Convergence Analysis

Analysis of Convergence of Sparse Time-Frequency Method

5.1. Convergence Analysis

CHAPTER 5

Domain Symbol Analysis Synthesis θ F_θ(.) (F_θ(g))_k=

´_θend

θ0 g(θ) exp

−i2πkθ

|_θend^−θ0|

dθ

|θ_end−θ0|

g(θ) = ^P∞

k=−∞(F_θ(g))_kexp_|θend^i2πkθ_−θ0| t∈[0,1] F(.) ´1(F(g))_k =

0 g(t)e^−i2πktdt

g(t) = ^P∞

k=−∞(F(g))_ke^i2πkt θ¯ ˆ()_θ^¯ ´₁ fˆθ¯(k) =

0 fθ¯e^−i2πkθ¯dθ¯

fθ¯= ^P∞

k=−∞

fˆθ¯(k)e^i2πkθ¯ t P_VM

0 (.) P_VM

0(g) =

M0

P

k=−M₀(F(g))_ke^i2πkt

R_VM

0 (g) = ^P

|k|>M0

(F(g))ke^i2πkt

k ∈Z k.k_1,M

0 kzk_1,M

0 =

P

|k|≤M0

|z(k)|

Table 5.1. Coordinates and Symbols for n∈N.

Proof. By definition, we have

θ¯^m(n)(t)=

X

|k|≤M0

F

θ¯^m(n)

k

e^i2πkt

≤ ^X

|k|≤M₀

F

θ¯^m(n)

k

≤ ^X

|k|≤M₀

F

θ¯^m(n)

k

= ^X

|k|≤M₀

|i2πk|ⁿ⁻¹

F

θ¯^m0

k

= (2πM₀)ⁿ⁻¹

F

θ¯^m0

₁.

Lemma 2 leads to Lemma 3, which bounds integrals like

´₁

0 e^iδ4θe^−iθ¯mdθ¯^mthat occur frequently in the main theorem. These integrals would be bounded by the norm of the Fourier transform of the phase correction 4θ⁰. In fact, this bound would help us construct a contraction in the main theorem.

Lemma 3. If θ¯^m0 > 0, t ∈ [0,1], θ¯^m(0) = 0, θ¯^m(1) = 1, and θ¯^m0,4θ⁰ ∈ V_M0. Also if e^iδ4θe^−iθ¯m is periodic, then for 6= 0, we have

(5.1.3)

ˆ 1 0

e^iδ4θe^−iθ¯mdθ¯^m

≤ P_mⁿM₀ⁿ

||ⁿ

n

X

j=1

|δ|^j(2πM₀)^−j(kF(4θ⁰)k₁)^j.

Proof. Using integration by parts, we haveˆ 1 0

e^iδ4θe^−iθ¯mdθ¯^m = 1 (i)ⁿ

ˆ 1 0





dⁿ

dθ¯^mne^iδ4θ



e^−iθ¯mdθ¯^m. Now, using Lemma 2, we have

ˆ 1 0

e^iδ4θe^−iθ¯mdθ¯^m

≤

1

||ⁿ ˆ 1

0





dⁿ

dθ¯^mne^iδ4θ



dθ¯^m

≤ P



 ^F

(^θ¯m)⁰

1

min(^θ¯m)⁰ , n



M₀ⁿ

minθ¯^m0

n

||ⁿ

n

X

j=1

|δ|^j(2πM₀)^−j(kF(4θ⁰)k₁)^j

=P_mⁿM₀ⁿ

||ⁿ

n

X

j=1

|δ|^j(2πM₀)^−j(kF(4θ⁰)k₁)^j.

Here P(x, n) is a polynomial of degree n−1 and P_mⁿ = P



 ^F

(^θ¯m)⁰

1

min(^θ¯m)⁰ , n



. This

completes the proof.

Remark 1. Here we present a simple calculation on how to compute the polynomial P(x, n) for smalln. For example for n= 2, we have

d² dθ¯^m2

e^iδ4θ

=

i











(δ4θ)⁰⁰

θ¯^m0

2 − (δ4θ)⁰θ¯^m00

θ¯^m0

3 +i

(δ4θ)⁰²

θ¯^m0

2











e^iδ4θ

≤

(δ4θ)⁰⁰

θ¯^m0

2

+

(δ4θ)⁰θ¯^m00

θ¯^m0

3

+

(δ4θ)⁰²

θ¯^m0

2

≤max(δ4θ)⁰⁰

minθ¯^m0

2 + max(δ4θ)⁰max

θ¯^m00

minθ¯^m0

3 + max

(δ4θ)⁰²

minθ¯^m0

2

≤







1 +

^F

(^θ¯m)⁰

1

min(^θ¯m)⁰



2πM₀F(δ4θ)⁰₁+F(δ4θ)⁰

2 1





minθ¯^m0

2 ,

where we have used θ¯^m0,4θ⁰ ∈V_M0. In other words, we have P (x,2) = K(x+ 1) for some positive constant K.

