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Amplitudes of mono-component signals and the generalized sampling functions
Signal Processing 94 (2014) 255–263
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Amplitudes of mono-component signals and the generalized sampling functions Qiuhui Chen a,1, Luoqing Li b,2, Yi Wang c,n a b c
Cisco School of Informatics, Guangdong University of Foreign Studies, Guangzhou, China Faculty of Mathematics and Computer Science, Hubei University, Wuhan 430062, China Department of Mathematics, Auburn University at Montgomery, P.O. Box 244023, Montgomery, AL 36124-4023, USA
a r t i c l e i n f o
abstract
Article history: Received 20 December 2012 Received in revised form 25 June 2013 Accepted 26 June 2013 Available online 6 July 2013
There is a recent trend to use mono-components to represent nonlinear and non-stationary signals rather than the usual Fourier basis with linear phase, such as the intrinsic mode functions used in Norden Huang's empirical mode decomposition [12]. A mono-component is a real-valued signal of finite energy that has non-negative instantaneous frequencies, which may be defined as the derivative of the phase function of the given real-valued signal through the approach of canonical amplitude-phase modulation. We study in this paper how the amplitude is determined by its phase for a class of signals, of which the instantaneous frequency is periodic and described by the Poisson kernel. Our finding is that such an amplitude can be perfectly represented by a sampling formula using the so-called generalized sampling functions that are related to the phase. The regularity of such an amplitude is identified to be at least continuous. Such characterization of mono-components provides the theory to adaptively decompose non-stationary signals. Meanwhile, we also make a very interesting and new characterization of the band-limited functions. & 2013 Elsevier B.V. All rights reserved.
Keywords: Mono-component Generalized sampling function Analytic signal Nonlinear phase Amplitude-phase modulation Hilbert transform Blaschke product Poisson kernel
1. Introduction Any real-valued non-stationary signal f of finite energy, that is, f is in the space L2 ðRÞ of square integrable functions on the set R of real numbers, may be represented as an amplitude-phase modulation with a time-varying amplitude ρ and a time-varying phase ϕ where phase ϕ is, in general, nonlinear [14]. Specifically, the value of f at t∈R may be represented as f ðtÞ ¼ ρðtÞ cos ϕðtÞ:
ð1:1Þ
Unfortunately, this type of representation is not unique because the modulation is obtained through a complex signal that can have various choices of the imaginary part.
However, one can determine a unique such factorization (1.1) by using the approach of analytic signals [10]. Indeed, let Aðf Þ be the analytic signal associated with f with the characteristic property ( 2f^ ðωÞ if ω≥0 ∧ ðAðf ÞÞ ðωÞ ¼ ð1:2Þ 0 if ω o0; where for any signal g∈L2 ðRÞ, g^ ¼ F g is the Fourier transform of g defined at ξ∈R by the equation Z 1 g^ ðξÞ ¼ ðF g ÞðξÞ≔ pffiffiffiffiffiffi g ðt Þeiξt dt: ð1:3Þ 2π R Eq. (1.2) is equivalent to for t∈R, Aðf ÞðtÞ ¼ f ðtÞ þ iHf ðtÞ;
n
Corresponding author. Tel.: +1 3342443318. E-mail addresses: [email protected] (Q. Chen), [email protected] (L. Li), [email protected] (Y. Wang). 1 The author is partially supported by the Natural Science Foundation of Guangdong Province under Grant no. S2011010004986. 2 The author is partially supported by the National Natural Science Foundation of China under Grant no. 11071058. 0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.06.034
where the operator H : L2 ðRÞ-L2 ðRÞ stands for the Hilbert transform, and for f ∈L2 ðRÞ, Hf at t∈R is defined through the principal value integral Z Z 1 f ðxÞ f ðxÞ dx ¼ lim dx: Hf ðtÞ≔p:v: π R tx ϵ-0 jxtj 4 ϵ tx
256
Q. Chen et al. / Signal Processing 94 (2014) 255–263
The value Aðf ÞðtÞ at t∈R is complex which can be written into the quadrature form Aðf ÞðtÞ ¼ ρðtÞeiϕðtÞ : Under the conditions that the derivative value ϕ′ðtÞ is nonnegative for all t∈R, the quantities ρðtÞ and ϕ′ ðtÞ are called the instantaneous amplitude and instantaneous frequency at t∈R, of the real signal f, respectively. The corresponding modulation (1.1) is then called the canonical amplitude-phase modulation, or canonical modulation for short [8]. The signal f with such defined non-negative instantaneous frequencies is thus called a mono-component. A large body of literature addresses this problem, see for example, [1,2,16,17,9,21]. The notions of instantaneous amplitude and frequency are fundamental in many applications involving modulations of signals that appear especially in communications or information processing. Thus constructing the canonical pair ðρ; ϕÞ of the instantaneous amplitude and phase is important in the theories of analytic signals and in order to facilitate the modulation and demodulation techniques such as processing speech signals [15] or signals in electrical and radio engineering [11]. It is equivalent to the problem of seeking the function pair ðρ; ϕÞ such that for t∈R, the following equation holds true: HðρðÞ cos ϕðÞÞðtÞ ¼ ρðtÞ sin ϕðtÞ:
ð1:4Þ
We remark that (1.4) can be apparently considered as a special case of the Bedrosian identity
We shall in this paper characterize the amplitude function ρ when the phase function ϕ is chosen at t∈R by Z ϕðtÞ ¼ θa ðtÞ ¼ pa ðxÞ dx ½0;t
such that Eq. (1.4) is satisfied. Our main result indicates that such kind of amplitude can be perfectly reconstructed in terms of a sampling formula using the generalized sampling function whose value at t∈R is given by sinca ðt Þ≔
sin θa ðtÞ : t
ð1:8Þ
In Section 2, we review the construction of the generalized sampling function and discuss some properties pertaining to it. In Section 3, we introduce the concept of Bedrosian subspace of the Hilbert transform and investigate some properties of functions in this space. In Section 4, we make an important observation when a linear phase is chosen, the amplitude function must be bandlimited in order to satisfy equation (1.4). In Section 5, we present our main result in Theorem 5.7.
