Slepian Functions – Why, What, How? · between concentrating a signal in the time (or, space) domain versus the frequency domain − principal applications in communications theory

Slepian Functions – Why, What, How?

Lecture 1: Basics of Fourier and Fourier-Legendre T ransforms

Lecture 2: Spatiospectral Concentration Problem – Car tesian & Spherical Cases

Lecture 3: Slepian Functions and Sequences, and Appl ications

• In general, one may think of the Slepian functions in connection with the tradeoff between concentrating a signal in the time (or, space) domain versus the frequency domain

− principal applications in communications theory (D. Slepian, Bell Labs, and others)

• A brief review of spectral methods prepares for the fundamental setup of the “spatiospectral concentration problem”

• Instead of the time variable, we consider a spatial variable, primarily in one dimension, but with generalization to two dimensions, in particular the sphere, on which many geodetic and geophysical signals reside

1.1

− geophysical applications (F. Simons, Princeton U., and others)

( ) ( ) 2 ,i f xg x G f e df xπ∞

−∞

= − ∞ < < ∞∫

( ) ( ) 2 ,i f xG f g x e dx fπ∞

−

−∞

= − ∞ < < ∞∫

• Fourier transform pair for continuous, integrable functions, ( ) ( )g x G f↔

• Discrete Fourier transform pair for finite sequences, kg G↔ℓ

ɶɶ

1 2

0

1, 0 1

Ni k

Nk

k

g G e NN x

π

∆

−

=

= ≤ ≤ −∑ℓ

ℓɶɶ ℓ

1 2

0

, 0 1N

i kN

kG x g e k Nπ

∆−

−

=

= ≤ ≤ −∑ℓ

ℓ

ℓ

ɶ ɶ

− G(f) = F(g(x)) is the Fourier transform (spectrum) of g(x)

− g(x) = F −1(G(f)) is the inverse Fourier transform of G(f)

1.2

• Fourier transforms and their inverses have many properties

− similarity, or scaling, property (Fourier integral transform)

( )1 2

1 2

1 fa g ax G

aa

↔

0 is a constanta ≠

( ) ( ) 2 21,i fax i f xf

g ax G f e df G e df f afa a

π π∞ ∞

′

−∞ −∞

′ ′ ′= = = ∫ ∫Proof:

⋅ an expansion of the spatial domain implies a contraction of the frequency domain, and vice versa

− Parseval’s theorem: ( ) ( )2 2g x dx G f df

∞ ∞

−∞ −∞

=∫ ∫Proof:

( ) ( ) ( ) ( ) ( ) ( )2 2* 2 * 2i f x i f xg x dx g x G f e df dx G f g x e dxdf G f dfπ π∞ ∞ ∞ ∞ ∞ ∞

−∞ −∞ −∞ −∞ −∞ −∞

= = =∫ ∫ ∫ ∫ ∫ ∫⋅ total energy of a function is the same in space domain as in frequency domain

1.3

• Convolution: ( ) ( ) ( ) ( ) ( )*y x g x h x g x h x x dx

∞

−∞

′ ′ ′= = −∫

− Fourier transform (convolution theorem): ( ) ( ) ( )Y f G f H f=

Proof:

( ) ( ) ( ) ( ) ( ) ( )22 2 i f x xi f x i f xY f g x h x x dx e dx g x e h x x e dx dxππ π∞ ∞ ∞ ∞

′− −′− −

−∞ −∞ −∞ −∞

′ ′ ′ ′ ′ ′= − = − ∫ ∫ ∫ ∫

− useful also for numerical calculation of convolution:( ) ( )( ) ( )( )( )1y x g x h x−=F F F

1.4

• Also, Fourier transform of a product is the convolution of Fourier transforms

− dual convolution theorem: ( ) ( ) ( ) ( ) ( ) ( )*w x g x h x W f G f H f= ↔ =

• Example 1: Rectangle function

( )1, 1 2

0.5, 1 2

0, 1 2

x

b x x

x

<= = >

( ) ( ) ( )sinsinc

fB f f

f

ππ

= ≡

− b(x) is “extent-limited”; its spectrum is different from zero almost everywhere

