Windowed Formulation of Phase--Spectral-Shift--DOS--Cosmological-Constant

Windowed Fo mula ion o
PhaseSpec al-Shi DOSCosmological-Cons an
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
We es ablish equi alence chain cen e ed on
gene alized sca e ing phase
, connec ing
Kon se ichVishik (KV) de e minan phase, gene alized K en spec al shi , densi y-o -s a es
die ence (DOS die ence), and Wigne Smi h (WS) ace iden i y, p o iding igo ous
a i-
able and ac o accoun ing
unde e en-dimensional asymp o ically hype bolic/con o mally
compac (AH/CCM) and s a ic-pa ch de Si e (dS) geome y. Unde s ic Li shi sK en (LK)
ace o mula and ela i e ace-class assump ions, we p o e subs i u ion ela ion be ween hea
ke nel die ence Laplace ep esen a ion and DOS slope unde
equency a iable
∆K(s) = Z∞
0
e−sω2Θ′(ω)dω,
whe e
Θ(ω) = 1
2πa g de KV S(ω)
,
Θ′= ∆ρω=−∂ωξω
. We p opose amily o
loga i hmic
equency windows
W
wi h Mellin-nullica ion condi ions, es ablishing
windowed Taube-
ian heo em
: unde non- apping, no ze o-ene gy esonance, analy ic F edholm and ope a o -
Lipschi z assump ions, small-
s
hea ke nel ni e pa equi alen o loga i hmic window a e age
o
Θ′
a scale
µ∼s−1/2
, wi h e o uppe bound. This denes
windowed in eg a ion law
∂ln µΛϕ,W (µ) = κΛΞW(µ),ΞW(µ) = ZR
ωΘ′′(ω)Wln(ω/µ)dln ω=1
2πZω ∂ωT Q(ω)W d ln ω,
p o iding dimensionally consis en cons an sepa a ion:
κHK
(hea ke nelwindowing a-
io) and
κΛ
(dimensional ac o mapping
⟨Θ′⟩W
o cosmological cons an ). Fo
open channels
(ho izon/abso p ion), espec i ely es ablish p o able condi ions o
∂ωa g de KV b
S=−iT b
S†∂ωb
S
ia
ex ended uni a iza ion
and
ela i e de e minan
, p ese ing WS- ace equali y. Demon-
s a e
κ
ex ac ion p ocedu e wi h one-dimensional
δ
po en ial as sol able empla e and s a ic-
pa ch dS scala as compu able empla e. P opose minimal ep oducible obse a ion pipeline
based on
FRB baseband
complex phase, p o iding closed- o m o mulas o second-o de
phase ke nel a iance scaling, dispe sion/mul ipa h leakage ke nel, and injec ion eco e y e-
cacy analysis. Abo e heo ems and enginee ing schemes join ly cons i u e e iable windowed
o mula ion.
Keywo ds
: Gene alized sca e ing phase; KV de e minan ; gene alized K en spec al shi ;
Li shi sK en ace o mula; Wigne Smi h ace; hea ke nel ni e pa ; Taube ian; loga i h-
mic window; ela i e de e minan ; s a ic-pa ch de Si e ; FRB baseband
1 In oduc ion and His o ical Backg ound
On e en-dimensional AH/CCM geome y, Guilla mou denes KV de e minan ia eno malized
ace (Kon se ichVishik ace, TR) yielding
1
a g de
KV Sn
2+iω=−2π ξ(ω) (mod 2π),
whe e
ξ
is gene alized K en spec al unc ion; i s loga i hmic de i a i e couples o TR- ace
o sca e ing ope a o
S
, p o iding geome ized e sion o phase = spec al shi . This ame-
wo k compa ible wi h F iedelLloydBi manK en (BK) ela ion, igo ously o mula ed in
e en-
dimensional
AH/CCM scena ios.
Sá Ba e oWang p o e: on non- apping AH,
S(ω)
is Fou ie in eg al ope a o (FIO) quan-
izing sca e ing ela ion, ensu ing die en iabili y, symbol and ke nel egula i y o
ω > 0
, laying
ounda ion o die en iable amewo k o WS- ace and KV-de .
Pelle cha ac e izes unc ion classes applicable o Li shi sK en ace o mula: o ope a o -
Lipschi z
,
T ( (H)− (H0)) = Z ′(λ)ξE(λ)dλ,
holds unde ela i e ace-class (o weake ela i e class- ace) condi ions. This pape places
(λ) = e−sλ
, h ough subs i u ion
λ=ω2
, connec ing hea ke nel die ence wi h equency-domain
DOS slope.
In elec omagne ic mul ipo sys ems, ace o WS ime-delay ma ix
Q=−i S†∂ωS
equi alen
o o al phase de i a i e:
∂ωa g de S= T Q
. This equali y measu able a bo h expe imen al and
algo i hmic le els, cons i u ing obse a ion-end in e ace.
In black hole/s a ic-pa ch dS scena ios, 
ela i e DOSpa i ion unc ionphase shi
 p o-
ides compu able pa adigm emo ing con inuous spec um di e gence, combining sca e ing phase
wi h one-loop ee ene gy/pa i ion unc ion. This pape uses his as empla e o
acyclic calib a-
ion
o absolu e cons an s.
This pape 's goal is comple e alignmen o abo e chain on
equency a iable
, es ablishing
Taube ian heo em o
loga i hmic equency windowing
, igo ously s a ing die en iabili y o
KV-de unde
open channels
and ace o mula applicabili y domain, p oposing implemen able
FRB baseband obse a ion es .
2 Model and Assump ions
2.1 Geome y and Ope a o s
We wo k on wo ypes o backg ounds:

