E o Con ol and Spec al Windowing Readou
in Compu a ional Uni e se:
Time–F equency–Complexi y Role o PSWF/DPSS
Window Func ions
Unde Uni ied Time Scale
Haobo Ma1Wenlin Zhang2
1Independen Resea che
2Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
In p e ious se ies wo ks on compu a ional uni e se Ucomp = (X, T,C,I), we ha e
es ablished disc e e complexi y geome y (complexi y dis ance, olume g ow h, and
disc e e Ricci cu a u e), disc e e in o ma ion geome y ( ask in o ma ion mani old
(SQ, gQ) and embedding ΦQ), con ol mani old (M, G) induced by uni ied ime
scale, as well as ime–in o ma ion–complexi y join a ia ional p inciple. The ein
uni ied ime scale gi en by sca e ing mas e scale
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
uni ying physical ime densi y, spec al shi unc ion de i a i e, and Wigne –
Smi h g oup delay ace.
Howe e , abo e s uc u es a e s ill “ideal limi s”: adius To complexi y ball
BT(x0) can be a bi a ily la ge, geodesics on con ol mani old can ex end a bi-
a ily, Fishe s uc u e on in o ma ion mani old can be pe ec ly iden i ied unde
in ini e da a. In ac ual compu a ional uni e se, all eadou s and decisions p oceed
unde ini e ime, ini e complexi y budge , and ini e equency band cons ain s,
hus necessa ily ca ying e o s. To igo ously con ol e o s wi hin uni ied ime
scale–complexi y geome y–in o ma ion geome y amewo k, equi es sys ema ic
“spec al windowing eadou ” heo y.
This pape in oduces eadou ope a o s and e o models wi hin compu a ional
uni e se amewo k, uni ying hem as window unc ion p oblem in uni ied ime scale
equency domain. We p o e: unde uni ied ime scale, w i ing eadou ope a o as
in eg al o e equency domain objec s
R( ) = ZΩ
W(ω) (ω) dω,
whe e W(ω) is window unc ion, (ω) is equency domain quan i y ela ed o
uni e se e olu ion (e.g., κ(ω) o i s weigh ing), can na u ally in oduce class o
ime–band-limi ed–complexi y-limi ed join ex emal p oblems.
1
In con inuous case, we p o e: unde cons ain s o gi en ime unca ion in e al
[−T, T] and equency band [−W, W], P ola e Sphe oidal Wa e Func ions (PSWF)
a e op imal window unc ion amily: hey maximize ene gy concen a ion unde
dual es ic ions o [−T, T] and [−W, W], he eby minimizing wo s -case e o o
“ene gy leakage ou side complexi y ball” when uni ied ime scale–complexi y budge
gi en.
In disc e e case, we in oduce co esponding Disc e e P ola e Sphe oidal Se-
quences (DPSS), de ining window sequences on ini e-leng h complexi y chains,
p o ing hey maximize ene gy concen a ion unde disc e e ime– equency es ic-
ions, he eby gi ing op imal e o con ol s uc u e o “ ini e-o de eadou ” unde
cons ain s o ini e complexi y s eps Nand ini e bandwid h W.
Unde language o uni ied ime scale–complexi y geome y, we ob ain ollowing
conclusions:
1. Fo compu a ional uni e se eadou s wi h band-limi ed uni ied ime scale e-
quency (e.g., sca e ing delay spec um), i complexi y budge only allows
2TW/π le el deg ees o eedom, hen PSWF window unc ions gi e op imal
e o –complexi y adeo unde his budge ;
2. On disc e e complexi y g aph Gcomp, DPSS p o ides op imal eadou sequence
unde ini e s ep leng h Nand ini e bandwid h W, whose e o decay and
spec al concen a ion cons an s con olled by DPSS eigen alues;
3. These esul s can be embedded in o ime–in o ma ion–complexi y join a ia-
ional p inciple, iewing “choosing eadou window unc ion” as adding “spec-
al windowing con ol dimension” laye on join mani old EQ, he eby gi ing
a ia ional cha ac e iza ion o “op imal obse a ion s a egy unde ini e e-
sou ces”.