1.5

• Digression: Dirac delta function

( )0

1limT

xx b

T Tδ

→

=

− hence, ( ) 0, 0x xδ = ≠

( ) 1x dxδ∞

−∞

=∫

− Fourier transform: ( ) 1, for all f f∆ =

Proof: ( )( ) ( ) ( )( )0 0

lim lim sinc 1T T

x B Tf Tfδ→ →

= = =F

− by convolution theorem ( ) ( ) ( )g x x x dx g xδ∞

−∞

′ ′ ′− =∫1.6

MathCad – Ch2_rectangle_fncs.xmcd

• Example 2: Gaussian function ( ) ( ) ( )21 xx eβ π βγβ

−= 0, a constantβ >

− Fourier transform: ( ) ( )2

( ) ff eβ π βΓ −=

− varying β demonstrates similarity (scaling) property

Figures.ppt Figures.ppt

1.7

• Fundamental result:

A (non-trivial) function cannot be both extent-limited and band-limited

• Extent-limited function: ( ) 0, 2g x x T= >

Proof (heuristic): If band-limited, then extended over the complex plane it is an “entire” function (all derivatives exist everywhere). Thus, if then also extent-limited, i.e., zero on some interval, then it must be zero everywhere.

• Band-limited function: ( ) 00,G f f f= > “bandwidth” = 2f0

1.8

Gibbs’s Effect

( ) ( ) ( )

( ) ( ) ( ) ( )( )

0 0

0

0 0

0

0

2 2 2

02 sin 21

f f

i fx i fx i fxf

f f

f

i f x x

f

g x G f e df g x e dx e df

f x xg x e df dx g x dx

x x

π π π

π ππ

∞′−

− − −∞

∞ ∞′− −

−∞ − −∞

′ ′= =

′− ′ ′ ′ ′= =

′ −

∫ ∫ ∫

∫ ∫ ∫

• Band-limited approximation of g(x):

1.9

( ) ( )g x b x→

x

( )b x

( )0 2fb x=

( )0 10fb x=

MathCad: Ch3_Gibbs_phenom.xmcd

• Questions: In the space of extent-limited functions (given T), which one has most of its energy concentrated in a given spectral band?

In the space of band-limited functions (given f0), which one has most of its energy concentrated in a given spatial extent?

• Answers: Series of 5 papers, 1960 – 1978, by D. Slepian, H.O. Pollak, H.J. Landau (Bell Labs)

“Prolate Spheroidal Wave Functions – Fourier Analysis and Uncertainty I, II, III, IV, V”

“I am going to use this occasion to tell you in detail about a problem in Fourier analysis that arose in a quite natural manner in a corner of electrical engineering known as Communication Theory. The problem was first attacked more than 20 years ago … jointly by me and two colleagues at Bell Labs-Henry Pollak and Henry Landau. It differed from other problems I have worked on in two fundamental ways. First, we solved it--completely, easily and quickly. Second, the answer was interesting-even elegant and beautiful. … [It] had so much unexpected structure that we soon saw that we had solved many other problems as well.”

• Sleptian, D. (1983), SIAM Review, 5(3):379-393 –

1.10

• Spherical domain: ( ) , | 0 ,0 2Ω θ λ θ π λ π= ≤ ≤ ≤ ≤

− Fourier-Legendre transform pair

( ) ( ) ( ), ,

0

, , , ,n

n m n m

n m n

g G Yθ λ θ λ θ λ Ω∞

= =−

= ∈∑∑

( ) ( ), ,

1, , , , 0

4n m n mG g Y d n m n nΩ

θ λ θ λ Ωπ

= − ≤ ≤ ≥∫∫− Parseval’s Theorem

( )2 2,

0

1,

4

n

n m

n m n

g d GΩ

θ λ Ωπ

∞

= =−

=∑∑∫∫− Extent-limited function: ( ) ( ), 0, , Rg θ λ θ λ Ω Ω= ∉ ⊂

− Band-limited function: , max0,n mG n n= >

− Same questions may be posed

1.11

• First studied to some extent by (Gilbert and) Slepian in 1970s, Albertella et al. (1999), and a few others in other fields; see references in (Simons et al. 2006)

• Comprehensively developed by F. Simons, F. Dahlen, and colleagues in the 2000s





• Questions: In the space of extent-limited functions (given T), which one has most of its energy concentrated in a given spectral band, |f| < f0?

