Instantaneous Frequency Estimation Based on

Synchrosqueezing Wavelet Transform

Qingtang Jiang and Bruce W. Suter ∗

February 2016

1st revision in July 2016

2nd revision in February 2017

Abstract

Recently, the synchrosqueezing transform (SST) was developed as an alternative to the

empirical mode decomposition scheme to separate a non-stationary signal with time-varying

amplitudes and instantaneous frequencies (IFs) into a superposition of frequency components

that each have well-defined IFs. The continuous wavelet transform (CWT)-based SST sharpens

the time-frequency representation of a non-stationary signal by assigning the scale variable of

the signal’s CWT to the frequency variable by a reference IF function. Since the SST method is

applied to estimate the IFs of all frequency components of a signal based on one single reference

IF function, it may yield not very accurate results. In this paper we introduce the instantaneous

frequency-embedded synchrosqueezing wavelet transform (IFE-SST). IFE-SST uses a rough

estimation of the IF of a targeted component to produce accurate IF estimation. The reference

IF function of IFE-SST is associated with the targeted component. Our numerical experiments

show that IFE-SST outperforms the CWT-based SST in IF estimation and separation of

multicomponent signals.

Keywords: Instantaneous frequency, Empirical mode decomposition (EMD), Synchrosqueezing

transform (SST), Signal separation

1 Introduction

Recently the study of modeling a non-stationary signal as a superposition of Fourier-like oscillatory

modes has been an active research area. To model a non-stationary signal x(t) as

x(t) = A0(t) +

K∑k=1

Ak(t) cos(2πφk(t)

)(1)

∗Qingtang Jiang is with the Department of Mathematics and Computer Science, University of Missouri-St.

Louis, MO 63141, USA, e-mail: jiangq@umsl.edu; Bruce W. Suter is with The Air Force Research Laboratory,

AFRL/RITB, Rome, NY 13441, USA, e-mail: bruce.suter@us.af.mil.

1

is important to extract information, such as the underlying dynamics, hidden in x(t). The rep-

resentation of x(t) in (1) with Ak(t), φ′k(t) > 0 and Ak(t), φ

′k(t) varying more slowly than φk(t),

is called an adaptive harmonic model (AHM) representation of x(t), where Ak(t) are called the

instantaneous amplitudes (IAs) and φ′k(t) the instantaneous frequencies (IFs), which can be used

to describe the underlying dynamics. AHM representations of non-stationary signals have been

used in many applications including geophysics (seismic wave), atmospheric and climate stud-

ies, oceanographic studies, medical data analysis, speech recognition, non-stationary dynamics in

financial system, see for example, [1]-[4].

The empirical mode decomposition (EMD) introduced by Huang et al is a popular method to

decompose a non-stationary signal as a superposition of intrinsic mode functions (IMFs) [5]. This

is an efficient data-driven approach and no basis of functions is used. It has been widely used in

many applications, see [1] and the references therein.

Many time-frequency methods have been developed to study the time-varying spectral prop-

erties of a given signal x(t) [6]. Recently the synchrosqueezing transform (SST), also called

the synchrosqueezed wavelet transform, was developed by Daubechies, Lu and Wu [7] to provide

mathematical theorems to guarantee the recovery of oscillatory modes from the SST of x(t). SST,

which was first introduced by Daubechies and Maes in 1996 for the consideration of speech signal

separation [8], is based on the continuous wavelet transform (CWT), which has scale and time

variables. SST re-assigns the scale variable to the frequency variable to sharpen the time-frequency

representation of a signal as the method of both time and frequency re-assignments studied by

Auger and Flandrin in 1995 [9] (see also [10] for the time-frequency and time-scale representations

of signals by the re-assignment method). In addition, the original signal can be recovered from its

SST. SST provides an alternative to the EMD method and its variants such as the ensemble EMD

(EEMD) scheme [11], and it overcomes some limitations of the EMD and EEMD schemes such as

mode-mixing and the possible negativeness of the IFs which arise in EMD and EEMD schemes.

See [12]-[14] for a comparison between EMD and SST. Generalized SST was introduced in [15] for

the time-frequency representation of signals with significantly oscillating IFs. The stability of SST

was studied in [16]. The SST with vanishing moment wavelets with stacked knots was introduced

in [17] to process signals on bounded or half-infinite time intervals for real-time signal process.

[18] introduced the hybrid EMT-SST computational scheme by applying the modified SST to

the IMFs of the EMD. [19] provided the AHM representation of oscillatory signals composed of

multiple components with fast-varying instantaneous frequency by optimization. [20] proposed

a new method to determine the time-frequency content of time-dependent signals consisting of

multiple oscillatory components.

The SST introduced in [8] and studied in above papers is also referred to the wavelet-based

SST. The short-time Fourier transform (STFT)-based SST was studied in [21]-[24]. Also, the S-

2

transform-based SST is introduced in [25] and has been applied in seismic spectral decomposition.

Other methods to decompose a non-stationary signal as the superposition of adaptive IMFs/subbands

include optimization methods [26]-[31], the empirical wavelet transform to produce an adaptive

wavelet frame system for the decomposition of a given signal [32], an alternative algorithm to

EMD with iterative filtering replacing the sifting process in EMD [33, 34], and a STFT-based

signal separation method in [35].

In this paper we will introduce instantaneous frequency-based synchrosqueezing wavelet trans-

form. Our work is motived by [23] on the demodulation transform with STFT. Our approach

gives IFs more straightly.

This paper is organized as follows. In Section 2, we review the SST. In Section 3, we introduce

the instantaneous frequency-embedded wavelet synchrosqueezing transform (IFE-SST) and study

its property. In Section 4, we consider the implementation issue. In Section 5, we use IFE-

SST for the separation of multicomponent signals. Our experimental results show that IFE-SST

outperforms wavelet-based SST in estimation of IFs and signal separation. The conclusion is

given in Section 6.

2 Synchrosqueezed wavelet transform

The synchrosqueezed wavelet transform (SST) separates Fourier-like oscillatory mode

Ak(t) cos(2πφk(t)

)from a superposition in (1), with A(t) and φ′(t) positive and slow-varying,

compared to φ(t), and they satisfying certain conditions. The SST approach in [8, 7] is based on

the continuous wavelet transform (CWT).

2.1 Continuous wavelet transform (CWT)

A function ψ(t) ∈ L2(R) is called a continuous (or an admissible) wavelet if it satisfies (see e.g.

[36, 37]) the admissible condition:

0 < Cψ =

∫ ∞−∞|ψ(ξ)|2 dξ

|ξ|<∞.

In this paper the Fourier transform of a function x(t) ∈ L1(R) is defined by

x(ξ) =

∫ ∞−∞

x(t)e−i2πξtdt,

which can be extended to functions in L2(R). Denote ψa,b(t) = 1aψ(t−ba

). The continuous wavelet

transform (CWT) of a signal x(t) ∈ L2(R) with a continuous wavelet ψ is defined by

Wx(a, b) = 〈x, ψa,b〉 =

∫ ∞−∞

x(t)1

aψ( t− b

a

)dt.