Here we present the convergence theorem. The essence of the algorithm is as follows. We try to construct a contraction iterative scheme on F(θ−θ^m)⁰₁, where θ^m is the approximate value of θ at the m^th step. This contraction is built upon the error bounds of the extracted envelopes at each iteration of the algorithm. The notations of this proof follow the notations of Algorithm 3.

Theorem 5. [Convergence Theorem] Assume that the instantaneous frequency in equation (5.1.1) is M0-sparse; i.e. θ⁰ ∈VM0. Furthermore, assume that

fˆ_0,θ¯(k)≤ C₀

|k|^p, fˆ_1,θ¯(k)≤ C₀

|k|^p,

for C0 > 0 and p≥ 4. If the initial guess satisfies

^F(^4θ0)⁰

1

2πM0 ≤ ¹₄, then there exists an η₀ >0 such that forL > η₀ we have

F(θ−θ^m+1)⁰₁ ≤ Γ1λ^2p−2L^−p+2+¹₂F(θ−θ^m)⁰₁ , for λ >1 and Γ1 >0.

Proof. We know that if 4θ^m = θ − θ^m, then a^m = f₁cos ∆θ^m, and b^m =

−f₁sin ∆θ^m. Let ˜a^m, ˜b^m be approximate envelope functions. Set the error in en- velopes as 4a^m = a − ˜a^m, and 4b^m = b −˜b^m. Using these, we get f = f₀ +

a^mcosθ^m + b^msinθ^m. Let L_m = ^θm(1)−θ_2π^m(0) and ¯θ^m = _2πL^θm_m. Then, we have f = f₀ +a^mcos 2πL_mθ¯^m +b^msin 2πL_mθ¯^m. If we take the Fourier transform in ¯θ^m coordinate (See Table (5.1)), we get

fˆθ¯^m(k) = ˆf_0,θ¯^m(k)+

ˆa^m_θ¯m(k−L_m) + ˆa^m_θ¯m(k+L_m)

2 +

ˆb^m_θ¯m(k−L_m)−ˆb^m_θ¯m(k+L_m)

2i .

Consequently, one can find ˆ

a^m_θ¯m(k) = ˆfθ¯^m(k+L_m) + ˆfθ¯^m(k−L_m)

−fˆ_0,θ¯^m(k+L_m)−fˆ_0,θ¯^m(k−L_m) +1

2(−ˆa^mθ¯^m(k+ 2Lm)−ˆa^mθ¯^m(k−2Lm)) +1

2i

ˆb^m_θ¯m(k+ 2L_m)−ˆb^m_θ¯m(k−2L_m),

and

ˆb^mθ¯^m(k) = ifˆθ¯^m(k+L_m)−ifˆθ¯^m(k−L_m)

−ifˆ_0,θ¯^m(k+L_m) +ifˆ_0,θ¯^m(k−L_m) +i

2(−ˆa^m_θ¯m(k+ 2L_m) + ˆa^m_θ¯m(k−2L_m)) +1

2

ˆb^m_θ¯m(k+ 2L_m) + ˆb^m_θ¯m(k−2L_m). In the periodic STFR algorithm, ˆ˜a^m_θ¯m, ˆ˜b^m_θ¯mare approximated as

ˆ˜

a^mθ¯^m(k) =







fˆθ¯^m(k+L_m) + ˆfθ¯^m(k−L_m), −^L_λ^m ≤k ≤ ^L_λ^m,

0, otherwise,

ˆ˜b^mθ¯^m(k) =







ifˆθ¯^m(k+L_m)−fˆθ¯^m(k−L_m), −^L_λ^m ≤k ≤ ^L_λ^m,

0, otherwise.