2. Generalized sampling functions Not very surprisingly the function sinca has many properties that are similar to the classic sinc defined at t∈R by the equation sin t : t
HðfgÞ ¼ f HðgÞ:
sincðt Þ≔
In [1], the author proved that, if both f ; g belong to L2 ðRÞ, f is of lower frequency, g is of higher frequency and f ; g have no overlapping frequency, then HðfgÞ ¼ f HðgÞ. This classic result of Bedrosian is not useful for constructing a mono-component. The reason lies in that the requirement of both f and g in L2 ðRÞ is invalid. Recently, an important phase function that renders monocomponents was given in [18]. The phase function is defined through the boundary values of a Blaschke product on a unit disk Δ≔fz : z∈C; jzj≤1g, where C indicates the set of complex numbers. Specifically, for a∈ð1; 1Þ, the Blaschke product at z∈C\f1=ag is given by
ð1:7Þ
Those properties include cardinality, orthogonality, decaying rate, among others. In the special case a ¼0, the function sinca reduces to the classic sinc, which will become clear later. Let us first review the approach to obtain an explicit form of sinca . The classic sinc function is fundamentally significant in digital signal processing due to the Shannon sampling theorem [19,20,3]. The Shannon sampling theorem enables to reconstruct a bandlimited signal from shifts of sinc functions weighted by the uniformly spaced samples of that signal. Recently efforts have been made to extend the classic sinc to generalized sampling functions, for example, in [5–7]. Intuitively, the spectrum of the sinc function is just the indicator function of a symmetric interval of finite measure. Hence, authors in [5] are inspired to consider functions with piecewise polynomial spectra to replace the usual sinc function for the purpose of sampling non-bandlimited signals. One kind of generalized sampling functions given in [5], denoted by sinca that is related to a constant a∈ð1; 1Þ, is defined as the inverse Fourier transform of a so-called symmetric cascade filter, denoted by H a . Specifically, rffiffiffi π ð1 þ aÞF 1 H a : sinca ≔ ð2:1Þ 2
then by taking the derivative of both sides of Eq. (1.6), we find that the phase θa is an anti-derivative of pa, and its derivative is always positive, that is,
Let N be the set of natural numbers, Z be the set of integers, and Zþ ≔f0g∪N. Let X be a subset of R, and for q∈N, we say a function f is in Lq ðXÞ if and only if the Lq ðXÞ norm
d θa ðt Þ ¼ pa ðt Þ 40: dt
∥f ∥q;X ≔
Ba ðzÞ ¼
za : 1az
ð1:5Þ
Subsequently the non-linear phase function, denoted by θa , is defined at t∈R by the equation eiθa ðtÞ ≔Ba ðeit Þ:
ð1:6Þ
If we recall that the periodic Poisson kernel pa whose value at t∈R is given by pa ðt Þ≔
1a2 ; 12a cos t þ a2
Z
1=q jf ðtÞjq dt X
o1:
Q. Chen et al. / Signal Processing 94 (2014) 255–263
Similarly, let Z be a subset of Z, a sequence y≔ðyk : k∈ZÞ is q q said to be in l ðZÞ if and only if the l ðZÞ norm !1=q ∥y∥q;Z ≔ ∑ jyk jq
and ð1 þ a2 Þ cos t2a : ð2:7Þ 12a cos t þ a2 We next list some properties of the function sinca .