3

The variables a and b are called the scale and time variables respectively. The signal x(t) can be

recovered by the inverse wavelet transform (see e.g. [36, 37, 38])

x(t) =1

Cψ

∫ ∞−∞

∫ ∞−∞

Wx(a, b)ψa,b(t)dbda

|a|.

The Fourier transform and the CWT given above can be extended to x(t) in the class of tem-

pered distributions, denoted by S ′, which includes the Dirac delta function, sinusoidal functions

or polynomials on R. For x(t) ∈ S ′, x is defined as the Fourier transform of x(t) if x statifies∫ ∞−∞

x(ξ) y(ξ) dξ =

∫ ∞−∞

x(ξ) y(ξ) dξ, ∀y(t) ∈ S,

where S is the Schwartz space or the space of testing functions all of whose derivatives of any

order exist and are rapidly decreasing. Refer to [39] for mathematically rigorous definitions of Sand S ′. For x(t) = ei2πct with frequency c, its Fourier transform is δ(ξ − c) (see [40]).

If the continuous wavelet ψ is in S, then the CWT Wx(a, b) of x ∈ S ′ with ψ is well defined.

For x(t) = A(t)ei2πφ(t), its CWT Wx(a, b) is well defined as long as ψ has certain decay as |t| → ∞to assure A(t)ψ(t) ∈ L1(R). In addition, if ψ(ξ) has enough decay as |ξ| → ∞, we have

Wx(a, b) = 〈x, ψa,b〉 =

∫ ∞−∞

x(ξ)ψ(aξ)ei2πbξdξ.

A function x(t) is called an analytic signal if it satisfies x(ξ) = 0 for ξ < 0. In this paper, we

consider analytic continuous wavelets. In addition, we assume ψ also satisfies

0 6= cψ =

∫ ∞0

ψ(ξ)dξ

ξ<∞. (2)

For an analytic signal x(t) ∈ L2(R), it can be recovered by another inverse wavelet transform

which does not involve the time variable b (refer to [8, 7]):

x(b) =1

cψ

∫ ∞0

Wx(a, b)da

a,

where cψ is defined by (2). Furthermore, for a real signal x(t) ∈ L2(R), it can be recovered by

the following formula (see [7]):

x(b) = Re( 2

cψ

∫ ∞0

Wx(a, b)da

a

).

Again, the above two formulas hold for x(t) = A(t)ei2πφ(t) as long as ψ has certain decay as

|t| → ∞.

The “bump wavelet” defined by

ψ(ξ) = e1− 1

1−σ2(ξ−µ)2 χ(µ− 1σ,µ+ 1

σ), (3)

4

and the (scaled) Morlet’s wavelet defined by

ψ(ξ) = e−2σ2π2(ξ−µ)2 − e−2σ2π2(ξ2+µ2), (4)

where σ > 0, µ > 0, are commonly used continuous wavelets. The parameter σ in (3) and

(4) controls the shape of ψ and has the effect on the CWT of a signal. For a multicomponent

signal x(t) =∑K

k=1Ak cos(2πφkt)

with positive constants Ak, φk, 0 < φk < φk+1, there are no

interference among the components Ak cos(2πφkt

)in |Wx(b, a)| with the bump wavelet as long

as the parameter σ is large enough. For a superposition (1) of AHMs with φk satisfying the

separation condition, the larger σ does not necessarily provide a better separation of AHMs. See

[13] for the detailed discussion on the effect of σ on the CWT of a signal with the bump wavelet.

Here we illustrate that for Morlet’s wavelet, a larger σ of Morlet’s wavelet does not necessarily

result in a sharper representation of the CWT in the time-scale plane as the “bump wavelet”

illustrated in [13]. Let

y(t) = ei2π(9t+5t2) + ei2π(13t+10t2), 0 ≤ t ≤ 1, (5)

which is sampled uniformly with 128 sample points. The CWT of y(t) with Morlet’s wavelet with

σ = 1, µ = 1 and σ =√

5, µ = 1 are shown Fig.1. Observe that the wavelet with a larger σ results

in more blurring representation of y(t) in the time-scale plane.

Figure 1: |Wy(a, b)|: CWT of y(t) = ei2π(9t+5t2) + ei2π(13t+10t2), 0 ≤ t ≤ 1 by using Morelet’s wavelet ψ

with σ = 1, µ = 1 (Left picture) and with σ =√

5, µ = 1 (Middle picture); CWT of s(t) = cos(2π(10t)

), 0 ≤

t ≤ 1 with wavelet ψ given by (6) (Right picture)

The “bump wavelet” is bandlimited, and hence it has a better frequency localization than

Morlet’s wavelet. On the other hand, Morlet’s wavelet ψ(t) is

ψ(t) =1

σ√

2πe−( t√

2σ)2

(ei2πµt − e−2π2σ2µ2).

Thus Morlet’s wavelet enjoys nice localization in both the time and frequency domains. The

“bump wavelet” is analytic, but Morlet’s wavelet is not. Observe that the second term in (4) is very

5

small for σ ≥ 1 and µ ≥ 1, e.g., with µ = 1, σ = 1, e−2σ2π2(ξ2+µ2) ≤ exp(−2π2) = 2.6753× 10−9.

Thus the second term in (4) could be dropped in practice. In addition, the first term of ψ(ξ) in

(4) is also very small for any ξ ≤ 0 if σ ≥ 1, µ ≥ 1. Thus in practice one may use ψ defined by

ψ(ξ) =

e−2π2(ξ−1)2

, for ξ > 0,

0, for ξ ≤ 0.(6)

ψ defined by (6) is one of the three wavelets used in [16]. In this paper, unless it is specifically

stated, we will use this ψ and we also call it Morlet’s wavelet.

Observe that ψ(ξ) of Morlet’s wavelet ψ given in (6) concentrates at ξ = 1. If an input signal

x(t) concentrates around ξ = c in the frequency domain, then its CWT concentrates around the

line a = 1c in the scale-time plane. For example, let us consider x(t) = A cos(2πct), where c > 0

is a constant. Then x(ξ) = A2 (δ(c) + δ(−c)). Thus for a > 0,

Wx(a, b) =

∫ ∞−∞

x(ξ)ψ(aξ)ei2πbξdξ =

1

2A ψ

(ac)ei2πbc.