Here, 1< λ defines the width of the filter. Hence, for4aˆ^m_θ¯m and 4ˆb^m_θ¯m we have

4aˆ^m_θ¯m(k) =























{−fˆ_0,θ¯^m(k+L_m)−fˆ_0,θ¯^m(k−L_m)

+¹₂−ˆa^m_θ¯m(k+ 2Lm)−ˆa^m_θ¯m(k−2Lm) |k| ≤ ^L_λ^m, +_2i¹ ˆb^m_θ¯m(k+ 2L_m)−ˆb^m_θ¯m(k−2L_m)},

ˆ

a^m_θ¯m(k), |k|> ^L_λ^m,

4ˆb^m_θ¯m(k) =























{−ifˆ_0,θ¯^m(k+L_m) +ifˆ_0,θ¯^m(k−L_m)

+₂ⁱ −ˆa^m_θ¯m(k+ 2Lm) + ˆa^m_θ¯m(k−2Lm) |k| ≤ ^L_λ^m, +¹₂ˆb^m_θ¯m(k+ 2L_m) + ˆb^m_θ¯m(k−2L_m)},

ˆb^m_θ¯m(k), |k|> ^L_λ^m.

The ideal case for updating the phase function is ^dθ_dt^new = ^dθ_dt^m − _dt^d arctan^˜_˜a^bmm. How- ever, the algorithm works in a way that we must choose ^dθm+1_dt in V_M0. Hence,

dθ^m+1

dt = P_VM

0

dθnew

dt

. So, at each step, we force ^dθ_dt^m to be in V_M0. In other words, ^dθ_dt^m ∈ VM0 for all m ≥ 0, m ∈ Z. This short analysis tells us that ^dθm+1_dt =

dθ^m dt −P_VM

0

_d

dtarctan_a^˜b_˜^mm. Since θ ∈V_M0 is sufficiently differentiable, and ^dθ_dt^m ∈V_M0, then (θ−θ^m =4θ^m)∈V_M0. Having these in mind, we can find

d

dt4θ^m+1 =d dt

θ−θ^m+1= dθ

dt − dθ^m

dt +PVM0

d

dtarctan˜b^m

˜ a^m

!

=P_VM

0

d

dt arctan˜b^m

˜

a^m −arctan b^m a^m

!

+R_VM

0

d dt4θ^m

!

. We know arctan^˜_˜a^bmm −arctan_a^bmm is in C¹. For any g ∈ C¹, we have P_VM

0

d

dt(g) =

d dtP_VM

0 (g). Hence, the Fourier transform of the IF error is

F4θ^m+10

k = F P_VM

0

d

dt arctan˜b^m

˜

a^m −arctanb^m a^m

!!!

k

+FR_VM

0 (4θ^m)⁰

k

= (i2πk) F P_VM

0 arctan˜b^m

˜

a^m −arctanb^m a^m

!!!

k

+FR_VM

0 (4θ^m)⁰

k.

As FP_VM

0

arctan^˜_˜a^bmm −arctan^b_a^mm_k = 0 for |k|> M₀, then

F(4θ^m+1)⁰₁ ≤ (2πM₀)FP_VM

0

arctan^˜_˜a^bmm −arctan^b_a^mm

₁

+FRV_M0(4θ^m)⁰

1. Now, we know that for any functiong(t), we have

FP_VM

0(g)₁ = ^XM0

k=−M₀

|(F(g))_k|= ^XM0

k=−M₀

ˆ ₁

0

g(t)e^−i2πktdt

≤

M0

X

k=−M₀

ˆ ₁

0

|g(t)|dt≤

M0

X

k=−M₀

kgk_∞ = (2M₀+ 1)kgk_∞. Hence, the l₁ norm of the Fourier of the IF error is

F4θ^m+10

₁ ≤ (2πM0) (2M0+ 1)

arctan˜b^m

˜

a^m −arctanb^m a^m

_∞

+FR_VM

0(4θ^m)⁰₁.

Since θ⁰ is sparse in V_M0, then the last term vanishes. In fact, we previously showed that ^dθ_dt^m ∈VM0, hence

F(4θ^m)⁰_k =F(θ−θ^m)⁰_k =







(F(θ⁰))_k−F(θ^m)⁰_k, |k| ≤M0, (F(θ⁰))_k = 0, |k|> M₀, so,

R_VM

0 (4θ^m)⁰ = ^X

|k|>M0

F(4θ^m)⁰_ke^i2πkt = ^X

|k|>M0

(F(θ⁰))_ke^i2πkt = 0, and then FR_VM

0(4θ^m)⁰_k = 0. Finally, we have the following bound on the IF error:

F(4θ^m+1)⁰₁ ≤ (2πM₀) (2M₀+ 1)arctan^˜_˜a^bmm −arctan_a^bmm_∞.

