
0
byafunctionalseries,whosetermsareboundedexponentially
7 1 ∂2
towardzerobeyondsomelimit.Thislimitwasusedtodescribe Ψ(r,t)=0 (3)
0 △− c2∂t2 ·
/ a random MIMO channel in dense [8] multipath and provide (cid:18) (cid:19)
s
c capacity results. where =(∂2/∂x2,∂2/∂y2,∂2/∂z2) and c=3 108ms−1
: More recent work – including wide-band MIMO motivates is the s△peed of light. Thevectorr=(r,θ,φ) with×0 θ <π,
v ≤
i analysisofthecapabilityofspatiallydiversesignalstosupport 0 φ<2π denotes position. We now formally pose:
X multiplexing over significant bandwidths. ≤Problem 1: Given a function in space-time x(r,t) which
r Given a region, bounded by radius R, centre- is non-zero for r R and t [0,T] has a frequency
a k k ≤ ∈
frequency F, bandwidth 2W and observation time componentin [F W,F+W]andsatisfies (3); whatnumber
T, what is the number of wireless (electromag- of signals ϕ(r,−t) are required to parameterize x(r,t)?
netic) signals may be obDserved? DBeforeaddressingProblem1,observethatwemayrepresent
anysignalwhichisconstrainedto[0,T] [0,R]andsatisfying
In[9]anapproximatedimensionalityresultwasgiven.This
×
(3)byasolutionof(3).InCartesiancoordinatessuchsolutions
bound suffered was excessively complex – resulting in a
are exponentials [13]:
loose over-bound. In this work we provide exponential error
bounds – reflecting the results of [3], and provide a tighter Ψk(r,t)=exp( ιkct ιk r) (4)
bound on the dimensionality of 3D-spatial broadbandsignals. − | | − ·
In developing our result we will also show how the four where k is the vector wave-number and ι = √ 1. The
parameters (R,T,W,F) may be traded against each other. magnitude k = 2πf/c is the well-known scala−r wave-
| |
The remainder of this paper is arranged as follows: We number.
provide a truncation point and bound the error for electro- Any function represented in terms of (4) may be written
as a series expansion (24). The series itself is not important,
†L.W.HanlenandT.D.AbhayapalaalsoholdappointmentswiththeRe- simplythatitexists.Thisseriesmaybetruncatedatapoint
searchSchoolofInformationSciences andEngineering, ANU.NationalICT N
which is an increasing function of frequency,time and space:
AustraliaisfundedthroughtheAustralianGovernment’sBackingAustralia’s
Abilityinitiative, inpartthroughtheAustralian Research Council.
Lemma 1 (Truncation Point): The critical threshold (a
function of r, t) at frequency f is
| |
N(r,t;f)=NT(t;f)+NS(r;f) (5) 2500
(t;f)= eπ∆ t (6)
NT ⌈ f ⌉ 2000
r
S(r;f)= eπf| | (7) 1500
N c F
(cid:24) (cid:25) Do1000
Once we have chosen an appropriate threshold, increasing
N
reduces the truncation error exponentially. 500
Lemma 2 (Truncation Error): The truncation error is 0
1000
ǫ 2/e 0.74 (8) 0.2
N ≤ ≈ 500 0.15
0.1
and for any α,δ 1 W (Hz) 0.05 R(m)
≥ 0 0
ǫ <ǫ e2−δ−α (9) (a) DoFvsbandwidth W andregionradiusR
N+δ+α N
·
We may use (6) and (7) and the tightness of the truncation 1000
to define the dimesionality of the 3D spatial signals.