Therefore, Wx(a, b) concentrates around ac = 1, i.e. a = 1/c. See Fig.1 for |Ws(a, b)| with

s(t) = cos(2π(10t)

). Observe that |Ws(a, b)| does concentrate around a = 0.1, the reciprocal

of the IF=10 of s(t). However |Ws(a, b)| spreads out around a = 0.1 and what we see in the

scale-time plane is a zone, not a sharp line, around a = 0.1. This property will cause the problem

that we cannot separate the IFs from their CWTs when two signals have close IFs, though for

the superposition of signals such as A cos(2πct) or Aei2πct, there is no such an issue as long as we

choose the parameter σ of the wavelet to be large enough. For example, as demonstrated in Fig.1,

the CWTs of the two components of y(t) given in (5) are mixed. SST re-assigns the scale variable

a to the frequency variable so that it sharpens the time-frequency representation of a signal.

2.2 Synchrosqueezed wavelet transform (SST)

The idea of SST is to re-assign the scale variable a to the frequency varilable. As in [7], we first

look at the CWT of x(t) = A cos(2πct), where c is a positive constant. As shown above, the CWT

of x(t) is Wx(a, b) = 12A ψ

(ac)ei2πbc. Observe that the IF of x(t), which is c, can be obtained by

∂∂bWx(a, b)

2πiWx(a, b)= c.

Thus, for a general x(t), at (a, b) for which Wx(a, b) 6= 0, a good candidate for the instantaneous

frequency (IF) of x(t) is∂∂bWx(a,b)

2πiWx(a,b) . In the following, denote

ωx(a, b) =∂∂bWx(a, b)

2πiWx(a, b), for Wx(a, b) 6= 0.

6

ωx(a, b) is called the “reference IF function” in [18] and the “phase transform” in [16]. SST is

to transform the CWT Wx(a, b) of x(t) to a quantity, denoted by Tx(ξ, b), on the time-frequency

plane:

Tx(ξ, b) =

∫a:Wx(a,b)6=0

Wx(a, b)δ(ωx(a, b)− ξ

)daa, (7)

where ξ is the frequency variable.

Figure 2: Assignment of a to ξ in SST for x(t) = A cos(2π(ct)

)Fig.2 illustrates the definition of SST for the special case with x(t) = A cos

(2π(ct)

). See Fig.3

for the SSTs of s(t) = cos(2π(10t)

), 0 ≤ t ≤ 1 and y(t) given in (5).

Figure 3: Left: SST of s(t) = cos(2π(10t)

); Right: SST of y(t) = ei2π(9t+5t2) + ei2π(13t+10t2)

The input signal x(t) can be recovered from its SST as shown in the following theorem.

Theorem 1. ([7]) Let cψ be the constant defined by (2). Then for a real-valued x(t),

x(b) = Re( 2

cψ

∫ ∞0

Tx(ξ, b)dξ)

; (8)

7

for an analytic x(t),

x(b) =1

cψ

∫ ∞0

Tx(ξ, b)dξ. (9)

In practice, a, b, ξ are discretized. Suppose aj , bn, ξk, j, n, k = 1, · · · , are the sampling points

of a, b, ξ respectively. Here we assume ξk+1 − ξk = ∆ξ for all k. Then the SST of x(t) is given by

Tx(ξk, bn) =∑

j: |ωx(aj ,bn)−ξk|≤∆ξ/2,|Wx(aj ,bn)|≥γ

Wx(aj , bn)a−1j (∆a)j ,

where (∆a)j = aj+1−aj , and γ > 0 is a threshold for the condition |Wx(a, b)| > 0. The recovering

formula (8) for a real signal x(t) leads to

x(bn) = Re( 2

cψ

∑k

Tx(ξk, bn)), n = 1, 2, · · · ,

while for an analytic x(t), we have, from (9),

x(bn) =1

cψ

∑k

Tx(ξk, bn), n = 1, 2, · · · .

2.3 Generalized SST

Figure 4: Left:z(t); Right: IF of z(t)

As shown in the above examples, SST works well for some signals such as those of constant

frequency. However, SST does not work well for the signals whose frequencies change significantly

with the time. For example, let z(t) be the signal given by

z(t) =

0.8 cos 16πt, 0 ≤ t < 2.5,

cos 2π(15t+ cos(2πt)

), 2.5 ≤ t ≤ 4.

(10)

z(t) is a variant of a signal considered in [15]. Here t is uniformly sampled with sample rate 4511 .

z(t) and its IF are shown in Fig.4. The SST of z(t) is presented in Fig.5. Observe that the IF of

z(t) is blurring for t > 2.5.

8

The generalized SST was introduced by Li and Liang in [15]. The idea is to transform a signal

x(t) = A(t) cos(2πφ(t)) or x(t) = A(t)exp(2πiφ(t)) to a signal with a constant frequency by

x(t) −→ x(t) exp(−2πiφ0(t)),

where φ0(t) is a function such that φ′0(t) = φ′(t) − ξ0 with ξ0 being the target frequency. If we

choose φ0(t) = φ(t) − ξ0t, then x(t) exp(−2πiφ0(t)) is a signal with constant frequency ξ0. The

problem of this approach is that in practice φ′(t) is unknown, one needs to estimate φ′(t).

Figure 5: SST of z(t) given in (10)

3 Instantaneous frequency embedded SST

Motivated by the work of S. Wang et al [23] on the demodulation transform with STFT, we define

the instantaneous frequency-embedded CWT (IFE-CWT) as follows.

Let ϕ(t) be a differentiable function with ϕ′(t) > 0. For x(t) ∈ L2(R), we define

xϕ,b,ξ0(t) := x(t)e−i2π(ϕ(t)−ϕ(b)−ϕ′(b)(t−b)−ξ0t

), (11)

where ξ0 ≥ 0. Oberve that if x(t) = A(t) exp(i2πφ(t)) for some φ(t) with φ′(t) > 0, then xϕ,b,ξ0(t)

with ϕ(t) = φ(t) has IF φ′(b) + ξ0. Also note that in the definition of generalized SST in [15], the

frequency demodulation of x(t) is x(t) exp(− i2π(ϕ(t)− ξ0t)

).

Definition 1. Suppose ϕ(t) is a differentiable function with ϕ′(t) > 0. The IFE-CWT of x(t) ∈L2(R) with a continuous wavelet ψ is defined by

W IFEx (a, b) = 〈xϕ,b,ξ0 , ψa,b〉 =

∫ ∞−∞

x(t)e−i2π(ϕ(t)−ϕ(b)−ϕ′(b)(t−b)−ξ0t

)1

aψ( t− b

a

)dt. (12)

In the above definition, we assume x(t) ∈ L2(R). Actually, the definition of IFE-CWT can

be extended to slowly growing functions x(t). Next, we have the following property about the

IFE-CWT.

9

Proposition 1. Let W IFEx (a, b) be the IFE-CWT of x(t) defined by (12). Then

W IFEx (a, b) = ei2πϕ(b)

∫ ∞−∞

x(ξ)ψ(aξ + aϕ′(b)

)ei2πbξdξ, (13)

where

x(t) = x(t)e−i2πϕ(t)+i2πξ0t. (14)

Proof. Let ψ1(t) = ψ(t)e−i2πϕ′(b)at. Then the Fourier transform of ψ1(t) is: ψ1(ξ) = ψ

(ξ+aϕ′(b)

).