The last term on the right hand side of this bound can be simplified further. If we use the fact that x²+xy≤ ^x₂² −y² for real x and y, we have

arctan˜b^m

˜

a^m −arctan b^m a^m

=

arctan ˜b^ma^m−a˜^mb^m a^m˜a^m+b^m˜b^m

!

≤

˜b^ma^m−˜a^mb^m a^m˜a^m+b^m˜b^m

=

(a^m+4a^m)4b^m−(b^m+4b^m)4a^m (a^m)²+ (b^m)²+ (4a^m)a^m+ (4b^m)b^m

≤(|a^m|+|4a^m|)|4b^m|+ (|b^m|+|4b^m|)|4a^m|

(a^m)²+(b^m)²

2 −(4a^m)²+ (4b^m)²

≤D(|4a^m|+|4b^m|), for D = max f2 ^f1+|4am|

1

2 −(^{(4am)2+(4bm)2}), _f2 ^f1+|4bm| 1

2−(^{(4am)2+(4bm)2})

!

, in which we have taken into account that f1 >0. Now, consider the fact that

|4a^m(t)| = |4a^m(θ)|=4a^mθ¯^m=

∞

X

k=−∞

4ˆa^m_θ¯m(k)e^i2πkθ¯m

≤

∞

X

k=−∞

|4aˆ^mθ¯^m(k)|=k4ˆa^mθ¯^mk₁,

and as, in general ˆgθ¯(−k) = ˆgθ¯(k) for anyg, then |ˆgθ¯(−k)|=|gˆθ¯(k)|. Consequently, the bound on the envelopes errors can be expressed as

|4a^m| ≤ k4ˆa^mθ¯^mk₁

≤ 2 ^X

(¹⁻λ¹)^Lm≤k≤(¹⁺λ¹)^Lm

fˆ_0,θ¯^m(k)

+ ^X

(²⁻¹λ)^Lm≤k≤(²⁺¹λ)^Lm

|ˆa^mθ¯^m(k)|+ˆb^mθ¯^m(k)

+ ^X

|k|>^Lm

λ

|aˆ^m_θ¯m(k)|,

and

|4b^m| ≤ 4ˆb^m_θ¯m

₁

≤ 2 ^X

(¹⁻¹λ)^Lm≤k≤(¹⁺¹λ)^Lm

fˆ_0,θ¯^m(k)

+ ^X

(²⁻¹λ)^Lm≤k≤(²⁺¹λ)^Lm

|ˆa^mθ¯^m(k)|+ˆb^mθ¯^m(k)

+ ^X

|k|>^Lm

λ

ˆb^m_θ¯m(k).

Before we use approximations on these terms, recall that according to one of our as- sumptions, we have: “The observation is periodic with mean zero.” Hence, ˆf_0,θ¯^m(0) = 0. Furthermore, as e^i2πkθ¯e^{−i2πωθ¯m} is periodic, so ise^i2πθ¯m(^kLmL −ω)e^ik4θmL , and then we have the following estimate using these facts and Lemma 3:

(5.1.4)

fˆ_0,θ¯^m(ω) ≤ 2C₀^αλ_|ω|^p−1 +2C₀P_mⁿ_2π|εω|^{M0λ n}

p−2

P

j=1

_γ

L

j π²

3 + 2^Pn

j=p

ω αλ

j−p+1_γ

L

j!

+2C0P_mⁿ_2π|εω|^{M0λ n}

2_L^γp−11 + ^|ω|_αλ.

Here, γ = k^F(4θm)0k₁

2πM0 . The proof of this inequality can be found in in Appendix A.

To find the estimate on ˆa^m_θ¯m(k), we follow a similar approach. The approximation procedure is detailed in Appendix B.

|aˆ^mθ¯^m(ω)| ≤2C₀ αλ

|ω|

!p−1

+fˆ_1,θ¯(0)P_mⁿ M₀ 2π|ω|

!n n

X

j=1

γ^j +C₀P_mⁿ M₀λ

2π|εω|

!n

2p p−1

n

X

j=1

2^jγ^j

+C₀P_mⁿ M₀λ 2π|εω|

!n



p−2

X

j=1

2^jγ L

j π² 3 +^Xn

j=p

2^j+1 ω αλ

j−p+1γ L

j

(5.1.5) 

+C₀P_mⁿ M₀λ 2π|εω|

!n

2^pγ L

p−1

1 + |ω|

αλ

!!

.