E en-dimensional AH/CCM
:
(Xn+1, g)
e en-dimensional con o mally compac sca e ing
geome y, sa is ying
non- apping
and
no ze o-ene gy esonance/embedded eigen-
alue
. Laplace- ype ope a o
H=−∆g+V
(wi h egula po en ial) and e e ence ope a o
H0
o m ela i e pai .

S a ic-pa ch de Si e (dS)
: Take s a ic-pa ch egion wi h ho izon as bounda y, adop
physical in/ou  channels, cons uc
ex ended channels
S
o
ela i e ope a o
b
S:= SS−1
e
.
Fo AH/CCM,
S(ω)
is FIO wi h ke nel smoo h in
ω
; o dS, ex ends o uni a y a e ex ended
channels.
2
2.2 KV De e minan and Gene alized K en
Deno e
Φ(ω) := a g de KV S(ω)
,
Θ(ω) := Φ(ω)/(2π)
. Guilla mou es ablishes
Φ(ω) = −2π ξω(ω) (mod 2π),Θ′(ω) = −∂ωξω(ω).
whe e
ξω(ω) := ξE(λ)λ=ω2
.
2.3 DOS and Va iable T ans o ma ion
In ene gy a iable
λ
,
∆ρE(λ) = −ξ′
E(λ)
. In equency a iable
ω
,
∆ρω(ω)=2ω∆ρE(ω2) = −∂ωξω(ω).
Dene
Θ′(ω)=∆ρω(ω),T Q(ω) = ∂ωΦ(ω) = 2π∆ρω(ω).
Las equali y is WS- ace.
2.4 Window Family and Mellin-Nullica ion
Take compac ly suppo ed window
W∈C∞
0(R)
, dene loga i hmic window a e age
⟨ ⟩W(µ) = ZR
(µeu)W(u)du =Z∞
0
(ω)Wln(ω/µ)dln ω.
Deno e Mellin ans o m
c
W(z) = RRezuW(u)du
. I
(ω)∼Pkckωβk+Pm,j dm,j ω˜
βm(ln ω)j
has high- equency powe -loga i hmic asymp o ics, equi e window o sa is y
c
W(βk) = 0,dj
dzjc
W(z)z=˜
βm= 0 (0 ≤j≤Jm),
nulli ying powe and loga i hmic powe e ms.
3 Main Resul s
Theo em 1
(1: PhaseSpec al-Shi DOSWS Unica ion)
.
Unde assump ions o 2.1, o
ω > 0
,
Θ′(ω) = ∆ρω(ω) = −∂ωξω(ω),T Q(ω) = ∂ωΦ(ω)=2π∆ρω(ω).
Mo eo e ,
Φ(ω) = −2π ξω(ω) (mod 2π)
.
P oo in 4.1.
Theo em 2
(2: Hea Ke nelF equency Domain Subs i u ion, No Ex a
2ω
)
.
Assume
(λ) = e−sλ
sa is ying LK condi ions, hen hea ke nel die ence
∆K(s) := T e−sH −e−sH0=Z∞
0
e−sω2Θ′(ω)dω,
holds when
ξE(0+) = 0
o in e p e ing small-
λ
endpoin ia Hadama d ni e pa .
P oo in 4.2.
No e
: I w i ing in ene gy a iable as
R∞
0e−sλ ∆ρE(λ)dλ
, h ough subs i u ion
λ=ω2
and
∆ρω= 2ω∆ρE
canceling Jacobian, igh side
con ains no ex a
2ω
, dimensionally sel -consis en .
3
Theo em 3
(3: Windowed Taube ian Theo em: Small-
s
Fini e Pa
↔
Loga i hmic Window A -
e age)
.
Assume
Θ′(ω)
possesses ni e numbe o powe -loga i hmic asymp o ic e ms and con olled
emainde as
ω→ ∞
, wi h Mellin ans o m analy ic bounded in s ip
ℜz > −α
. Take window am-
ily
W
making
c
W
nulli y a abo e powe s and loga i hmic powe indices. Then he e exis cons an s
CW>0
and
α′>0
such ha
ps→0+∆K(s) = κHK CW·Θ′Wµ=1
√s+Osα′,
whe e
κHK
only depends on dimension, eld con en and chosen egula iza ion scheme.
P oo in 4.3.
Co olla y 4
(3.1: Windowed In eg a ion Law)
.
Dene
Λϕ,W (µ) := κΛ⟨Θ′⟩W(µ)
(
κΛ
has
M3
dimension, in
D= 4
making
Λ∼M2
sel -consis en ), hen
∂ln µΛϕ,W (µ) = κΛΞW(µ),ΞW(µ) = ZωΘ′′(ω)Wln(ω/µ)dln ω=1