With x(t) given by (14), we have

W IFEx (a, b) = ei2πϕ(b)

∫ ∞−∞

x(t)e−i2πϕ(t)+i2πξ0tei2πϕ′(b)(t−b) 1

aψ( t− b

a

)dt

= ei2πϕ(b)

∫ ∞−∞

x(t)1

aψ1

( t− ba

)dt = ei2πϕ(b)

∫ ∞−∞

x(ξ) ψ1(aξ) ei2πbξdξ

= ei2πϕ(b)

∫ ∞−∞

x(ξ)ψ(aξ + aϕ′(b)

)ei2πbξdξ,

as desired.

We note that the proof of (13) is straightforward. However, the formula (13) plays an impor-

tant role in our discussion and implementation of IFE-SST. Thus, we consider it in a proposition.

If x(t) = Cei2πφ(t) for a constant C and we choose ϕ(t) = φ(t), then x(ξ) = Cδ(ξ0). Thus

W IFEx (a, b) = Cei2πφ(b)ψ

(aξ0 + aφ′(b)

)ei2πbξ0 .

Observe that for Morlet’s wavelet given in (6), ψ(aξ0 + aφ′(b)

)(hence W IFE

x (a, b)) concentrates

along

a(ξ0 + φ′(b)

)= 1.

Thus IFE-CWT gives more straightforward scale-time representation of a signal.

Let u(t) be the signal given by

u(t) = ei2π(10t+10t2), 0 ≤ t ≤ 1. (15)

u(t) is a chirp considered in [24]. Here we uniformly sample u(t) with 128 sample points. In Fig.6,

with ϕ(t) = 9.6982t2 +10.5970t, ϕ′(t) = 19.3964t+10.5970, which can be estimated from CWT of

u(t), the IFE-CWT of u(t)with ξ0 = 20 is shown. Observe that the IF of u(t) is 10+20t, 0 < t < 1,

a line segment. The IFE-CWT of u(t) displayed in Fig.6 is indeed a zone concentrating a line

segment, while the CWT does not give a clear picture for the IF of u(t). The picture of IFE-CWT

of u(t) with a different ξ0 is similar to that with ξ0 = 20. In the following experiments, we simply

set ξ0 = 0.

Next, we show that x(t) can be recovered back from its IFE-CWT.

10

Figure 6: CWT (Left picture) and IFE-CWT (Right picture) with ξ0 = 20, ϕ(t) = 9.6982t2 +

10.5970t, ϕ′(t) = 19.3964t+ 10.5970 of u(t) = ei2π(10t+10t2), 0 ≤ t ≤ 1

Theorem 2. Let x(t) be a function in L2(R). Then

x(b) =1

cψexp(−i2πξ0b)

∫ ∞−∞

W IFEx (a, b)

da

|a|, (16)

where cψ is defined by (2).

When x(t) satisfies certain condition, x(t) can be recovered from its IFE-CWT with the scale

variable a restricted to a > 0.

Theorem 3. Let x(t) be a function in L2(R). Suppose there is ϕ with ϕ′(t) > 0 such that

y(t) defined by y(t) = x(t) exp(−i2πϕ(t)) satisfies y(ξ) = 0, ξ ≤ A for some constant A. Let

W IFEx (a, b) be the IFE-CWT of x(t) defined by (12) with ϕ(t) and ξ0 > −A. Then

x(b) =1

cψexp(−i2πξ0b)

∫ ∞0

W IFEx (a, b)

da

a, (17)

where cψ is defined by (2).

The proof of Theorem 2 is similar to that of Theorem 3. Here we give the proof of Theorem

3.

Proof of Theorem 3 Let x(t) be the function defined by (14). Observe that x(t) = y(t)ei2πξ0t.

Thus x(ξ) = y(ξ − ξ0). Hence, by (13) in Proposition 1, we have∫ ∞0

W IFEx (a, b)

da

a=

∫ ∞0

ei2πϕ(b)∫ ∞−∞

x(ξ)ψ(a(ξ + ϕ′(b)

))ei2πbξdξ

da

a

=

∫ ∞0

ei2πϕ(b)∫ ∞−∞

y(ξ − ξ0)ψ(a(ξ + ϕ′(b)

))ei2πbξdξ

da

a

= ei2πϕ(b)∫ ∞0

∫ ∞−∞

y(ξ)ψ(a(ξ + ξ0 + ϕ′(b)

))ei2πb(ξ+ξ0)dξ

da

a

= ei2πϕ(b)+i2πξ0b∫ ∞−∞

y(ξ)

∫ ∞0

ψ(a(ξ + ξ0 + ϕ′(b)

))daaei2πbξdξ

11

= ei2πϕ(b)+i2πξ0b∫ ∞A

y(ξ)

∫ ∞0

ψ(a(ξ + ξ0 + ϕ′(b)

))daaei2πbξdξ

= ei2πϕ(b)+i2πξ0b∫ ∞A

y(ξ)

∫ ∞0

ψ(a)da

aei2πbξdξ

(since ξ + ξ0 + ϕ′(b) > 0 when ξ > A)

= cψei2πϕ(b)+i2πξ0b

∫ ∞A

y(ξ)ei2πbξdξ = cψei2πϕ(b)+i2πξ0b

∫ ∞−∞

y(ξ)ei2πbξdξ

= cψei2πϕ(b)+i2πξ0by(b) = cψe

i2πξ0bx(b).

This shows (17).

Remark 1. If the condition y(ξ) = 0, ξ ≤ A is not satisfied, then for large ξ0, we have

x(b) ≈ 1

cψexp(−i2πξ0b)

∫ ∞0

W IFEx (a, b)

da

a. (18)

This can be obtained as follows. Following the proof of Theorem 3 and noting that

y(ξ)ψ(a(ξ + ξ0 + ϕ′(b)

))= 0 if ξ + ξ0 + ϕ′(b) ≤ 0,

we have ∫ ∞0

W IFEx (a, b)

da

a= ei2πϕ(b)+i2πξ0b

∫ ∞−∞

y(ξ)

∫ ∞0

ψ(a(ξ + ξ0 + ϕ′(b)

))daaei2πbξdξ

= ei2πϕ(b)+i2πξ0b∫ ∞−ξ0−ϕ′(b)

y(ξ)

∫ ∞0

ψ(a(ξ + ξ0 + ϕ′(b)

))daaei2πbξdξ

= ei2πϕ(b)+i2πξ0b∫ ∞−ξ0−ϕ′(b)

y(ξ)

∫ ∞0

ψ(a)da

aei2πbξdξ

= cψei2πϕ(b)+i2πξ0b

∫ ∞−ξ0−ϕ′(b)

y(ξ)ei2πbξdξ

= cψei2πϕ(b)+i2πξ0b

(∫ ∞−∞

y(ξ)ei2πbξdξ −∫ −ξ0−ϕ′(b)

−∞y(ξ)ei2πbξdξ

)= cψe

i2πϕ(b)+i2πξ0by(b)− cψei2πϕ(b)+i2πξ0b∫ −ξ0−ϕ′(b)

−∞y(ξ)ei2πbξdξ

≈ cψei2πξ0bx(b).