In the space of band-limited functions (given f0), which one has most of its energy concentrated in a given spatial extent, |x| < T/2?

( ) ( ) ( )21 xx eβ π βγβ

−=

0, a constantβ >

( ) ( )2

( ) ff eβ π βΓ −=

Figures.ppt

• Note: Gaussian is practically extent- and band-limited, but we seek functions of two parameters

2.0

• Recall -

• Spectralconcentration problem

− we seek an extent-limited function whose spectral energy is maximally concentrated in a finite bandwidth

• The finite extent and bandwidth are independently given

− finite-extent function: ( ) ( ), 2

0, 2T

g x x Tg x

x T

≤= >

− define finite bandwidth: 0f f≤

• Consider the ratio: ( )( )

( )

0

0

2

02

f

T

fT

T

G f df

f

G f df

λ −∞

−∞

=∫

∫

− Fourier transform: ( ) ( )2

2

2

,

T

i fxT T

T

G f g x e dx fπ−

−

= − ∞ < < ∞∫

• Find the function that maximizes this ratio2.1

( )( ) ( ) ( )

( )

( ) ( ) ( )

( )

0

0

2 2

2

0

2 20 2

2 2

2

;

f T T

i x x fT T T T

f T TT T

T T

T

g x g x e df dxdx g x g x D x x f dxdx

f

g x dx g x dx

π

λ

∞ ∞′−

−∞ −∞ − − −∞

−∞ −

′ ′ ′ ′ ′− = =

∫ ∫ ∫ ∫ ∫

∫ ∫

• Numerator: substituting the Fourier transform

• Denominator: substitute Parseval’s relationship

• Variational problem: Find gT(x) such that small variations produce no variation in λT(f0) in the limit.

• Use ∂ to denote a small variation and consider

( ) ( ) ( ) ( ) ( )2 2 2

20 0

2 2 2

;

T T T

T T T T

T T T

f g x dx g x g x D x x f dxdxλ− − −

′ ′ ′∂ = ∂ −

∫ ∫ ∫2.2

( )( )

( )

0

0

2

02

f

T

fT

T

G f df

f

G f df

λ −∞

−∞

=∫

∫

( ) ( ) ( )( )( )

0

0

020

sin 2;

f

i x x f

f

x x fD x x f e df

x xπ π

π′−

−

′−′− = =

′−∫− where

• After some manipulations, we can arrive at (details provided in supplemental notes)

( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( )

2

20

2

2 2

0 0

2 2

2 ;

T

T T

T

T T

T T T T

T T

f g x dx

g x D x x f f g x x x dx g x dx

λ

λ δ

−

− −

∂

′ ′ ′ ′ ′= − − − ∂

∫

∫ ∫

• The left side goes to zero for arbitrarily small variations, ∂gT(x), only if the parenthetical integral vanishes:

( ) ( ) ( ) ( )2

0 0

2

;

T

T T T

T

g x D x x f dx f g xλ−

′ ′ ′− =∫

− this is a homogeneous Fredholm integral equation of the second kind

• Before solving this equation, we consider the complementary spatial concentration problem

2.3

• We seek a band-limited function whose spectral energy is maximally concentrated in a finite extent

− band-limited function: ( ) ( )0

0

0

,

0,f

G f f fG f

f f

≤= >

− define finite extent: 2x T≤

• Consider the ratio:

( )( )

( )

( ) ( ) ( )

( )

0 0

0 00

0 0

0 0

0 0

0

2

*2

2

22

; 2

f fT

f ff

f fTf f

f f

f

G f G f D f f T dfdfg x dx

T

g x dx G f df

µ − −−∞

−∞ −

′ ′ ′−

= =∫ ∫∫

∫ ∫

− inverse Fourier transform: ( ) ( )0

0 0

0

2 ,

f

i fxf f

f

g x G f e df xπ

−

= − ∞ < < ∞∫

2.4

• The variational problem is solved, as before, by a solution to the integral equation