2πZω ∂ωT Q W d ln ω .
P oo in 4.4.
Theo em 5
(4: Open Channels: Ex ended Uni a iza ion and Rela i e De e minan )
.
Assume
s a ic-pa ch dS/black hole sca e ing ope a o h ough
channel ex ension
becomes
S(ω)
uni a y
and die en iable, o he e exis s e e ence p opaga o
S e
making
b
S=SS−1
e
sa is y
b
S−1∈S1
,
∂ωb
S∈S1
. Then
∂ωa g de
KV b
S(ω) = −iT b
S†∂ωb
S,
die ing om phase gi en by
S
ou e only by cons an , hus equi alen a
ΞW
le el.
P oo in 4.5.
Theo em 6
(5: Th eshold Fini e Pa and E en-Dimensional Loga i hmic Te ms)
.
Unde assump-
ions o 2.1 and
no ze o-ene gy esonance
condi ion,
Θ′(ω)
possesses Hadama d ni e pa a
ω→0+
; e en-dimensional
ωmlog ω
ype e ms can be nullied by abo e window amily, wi h ni e
pa and windowed limi commu ing.
P oo in 4.6.
4 P oo s
4.1 P oo o Theo em 1
(i)
KVK en equi alence
: Guilla mou p o es on e en-dimensional AH/CCM
−2πi ∂zξ(z) = TR∂zSn
2+izS−1
n
2+iz,
hus
a g de KV S(n
2+iω) = −2π ξ(ω) (mod 2π)
. This yields
Θ′=−∂ωξω
.
(ii)
DOSspec al shi
: Li shi sK en denes
∆ρE=−ξ′
E
, a iable subs i u ion yields
∆ρω= 2ω∆ρE(ω2) = −∂ωξω
.
(iii)
WS- ace
: Fo uni a y
S
,
Q=−iS†∂ωS
,
T Q=∂ωa g de S
. Subs i u ing
Φ=2πΘ
yields
T Q= 2πΘ′
. Elec omagne ic mul ipo e sion holds in measu able amewo k.
Th ee s eps combined yield p oposi ion.
4
4.2 P oo o Theo em 2
By LK ace o mula (Pelle ) o ela i e pai
(H, H0)
wi h
(λ) = e−sλ
,
∆K(s) = T  (H)− (H0)=Z∞
0
′(λ)ξE(λ)dλ.
In eg a ion by pa s yields
∆K(s) = h−e−sλξE(λ)i∞
0++Z∞
0
e−sλ ∆ρE(λ)dλ.
High-ene gy end
e−sλ →0
anishes; low-ene gy end equi es
ξE(0+)=0
o Hadama d ni e pa
in e p e a ion (Theo em 5). Subs i u ing
λ=ω2
,
dλ = 2ω dω
wi h
∆ρω= 2ω∆ρE(ω2)
canceling
Jacobian yields
∆K(s) = Z∞
0
e−sω2∆ρω(ω)dω =Z∞
0
e−sω2Θ′(ω)dω.
P oo comple e.
4.3 P oo o Theo em 3 (Windowed Taube ian)
Th ee s eps.
(a)
F equency domain decomposi ion and window Mellin-nullica ion
. Assume
Θ′(ω) =
K
X
k=1
ckωβk+X
m
Jm
X
j=0
dm,j ω˜
βm(ln ω)j+ (ω),
's Mellin ans o m analy ic bounded in
ℜz > −α
. Take
W∈C∞
0
making
c
W(βk)=0
,
dj
dzjc
W(z)|z=˜
βm= 0
. Then
⟨Θ′⟩W(µ) = ⟨ ⟩W(µ).
(b)
Laplace saddle and scale ma ching
. Deno e
u= ln(ω/µ)
, hen
∆K(s) = ZR
e−sµ2e2uΘ′(µeu)du ≈Θ′(µ)ZR
e−sµ2e2udu +···
Take
µ=s−1/2
placing saddle a
u= 0
. By s a iona y phase/saddle app oxima ion, he e exis
CW
and
α′>0
such ha
ps→0∆K(s) = κHK CW⟨Θ′⟩W(µ=s−1/2) + O(sα′).
(
CW
can w i e as cons an mul iple o
Re−e2uW(u)du
; exac coecien de e mined by egula -
iza ion and window no maliza ion.)
(c)
OL cons an and ela i e class- ace s abili y
. By Pelle 's OL es ima e, OL cons an
o
(λ) = e−sλ
bounded as
s↓0
(specically depends on Beso no m), combined wi h ela i e
class- ace assump ion yielding e o bound consis ency.
Combined abo e p o es heo em.
5