900
Theorem 1 (3D dimensionality ): The number of or-
D3D 800
thogonal electromagnetic waves which may be observed in
a three-dimensionalspatial region bounded by radius R, over 700
frequency range F W and time interval [0,T] is 600
± z)
H
2 (500
eπR(F W) W
(eπ2WT +1) − +1 400
3D
D ≈ c
(cid:18) (cid:19) 300
2
eπR 2
+eπ2WT 2FW W2 200
c − 3
(cid:18) (cid:19) (cid:20) (cid:21) 100
2
eπR eπR 1
+ 4FW +eπ2WT F + (10) 0
c c 6 0 0.05 0.1 0.15 0.2
(cid:18) (cid:19) (cid:20)(cid:18) (cid:19) (cid:21) R(m)
NoteF W bydefinition,sobothtermsin(10)arealways
(b) Contours
≥
positive.Thesecondtermin(10)givesanestimateoftheeffect
of signal bandwidth on the degrees of freedom in space. For Fig. 1. Number of degrees of freedom for moderate W and T =0.5ms,
F =2.4GHz,λ=0.125m: Curvature toward bottom ofFig.1(b)denotes
practical applications we expect R c and TW to be small
saturation wrt.radius.
≪
(TW < 10). Thus, for the second term in (10) to be non-
negligible we require √FW c, where we are interested
≈
only in magnitudes.
Figure 2 shows the DoF for a broadband signal W F
≤
A. Asymptotic Results with centre frequency F = 2.4MHz. In this case the centre
wave-lengthis 125m(andatF+W, λ =62.5m)so R
For R 0 (10) reduces to 7eπTW/3+ 1 which over- min ≪
→ λ/2. In this case at R 0 we see the usual 2WT linear
bounds (1). For T,W 0, (10) reduces to (2) while for →
W =0,T 0 (10) red→uces to (2) with F F +W. For growthinDoF,atR≈λmin/4weseeaknee-pointintheDoF
6 → → surface,correspondingtospatialdegreesoffreedombecoming
extreme broadband signals F =W, and
effective. This can be seen by the curvature of the contours
eπR 2 4 near R=5m in Figure 2(b)
2eπTW W2 (11)
3D
D ∝ c 3
(cid:18) (cid:19) IV. EXAMPLE:MUTUAL INFORMATION
I
and when all parameters are non-trivial
We assume all channel eigenvalues are equal magnitude
eπR 2 W2 in space and frequency, up to T and S respectively. The
3D 2eπTW F2+ (12) transmitteruses(4)asmatchedfiNltersforNthechannelandsends
D ∝ c 3
(cid:18) (cid:19) (cid:18) (cid:19) uniform power ρ on the subset of modes with non-neglible
III. PLOTS magnitude. This is a reasonable capacity approximation [14].
We have considered two common spatially diverse scenar- Ana¨ıveapplicationof(10)tomutualinformationwouldbe
ios.InFigure1wehaveusedacentrefrequencyF =2.4GHz, = log(1+ρ/ ) ρ. This ignores the random nature
I D D ≤
1kHz bandwidth and R < 2λ. Figure 1(a) shows a mesh of of the MIMO channel: spatial signals are mixed through a
thedegreesoffreedom,whileFigure1(b)givesacontourplot. random scattering channel, while frequency signals are not.
The super-linear growth in DoF can be seen as both R and Assume both transmitter and receiver have identical geome-
W increase. tries and are situated in dense scatter.
=ρN .ForN (f)anincreasingfunctionoff,thesum(14)
t t
I
is increased, thus
2
200 eπR(F W)
ρ − +1 (15)
I ≥ c
150 (cid:18) (cid:19)
Due to random scattering, spatial modes provide a linear
F100 increase in capacity, while frequency modes provide parallel
o
D
channels.
50
0 V. CONCLUSION
3
2 15 Wehaveshownthatthedegreesoffreedomforaspherically
x 106 W (Hz)1 5 R(1m0) restricted broadband wireless signal is proportional to the
surface area of the spatial region, the square of the frequency
0 0
and bandwidth and the DoF of the broadband signal itself.
(a) DoFvsbandwidthW andregionradius R
x 106 We have shown that the error associated with truncating such
a signal at N terms decreases exponentially as N increases.
As an example we have shown that broadband spatial
2
communication systems may have a capacity beyond that ex-
pected by combining MIMO capacity results by with parallel
1.5 frequency channels.
z)
(H APPENDIX I
W 1 OPERATORMATERIAL
Definition 1 (Truncation Projection D): Given the spatial
0.5 interval S and time interval [0,T], the truncation operator
D sets a field f(r,t) to zero outside the time- and space-
intervals.