( ) ( ) ( ) ( )0

0 0 0

0

; 2

f

f f f

f

G f D f f T df T G fµ−

′ ′ ′− =∫

• A change in variable reduces each integral equation to a more convenient form

( ) ( ) ( ) ( ) ( ) ( ) ( )1

1

c c c cy D y y dy yψ λ ψ−

′ ′ ′− =∫

( ) ( ) ( ) ( ) ( )02 , 2 ,c cT Ty x T y g Ty fψ λ λ= = =

− or

( ) ( ) ( ) ( ) ( )0 00 0, ,c c

f fy f f y G f y Tψ λ µ= = =

− where 0c Tfπ=

( ) ( ) ( )( )( ) ( )( )sin

sincc c y y cD y y c y y

y yπ

π π′−

′ ′− = = −′−

− and

2.5

− the left side is almost a convolution of ψ(c)(y) with the sinc function

− therefore, in the dual domain, this is almost a product of the transform of ψ(c)(y) and a rectangle function

⋅ e.g., for the first (spectral) concentration problem, this “almost-convolution”, transformed to the frequency domain, is an “almost-product” of the transform of ψ(c)(y) and the rectangle function of frequency

⋅ i.e., the right side is almost band-limited

• Finding the solution(s) to the integral equation is an eigenfunction/eigenvalue problem:

• This is explored in more detail in the next lecture

2.6

( ) ( ) ( ) ( ) ( ) ( ) ( )1

1


′ ′ ′− =∫• Repeating:

− the solution to the concentration problem is the eigenfunction with the maximumeigenvalue

ψ λψ=D

• Instead of a function on the continuous domain, the spatiospectral concentration problem can be formulated also for sequences, which addresses more practical applications

− extent-limited sequence ( ) , 2 2 1

0, 2 or 2 1N

g N Ng

N N

− ≤ ≤ −= < − > −

ℓ

ℓ

ℓ

ℓ ℓ

− Fourier transform: ( ) ( )2 1

2

2

,N

i f xN N

N

G f x g e f f fπ ∆∆−

−

=−

= − ≤ <∑ ℓ

ℓ

ℓ

ɶN N

• Concentration ratio

( )( )

( )

( ) ( ) ( )( )

( )

( )

0

0

2 1 2 12

0T

2 20 2 1 T

22

2

;

fN N

N N cf N N

N f N

N

Nf

G f df x g g D x f

f

gG f df

∆ ∆λ

− −

′′− =− =−

−

=−−

′−

= = =∫ ∑ ∑

∑∫

Dg gg g

ℓ ℓ

ℓ ℓ

ℓ

ℓ

ɶℓ ℓ

ɶ

N

N

2.7

( ) ( ) ( )( )( ),

sin 2c c c ND

π′

′− = = ′−

Dℓ ℓ

ℓ ℓ

ℓ ℓ− where ( )Ng = g

ℓ

− in the case of spectral concentration …

• Maximizing the concentration ratio is equivalent to find the eigenvector (the finite-extent sequence) with maximum eigenvalue for the vector-matrix equation:

( ) ( )c cλ=D g g

− this is investigated in more detail in the next lecture

• An illustrative example

− rectangle sequence: ( ) 1, 12 2

0, otherwise

NN N

b − ≤ ≤ −=

ℓ

ℓ

− Fourier transform: ( ) ( ) ( )( )

2 12

2

sin

sin

NN i x f i xf

N

N xfB f x e x e

xfπ∆ π∆π∆

∆ ∆π∆

−−

=−

= =∑ ℓ

ℓ

ɶ

2.8

− compare spectral energy distribution of this to that of solutionto spectral concentration problem

• Choose parameters: 1x∆ = 50N = 0

10.02f

N x∆= = (first zero of ( ) ( ) )NB fɶ

− the energy of the rectangle sequence is not well concentrated in the band, |f |< f0

− spectral energy of eigensequence is well concentrated in the spectral band, |f| < f0

2.9

0 0.05 0.1 0.15 0.20.01

0.1

1rectangle sequence

MathCad: discrete_prolate_spheroidal_sequence.xmcd

frequency, f0 0.05 0.1 0.15 0.2

0.01

0.1

1rectangle sequencedpss

25− 20− 15− 10− 5− 0 5 10 15 20 250

0.05

0.1

0.15

0.2

sequence index

MathCad: discrete_prolate_spheroidal_sequence.xmcd

eigensequence* with the maximum eigenvalue

*called discrete prolate spheroidal sequence, dpss

• Finally, we can set up the spatiospectral concentration problem for the sphere

− consider only the second problem of finding the band-limited function whose energy is most concentrated on a given patch of the sphere