cos θa ðt Þ ¼
o1:
k∈Z
Proposition 2.1. Let the generalized sampling function sinca be defined by Eq. (2.4) or Eq. (1.8). Then the following statements hold:
The symmetric cascade filter Ha is a piecewise constant function whose value at ξ∈R is given by H a ðξÞ≔ ∑ an χ In ðξÞ;
ð2:2Þ
n∈Zþ
1. For t∈R,
where χ I is the indicator function of the set I, and the interval In, n∈Zþ , is the union of two symmetric intervals given by the equation
1
ð1a2 Þ sin t ¼ pa ðt Þ sincðt Þ: 12a cos t þ a2 t
ð2:8Þ
F sinca ¼
rffiffiffi π ð1 þ aÞH a : 2
ð2:9Þ
2
Of course, we have that H a ∈L ðRÞ∩L ðRÞ because the 1 2 sequence b≔ðan : n∈Zþ Þ∈l ðZþ Þ∩l ðZþ Þ, and hence sinca ∈L2 ðRÞ since the Fourier operator is closed in L2 ðRÞ and sinca is continuous because H a ∈L1 ðRÞ. The symmetric cascade filter H a can be associated with an analytic function F on the open unit disk Δ defined at z∈Δ by FðzÞ≔ ∑ an zn :
3. sinca ðnπÞ ¼ ðð1 þ aÞ=ð1aÞÞδn;0 , where δn;0 ¼ 1 if n ¼0 and δn;0 ¼ 0 if n∈Z\f0g. 4. sinca is even, bounded, infinitely differentiable. 5. jsinca ðtÞj≤ðð1 þ jajÞ=ð1jajÞÞ2=ð1 þ jtjÞ for t∈R, and sinca ∈L2 ðRÞ. 6. The set fsinca ðnπÞ : n∈Zg is an orthogonal set, that is
ð2:3Þ
n∈Zþ
〈sinca ; sinca ðnπ Þ〉 ¼
Thus, by substituting Eq. (2.2) into Eq. (2.1) and making use of Eq. (2.3) an alternative form of sinca ðtÞ, t∈R in terms of F can be found as t RefFðeit Þeð1=2Þit g; ð2:4Þ sinca ðt Þ ¼ ð1 þ aÞ sinc 2
Ba ðzÞ1 : z1
Proof. Eq. (2.8) directly follows by substituting Eq. (2.6) into Eq. (1.8). Eq. (2.9) is obtained by taking the inverse Fourier transform of both sides of Eq. (2.1). The third and fourth statements directly follow from Eq. (2.8). The fifth statement follows from Eq. (2.8) and noticing sincðtÞ≤ 2=ð1 þ jtjÞ and pa ðtÞ≤ð1 þ jajÞ=ð1jajÞ for any t∈R. The last statement is a special case of Corollary 3.2 of [5]. For the convenience of readers, we provide a direct proof here. By Parseval's theorem and Eq. (2.9) we have Z Z π sinca ðt Þ sinca ðtnπ Þ dt ¼ ð1 þ aÞ2 H 2a ðxÞeinπx dx 2 R R Z π 2 2k inπx ¼ ð1 þ aÞ ∑ a χ Ik ðxÞe dx 2 R k∈Zþ Z π 1þa πδn;0 ; ¼ ð1 þ aÞ2 ∑ a2k einπx dx ¼ 2 1a Ik k∈Zþ
ð2:5Þ
Plugging the formula (2.5) into Eq. (2.4) we readily obtain the explicit expression of sinca given earlier in Eq. (1.8). The next two formulas shall be used later. Expanding the left-hand side of Eq. (1.6) using Euler's formula and separating the real part from the imaginary part of the right-hand side, one obtains that for t∈R ð1a2 Þ sin t ¼ pa ðt Þ sin t 12a cos t þ a2
ð1 þ aÞ πδn;0 ; 1a
R where 〈u; v〉 ¼ R uðtÞvn ðtÞ dt denotes the usual inner product of the two functions u; v∈L2 ðRÞ, and vn is the complex conjugate of v.
where ReðzÞ is the real part of a complex number z. A very interesting fact, as discovered in the paper [5], is that the function F is linked to the Blashke Product Ba by the equation
sin θa ðt Þ ¼
sinca ðt Þ ¼ 2.
I n ≔ððn þ 1Þ; n∪½n; ðn þ 1ÞÞ:
ð1 þ aÞF ðzÞ≔
257
ð2:6Þ
3
2 1.8
sinca(t) sinc(t)
2.5
1.6
2
1.4 1.2
1.5
1 1
0.8 0.6
0.5
0.4
0
0.2 0 −20
−15
−10
−5
0
5
10
Fig. 1. Left: the graph of ð
15
20
−0.5 −20
−15
−10
−5
0
5
10
pffiffiffiffiffiffiffiffi π=2Þð1 þ aÞHa , a ¼0.5. Right: the graph of sinca , a¼ 0.5.