Thus (18) holds with error |cψ|∫ −ξ0−ϕ′(b)

−∞ |y(ξ)|dξ.

For x(t), at (a, b) for which W IFEx (a, b) 6= 0, we need to define the reference IF function

ωIFEx (a, b). Following the definition of ωx(a, b),

∂∂bW

IFEx (a, b)

2πiW IFEx (a, b)

(19)

may be a good candidate for the reference IF function. First we look at ∂∂bW

IFEx (a, b). From (13),

12

we have

∂

∂bW IFEx (a, b) = i2πϕ′(b)ei2πϕ(b)

∫ ∞−∞

x(ξ)ψ(aξ + aϕ′(b)

)ei2πbξdξ

+ei2πϕ(b)

∫ ∞−∞

x(ξ)ψ(aξ + aϕ′(b)

)ei2πbξi2πξdξ

+ei2πϕ(b)

∫ ∞−∞

x(ξ)ψ′(aξ + aϕ′(b)

)aϕ′′(a)ei2πbξdξ

=: I1 + I2 + I3. (20)

where x is given by (14). Clearly, I1 = i2πϕ′(b)W IFEx (a, b).

Consider the case x(t) = Cei2πφ(t) again. As shown above, with ϕ(t) = φ(t), we have x(t) =

C exp(i2πξ0), and hence x(ξ) = Cδ(ξ0), and

W IFEx (a, b) = Cei2πϕ(b)ψ

(aξ0 + aφ′(b)

)ei2πbξ0 .

Note that in this case

I1 + I2 = i2πφ′(b)W IFEx (a, b) + Cei2πφ(b)ψ

(aξ0 + aφ′(b)

)ei2πbξ0i2πξ0

= i2πφ′(b)W IFEx (a, b) + i2πξ0W

IFEx (a, b)

= i2π(φ′(b) + ξ0

)W IFEx (a, b).

Therefore,I1 + I2

2πiW IFEx (a, b)

= φ′(b) + ξ0

is the IF of x(t) plus the target frequency ξ0. Hence, for a general x(t), we define the reference

IF function ωIFEx (a, b) of the IFE-CWT of x(t) to be

ωIFEx (a, b) :=

I1 + I2

2πiW IFEx (a, b)

= ϕ′(b) +I2

2πiW IFEx (a, b)

, (21)

where I1 and I2 are defined by (20). Observe that the reference IF function ωIFEx (a, b), defined by

(21), is not the quantity defined by (19). Instead, it is

∂∂bW

IFEx (a, b)− I3

2πiW IFEx (a, b)

.

Clearly, ωIFEx (a, b) depends on ϕ.

Definition 2. The instantaneous frequency-embedded wavelet synchrosqueezing transform (IFE-

SST) of a signal x(t) with ϕ and ξ0 is defined by

T IFEx (ξ, b) =

∫a:W IFE

x (a,b)6=0W IFEx (a, b)δ

(ωIFEx (a, b)− ξ

)daa,

where W IFEx is IFE-CWT of x(t) defined by (12) and ωIFE

x (a, b) is defined by (21).

13

By Theorem 3, we know the input signal x(t) can be recovered from its IFE-SST as shown in

the following theorem. The author refers to Theorem 3.3 in [7] for the recovery of components of

a signal from the orginal SST.

Theorem 4. Let x(t) be a function in Theorem 3 with A = 0. Then

x(b) =1

cψexp(−i2πξ0b)

∫ ∞0

T IFEx (ξ, b)dξ. (22)

In practice, a, b, ξ are discretized. Suppose aj , bn, ξk, j, n, k = 1, · · · , are the sampling points

of a, b, ξ respectively. Again, we assume ξk+1 − ξk = ∆ξ for all k. Then the IFE-SST of x(t) is

given by

T IFEx (ξk, bn) =

∑j: |ωIFE

x (aj ,bn)−ξk|≤∆ξ/2, |W IFEx (aj ,bn)|>γ

W IFEx (aj , bn)a−1

j (∆a)j ,

where (∆a)j = aj+1 − aj , and γ > 0 is parameter to set the condition |Wx(a, b)| 6= 0. The

recovering formula (22) for x(t) implies

x(bn) =1

cψexp(−i2πξ0bn)

∑k

T IFEx (ξk, bn), n = 1, 2, · · · . (23)

We will discuss more about the discretization and the implementation of IFE-SST in the next

section.

Another issue we need to consider about IFE-CWT and IFE-SST is that, in practice, to

estimate IFs of x(t) from its IFE-SST, we need ϕ(t) and ϕ′(t) which should be close to φ(t) and

φ′(t) respectively. We will first use (regular) CWT/STFT to have a rough estimate of φ, φ′ to be

used as the input ϕ,ϕ′ for IFE-CWT and IFE-SST. Then we use IFE-CWT or IFE-SST to get

more accurate estimate of φ, φ′. See the next section for more details.

4 Implementation

For the implementation of the IFE-SST, one may modify the procedures in [16]. Suppose x(t) is

discretized uniformly at points

tn = t0 + n∆t, n = 0, 1, · · · , N − 1.

Let bn = n∆t, n = 0, 1, · · · , N − 1. Let x ∈ CN denote the discretization of x in (14):

x =[x0, x1, · · · , xN−1

]T,

where T denotes the transpose of a vector/matrix, and

xn = x(tn) = x(tn)e−i2πϕ(tn)+i2πξ0tn .

14

Let ∆η = 1N∆t and ηk =

k∆η, for 0 ≤ k ≤ [N2 ],

(k −N)∆η, for [N2 ] + 1 ≤ k ≤ N − 1,be the sampling points

for the frequency variable η. Let xk = x(ηk), 0 ≤ k ≤ N − 1 denote the discretization of the

Fourier transform x of x. One may obtain x =[x0, x1, · · · , xN−1

]Tby applying the FFT to x:x = ∆t FFTx.

The scale variable can be discretized as

aj = νj∆t, νj = 2j/nν , j = 1, 2, · · · , nν([

log2N]− 1),

where nν is a parameter which user can choose. One may choose nν = 32 or nν = 64 as suggested

in [16]. Then we have∫ ∞−∞

x(η)ψ(aη + aϕ′(b)

)ei2πbηdη ≈

N−1∑k=0

∆ηx(ηk)ψ(ajηk + ajϕ′(bn)

)ei2πbnηk

=1

N

N−1∑k=0

(FFTx)(k)ψ(ajηk + ajϕ′(bn)

)ei2πbnηk .