− this leads to the problem of how to represent most efficiently a given spherical harmonic model of the potential in a given area

− band-limited function: ( ) ( ), ,

0

, ,K n

K n m n m

n m n

g G Yθ λ θ λ= =−

=∑∑

• Concentration ratio:

− limited extent: RΩ Ω⊂

( )( )

( )

( ) ( )2, , , ,

0 0

2 2,

0

1 1, , ,

4 4

1,

4

R R

K n K n

K n m n m n m n m

n m n n m n

K R K n

K n m

n m n

g d G G Y Y d

g d G

Ω Ω

Ω

θ λ Ω θ λ θ λ Ωπ π

µ Ωθ λ Ω

π

′

′ ′ ′ ′′ ′ ′= =− = =−

= =−

= =∑∑ ∑∑∫∫ ∫∫

∫∫ ∑∑2.11

( )( ) ( ), , , , , T

0 0T

2,

0

R

K n K n

n m n m n m n m R

n m n n m nK R K n

n m

n m n

G G D

G

ΩΩ

µ Ω

′

′ ′ ′ ′′ ′= =− = =−

= =−

= =∑∑∑∑

∑∑DG GG G

• That is,

• Maximizing this ratio leads to the matrix equation,( ) ( )R RΩ Ωµ=D G G

• The difficult part is the integration of the product of spherical harmonics over ΩR

( ) ( ) ( ), , , , ,

1, ,

4R

n m n m R n m n mD Y Y dΩ

Ω θ λ θ λ Ωπ′ ′ ′ ′= ∫∫− where

( ) ( ) ( ) ( ), , ,

0

cos cos sin2

s

m mn n s n m n mD P P d

θ

θ

εθ θ θ θ θ′ ′

=

= ∫− then

− if ΩR is a polar cap, ( ) , | 0 2 ,0R s sΩ Ω θ λ λ π θ θ= = ≤ ≤ ≤ ≤

2.12

• Solution is the eigenvector, G, with maximum eigenvalue; it contains the spectrum of the band-limited function with maximally concentrated energy in the polar cap

Legendre spectrum of g(θ)

• Compare the truncated (band-limited) Dirac delta function on the sphere with the solution to the spatial concentration problem

− band-limited Dirac delta function: ( ) ( ) ( )0

2 1 cosK

K n

n

n Pδ θ θ=

= +∑

⋅ Legendre spectrum is “rectangle” in frequency domain

2.13

− maximally concentrated function on polar cap: g(θ), θs = 10º

⋅ both δK(θ) and g(θ) are band-limited, K = 60





• From the theory of integral equations, it is known that for this particular kernel, the integral operator has eigenvalues, λ(c), that are real, distinct, and countably infinite

( ) ( )0 11 c cλ λ> > >⋯

− the corresponding eigenfunctions are orthogonal on both [−1,1], as well as (−∞, ∞), and can be normalized so that they are orthonormal on [−1,1]

( ) ( ) ( ) ( )1

1

c ck ky y dyψ ψ δ −

−

=∫ ℓ ℓ

− the eigenvalues, being concentration ratios, must all be less than unity

• Interestingly, like the Gaussian function, the eigenfunction and its Fourier transform have the same form, one a scaled version of the other – see the supplemental notes

3.1

( ) ( ) ( ) ( ) ( ) ( ) ( )1

1


′ ′ ′− =∫

• Recall the equation to be solved for the spatiospectral concentration problem:

− where the kernel function is ( ) ( ) ( )( )( )

sinc c y yD y y

y yπ′−

′− =′−

0c Tfπ=

• The eigenfunction for the maximum eigenvalue solves the concentration problem

• The eigenfunction for the next largest eigenvalue is next best in concentrating its energy, and so on

• The total set of eigenfunctions is a basis for the space of extent- (or band-) limited functions

− however, not all eigenfunctions are needed in practice to represent an extent- (or band-) limited function

• Analogous to the eigenvalues of a matrix operator, the sum of the eigenvalues equals the trace of the kernel function of the integral equation

( ) ( ) ( )1 1

0

0 1 1

22c c

k

k

c cD y y dy dy f Tλ

π π

∞

= − −

= − = = =∑ ∫ ∫− it can be shown that the values of the eigenvalues transition sharply from near unity to

near zero

− therefore, the number of significant eigenvalues, and correspondingly significant eigenfunctions, is approximately