15
20
258
Q. Chen et al. / Signal Processing 94 (2014) 255–263
where, in the last equality we have used the orthogonality R identity Ik einπx dx ¼ 2δn;0 , k∈Zþ . The interchange of the integral operator and the infinite sum is guaranteed by the absolute convergence of the series. □ In Fig. 1, we show pffiffiffiffiffiffiffiffiffi ffi an example of the Fourier transform pair sinca and ðπ=2Þð1 þ aÞH a with a ¼0.5. In the plot of sinca , the graph of the standard sinc is also shown, which corresponds to the case a ¼0. Next we review several basic properties of the Hilbert transform which we will need frequently later. These properties can be found, for example, in the book [13]. First the Hilbert transform is an anti-involution, that is, H2 ¼ I
ð2:10Þ
where I is the identity operator. Second the operator H is anti-self adjoint, that is, 〈Hu; v〉 ¼ 〈u; Hv〉:
ð2:11Þ
Third, for any f ∈L2 ðRÞ and t∈R, the composition of the Fourier transform and the Hilbert transform is given by F ðHf ÞðtÞ ¼ i sgnðtÞF f ðtÞ;
ð2:12Þ
where sgnðÞ is the signum function having values defined by sgnðxÞ ¼ 1 if x∈Rþ ≔ft∈R : t 4 0g, sgnðxÞ ¼ 1 if x∈R ≔ ft∈R : t o0g, and sgnð0Þ ¼ 0. Theorem 2.2. The system Φ≔fsinca ð2kπÞ; H sinca ð2kπÞ : k∈Zg
ð2:13Þ
is an orthogonal system in L2 ðRÞ. Proof. By the third statement of Proposition 2.1, we have that 〈sinca ; sinca ð2kπ Þ〉 ¼
1þa δ : 1a 0k
〈H sinca ; H sinca ð2kπ Þ〉 ¼ 〈sinca ; H2 sinca ð2kπ Þ〉 1þa δ : ¼ 〈sinca ; sinca ð2kπ Þ〉 ¼ 1a 0k By Parseval's theorem and Eq. (2.9) we have
R
sinca ðt ÞH sinca ðt2kπ Þ dt ¼
ð1 þ aÞ2 πi 2
Z R
Invoking the expression (2.2) of Ha, Eq. (2.14) becomes Z sinca ðt ÞH sinca ðt2kπ Þ dt 2
¼ ¼
ð1 þ aÞ πi 2
Z ∑ a2n χ In ðt Þ sgnðt Þei2kπt dt
R n∈Zþ
ð1 þ aÞ2 πi ∑ a2n 2 n∈Zþ
ð1 þ aÞ2 πi ∑ a2n ¼ 2 n∈Zþ
Z Z i2kπt
e ½n;nþ1Þ
□
3. Bedrosian subspace of the Hilbert transform In this section, we pay attention to the set S a ≔ff : f ∈L2 ðRÞ; Hðf cos θa ðÞÞ ¼ f sin θa ðÞg:
ð3:1Þ
2
It is clear that the set S a is a subspace of L ðRÞ due to the linearity of the Hilbert transform. The subspace S a shall be called the Bedrosian subspace of the Hilbert transform. We first make a simple observation that we shall need frequently later. Lemma 3.1. Eq. (1.4) is true if and only if for t∈R, HðρðÞeiϕ ÞðtÞ ¼ iρðtÞeiϕt :
ð3:2Þ
Proof. Applying the Hilbert transform H to both sides of Eq. (1.4) and utilizing Eq. (2.10) yields HðρðÞ sin ðϕÞÞðtÞ ¼ ρðtÞ cos ðϕtÞ:
ð3:3Þ
Combining Eqs. (1.4) and (3.3) produces Eq. (3.2). Thus Eq. (1.4) is equivalent to Eq. (3.2). □ Lemmas 3.2–3.4 are several technical lemmas we need in the sequel. Lemma 3.2 concerns the Fourier transform of the product of a L2 ðRÞ function and a periodic function. It is an immediate result of the assumption of the absolute convergence of the series ∑k∈Z ck and the definition (1.3) of the Fourier transform. We state it below by omitting its proof.
We say that a complex τ-periodic signal g is circularly analytic if its Fourier expansion is the one-sided series gðtÞ ¼ ∑k∈Zþ ck eið2π=τÞkt ; t∈R. Lemma 3.3. Suppose that f ∈L2 ðRÞ is a real-valued function and the complex function g is τ-periodic and circularly analytic with real Fourier coefficients ck, k∈Zþ . Then HðfgÞðtÞ ¼ if ðtÞgðtÞ
sgnðt Þei2kπt dt In
for any k∈Z.