This pape as “e o con ol” chap e in compu a ional uni e se se ies, a in e -
ace o uni ied ime scale– equency domain–complexi y geome y, ele a es classi-
cal ime– equency concen a ion esul s o PSWF/DPSS o e o con ol and ob-
se abili y heo y in compu a ional uni e se, p o iding heo e ical ounda ion o
subsequen cons uc ion o uni ied eadou design on speci ic physical–enginee ing
es beds such as FRB/δ- ing.
Keywo ds: Compu a ional uni e se; E o con ol; Spec al windowing; PSWF; DPSS;
Time- equency concen a ion; Uni ied ime scale; Complexi y geome y
1 In oduc ion
In any ac ual physical o compu a ional sys em, eadou s and decisions canno p oceed
o e in ini e ime, in ini e complexi y budge , and in ini e equency band. Uni ied ime
scale–complexi y geome y–in o ma ion geome y gi e geome ic s uc u es o “in ini e
ideal uni e se”:
Complexi y ball BT(x0) can expand as T→ ∞;
Geodesic wo ldlines on con ol mani old (M, G) can ex end o in ini y;
Uni ied ime scale densi y κ(ω) in equency domain can be obse ed o e in ini e
equency band;
2
Fishe s uc u e o in o ma ion mani old (SQ, gQ) can be comple ely iden i ied unde
in ini e da a.
Bu in eali y, we mus wo k unde ollowing es ic ions:
1. Time–complexi y limi a ion: To al complexi y budge T ini e, egions ou side
complexi y ball un eachable in ini e ime;
2. F equency band limi a ion: E ec i e equency band o uni ied ime scale den-
si y κ(ω) ini e, o ac ual eadou de ice can only espond wi hin ini e equency
band;
3. Readou o de limi a ion: Realizable eadou ope a o dimension ini e, e.g.,
can only sample ini e ime momen s o ini e equency poin s, o can only execu e
ini e-o de momen – il e ing ope a ions;
4. E o ole ance: Sys em mus ensu e e o unde hese limi a ions does no exceed
some admissible h eshold.
In signal p ocessing and ime– equency analysis, PSWF/DPSS become classical ools
wi h hei p ope y o “op imal ene gy concen a ion unde ini e ime and ini e band”.
Howe e , hese esul s mos ly discussed in pu e signal spaces (e.g., in L2([−T, T]) and
L2(R)), no ye sys ema ically embedded in o uni ied ime scale–complexi y geome y
amewo k.
Pu pose o his pape h ee old:
1. Fo malize eadou p ocess in compu a ional uni e se as window unc ion p oblem
in uni ied ime scale equency domain;
2. Unde uni ied ime scale–complexi y budge cons ain s, gi e op imali y heo ems
o PSWF/DPSS window unc ions in e o con ol sense;
3. Embed hese esul s in o ime–in o ma ion–complexi y join a ia ional p inciple,
cons uc ing heo e ical amewo k o “op imal obse a ion unde ini e esou ces”.
Pape o ganiza ion: Sec ion 2 de ines eadou ope a o s and e o models in compu-
a ional uni e se. Sec ion 3 in oduces spec al windowing eadou in uni ied ime scale
equency domain. Sec ion 4 e iews and es a es ene gy concen a ion and ini e ime–
band op imali y p ope ies o PSWF/DPSS, ansla ing hem o e o uppe bounds o
“ ini e complexi y eadou ”. Sec ion 5 discusses “complexi y– ime–bandwid h” iple
uni ied cons ain s in compu a ional uni e se. Sec ion 6 inco po a es window unc ion
choice in o join a ia ional p inciple, gi ing a ia ional o m o “op imal obse a ion
s a egy”. Appendices gi e de ailed p oo s o PSWF/DPSS de ini ions, key eigen alue
p ope ies, and main e o uppe bound heo ems.
2 Readou Ope a o s and E o Models in Compu-
a ional Uni e se
This sec ion o malizes eadou ope a o s on compu a ional uni e se Ucomp = (X, T,C,I),
de ining e o and e o budge .
3
2.1 Pa h-Le el Readou Ope a o s
Conside e olu ion pa h s a ing om ini ial s a e x0
Γ = (x0, x1, x2, . . . ),(xk, xk+1)∈T.