0
0 5 10 15
f(r,t) r S and t [0,T]
R(m) Df(r,t)= ∈ ∈ (16)
(b) Contours (0 else
Fig.2. NumberofdegreesoffreedomforlargeW,smallRandT =1µs,
F =2.4MHz,λ=125m Definition 2 (Wavefield Projection W): The wavefield pro-
jectionW projectsafieldf(r,t)ontosolutionsΨ(r,t)of (3).
ConsiderFigure3,everyhorizontaldashedlinemaybecon- Wf(r,t)= fl,m,nΨl,m,n(r,t) (17)
sideredasaninputelementwitheachelementoperatingat2W lX,m,n
different frequency taps (eg. through OFDM). At frequency f = f(r,t)Ψ (r,t)dtdr
l,m,n l,m,n
F W f F + W there are N = (eπRf/c + 1)2,
− ≤ ≤ t ZZ
independent input signals from (2). Then there are k is the wave vector in three dimensions k = (k ,k ,k )
x y z
F+W with scalar wave number k = k = (k2 +k2+k2)1/2 [13,
N = Nt(f)= (eπRf/c+1)2 (13) eqn.6.94 p.759]. Ψl,m,n is give|n|by (4)xwithythe vazlues of k
f f=F−W chosen discretely:
X X
parallel input channels and mutual information at f is f 2π 2π 2π 2π
I k= p k = l k = m k = m (18)
x y z
ρ N cT L L L
=logdet I + XX∗ N log 1+P t
If Nt N → t N
(cid:16) (cid:17) (cid:18) (cid:19) Using (16) and (17) we may write the signal observed in
Where X is a Gaussian random matrix of dimension N (f), S T as:
t
×
see forexample[6].Sinceeachfrequencychannelisindepen- g(r,t)=WDf(r,t) (19)
dent,the totalmutualinformationis foundby combining(13)
and Lemma 3: WD is a compact operator.
F+W N (f) Lemma 3 emphasizes that although there are infinitely many
= N (f)log 1+ρ t (14)
t independent electromagnetic waves, only a finite number of
I N
f=XF−W (cid:18) (cid:19) electromagnetic signals may be resolved within the region
which may be calculated numerically. Note, in the case of S T. The implication of this is that any approximation for
×
N (f)=constwe returnto the classic parallelchannelresult a given signal has a bounded error.
t
APPENDIX II
PROOFS e π R f
Proof: [Lemma 1] = c
n
From (4)
R
Ψk(r,t)=exp( ιkminct)exp ιk´ct ιk r (20)
− − − ·
=exp( ιkminct)Ψˆk(r(cid:16),t) (cid:17) (21)
−
wherekrdenotesthevectordotproductand0 k´ 4πW/c. n=1
· ≤ ≤
We wish to bound the number of terms required to approxi- f
matethisfunction.Noteexp( ιkminct)hasexactlyonedegree F W F +W
of freedom, so we may equiv−alently calculate the DoF for Ψˆ. −
Fig. 3. Geometry for spatial functions. F −W ≤ f ≤ F +W and
Using the Jacobi-Anger expansion [15, eqn2.45, pp.32] and 0≤|r|≤R.Timeformsathirddimension(intothepage).
summation theorem [15, eqn2.29, pp.27]
∞ n
e−ιk·r=4π ιnjn(kkrk) Ynm(ˆr)Ynm(kˆ) (22) Both sums (28) converge if P +1>eck´t/2 and
n=0 m=−n N +1>ek r/2.