4.4 P oo o Co olla y 3.1 (Windowed In eg a ion Law)
Die en ia ing
⟨Θ′⟩W(µ) = RΘ′(ω)W(ln(ω/µ)) dln ω
:
∂ln µ⟨Θ′⟩W=−ZΘ′(ω)∂ln ωW d ln ω=ZωΘ′′(ω)W d ln ω.
Using
∂ωT Q= 2πΘ′′
yields s a emen .
4.5 P oo o Theo em 4 (Open Channels)
(i)
Ex ended channels
: In s a ic-pa ch dS, iew ho izon as in/ou sca e ing channels ex ending
o uni a y
S
. Uni a i y and die en iabili y ensu e
∂ωa g de S= T Qex
.
(ii)
Rela i e de e minan
: Choose e e ence
S e
(Rindle /ou e egion) making
b
S−1
,
∂ωb
S∈
S1
. KV de e minan die en iable wi h
∂ωlog de
KV b
S= TRb
S−1∂ωb
S=−iT b
S†∂ωb
S,
equali y holding elies on
b
S
's quasi-uni a i y and ideal class condi ions. Die s om phase gi en
by
S
only by cons an , equi alen a
ΞW
le el.
4.6 P oo o Theo em 5 (Th eshold Fini e Pa )
AH/CCM spec um a
ω→0
con ains
ωmlog ω
o m e ms. Non- apping and no ze o-ene gy es-
onance ensu e esol en h eshold con ol, FIO s uc u e yields ke nel singula i y o de ; acco dingly
Θ′(ω)
possesses Hadama d ni e pa . A e window amily sa ises loga i hmic nullica ion, ni e
pa and windowed limi commu e, p oo comple e.
5 Modeled Examples
5.1 One-Dimensional
δ
Po en ial (Sol able Ve ica ion)
Le
V(x) = α δ(x)
. Pa ial wa e degene a es, sca e ing phase
δ(ω) = a c an(−α/2ω)
. Thus
Φ(ω)=2δ(ω),Θ′(ω) = 1
2π∂ωΦ = 1
π
α
4ω2+α2.
Subs i u ing in o Theo em 2 e ies closed- o m compu abili y consis ency o
∆K(s) = R∞
0e−sω2Θ′(ω)dω
;
ake window wi h
c
W(0) = 1
,
c
W(1) = 0
, nume ically e i y windowed Taube ian e o o de
O(sα′)
.
5.2 S a ic-Pa ch dS Scala Templa e (
κ
Ex ac ion)
Following Alb ychiewiczNeiman, o massless scala g eybody ac o / ansmission phase
δℓ(ω)
and
ela i e DOS w i en as pa ial wa e sum,
Θ′(ω) = Pℓ(2ℓ+ 1)δ′
ℓ(ω)/π
. LawPa men ie 's  ela-
i e DOSpa i ion unc ion yields consis ency wi h one-loop ee ene gy. Nume ically compu e
⟨Θ′⟩W(µ)
on low- equency cu o
ω≤µ
, ma ching wi h small-
s
hea ke nel ni e pa (Seeley
DeWi bulk e m), ex ac
κHK = ps→0∆K(s)
CW⟨Θ′⟩W(s−1/2), κΛ
de e mined by dimensional ma ching
,
as nume ical demons a ion o acyclic calib a ion.
6
6 Enginee ing Scheme (FRB Baseband)
6.1 Obse able Ke nel
Recons uc sys em ans e
b
S(ω) = Hsys(ω)H e (ω)−1
om c oss-spec um/mul ipo ne wo k, ake
Φ(ω) = a g de KV b
S
, dene
b
Θ(ω) = Φ(ω)
2π,b
ΞW(µ) = Zω ∂2
ωb
Θ(ω)W(ln(ω/µ)) dln ω=1
2πZω ∂ωT b
Q W d ln ω.
WS- ace's elec omagne ic measu abili y p o ides di ec es ima o o his cons uc ion.
6.2 Leakage Ke nel and Va iance
Phase-le el esidual dispe sion
ϕDM =KDM ω−1
and hin-sc een b oadening
ϕsca =Ksca ω−3
induce
ΞDM =ω ∂2
ωϕDM = + 2KDM ω−2,Ξsca = + 12Ksca ω−3.
Second-o de de i a i e noise amplica ion: i phase noise spec um nea -whi e wi h channel
wid h
∆ω
, disc e e second-o de die ence ope a o yields
Va b
Ξ(ω)≃Cω2
∆ω4σ2
ϕ(ω),
C
de e mined by disc e e ke nel spec al no m. A e windowing p o ides closed- o m uppe
bound o
Va [b
ΞW]
by
W
's
L2
no m.
6.3 Da a, Pipeline, and Ecacy
CHIME/FRB published app oxima ely 140 baseband e en s con aining cohe en dedispe sion and
pola iza ion in o ma ion, sa is ying phase-le el access; we p o ide minimal ep oducible pipeline:
ead and calib a e