Unde uni ied ime scale, each s ep has physical ime inc emen
∆ k=C(xk, xk+1)/λ,
whe e λis uni con e sion cons an (can be abso bed in o Cbelow).
De ini ion 2.1 (Pa h Readou Ope a o ).A pa h eadou ope a o is map
R:{pa h Γ} → Cm,
w i able as composi ion o ini e- ime linea unc ionals, e.g.,
R(Γ) = ⟨ 1,Γ⟩,...,⟨ m,Γ⟩,
whe e each jis ini ely-suppo ed “ke nel”, e.g.,
⟨ j,Γ⟩=X
k
j(k, xk).
In con inuous limi , can iew Γ as cu e (θ( ), ϕ( )) on con ol mani old and in o ma-
ion mani old, eadou becomes ime in eg al
⟨ j,Γ⟩=ZT
0
Rj(θ( ), ϕ( )) wj( ) d ,
whe e wj( ) is weigh unc ion (window unc ion).
2.2 Ideal Readou and T unca ed Readou
Ideally, we wan eadou o e in ini e-leng h pa h o in ini e ime window:
Rideal(Γ) = Z∞
0
R(θ( ), ϕ( )) d .
Howe e unde ini e complexi y budge Tmax, can only eadou in ini e ime:
R unc(Γ) = ZTmax
0
R(θ( ), ϕ( )) W( ) d ,
whe e W( ) is window unc ion suppo ed on [0, Tmax], used o smoo hly unca e a
ime bounda y.
Di e ence be ween hem de ines eadou e o :
E (Γ; W) = Rideal(Γ) − R unc(Γ).
In equency domain desc ip ion, choice o Wdi ec ly de e mines e o decay and
ene gy leakage p ope ies, hus choosing op imal window unc ion Wis co e o e o
con ol.
4
2.3 E o No m and Wo s -Case E o
Fo uni ied discussion, we in oduce semi-no m on pa h space (e.g., L2no m induced by
uni ied ime scale equency spec um).
Suppose o each pa h Γ he e exis s co esponding equency domain objec Γ(ω)
(e.g., combined om sca e ing da a and uni ied ime scale densi y on con ol mani old),
sa is ying
|Γ|2=ZΩ
| Γ(ω)|2dµ(ω).
Readou e o can be w i en as equency domain window o m (see Sec ion 3), whose
wo s -case no m de ined as
E(W) = sup
Γ=0
|E (Γ; W)|
|Γ|.
We ca e abou inding window unc ion amily {W}making E(W) as small as possible
unde gi en ime–bandwid h and complexi y budge cons ain s.
3 Spec al Windowing Readou in Uni ied Time Scale
F equency Domain
This sec ion in oduces equency domain desc ip ion unde uni ied ime scale mas e
scale, w i ing eadou ope a o as inne p oduc o window unc ion and equency spec-
um.
3.1 Uni ied Time Scale F equency Domain Rep esen a ion
In p e ious wo k, we ha e connec ed uni ied ime scale wi h sca e ing da a on physical
uni e se side. Pulling his s uc u e back o compu a ional uni e se, can conside each
pa h Γ co esponds o equency domain objec
Γ(ω) = κ(ω) Φ(Γ; ω),
whe e κ(ω) is uni ied ime scale densi y, Φ(Γ; ω) encodes pa h esponse o his e-
quency mode h ough con ol–sca e ing s uc u e.
Fo gi en eadou ke nel, can de ine window in equency domain
W(ω)
such ha
R(Γ) = ZΩ
W(ω) Γ(ω) dω.
Ideal eadou co esponds o Wideal(ω)≡1 (o some ixed weigh ), while ini e com-
plexi y eadou co esponds o es ic ing W(ω) o ini e bandwid h [−W, W] o ini e
deg ee-o - eedom space.
5
3.2 Time–F equency Dual Res ic ion and Window Func ion
Design P oblem
Suppose we can only eadou wi hin ime in e al [−T, T], co esponding o ime window
wT( ) (e.g., wT( ) = 1 on | | ≤ T, o he wise 0), and only in e es ed in equency band
[−W, W].
In equency domain, eadou sensi i i y o ene gy ou side equency band de e mines
e o : i pa h spec al componen s lea e [−W, W], hen window unc ion W(ω) needs o
supp ess hem as much as possible; bu simul aneously should main ain as good pass
cha ac e is ics as possible wi hin [−W, W].