Thus the IFE-CWT of x(t) can be discretized as

W IFEx (aj , bn) = ei2πϕ(bn) 1

N

N−1∑k=0

(FFTx)(k)ψ(ajηk + ajϕ′(bn)

)ei2πbnηk . (24)

Similarly, the integral I2 in (20) can be discretized as

I2(aj , bn) = i2πei2πϕ(bn) 1

N

N−1∑k=0

(FFTx)(k)ψ(ajηk + ajϕ′(bn)

)ei2πbnηkηk.

Therefore, ωIFEx (a, b) defined by (21) can be approximated by, for W IFE

x (aj , bn) 6= 0,

ωIFEx (aj , bn) = ϕ′(bn) +

∑N−1k=0 (FFTx)(k)ψ

(ajηk + ajϕ′(bn)

)ei2πbnηkηk∑N−1

k=0 (FFTx)(k)ψ(ajηk + ajϕ′(bn)

)ei2πbnηk

. (25)

The frequency variable ξ > 0 of the IFE-SST can be discretized as follows. Let ∆ξ be the

frequency resolution parameter (one may set ∆ξ = 1nν

12(log2N−1)∆t). Partition the time-frequency

region (ξ, b) : 0 < ξ ≤ 12∆t , b ≥ 0 into K0 :=

[1

2∆t∆ξ

]nonoverlap zones:

Ωk :=

(ξ, b) : ξk −∆ξ

2< ξ ≤ ξk +

∆ξ

2, b ≥ 0

, k = 1, 2, · · · ,K0,

where ξk = k∆ξ. Let γ > 0 be parameter to set the condition |Wx(a, b)| 6= 0. One may choose γ

to be a number between 10−8 and 10−4. Then we obtain the IFE-SST of x(t):

T IFEx (ξk, bn) =

∑j: ξk−∆ξ

2<ωIFE

x (aj ,bn)≤ξk+ ∆ξ2, |W IFE

x (aj ,bn)|>γ

W IFEx (aj , bn)

log 2

nν, (26)

15

where we have used the fact a−1j (∆a)j ≈ log 2

nν. Finally, x(b) can be recovered by (23). In the

following we summarize our calculation of IFE-SST as Algorithm 1.

Algorithm 1. (Calculation of IFE-SST of monocomponent signal)

Input: x(tn), ϕ(tn), ϕ′(tn), 0 ≤ n ≤ N − 1, ξ0, and γ > 0.

Step 1. Calculate W IFEx (aj , bn) by (24).

Step 2. Calculate ωIFEx (aj , bn) by (25) for all j, n with |W IFE

x (aj , bn)| > γ.

Step 3. Calculate T IFEx (ξj , bn) by (26).

Next we give the procedures to estimate IF of a monocomponent signal with IFE-SST.

Algorithm 2. (IFE-SST-based IF estimation of monocomponent signal)

Input: x(tn), ξ0, γ > 0, and initial ϕ(tn), ϕ′(tn), 0 ≤ n ≤ N−1 estimated from CWT/SST

of x(tn).

Step 1. Calculate IFE-SST by Algorithm 1.

Step 2. Estimate IF φ′(tn) from IFE-SST and set φ(tn) =∑n

k=0 φ′(tk)∆t.

Step 3. Repeat Step1 (with φ(tn), φ′(tn) obtained in Step 2 as the initial ϕ(tn), ϕ′(tn)) and Step 2

till the error criterion is reached.

Figure 7: Left: SST of u(t) = ei2π(10t+10t2), 0 ≤ t ≤ 1; Middle: IFE-SST of u(t) with ξ0 = 0, ϕ(t) =

9.8305t2 + 10.3153t, ϕ′(t) = 19.6610t + 10.3153; Right: IFE-SST of u(t) with ξ0 = 0, ϕ(t) = 10.0061t2 +

9.9960t, ϕ′(t) = 20.0122t+ 9.9960

In Algorithms 1 and 2, we need initial estimate ϕ(tn), ϕ′(tn) for the phase function φ and IF

φ′. As mentioned above, we may use CWT or SST to obtain a rough estimation of them. For

16

a monocomponent signal x(tn), 0 ≤ n ≤ N − 1, one may give an estimation of its IF from its

CWT/SST as follows. Let

Mn = maxjωx(aj , bn), 0 ≤ n ≤ N − 1,

(Mn will be maxj|Wx(aj , bn)| if we use CWT for the estimation). Then a curve obtained by

approximating (bn,Mn), 0 ≤ n ≤ N − 1 gives the SST-based estimation of IF of x(tn). For the

monocomponent signal, one may simply use polynomial fitting least-squares polynomial fitting

to obtain the IF estimation. The IF estimation from IFE-SST in Step 2 of Algorithm 2 can be

carried out similarly. More precisely, denote,

M IFEn = max

jωIFE

x (aj , bn), 0 ≤ n ≤ N − 1.

Then a curve obtained by approximating (bn,MIFEn ), 0 ≤ n ≤ N − 1 gives the estimated IF of

x(tn) with IFE-SST.

Next let us look at u(t) given in (15) as an example about how to obtain its IF by IFE-SST.

Throughout this paper, we set γ = 10−5 and nν = 132 for SST and IFE-SST. From the SST of

u(t), which is shown in the left picture of Fig.7, we obtain Mn, 0 ≤ n ≤ 127. Then we obtain

linear least square approximation with (bn,Mn), 4 ≤ n ≤ 123:

ϕ′(t) = 19.3426t+ 10.7704. (27)

One could use higher order polynomial least square approximation. Here we use the linear least

square approximation for the purpose to compare the estimation to the true IF of u(t):

φ′(t) = 20t+ 10.

Also we consider n from 4 to 123 to reduce the boundary effect. Then we use ϕ′(t) in (27) and its

integral as the initial input IF and phase function to calculate IFE-SST of u(t). From the IFE-SST,

we then obtain M IFEn and IF estimation, denoted by φ′(t), by linear least square approximation

with (bn,MIFEn ), 4 ≤ n ≤ 123. We continue this procedure iteratively as in Algorithm 2. In

Table 1, we list the estimated φ′(t). Observe that the best estimation we can get is φ′(t) =

20.0122t+ 9.9960 which is quite close to the true IF φ′(t).

IFE-SST of u(t) with ϕ′(t) = 19.6610t+ 10.3153 and that with ϕ′(t) = 20.0122t+ 9.9960 are

shown in Fig.7. Note that IFE-SST of u(t) with ϕ′(t) = 19.6610t+10.3153 already gives a sharper

representation of IF than SST. Here we also provide the differences |Mn−φ′(tn)| and |M IFEn −φ′(tn)|

to show the performance of SST and IFE-SST, where φ′(t) is true IF of u(t). The comparison

between |Mn−φ′(tn)|, n = 5, · · · , 124 and |M IFEn −φ′(tn)| with ξ0 = 0, ϕ′(t) = 19.6610t+ 10.3153

and that with ξ0 = 0, ϕ′(t) = 20.0122t+ 9.9960 are shown in the left and middle pictures of Fig.8.