ela i e de e minan

phase unw apping

egula ized die en ia ion
(Tikhono /TV on
ln ω
axis)

windowing

shape consis ency es /uppe limi . Injec ion eco e y
expe imen : injec
Ξinj(ω) = A ω−1ψ(ln ω)
and
ω−2
,
ω−3
empla es, compa e eco e y bias a iance
wi h Fishe -CR lowe bound, assess sample s acking ecacy cu e.
7 Discussion: Risks, Bounda ies, Rela ed Wo k
Ma hema ical co e o his amewo k is
e en-dimensional
KVK en equi alence and FIO s uc-
u e o AH/CCM; odd dimensions equi e al e na i e in oduc ion. Th eshold
log
e ms and open
channel die en iabili y equi e s ic ela i e class- ace and b anch con inui y. In black hole/s a ic-
pa ch dS,  ela i e DOSpa i ion unc ion p o ides compu able ancho . Obse a ion-wise,
Ξ
's
second-o de de i a i e noise amplica ion and dispe sion/mul ipa h leakage need windowing and
egula iza ion con ol. This pape 's windowing law is
s uc u al equali y
, absolu e nume ical
mapping depends on
κHK
,
κΛ
calib a ion.
7
8 Conclusion
This pape igo ously aligns phasespec al-shi DOSWShea -ke nel chain on equency a i-
able, p o iding subs i u ion heo em wi hou ex a
2ω
and
loga i hmic equency windowing
Taube ian heo em, es ablishing windowed in eg a ion law main aining measu able WS- ace equal-
i y unde open channels. Demons a e
κ
ex ac ion ou e wi h
δ
po en ial and s a ic-pa ch dS
empla es, p o iding minimal ep oducible scheme o FRB baseband. This amewo k connec s
spec al geome y wi h obse able phase analysis as e iable me hodology.
A No a ion and Fac o Accoun ing

F equency/ene gy:
λ=ω2
,
dλ = 2ω dω
.

Spec al shi /DOS:
∆ρE=−ξ′
E
,
∆ρω(ω)=2ω∆ρE(ω2)
.

Sca e ing phase:
Φ = a g de KV S
,
Θ=Φ/2π
.

Co e iden i ies:
Θ′= ∆ρω=−∂ωξω,T Q=∂ωΦ=2π∆ρω,∆K(s) = Z∞
0
e−sω2Θ′(ω)dω.