Thus we ob ain ypical dual- es ic ion window unc ion design p oblem:
Time es ic ion: Readou window wT( ) suppo ed on [−T, T];
F equency es ic ion: Readou window spec um bwT(ω) concen a ed on [−W, W];
Objec i e: Unde gi en cons ain s, minimize ou -o -band ene gy o wo s -case
e o .
In signal analysis, his p ecisely classical p oblem o PSWF/DPSS; his pape in e -
p e s i as “bes ini e complexi y eadou ” unde uni ied ime scale–complexi y geome y.
4 PSWF/DPSS and Ene gy Concen a ion: F om
Time–F equency o Time–Complexi y
This sec ion e iews de ini ions and ene gy concen a ion o con inuous PSWF and dis-
c e e DPSS, ansla ing hem o e o con ol esul s in compu a ional uni e se.
4.1 De ini ion and Time–Band Concen a ion o Con inuous
PSWF
De ini ion 4.1 (Con inuous PSWF).Fix ime window T > 0 and bandwid h W > 0.
De ine in eg al ope a o
(K )( ) = ZT
−T
sin W( −s)
π( −s) (s) ds, | | ≤ T.
This ope a o equi alen o composi ion o “ i s ime-limi o [−T, T], hen band-limi
o [−W, W]”. I s eigen unc ions ψn( ) and eigen alues λnsa is y
ZT
−T
sin W( −s)
π( −s)ψn(s) ds=λnψn( ),| | ≤ T.
ψncalled P ola e Sphe oidal Wa e Func ions unde ime window [−T, T] and band-
wid h [−W, W].
P oposi ion 4.2 (Ene gy Concen a ion o PSWF).PSWF sa is y:
1. They cons i u e o hogonal basis on [−T, T];
6
2. Fo each ψn, de ine equency domain ene gy concen a ion
αn=RW
−W|b
ψn(ω)|2dω
R∞
−∞ |b
ψn(ω)|2dω,
hen αn=λn, and λ0≥λ1≥. . . ;
3. Fo gi en ime window and bandwid h, {ψn}a e op imal basis in sense o “ene gy
concen a ion unde simul aneous ime and equency es ic ions”: o al ene gy
concen a ion o any o he same-dimensional subspace does no exceed sum o PSWF
subspace.
P oo o P oposi ion ?? gi en in Appendix A.
4.2 De ini ion and Fini e S ep–Fini e Bandwid h o Disc e e
DPSS
In disc e e case, conside sequence o leng h N:x[0], . . . , x[N−1], whose disc e e ime–
equency es ic ion p oblem can be cha ac e ized h ough DPSS (Disc e e P ola e Sphe oidal
Sequences).
De ini ion 4.3 (DPSS).Fix sequence leng h Nand no malized bandwid h W∈(0,1/2).
De ine Toepli z ma ix
Kmn =sin 2πW(m−n)
π(m−n),0≤m, n ≤N−1,
on diagonal de ine Kmm = 2W. Sol e eigen alue p oblem
N−1
X
n=0
Kmn (k)
n=λk (k)
m.
No malized eigen ec o s (k)a e DPSS unde (N, W), eigen alues λka e ene gy con-
cen a ions:
λk=P|ω|≤W|b (k)(ω)|2
P|ω|≤1/2|b (k)(ω)|2.
P oposi ion 4.4 (Disc e e Ene gy Concen a ion o DPSS).DPSS { (k)}maximize
ene gy concen a ion unde disc e e ime– equency dual es ic ion: o any subspace
V⊂CNo dimension K, i s ene gy concen a ion sum wi hin equency band [−W, W]
does no exceed concen a ion sum o space spanned by i s KDPSS.
P oo see Appendix A.
7
4.3 In e p e a ion in Compu a ional Uni e se as “Fini e Com-
plexi y Readou ”
In compu a ional uni e se, pa h segmen o leng h N iewable as complexi y s ep numbe
limi a ion N; co esponds o ime window T≈N∆ unde uni ied ime scale. F equency
band Wco esponds o e ec i e suppo o uni ied ime scale densi y κ(ω).