H 2a ðt Þ sgnðt Þei2kπt dt:
ð2:14Þ
R
〈sinca ; ðH sinca Þð2kπÞ〉 ¼ 0
Lemma 3.2. Suppose that f ∈L2 ðRÞ and g∈L2 ðτ=2; τ=2Þ is a τperiodic function having Fourier series gðtÞ ¼ ∑k∈Z ck R eikð2π=τÞt with ck ¼ ð1=τÞ ðτ=2;τ=2Þ gðtÞeiðð2πÞ=τÞkt dt. Moreover, the series ∑k∈Z ck is absolutely convergent. Then fg∈L2 ðRÞ and its Fourier transform at ξ∈R is given by 2π ðfgÞ∧ ðξÞ ¼ ∑ ck f^ ξ k : τ k∈Z
Invoking Eqs. (2.10) and (2.11) yields
Z
brackets is zero. Therefore we have
Z dt ððnþ1Þ;n
i2kπt
e
dt ;
where the interchange of the integral and the infinite sum in the second equality is guaranteed by the absolute convergence of the series. When k ¼0, the difference in the pair of brackets is zero, while when k∈Z\f0g, each integral inside the pair of
if and only if for a.e. ξ∈R , 2π ∑ ck f^ ξ k ¼ 0: τ k∈Zþ
ð3:4Þ
ð3:5Þ
Proof. Eq. (3.4) indicates that fg is an analytic signal. This is equivalent to say that fg has no negative frequency, that is, ðfgÞ∧ ðξÞ ¼ 0; ξ∈R . By Lemma 3.2 and the fact that g is circularly analytic, we obtain Eq. (3.5). □
Q. Chen et al. / Signal Processing 94 (2014) 255–263
259
Lemma 3.4. The 2π-periodic function g≔eiθa ðÞ is circularly analytic and the Fourier expansion of eiθa ðÞ at t∈R is given by
Lemma 3.6. The real-valued signal ρ∈S a if and only if for a.e. ξ∈R\f0g, n∈Zþ ,
eiθa ðtÞ ¼ a þ ∑ ð1a2 Þak1 eikt :
^ þ sgnðξÞnÞ ¼ an ρðξÞ: ^ ρðξ
ð3:6Þ
ð3:8Þ
k∈N
Proof. Recalling Eq. (1.6), the Fourier coefficients ck, k∈Z, of g is given by Z 1 eit a ikt ck ¼ e dt: 2π ðπ;πÞ 1aeit Let Rezðf ðzÞ; cÞ indicate the residue of the function f at the point c∈C. Denote the boundary of the unit disc Δ by ∂Δ. For k ¼0, we have c0 ¼ ð1=2πiÞ∮∂Δ ððzaÞ=ð1azÞÞdz=z ¼ ResððzaÞ=ð1azÞz; 0Þ ¼ a. In the case k is a negative integer, by using the Cauchy theorem we get that Z 1 eit a ijkjt 1 ðzaÞzjkj1 ck ¼ ∮∂Δ dz ¼ 0; e dt ¼ 2π ðπ;πÞ 1aeit 2πi 1az because the integrand is analytic in the unit disk Δ as jajo 1. We are left to consider the case of k∈N. Using the residue theorem and the formula for j∈N,
d za ¼ 1a2 aj1 j!ð1azÞj1 ; dzj 1az j
Proof. By Lemma 3.1 ρ∈S a is equivalent to for t∈R, HðρðÞeiθa ðÞ ÞðtÞ ¼ iρðtÞeiθa ðtÞ ;
which is equivalent to saying that the signal ρðÞeiθa ðÞ is an analytic signal. Applying Lemmas 3.3 and 3.4 to the signal ρðÞeiθa ðÞ yields that ρ∈S a holds if and only if for a.e. ξ∈R , ^ ¼ ð1a2 Þ ∑ ak1 ρðξkÞ: ^ aρðξÞ
In Eq. (3.10), separating the first term from the rest in the sum on the right-hand side, we have ! ^ ^ ^ ¼ ð1a2 Þ ρðξ1Þ þ ∑ ak1 ρðξkÞ aρðξÞ k∈1þN 2 ^ ^ ^ ¼ ρðξ1Þa ρðξ1Þ þ ð1a2 Þ ∑ ak1 ρðξkÞ:
On the other hand, multiplying both sides of Eq. (3.10) by a and replacing ξ by ξ1 we also have k∈N
Combining Eqs. (3.11) and (3.12) deduces that for a:e: ξ∈R ,
1 d za lim k ¼ 1a2 ak1 : k! z-0 dz 1az
^ ^ ρðξ1Þ ¼ aρðξÞ: By induction, we can get that for n∈Zþ and a:e: ξ∈R ,
□
Next proposition indicates that the subspace S a is invariant under the Hilbert transform. Proposition 3.5. Suppose that ρ∈S a , then Hρ∈S a . Proof. We first claim that the complex-valued signal ðρ þ iHρÞeiθa is an analytic signal. Indeed, by Lemma 3.4 the signal eiθa is circularly analytic, and we write eiθa ðtÞ ¼ ∑k∈Zþ g k eikt . Then by Lemma 3.2, we have for ξ∈R, F ððρ þ iHρÞeiθa ÞðξÞ ¼ ∑ g k F ðρ þ iHρÞðξkÞ:
ð3:7Þ
k∈Zþ
Note that ðρ þ iHρÞ is an analytic signal thus it has no negative spectrum, that is F ðρ þ iHρÞðξÞ ¼ 0 when ξ∈R . Consequently, when ξ∈R , ∑ g k F ðρ þ iHρÞðξkÞ ¼ 0;
^ ^ ρðξnÞ ¼ an ρðξÞ:
ð3:13Þ
Reversing the above calculations reveals that condition (3.13) is also sufficient for (3.10). Taking the conjugate of both sides of Eq. (3.10) and using ^ we obtain an equivalent the Hermitian property of ρ, equation ^ ¼ ð1a2 Þ ∑ ak1 ρðξ ^ þ kÞ aρðξÞ
ð3:14Þ
k∈N
for a.e. ξ∈R . A similar discussion to the case ξ∈R gives that Eq. (3.14) is equivalent to ^ þ nÞ ¼ an ρðξÞ ^ ρðξ
ð3:15Þ
for n∈Zþ and a.e. ξ∈R . Finally combining Eqs. (3.13) and (3.15) produces (3.8). □ Proposition 3.7.