There is a similar bound for ˆb^m_θ¯m(ω) as well:

ˆb^mθ¯^m(ω)≤2C0

αλ

|ω|

!p−1

+fˆ_1,θ¯(0)P_mⁿ M₀ 2π|ω|

!n n

X

j=1

γ^j +C₀P_mⁿ M0λ

2π|εω|

!n

2p p−1

n

X

j=1

2^jγ^j

+C₀P_mⁿ M₀λ 2π|εω|

!n



p−2

X

j=1

2^jγ L

j π² 3 +^Xn

j=p

2^j+1

ω αλ

j−p+1γ L

j



+C₀P_mⁿ M₀λ 2π|εω|

!n

2^pγ L

p−1

1 + |ω|

αλ

!!

.

Now we approximate the terms in the inequality for |4a^m|. One can find the detail of this approximation in Appendix C. We have a similar inequality for |4b^m|. These can be expressed in the following way

(5.1.6) |4a^m| ≤ C₁λ^2p−2L^−p+2+C₂λ²ⁿLmax(−n,−2p+1,−n−p+2,−2p)γ ,

(5.1.7) |4b^m| ≤ C₁λ^2p−2L^−p+2+C₂λ²ⁿLmax(−n,−2p+1,−n−p+2,−2p)γ ,

where C₁ and C₂ depend on C₀,M₀, ε, α, P_mⁿ, n, p. Before moving forward, we need to show that parameters like αand P_mⁿ are uniformly bounded. We start with α. We have

|1−α|=

1− L_m L

=

1−

θ^m(1)−θ^m(0) 2π θ¯^m(1)−θ¯^m(0)

2π

=

∆θ^m(1)−∆θ^m(0) 2πL

≤

(∆θ^m)⁰ 2πL

_∞

≤

F(∆θ^m)⁰ 2πL

₁

= γM₀

L ≤ M₀ 4L ≤ 1

8. (5.1.8)

Here, we have used the fact that γ ≤ ¹₄. At the last part of the proof, we will show that this condition remains intact for all iterations. The last inequality in (5.1.8) can be true for the condition L ≥ 2M₀. Hence, ⁷₈ ≤ α ≤ ⁹₈. In order to prove the boundedness of P_mⁿ at every step, we need to find bounds on

F

θ¯^m0

₁ and minθ¯^m0. If we take the condition minθ¯⁰≥ ^M_2L⁰, we get ¯θ⁰ ≥ ^M_2L⁰ > ^M_4L⁰ ≥ ^γM_L⁰ =

F(∆θ^m)⁰ 2πL

₁ ≥^(∆θ_2πL^m)0. Using this, we have

θ¯^m0

=

(θ^m)⁰ 2πL_m

=

(θ−∆θ^m)⁰ 2παL

≥ 8 9

(θ−∆θ^m)⁰ 2πL

=8 9

θ¯⁰− (∆θ^m)⁰ 2πL

≥ 8

9 θ¯⁰− (∆θ^m)⁰ 2πL

!

≥ 8 9

θ¯⁰ −M₀ 4L

≥8 9

θ¯⁰− M0

4L

≥ 8

9 θ¯⁰− θ¯⁰ 2

!

≥ 4

9minθ¯⁰. We also have

F

θ¯^m0

₁ =1

α

F (θ−∆θ^m)⁰ 2πL

! ₁

= 1 α

Fθ¯⁰− F(∆θ^m)⁰ 2πL

₁

≤8 7





Fθ¯⁰

1+

F(∆θ^m)⁰₁

2πL



= 8 7

Fθ¯⁰

1+M₀γ L

≤8 7

Fθ¯⁰

1+M₀ 4L

.

These two estimates pave the way to rigorously prove that P_mⁿ is bounded at every single step. Now it is time for a bound on D. In fact, if we bound D uniformly over all steps, we can set a contraction. Assuming that 1 < λ < λ₀, and since f₁ > 0, taking thecondition

C₁λ^2p−2₀

L^−p+1 + C₂λ²ⁿ₀

4L⁻max(−n,−2p+1,−n−p+2,−2p) ≤

√2

4 minf₁, would result in¹

|4a^m| ≤C1λ^2p−2L^−p+2+C2λ²ⁿLmax(−n,−2p+1,−n−p+2,−2p)

γ

≤C₁λ^2p−2₀

L^p−2 + C₂λ²ⁿ₀

4L⁻max(−n,−2p+1,−n−p+2,−2p) ≤

√2

4 minf₁.

1Remember that if f1 has a zero crossing, this condition would never be satisfied. Hence, the rest of the proof will not be valid.