17

Iteration φ′(t)

1 19.6610t+ 10.3153

2 19.8390t+ 10.1597

3 19.9180t+ 10.0764

4 19.9648t+ 10.0322

5 19.9890t+ 10.0117

6 19.9997t+ 10.0043

7 20.0122t+ 9.9960

8 20.0122t+ 9.9960

Table 1: Estimated IF of u(t) = ei2π(10t+10t2) by IFE-SST with Algorithm 2

Figure 8: Left: IF estimation errors with SST and with IFE-SST for u(t) = ei2π(10t+10t2), 0 ≤ t ≤ 1

with ξ0 = 0, ϕ′(t) = 19.6610t + 10.3153; Middle: IF estimation errors with SST and with IFE-SST for

u(t) = ei2π(10t+10t2), 0 ≤ t ≤ 1 with ξ0 = 0, ϕ′(t) = 20.0122t + 9.9960; Right: IF estimation errors with

SST and with IFE-SST for x(t) = ei2π(10t+100t2), 0 ≤ t ≤ 1

Our IFE-SST-based IF estimation works well with chirps of high frequency. As an example,

we show in Fig.8, IF estimation errors |Mn − φ′(tn)| and |M IFEn − φ′(tn)| for n = 13, · · · , 500

for x(t) = ei2π(10t+100t2), 0 ≤ t ≤ 1, which is uniformly sampled with 512 sample points, where

φ′(t) = 10 + 200t is the true IF.

5 IFE-SST based signal separation

We will apply the IFE-SST for signal separation. We consider the adaptive harmonic model

(AHM) after the trend removal process:

x(t) =K∑k=1

xk(t) + ε(t), xk(t) = Ak(t) cos(2πφk(t)

),

18

where ε(t) is the noise. We may separate the components of x(t) as follows. Use CWT/STFT to

identify the highest frequency component, say x1(t), and estimate initial φ′1(t) and φ1(t). Then

we use Algorithm 2 to have accurate estimate of φ′1(t), φ1(t) and recover x1(t). After that we

remove x1(t) from x(t) and repeat the same procedures to the new signal to recover the component

of the 2nd highest frequency and other components. Our method is different from that in [13],

where optimization method is used to reconstruct the components of a multicomponent signal

simultaneously.

Next we modify the definition of IFE-CWT and IFE-SST for the purpose to estimate the

IF of a particular component of a multicomponent signal. Consider the case xk(t) = Akei2πφk(t)

and ξ0 = 0. We assume the IFs of different xk(t) lie nonoverlap different zones in the time-

frequency plane. Suppose we want to estimate the IF of the `th component x`(t). We should

choose ϕ′(t) close to φ′`(t). Assume it happens that ϕ(t) = φ`(t). Then xk(t) = xk(t)e−i2πϕ(t) =

Akei2π(φk(t)−φ`(t)

). Thus x`(ξ) = A` δ(ξ), and for k 6= `, the IF of xk(t) lies in a zone away from

the line ξ = 0, and we have xk(ξ) ≈ 0 for |ξ| ≤ ε, where ε is a small positive number. Therefore,

if we define the modifies IFE-CWT W IFEx (a, b) by

W IFEx (a, b) := ei2πφ`(b)

∫|ξ|≤ε

x(ξ)ψ(a(ξ + φ′`(b)

))ei2πbξdξ,

then

W IFEx (a, b) = ei2πφ`(b)

K∑k=1

∫|ξ|≤ε

xk(ξ)ψ(a(ξ + φ′`(b)))ei2πbξdξ

≈ ei2πφ`(b)∫|ξ|≤ε

A`δ(ξ)ψ(a(ξ + φ′`(b)

))ei2πbξdξ

= A`ei2πφ`(b)ψ

(aφ′`(b)

).

Hence, W IFEx (a, b) concentrates along aφ′`(b) = 1, the IF of x`(t) in the time-scale plane.

Numerically, we consider

W IFEx (aj , bn) =

ei2πϕ(bn)

N

( U∑k=0

+N−1∑

k=N−L

)(FFTx)(k)ψ

(ajηk + ajϕ′(bn)

)ei2πbnηk , (28)

and

ωIFEx (aj , bn) = ϕ′(bn) +

(∑Uk=0 +

∑N−1k=N−L

)(FFTx)(k)ψ

(ajηk + ajϕ′(bn)

)ei2πbnηkηk(∑U

k=0 +∑N−1

k=N−L

)(FFTx)(k)ψ

(ajηk + ajϕ′(bn)

)ei2πbnηk

(29)

instead of W IFEx (aj , bn) and ωIFE

x (aj , bn) defined by (24) and (25) resp. Then we define the

modified IFE-SST of x(t):

T IFEx (ξk, bn) =

∑j: ξk−∆ξ

2<ωIFE

x (aj ,bn)≤ξk+ ∆ξ2, |W IFE

x (aj ,bn)|>γ

W IFEx (aj , bn)

log 2

nν. (30)

19

Here U and L are some nonnegative integers, which are not large. In addition, if φ′` to be estimated

is the IF of the component with the highest frequency, one may choose L = 0, while if φ′` is the

IF of the component with the lowest frequency, one may choose U = 0.

Figure 9: Right: IFE-SST of z(t) with Algorithm 3 and U = L = 20; Middle: v(t) = ei2π(10t+10t2) +

ei2π(9t+5t2), 0 ≤ t ≤ 1 (real part); Right: SST of v(t)

In the following, we describe the procedures to estimate the IF of a particular component and

separate components of multicomponent signals.

Algorithm 3. (IFE-SST-based IF estimation of `th component of multicomponent

signal)

Input: x(tn), γ > 0, ξ0, and initial ϕ`(tn), ϕ′`(tn) for `th component estimated from

CWT/SST of x(tn). Choose integers U,L ≥ 0

Step 1. Calculate the modified IFE-SST T IFEx,` (ξj , bn) of `th component by (30).

Step 2. Estimate IF φ′`(tn) from IFE-SST T IFEx,` (ξj , bn) and set φ`(tn) =

∑nk=0 φ

′`(tk)∆t.

Step 3. Repeat Step 1 (with φ`(tn), φ′`(tn) obtained in Step 2 as the initial ϕ`(tn), ϕ′`(tn)) and Step

2 till the error criterion is reached.

Algorithm 4. (IFE-SST based signal separation)

Step 1. Use CWT/STFT/SST to identify the number K of frequency components. Choose a (tar-

geted) frequency component, say x`(t). Use CWT/STFT/SST to obtain initial estimation

φ′`, φ` of φ′`, φ`.