k∈Zþ
which implies by Lemma 3.3 that iθa
k∈1þN
ð3:12Þ
1 za za ∮∂Δ dz ¼ Res ;0 kþ1 kþ1 2πi ð1azÞz ð1azÞz
The proof of this lemma is completed.
ð3:11Þ
k∈1þN
a2 ρðξ1Þ ^ ¼ ð1a2 Þ ∑ ak ρðξ1kÞ ^ ¼ ð1a2 Þ ∑ ak1 ρðξkÞ: ^
k
¼
ð3:10Þ
k∈N
leads to when k∈N, ck ¼
ð3:9Þ
sinca ∈S a
and
H sinca ∈S a :
iθa
H½ðρ þ iHρÞe ¼ iðρ þ iHρÞe : Simplifying both sides leads to H½ðHρÞeiθa ¼ i½ðHρÞeiθa , which indicates that Hρ∈S a . □ The next lemma indicates that any real-valued function ρ∈S a is completely determined by its spectrum ρχ ^ ð1;1Þ on the interval ð1; 1Þ.
Proof. By Lemma 3.5, it suffices to show sinca ∈S a . Observing the identity H a ð þ nÞ ¼ an H a ðÞ for a.e. ξ∈R\f0g, n∈Zþ and recall Eq. (2.9) we thus have F ðsinca Þðξ þ jnjsgnðξÞÞ ¼ ajnj F ðsinca ÞðξÞ: Hence by Lemma 3.6, we conclude that sinca ∈S a .
□
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Q. Chen et al. / Signal Processing 94 (2014) 255–263
4. An observation – how a linear phase determines the amplitude In this section, we specifically consider the case when the phase ϕ in Eq. (1.4) is a linear phase and investigate the representation of the corresponding amplitude. The result is given in the following theorem. We use the notation suppðf Þ for the set of real numbers on which the values f(x) at x∈R are nonzero. Theorem 4.1. Suppose that γ is a positive real number and ρ is a non-zero real signal in L2 ðRÞ. Then the following equation: HðρðÞ cos ðγÞÞðtÞ ¼ ρðtÞ sin ðγtÞ
ð4:1Þ
^ holds if and only if ρ is bandlimited with suppðρÞ⊂½γ; γ.
and 1 1 1eiξ ∧ ðH : a ðÞÞ ðξÞ ¼ pffiffiffiffiffiffi 2π 1aeiξ iξ Proof. We start with the observation k k1 Hþ χ ½0;kÞ ðtÞ: a ðtÞ ¼ ∑ a χ ½k;kþ1Þ ðtÞ ¼ ð1aÞ ∑ a k∈Zþ
1a eikξ 1 ; ¼ pffiffiffiffiffiffi ∑ ak1 iξ 2π k∈N
Proof. Lemma 3.1 implies that Eq. (4.1) is equivalent to HðρðÞeiγ ÞðtÞ ¼ iρðtÞeiγt ;
ð4:2Þ
or equivalently, the complex signal ρðtÞeiγt is an analytic signal. Note that the Fourier transform of the function on ^ the left-hand side of Eq. (4.2) is given by i sgnðξÞρðξγÞ and the Fourier transform of the right-hand side of ^ Eq. (4.2) is iρðξγÞ. This leads to the equivalent equation of (4.1) in the frequency domain ðsgnðξÞ1ÞρðξγÞ ^ ¼ 0;
ð4:3Þ
i.e. ρðξÞ ^ ¼0
for
ξ∈ð1; γ:
ð4:4Þ
Thus if ρ is bandlimited with supp ρ⊂½γ; ^ γ, then clearly Eq. (4.4) is true, hence Eq. (4.1) is true. On the other hand, assuming that Eq. (4.1) is true, we obtain that ρðξÞ ^ ¼ 0 for ξ∈ð1; γ. Since ρ is real-valued, by the Hermitian ^ ¼0 property of the Fourier transform of a real signal, ρðξÞ for ξ∈½γ; þ 1Þ. Consequently we conclude that ρ is a bandlimited function with its support in the frequency domain belongs to ½γ; γ. □ Remark. From this theorem we know that the amplitude function ρ can be represented by shifts of sinc function if and only if Eq. (4.1) is true.