( )02 1

0

0

2f T

cc k

k

E f T λ−

=

= ∑≃• This is known as the Shannon number (also the extent-bandwidth product)

3.2

0c Tfπ=

• By a “seemingly lucky accident” (Slepian 1983), the eigenfunctions are also certain solutions to the Helmholz wave equation in prolate spheroidal coordinates

( ) ( ) ( )2

2 2 221 2 0

d u duy y y u y

dydyζ χ− − + − = ζ, χ are constants

− thus, the eigenfunctions of the integral operator are also known as prolate spheroidal wave functions

• Indeed, writing the integral and differential equations in terms of operators, that are operating on an arbitrary function, h(y),

( ) ( ) ( ) ( ) ( ) ( )1

1

c ch y D y y h y dy h yλ−

′ ′ ′= − =∫D

( ) ( ) ( ) ( ) ( )2 2 21 cd dhh y y c y h y h y

dy dyλ

= − − + =

L

− it can be shown that these operators commute – see the supplemental notes

( )( ) ( )( )h y h y=D L L D

• Hence they have the same eigenfunctions – proved on next slide3.3

( ) ( )0 0 0h y h yλ=D

• To prove that D and L share eigenfunctions, let h0(y) be an eigenfunction of D

− then ( )( ) ( ) ( )( ) ( )0 0 0 0 0 0h y h y h y h yλ λ= ⇒ =L D L D L L

− and, Lh0(y) is an eigenfunction of D with the same eigenvalue

− this means Lh0(y) and h0(y) are linearly related, i.e., the latter is an eigenfunction of L

− it turns out that finding the eigenfunctions of L is easier than for D

• Once the eigenfunction, ψk(y), is obtained, the eigenvalue with respect to D is simply

( )( )k

kk

y

y

ψλ

ψ=D

• These ideas considerably simplify also the practical application with sequences, i.e., functions defined on a discrete domain

3.4

3.5

• For sequences, recall that the spectral concentration problem issolved by finding the eigenvector with maximum eigenvalue for the matrix equation,

• Slepian (1978) found a “difference” equation whose operator commutes with D(c)

( ) ( )( )2

1 1

2 1 1cos 1 1 0

2 2 2n n n

n c NN n g n g n N n g

Nζ− +

− − + − − + + − − =

( )( )

( )( )

( )( )

( )( )

2

0 0

1 12

2 2

1 1

2

2 1 1cos 1 1 0 0

2 2

11 1

2

2 10 cos 0

2

11 1

2

1 2 10 0 1 1 cos 1

2 2

N N

N N

c NN

N

g gNg g

c Nn

Ng g

N g g

c NN N

N

ζ

− −

− −

− −

− − =− −

− − − +

⋯

⋱ ⋱ ⋱ ⋮

⋮ ⋮⋱ ⋱

⋮ ⋱ ⋱ ⋱

⋯

( ) ( ) ( )( )( ),

sin 2c c c ND

π′

′− = = ′−

Dℓ ℓ

ℓ ℓ

ℓ ℓ− where [ ]g=g

ℓ

( ) ( )c cλ=D g g

( )c ζ=S g g− or, − and: ( ) ( ) ( ) ( )c c c c=S D D S

• Thus, S(c) and D(c) commute and have the same eigenvectors, but not the same eigenvalues.

( )( ) ( )

( ) , 0, , 1

c ckc

k ck

k Nλ = = −D g

g…

• Having obtained the eigenvectors for S(c), hence D(c), the eigenvalues with respect to D(c) are

• For example, N = 60 (extent-limited sequence), f0 = 0.05 (Shannon number = 2N∆xf0 = 6)

3.6

frequency

( ) ( )0cG f

( ) ( )2cG f

( ) ( )4cG f

( ) ( )6cG f

• The Fourier spectra of the eigensequences show good concentration of energy for kless than the Shannon number, k < 6

3.7

• Similar results are obtained for the spherical case – instead, later, we consider the spatial concentration problem

Estimating the Power Spectral Density

• Definition of power spectral density (PSD) – Fourier transform of the covariance function of a stationary stochastic process

( ) ( ) 2, ,

i fg g g gC f c e dπξξ ξ

∞−

−∞

= ∫ ( ) ( ),g g x xc g g ξξ += Ecovariance function:

• Estimate of the covariance function: ( )2 1

,

2

1, 0, , 1

N

g g n n

n N

c g g NN

− −

+=−

= = −∑ℓ

ℓℓ

ℓ …

• It can be shown that an estimator of the PSD at discrete frequencies is

( ) ( ) ( )2 2

,

1 1DFT ,g g k k n kk

C f G g f k N xN x N x

∆∆ ∆

= = =

ɶ

− this is known as the periodogram

− it is a biased estimator due to aliasing and spectral leakage

3.8

• Aliasing: due to sampling of a function with spectral content beyond the spatial resolution defined by the sampling; i.e., content beyond the Nyquist frequency, |f| > f

N= 1/(2∆x)

• Spectral leakage: blurring and/or biasing of a spectrum due to truncation of the function.

− using finite extent of a function, i.e., “windowing” a function, is the same as multiplying by the rectangle function

( ) ( ) ( ), 2 2

0, otherwiseT

g x T x T xg x b g x

T

− ≤ ≤ = =

− by the dual convolution theorem, the Fourier transform of gT(x) is the convolution of the Fourier transforms of b(x/T) and g(x)

( ) ( ) ( ) ( ) ( )( )* sincTG f TB fT G f T G f f f T df

∞

−∞

′ ′ ′= = −∫

3.9

3.10

MathCad – Ch1_polar_motion.xmcd

6 years of data


8 years of data


12 years of data

x pam

plitu

de s

pect

rum

[arc

sec/

(cy/

yr)]

frequency [cy/yr]


50 years of data

Example of Spectral Leakage – polar motion

Window Function, Data Taper

frequency

MathCad – Ch3_windows_comparison.xmcd

extent

MathCad – Ch3_windows_comparison.xmcd

3.11

x pam

plitu

de s

pect

rum

[arc

sec/

(cy/

yr)]

frequency [cy/yr]


12 years of data

Another Example – PSD DeterminationMathCad: Magnetic data periodograms with tapers.xmcd

MathCad: Magnetic data periodograms with tapers.xmcd

MathCad: Magnetic data periodograms with tapers.xmcd

3.12

( ) ( )R RΩ Ωµ=D G G where ( ) ( ) ( ) ( ), , , , ,

1, ,

4R

R

n m n m R n m n mD Y Y dΩ

Ω

Ω θ λ θ λ Ωπ′ ′ ′ ′

= = ∫∫D

( ) ( )( ) ( ) ( )

( ) ( )

( ) ( )

, , ,

0 0

0

2

1tr , ,

4

12 1 cos0

4

14

R R

R

R

K n K n

n m n m n m

n m n n m n

K

n

n

R

E Y Y d

n P d

AK

Ω Ω

Ω

Ω

µ θ λ θ λ Ωπ

Ωπ

Ωπ

= =− = =−

=

= =

= +

= +

∑∑ ∑∑∫∫

∑ ∫∫

D≃

− where A(ΩR) is the area of ΩR

• Recall the matrix equation for the spherical spatial concentration problem

• The eigenvalues, µ(ΩR), are real and distinct; the eigenvectors, G, are orthonormal

• The sum of eigenvalues is the trace of D(ΩR) and approximates the Shannon number

and form a basis for band-limited functions; they are also orthonormal on Ω

• The eigenfunctions, ( ) ( ) ( ), ,

0

, ,K n

j n m n mjn m n

g G Yθ λ θ λ= =−

=∑∑ , are orthogonal on ΩR;

( ) ( )( )T

0,0 ,j K Kj jG G=G ⋯

Tj k j kδ −=G G

3.13

• For the special case of a polar cap for the concentration region, ( ) , | 0 2 ,0R s sΩ Ω θ λ λ π θ θ= = ≤ ≤ ≤ ≤

• For m = −K,…,K, the matrix, D(m)(θs), commutes with T(m)(θs), defined by

( ) ( )( ) ( )

( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( )( )

( ) ( )( )

1, 1

2 2

1, 2 2, 1

,

1 cos , , ,

12 2 , , , 1

2 1 2 3

0, otherwise

ms s

n m n m

m ms s

n m n m n m n m

ms

n n

T n n n m K

n mT T n n K K n m K

n n

T

θ θ

θ θ

θ

− + − +

− + − + − + − +

′

= − + =

+ −= = + − + = −

+ +

=

…

…

• T(m)(θs) is tri-diagonal and symmetric

( ) ( )