Loga i hmic window a e age:
⟨ ⟩W(µ) = R (ω)W(ln(ω/µ)) dln ω
,
dln ω=dω/ω
.

Obse able ke nel:
ΞW(µ) = ∂ln µ⟨Θ′⟩W=RωΘ′′ W d ln ω=1
2πRω ∂ωT Q W d ln ω
.
B LK T ace Fo mula and Ope a o -Lipschi z
P oposi ion 7
(B.1 (LK))
.
Fo sel -adjoin pai
(H, H0)
sa is ying ela i e class- ace assump ion
making
(H)− (H0)∈S1
, wi h
ope a o -Lipschi z, hen
T  (H)− (H0)=R ′(λ)ξE(λ)dλ
.
P oo essen ials
: Pelle 's OL c i e ion (
∈B1
∞1
) combined wi h Hele Sjös and ep esen-
a ion; Hilbe Schmid es ima e o esol en die ence ensu es class- ace. Fo
(λ) = e−sλ
, i s
OL cons an bounded as
s↓0
.
C Window Family Cons uc ion and Mellin-Nullica ion
Take smoo h compac window
W
wi h
RW= 1
. To nulli y powe laws
ωβk
and
ω˜
βm(ln ω)j
, equi e
c
W(βk)=0,dj
dzjc
W(z)z=˜
βm= 0.
Cons uc ion me hod: s a wi h mo he window
W0
, o m ni e linea combina ion
W=
PℓaℓW0(·−uℓ)
, coecien s de e mined by nullica ion linea equa ions. Mellin-wa ele (loga i h-
mic axis pa i ion o uni y) amewo k ensu es nume ical s abili y.
8
D KV-de Die en iabili y o Open Channels
P oposi ion 8
(D.1)
.
Assume e e ence
S e
makes
b
S=SS−1
e
sa is y
b
S−1∈S1
,
∂ωb
S∈S1
, wi h
b
S
quasi-uni a y. Then KV de e minan exis s wi h
∂ωlog de
KV b
S= TRb
S−1∂ωb
S=−iT b
S†∂ωb
S.
P oo essen ials
: TR mul iplica i e p ope y and loga i hmic de i a i e deni ion; quasi-
uni a i y educes TR o ace. S a ic-pa ch dS wi h Rindle /ou e egion as e e ence sa ises ideal
class condi ions.
E Th eshold
ω→0
Fini e Pa
P oposi ion 9
(E.1)
.
Non- apping and no ze o-ene gy esonance imply esol en h eshold con-
ollabili y,
Θ′
's loga i hmic singula i y a mos ni e o de . Fo e en-dimensional
ωmlog ω
e ms,
choosing
c
W
nulli ying a co esponding indices and hei de i a i es yields
pω→0Θ′= limµ↓0⟨Θ′⟩W(µ)
.
P oo essen ials
: FIO s uc u e and analy ic F edholm heo y yield ke nel h eshold o m;
windowed limi commu es wi h ni e pa by domina ed con e gence and nullica ion condi ions.
F FRB Pipeline Disc e e Implemen a ion and E o P opaga ion
F.1 Phase unw apping and ela i e de e minan
: Mul i-beam die ence, c oss-pola iza ion
and injec ion noise gi e e e ence
H e
, ensu ing con inuous
Φ(ω)
ia p incipal alue phase and
b anch pa ching.
F.2 Second-de i a i e egula iza ion
: On
ln ω
axis use Tikhono /TV, egula iza ion pa-
ame e ake L-cu e o GCV. Second-o de die ence ke nel
D(2)
spec al no m
|D(2)| ∼ ∆ω−2
,
hus
Va b
Ξ(ω)≃C ω2∆ω−4σ2
ϕ(ω).
F.3 Leakage ke nel
: Dispe sion
ϕDM =KDMω−1⇒ΞDM = + 2KDMω−2
(posi i e sign);
Thin-sc een b oadening
ϕsca =Kscaω−3⇒Ξsca = + 12Kscaω−3
. A e windowing
b
ΞW= ΞW+⟨ΞDM⟩W+⟨Ξsca⟩W+ noise,
shape sepa abili y ensu ed by powe index die ence and windowed equency band decomposi-
ion.
F.4 Injec ion eco e y
: Injec
Ξinj(ω)
(
ω−1
,
ω−2
,
ω−3
) in o public baseband, eco e
b
ΞW
h ough ull pipeline, epo
|b
A/A −1|
e sus window wid h ela ionship and Fishe -CR lowe
bound.
End o Main Tex and Appendices
9