Unde his pe spec i e:
PSWF co esponds o op imal con inuous window unc ion unde gi en complexi y
window [−T, T] and equency band [−W, W];
DPSS co esponds o op imal disc e e eadou sequence unde disc e e complexi y
s ep leng h Nand equency band W.
The e o e, unde ini e complexi y budge any eadou ope a o hoping o ai h ully
cap u e in o ma ion in uni ied ime scale equency spec um, i s ime window o sequence
should app oxima e PSWF/DPSS as much as possible.
5 Complexi y–Time–Bandwid h T iple Uni ied Con-
s ain s
This sec ion discusses ela ionship among complexi y budge T, ime window T, and
equency band W, gi ing “compu a ional uni e se e sion o Landau–Pollak–Slepian
es ic ion”.
5.1 Time–F equency–Complexi y Deg ee-o -F eedom Coun ing
In classical ime– equency analysis, e ec i e deg ee-o - eedom numbe o signal subspace
wi h bandwid h Wand ime es ic ion Tis
Ne ≈2WT
π.
This esul can be ob ained h ough asymp o ic beha io o PSWF eigen alues: eigen-
alues λnclose o 1 when n < 2WT/π, apidly decline o 0 when n > 2WT/π.
In uni ied ime scale–complexi y geome y, we can in e p e his deg ee-o - eedom
coun as:
Fo gi en complexi y budge Tand uni ied ime scale equency band W, numbe
o independen modes ha can be eliably encoded o ead ou is app oxima ely
Ne ;
On ask in o ma ion mani old, his co esponds o numbe o Fishe modes iden i-
iable unde ini e complexi y budge .
5.2 Complexi y–Time–Bandwid h Cons ain Inequali y
We o malize compu a ional uni e se e sion:
8
Theo em 5.1 (Complexi y–Time–Bandwid h Deg ee-o -F eedom Uppe Bound).Sup-
pose compu a ional uni e se has uni ied ime scale equency domain ep esen a ion Γ(ω),
whose suppo o e ec i e ene gy concen a ed on [−W, W]. Fo complexi y budge T, e-
s ic ing pa h wi hin complexi y ball BT(x0), conside all eadou ope a o s
Rj(Γ) = ZW
−W
Wj(ω) Γ(ω) dω, j = 1, . . . , K,
whe e {Wj}a e o hogonal window unc ion amily in L2([−W, W]).
Then unde e o ole ance ε, he e can exis op imal window unc ion amily {W⋆
j}
(gi en by i s KPSWF spec a), such ha o all pa hs Γsa is ying
Γ(ω)−
K
X
j=1
⟨ Γ, W⋆
j⟩W⋆
j(ω)L2([−W,W ])
≤ε| Γ|L2([−W,W ])
only when
K≳2WT
π+Olog(1/ε).
Tha is, unde complexi y– ime–bandwid h iple cons ain , eliably dis inguishable
deg ee-o - eedom numbe does no exceed ≈2WT/π.
P oo see Appendix B.
This esul unde s andable as “compu a ional uni e se’s Nyquis –Slepian es ic ion”:
uni ied ime scale equency band W oge he wi h complexi y budge Tde e mine e ec-
i e mode numbe eadable in ini e ime.
6 Va ia ional P inciple o Window Func ion Choice:
Op imal Obse a ion S a egy
This sec ion in oduces window unc ion choice in o ime–in o ma ion–complexi y join
a ia ional p inciple, cons uc ing a ia ional o m o “op imal obse a ion s a egy”.
6.1 Ex ended Join Mani old and Window Func ion Deg ees o
F eedom
P e ious join mani old
EQ=M×SQ
whe e cu e z( )=(θ( ), ϕ( )) desc ibes con ol–in o ma ion s a e e olu ion.
Now add window unc ion deg ees o eedom: le Wbe some window unc ion space
(e.g., subspace in L2([−T, T]) sa is ying band-limi cons ain s), o each eadou channel
jchoose window unc ion Wj∈ W.
Ex ended join con igu a ion as
bz( )=(θ( ), ϕ( ),{Wj}K
j=1).
Window unc ion i sel can ha e no explici ime e olu ion ( iewed as s a ic pa o
s a egy), o can be upda ed on slow a iable scale.
9