Similarly, we have the same bound for |4b^m|, and then

D= max







f₁+|4a^m|

f₁²

2 −(4a^m)²+ (4b^m)², f₁+|4b^m|

f₁²

2 −(4a^m)²+ (4b^m)²







≤ maxf₁ +^√₄²minf₁

min^f₂¹² −(4a^m)²+ (4b^m)² ≤ maxf₁+^√₄²minf₁ min^f₂¹² − ¹₄minf₁²

=4 maxf₁+√

2 minf₁

minf₁² =E₀.

The latter shows thatD is also bounded. So, we get

F4θ^m+10

₁ ≤(2πM₀) (2M₀+ 1)

arctan˜b^m

˜

a^m −arctan b^m a^m

_∞

≤(2πM₀) (2M₀+ 1)D(|4a^m|+|4b^m|)

≤E₀(4πM₀) (2M₀+ 1)C₁λ^2p−2L^−p+2

+E₀(4πM₀) (2M₀+ 1)C₂λ²ⁿLmax(−n,−2p+1,−n−p+2,−2p)

γ.

The last inequality is nothing but

F(4θ^m+1)⁰₁ ≤ Γ1λ^2p−2L^−p+2 +Γ2λ²ⁿLmax(−n,−2p+1,−n−p+2,−2p)

F(4θ^m)⁰₁.

If we have the condition that Γ2λ²ⁿ₀ Lmax(−n,−2p+1,−n−p+2,−2p) ≤ ¹₂, we have the con- traction that we were looking for. Before finishing, we need to state the follow- ing: ∃η₀ > 0 such that L > η₀, then all the conditions L ≥ 2M₀, minθ¯⁰ ≥ ^M_2L⁰,

C1λ^2p−2₀

L^p−2 +_4L−max(−n,−2p+1,−n−p+2,−2p)^C2λ2n0 ≤

√2

4 minf₁, and Γ2λ²ⁿ₀ Lmax(−n,−2p+1,−n−p+2,−2p) ≤ ¹₂ would be satisfied. As a result, we have

F(θ−θ^m+1)⁰₁ ≤ Γ1λ^2p−2L^−p+2+¹₂F(θ−θ^m)⁰₁ .

Having this bound, the condition γ ≤ ¹₄ would remain intact for all iterations. In other words, when there is a contraction onF(θ−θ^m+1)⁰₁, this term would remain

bounded. Hence, if

^F(^4θ0)⁰

1

2πM0 ≤ ¹₄ for the first iteration, it will remain bounded by

1

4 for all iterations. This completes the proof.

A corollary of this is that if f₀ has only high-frequency components, there would be no interference between f₀ and f₁cosθ. Hence, the convergence proof would be true for λ > 0, as well. Furthermore, since it is possible that an Intrinsic Signal has multiple representations and this theorem states merely that the algorithm converges to an IMF in one of these representations, the algorithm’s result is not necessary unique. The theorem does not mention which representation the algorithm converges to. The theorem that we observed, in this part, says that if in a representation we havefˆ_0,θ¯(k)≤ _|k|^C0p,fˆ_1,θ¯(k)≤ _|k|^C0p, then increasing the width of the filter will reduce the error in extraction. More specifically, if in a representation we have a wide band signal (like an intrawave signal), we are more likely to capture it by widening the filter width. This approach helps us to find a sparser representation compared to the case in which one uses a normal envelope dictionary 1.2.3. From the algorithmic point of view, the initial guess θ₀ and the parameterλ define the representation to which we converge. In practice, if the IMFs that constitute the signal are separated enough in the time-frequency domain, the reduction of the parameter λ is always beneficial.

This idea is illustrated in the following example.

Example 15. Again consider our generic intrawave IMFx(t) = cosωt+^Mω_p sin (pt). We can have many representations for this signal. The first representation would be (5.1.9) x(t) = cos ωt+Mω

p sin (pt)

!

, for ¯θ=ωt+^Mω_p sin (pt). The second representation would be

x(θ) =

(

J₀ Mω p

!

+ 2^X∞

k=1

J_2k Mω p

!

cos (2kpt)

)

cos (ωt)

−

(

2^X∞

k=1

J2k−1 Mω p

!

sin ((2k−1)pt)

)

sin (ωt),

for ¯θ = ωt. So, with a constant initial guess for θ0 close to ¯θ = ωt, the second representation is the one that is seen by the algorithm at the first iteration. Hence, if a narrow-band filter (large value of λ) is used, only the first harmonic is extracted by the algorithm. In order to capture more terms in the envelope of the second representation, we need a smaller value of λ.