( ) ( )( )

( ) ( )( )( ) ( )( )

( ) ( )( )

( ) ( )( )

1 1

1

0

1 1

1

1 1

0

diag

0

Ks

sK K

mss

K K

sK K

Ks

θ

θ

θθ

θ

θ

−

−

+ +

=

D

D

DD

D

D

⋱

⋱

( )RΩ =D

( ) ( ) ( ) ( ), , ,

0

cos cos sin2

s

m mn n s n m n mD P P d

θ

θ

εθ θ θ θ θ′ ′

=

= ∫

3.14

( ) ( ), 1, , 1,mj s j K mθ = − +G …

( ) ( )msθT• The eigenvectors, of are eigenvectors of( ) ( )m

sθD

− eigenvalues are obtained from: ( ) ( )( ) ( ) ( ) ( )

( ) ( )

m ms j sm

j s mj s

θ θµ θ

θ=

D G

G

− eigenvalues can be ranked: ( ) ( ) ( ) ( ) ( )20 1 11 s s s sKαµ θ µ θ µ θ µ θ

+> > > > > >⋯ ⋯

• Example: K = 60, θs = 10º, E = 28; (K + 1)2 = 3721

⋅ each α corresponds to a particular m and j

3.15

• The corresponding eigenfunctions (basis functions for band-limited functions) are

( ) ( ) ( ), ,

cos , 0, cos

sin , 0

K

n m n mjn m

m mg G P

m mα

λθ λ θ

λ=

≥ = < ∑

• For λ = 0º

polar angle, θ [deg]

0g1g

3g

6g 10g15g

19g23g

30g

g α(θ,

λ=

0)

3.16

α m µα0 0 0.99999986311 1 0.99999474432 -1 0.99999474423 2 0.99990537754 -2 0.99990537435 0 0.99982272856 3 0.99894582957 -3 0.99894582728 1 0.99727603369 -1 0.997276009310 4 0.991993081711 -4 0.991993080312 2 0.975664996813 -2 0.975664847114 0 0.966894978215 5 0.957580285416 -5 0.957580257917 3 0.870559583418 -3 0.870559488419 6 0.846662951420 -6 0.846662898521 1 0.804367656222 -1 0.804367419323 7 0.627486583924 -7 0.627486059925 -4 0.607265631926 4 0.607265484927 2 0.449818031128 -2 0.449817135229 0 0.400509096730 8 0.364136583

• For θs = 10º, K = 100, E = 77 (L. Wang 2012, OSU Report 498)

3.17

• Given a band-limited representation of a function (signal) on the sphere,

( ) ( ), ,

0

, ,K n

n m n m

n m n

s S Yθ λ θ λ= =−

=∑∑− we wish to represent it with maximum efficiency in a local domain – i.e., fewer

basis functions, but with the full energy of the signal

( ) ( )( )

( )2

1

1 1

, , ,K E

s S g S gα α α αα α

θ λ θ λ θ λ+

= =

= ∑ ∑≃

− where E is the Shannon number

• Thus, ( )( )

( )2

1

, ,

1 0

, ,K K n

n m n m

n m n

S g S Yβ ββ

θ λ θ λ+

= = =−

=∑ ∑∑

( ), ,

0

K n

n m n m

n m n

S S Gα α= =−

=∑∑

− with ( ) ( ) ( ), ,

0

, ,K n

n m n m

n m n

g G Yα αθ λ θ λ

= =−

=∑∑ and orthonormality of gα, as well as Yn,m, on Ω

3.18

• Model-predicted co-seismic gravity changes for 2010 Chile earthquake (Mw 8.8), band-limited to spherical harmonic degree and order 100; ±8 microGal (L. Wang 2012; OSU Report 498)

full spherical harmonic series, 10201 spherical harmonics

Slepian series,77 basis functions

rms difference:0.14% of signal in cap

,n mS Sα

• For θs = 10º, K = 100, E = 77

3.19

Documents

Slepian Functions – Why, What, How? · between concentrating a signal in the time (or, space) domain versus the frequency domain − principal applications in communications theory