k∈N
By the definition of the Fourier transform (1.3), we obtain that Z 1a ∧ ðH þ ∑ ak1 χ ½0;kÞ ðt Þeiξt dt a ðÞÞ ðξÞ ¼ pffiffiffiffiffiffi 2π R k∈N Z 1a ¼ pffiffiffiffiffiffi ∑ ak1 eiξt dt 2π k∈N ½0;kÞ
where, the interchange of the order of the summation and the integral is justified by the absolute convergence of the series. Continue by noting ∑k∈N ak1 eikξ ¼ eiξ =ð1aeiξ Þ we have 1 1aeiξ ð1aÞeiξ ∧ ðH þ a ðÞÞ ðξÞ ¼ pffiffiffiffiffiffi iξð1aeiξ Þ 2π 1 1 1eiξ : ¼ pffiffiffiffiffiffi 2π 1aeiξ iξ þ Using the identity H a ðÞ ¼ H a ðÞ, we obtain that
1 1 1eiξ þ ∧ ∧ : ðH a ðÞÞ ðξÞ ¼ ðH a ðÞÞ ðξÞ ¼ pffiffiffiffiffiffi 2π 1aeiξ iξ
□
Lemma 5.2. The quasi-Bedrosian type identity
1þa p ðt ÞH sincðt Þ H pa ðÞ sincðÞ ðt Þ ¼ 1a a
ð5:3Þ
is true for t∈R and 1 eit 1 1 ¼ ðsinca ðt Þ þ iH sinca ðt ÞÞ: 1þa 1aeit it
ð5:4Þ
Proof. Let rðtÞ≔ð1=ð1aeit ÞÞðeit 1Þ=it. Separating the real part from the imaginary part of r(t) produces 1a sin t 1þa 1 cos t þi t 12a cos t þ a2 t 12a cos t þ a2 1 1 p ðt Þ sincðt Þ þ i p ðt ÞH sincðt Þ; ¼ ð5:5Þ 1þa a 1a a
r ðt Þ ¼ 5. How does non-linear phase determine amplitude? In this section, we shall completely characterize a realvalued function ρ∈S a . We begin with introducing two onesided filters that are related to the two-sided symmetric cascade filter Ha: Hþ a ðtÞ≔H a ðtÞχ Rþ ðtÞ;
H a ðtÞ≔H a ðtÞχ R ðtÞ;
ð5:1Þ
where t∈R. The next lemma provides the Fourier trans form pairs of H þ a and H a , respectively. Lemma 5.1. The Fourier transform of Hþ a and H a are given, respectively, by
1 1 1eiξ ∧ ; ðH þ a ðÞÞ ðξÞ ¼ pffiffiffiffiffiffi iξ iξ 2π 1ae
ð5:2Þ
where we have used the identity H sincðt Þ ¼
1 cos t : t
ð5:6Þ
pffiffiffiffiffiffi ∧ On the other hand, observepthat rðtÞ ¼ 2π ðH þ a Þ ðtÞ, so ffiffiffiffiffiffi þ ^ the Fourier transform r ðtÞ ¼ 2π H a ðtÞ has zero negative spectrum. This implies r is an analytic signal. Therefore the imaginary part of r(t) equals to the Hilbert transform of its real part, that is, 1 1 pa ðÞ sincðÞ ðt Þ ¼ p ðt ÞH sincðt Þ: H 1þa 1a a After rearranging the above equation we obtain Eq. (5.3). Now we turn to show Eq. (5.4). Applying Eq. (5.3) to the
Q. Chen et al. / Signal Processing 94 (2014) 255–263
that for t∈R,
imaginary part of Eq. (5.5) we continue to have that r ðt Þ ¼ ¼
1 p ðt Þ sincðt Þ þ iH pa ðÞ sincðÞ ðt Þ 1þa a
ρðtÞ ¼ ∑ r k sinca ðt2kπÞ þ ∑ sk ðH sinca Þðt2kπÞ: k∈Z
1 ðsinca ðt Þ þ iH sinca ðt ÞÞ; 1þa
which is Eq. (5.4).