X X | |
whFerroemjn[(1z3),=8.5342π.z1J]n+12(z) is a spherical Bessel function. e(N +1)(P +1) eck´t P ek|r|/c N
P+1 N+1
p ǫ < (29)
∞ P,N (2N +2 x)(2P(cid:16)+2(cid:17) y(cid:16))2N+P−(cid:17)1
e−ιck´t = ιp(2p+1)j (ck´t) (23) − −
p eπf´t P eπf r/c N
p=0
X <2e | | (30)
Combining (22) and (23) P +1! (cid:18) N +1 (cid:19)
P 0 f´ 2W and F W f F +W
Ψ(P,N)(r,t)=4π ιp(2p+1)j (ck´t) ≤ ≤ − ≤ ≤
p Proof: [Lemma 2] Define x = eckt and y = ekR then
Xp using (30)
N
ιnj (k r )Ym(ˆr)Ym(kˆ) (24) P+δ N+α
× n k k n n ǫN+α,P+δ P +1 N +1
=
n,m
X ǫ P +1+δ N +1+δ
We may truncate Ψ(r,t) and bound the error as in [3]: N,P (cid:18) (cid:19) (cid:18) (cid:19)
δ α
x y
ǫ(P,N) = Ψ(r,t) Ψ(P,N)(r,t) × P +1 N +1
− (cid:18) (cid:19) (cid:18) (cid:19)
(cid:12)(cid:12) ∞ (cid:12)(cid:12) Now α,δ 1, P +1>x and N +1>y by definition
=(cid:12)4π ιp(2p+1)j (c(cid:12)k´t) ≥
(cid:12) p ǫ P +1 P+δ N +1 N+α
(cid:12)(cid:12)pX>N N+α,P+δ e1−δe1−α
(cid:12)(cid:12)(cid:12) × ∞ ιnjn(kkrk) n Ynm(ˆr)Ynm(kˆ)(cid:12) UsingǫNth,Pe iden≤tit(cid:18)yP(1++δa(cid:19)/N)N(cid:18)<Nea+δ(cid:19) ≤
nX>N mX=−n (cid:12) Proof: [Theorem 1] From (30) P may found directly by
(cid:12)
Taking the absolute values inside the summation and usin(cid:12)g using the maximum values of t and k´:
n Ym(ˆr)Ym(kˆ) (2n+1)/(2π) [15, pp.27] give(cid:12)s
m=−n n n ≤ P =eπ2WT (31)
(cid:12)(cid:12)P ∞ (cid:12)(cid:12) ∞
(cid:12)ǫ 2 (2p+1)(cid:12)j (ck´t) (2n+1)j (k r) (25) Truncating(30)atP gives =2eπWT+1terms.Thesame
P,N ≤ p | n | | | DT
pX>N (cid:12) (cid:12)nX>N technique cannot be used for N, since the bound becomes
From [3,12] (cid:12)(cid:12) (cid:12)(cid:12) excessively loose. Instead we use a geometric argument:
Consider Figure 3. The dotted vertical lines give the fre-
√π 1 x n
jn(x) (26) quency constraints, while the top line n = eπRf/c gives
| |≤ 2 Γ(n+3/2) 2
the spatial constraint of (7). The constraint set defines a
(cid:16) (cid:17)
Using the identity (2n+1)/Γ(n+3/2)=1/Γ(n+1/2) trapezium in the space-frequency plane (or trapezoid when
and [16, pp.257] Γ(n+1/2)>e−n−1/2(n+1/2)n(2π)1/2 timeisincluded).Theheightsofthetrapeziumarefoundfrom
e (ck´t)e p (k r)e n (7)withf =F W.Notethatthefigureisnotdrawntoscale:
±
ǫP,N < | | (27) for any reasonable value of R, the slope of the line eπR/c is
2 2(p+1/2) 2(n+1/2)
p>P" # n>N(cid:20) (cid:21) almost zero.
X X
e (ck´t)e p (k r)e n Each horizontaldashedline representsa collectionof time-
< | | (28) frequencysignals which may be observedat a point in space.
2 2(P +1) 2(N +1)
p>P" # n>N(cid:20) (cid:21) Spatial diversity allows observation of multiple collections.
X X
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Helmholtz) projection G [8].
1 F+W ∞
Bf(r,t)= dωeιωt dte−ιωtf(r,t) (38)
2π
ZF−W Z−∞
Gf(r,t)= fkexp(ιk r) (39)
·
k
X
Write (19) as:
g(r,t)=GBDf(r,t) (40)
We know [11] the operator BD is compact, since it maps
a unit ball in (finite energy signals) to an essentially
2
L
finite dimensionalball (of approximatedimension2WT +1).
From[4]Gisaprojectionandthusboundedso[17,Lem.8.3-2
p.422] the product G BD is compact.
·