Step 2. With ϕ′ = φ′`, ϕ = φ`, use Algorithm 3 to have accurate estimation of φ′`(t), φ`(t) and

obtain x`(t), recovered x`(t).

Step 3. Remove x`(t) from x(t) and repeat Steps 1-2 to recover the second targeted component.

Step 4. Do Step 3 for other components.

20

Step 5. If time permits, set y`(t) = x(t) −∑

0≤k≤K,k 6=` xk(t). Apply Algorithm 3 to y`(t) to have

more accurate estimation of φ′`(t), φ`(t) and x`(t); and do the same to other components.

Figure 10: Left: IFE-SST of estimated 1st component with estimated IF: φ′1(t) = 19.9813t+ 10.0093, 0 ≤t ≤ 1; Right: IFE-SST of estimated 2nd component with estimated IF: φ′2(t) = 10.0337t+8.9831, 0 ≤ t ≤ 1

Before we consider multicomponent signals, we remark that we can also use Algorithm 3 to

estimate the IF of a monocomponent signal. For example, for the signal z(t) with significantly

changing frequency given by (10), we show in Fig.9 its IF estimation obtained by Algorithm 3

with U = L = 20.

Next we consider a signal consisting of two chirps:

v(t) = v1(t) + v2(t), v1(t) = ei2π(10t+10t2), v2(t) = ei2π(9t+5t2), 0 ≤ t ≤ 1. (31)

v(t) is uniformly sampled with 128 sample points. The real part of this signal v(t) is shown in the

middle picture of Fig.9 and the SST of v(t) is shown in the right picture of Fig.9. Fig.10 shows the

IFE-SSTs of the estimated 1st and 2nd components by Algorithm 4. From the FE-SSTs, we can

recover v1 and v2 by recovering formula (22) in Theorem 4. The error between v1 and recovered

v1 and that between v2 and recovered v2 are shown in Fig.11.

Figure 11: Left: Error between v1(t) = ei2π(10t+10t2), 0 ≤ t ≤ 1 and recovered v1(t); Right: Error between

v2(t) = ei2π(9t+5t2), 0 ≤ t ≤ 1 and recovered v2(t)

21

Figure 12: Left: v(t) and noised v(t) with noise (10dB) (real part); Right: SST of noised v(t)

We also consider signal separation in noisy environment. The signal to noise ratio (SNR) of a

noised signal x = x+ ε with noise ε is defined by

SNR = 20 log‖x− x‖2‖ε‖2

(dB),

where x is the mean of x.

We consider the case v(t) = v(t) + ε(t) with v(t) defined by (31) and ε(t) a Gaussian white

noise with 10dB. v(t) is shown in Fig.12, where its SST is also provided. With Algorithm 4 (we

choose L = 10, U = 0 for v2(t), the component with lower frequency, and choose L = 0, U = 10

for v1(t), the component with higher frequency, and we also do Step 5), estimated IFE-SSTs of

v1(t) and v2(t) in the noise environment are shown in the top row of Fig.13. We also provide the

recovered v1(t) and v2(t) in Fig.13.

Next example considered is w(t) = w1(t) + w2(t) with

w1(t) = log(2 + t/2) cos(2π(5t+ 0.1t2)

), w2(t) = exp(−0.1t) cos

(2π(4t+ 0.5 cos t)

), (32)

for 0 ≤ t ≤ 8. We sample w(t) uniformly with 1024 sample points. We discuss the IF estimation

of w(t) in noisy environment. Let w(t) = w(t) + ε(t), where ε(t) a Gaussian white noise with

10dB. w(t) and its SST are shown in Fig.14. With Algorithm 4, the estimated IFE-SSTs of w1(t)

and w2(t) in the noise environment are shown in the top row of Fig.15, and the recovered w1(t)

and w2(t) are shown in the bottom row of Fig.15.

From the SST and IFE-SST provided in Figs.13 and 15, we see IFE-SST gives a better IF

representation of a signal than SST. These examples also show that our IFE-SST works well in

the noise environment.

Our last example is to use IFE-SST to separate the components of a bat echolocation signal.

Fig.16 shows an echolocation pulse emitted by the Large Brown Bat (Eptesicus Fuscus). The

data can be downloaded from the website of DSP at Rich University:

http://dsp.rice.edu/software/bat-echolocation-chirp. There are 400 samples; the sam-

pling step is 7 microseconds. The IF representation of this bat signal has studied in [23] and

22

Figure 13: Top row: IFs of v1(t) and v2(t) recovered by Algorithm 4; Bottom row: Recovered v1(t) and

v2(t) in noise environment by Algorithm 4

[24] by matching-modulation-transform-SST (MDT-SST) and the second-order SST respectively.

Here we remark that since the IFE-SST is based on CWT, we only compare our method with

CWT-based SST in this paper, not with the SST based on STFT, including MDT-SST and the

second order SST. For the bat signal, we just show that using Algorithm 4, we can get the IF

estimation and separate the components. The SST of the bat signal is shown in Fig.16, where

Morlet’s wavelet with ψ(ξ) = e−20π2(1−ξ)2χ(0,∞)(ξ) is used. The IFE-SSTs of the four main

components obtained by Algorithm 4 are shown in Fig. 17, and the recovered components are

provided in Fig.18.

6 Conclusion

In this paper we introduce the instantaneous frequency-embedded continuous wavelet transform

(IFE-CWT). We establish that the original signal can be reconstructed from its IFT-CWT. Then

based on IFE-CWT, we introduce the instantaneous frequency-embedded synchrosqueezing trans-

form (IFE-SST). IFE-SST can preserve the IF of monocomponent signal. For each component of

a multicomponent signal, IFE-SST uses a reference IF function associated with that component.

Our numerical experiments show that IFE-SST has a better performance than the CWT-based

SST in the separation of multiple components of non-stationary signals. The experimental results

also show that IFE-SST works well in the noise environment.

23

Figure 14: Left: w(t) and noised w(t) with noise (10dB); Right: SST of noised w(t)

Figure 15: Top row: IFs of w1(t) and w2(t) recovered by Algorithm 4; Bottom row: w1(t) and w2(t)

recovered in noise environment by Algorithm 4

ACKNOWLEDGMENT OF SUPPORT AND DISCLAIMER: (a) Contractor acknowledges

Government’s support in the publication of this paper. This material is based upon work funded

by AFRL, under AFRL Contract No. FA8750-15-3-6000 and FA8750-15-3-6003. (b) Any opinions,

findings and conclusions or recommendations expressed in this material are those of the author(s)

and do not necessarily reflect the views of AFRL.

The authors thank the anonymous reviewers for their valuable comments. The authors wish

to thank Curtis Condon, Ken White, and Al Feng of the Beckman Institute of the University of

Illinois for the bat data in Fig.16 and for permission to use it in this paper.

24

Figure 16: Left: Bat echolocation chirp; Right: SST of bat signal

Figure 17: IFE-SSTs of four main components of bat signal

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