□
Corollary 5.3. Let cosinca ðtÞ≔ð1 cos θa ðtÞÞ=t and define cosinca ð0Þ ¼ limt-0 ð1 cos θa ðtÞÞ=t, then for t∈R, H sinca ðtÞ ¼ cosinca ðtÞ:
ð5:7Þ
Moreover, the function cosinca is odd, bounded, infinitely differentiable, cosinca ðtÞ≤ðð1 þ jajÞ=ð1jajÞÞð3=jtj þ 1Þ for t∈R, and cosinca ∈L2 ðRÞ. Proof. Eq. (5.7) is just a new form of Eq. (5.3). Indeed, the left-hand side of Eq. (5.3) is just H sinca ðtÞ. Utilizing Eqs. (1.7) and (5.9) then the right-hand side of Eq. (5.3) is simplified to cosinca ðtÞ. The second statement follows by rewriting cosinca as cosinca ðt Þ ¼
1þa 1 cos t p ðt Þ ; 1a a t
ð5:8Þ
1 cos t t
ð5:9Þ
is infinitely differentiable if we define H sincð0Þ ¼ limt-0 ð1 cos tÞ=t, and jð1 cos tÞ=tj≤ð3=1 þ jtjÞ for any t∈R. □ An example of the graph of cosinca with a¼0.5 is shown in Fig. 2. As a comparison, the graph of H sincðtÞ ¼ ð1 cos tÞ=t is also shown in Fig. 2. The following lemma appeared in [4]. However, a completely new proof by direct construction is given here for both sufficiency and necessity. The construction approach casts a new insight to understand the relation of the spectrum ρ^ of a function ρ∈S a to the symmetric cascade filter Ha. Lemma 5.4. A real signal ρ∈S a if and only if there are two 2 real sequences r ¼ fr k : k∈Zg and s ¼ fsk : k∈Zg in l ðZÞ such 2
Proof. We first show the necessity, that is, if ρ∈S a , then ρ is given by Eq. (5.10). From Lemma 3.6, we know that ρðξÞχ Rþ ðξÞ and ρðξÞχ R ðξÞ are completely determined by ρðξÞχ ð0;1Þ ðξÞ and ρðξÞχ ð1;0Þ ðξÞ, respectively. Expand ρðξÞχ ð0;1Þ ðξÞ and ρðξÞχ ð1;0Þ ðξÞ into 1-periodic functions γ 1 ðξÞ≔∑k∈Z uk ei2kπt and γ 2 ðξÞ≔∑k∈Z vk ei2kπt , where the two series 2 ðuk : k∈ZÞ and ðvk : k∈ZÞ are in l ðZÞ. By Lemma 3.6 we thus obtain that for a.e. ξ∈R, ρðξÞ ^ ¼ γ 1 ðξÞH þ a ðξÞ þ γ 2 ðξÞH a ðξÞ:
The Hermitian property of ρ^ implies that for ξ∈R\f0g, γ 2 ðξÞ ¼ ðγ 1 ðξÞÞn : For convenience let γ ¼ γ 1 . Consequently ρ^ has the representation n ρðξÞ ^ ¼ γðξÞH þ a ðξÞ þ ðγðξÞÞ H a ðξÞ;
for a.e. ξ∈R. Let ρA be the analytic signal associated with ρ, which may be defined by
or equivalently for t∈R, ρA ðtÞ ¼ ρðtÞ þ iHρðtÞ: Then we have ρðtÞ ¼ ReðρA ðtÞÞ. We need to investigate ρA . The inverse Fourier transform of ρ^ A yields Z 1 iξt ρA ðt Þ ¼ pffiffiffiffiffiffi 2γ ðξÞH þ dξ a ðξÞe 2π R Z 2 iξt ¼ pffiffiffiffiffiffi ∑ uk ei2kπξ Hþ dξ: a ðξÞe 2π R k∈Z Applying the Lebesgue dominated convergence theorem to interchange the order of the integral and the sum, and appealing to Eq. (5.2) we obtain that Z 2 iξðt2kπÞ ρA ðt Þ ¼ pffiffiffiffiffiffi ∑ uk H þ dξ a ðξÞe 2π k∈Z R 2 1 eiðt2kπÞ 1 : ¼ pffiffiffiffiffiffi ∑ uk iðt2kπÞ iðt2kπÞ 2π k∈Z 1ae Denote uk ¼ uð1Þ þ iuð2Þ . Recalling Eq. (5.4) we readily obtain k k that
cosinca(t) Hsinc(t)
1.5
2 ρðt Þ ¼ pffiffiffiffiffiffi 2π ð1 þ aÞ
1 0.5 0 −0.5 −1 −1.5 −2 −20
−15
−10
−5
ð5:10Þ
k∈Z
^ ρ^ A ðξÞ ¼ 2ρðξÞχ Rþ ðξÞ
and noting the function H sincðt Þ ¼
261
0
5
10
Fig. 2. The graph of cosinca , a ¼0.5.
15
20
! ∑ uð1Þ sinca ðt2kπÞ ∑ uð2Þ ðH sinca ðÞÞðt2kπÞ : k k
k∈Z
k∈Z
pffiffiffiffiffiffi pffiffiffiffiffiffi Let r k ¼ ð2=ð 2π ð1 þ aÞÞÞukð1Þ and sk ¼ ð2=ð 2π ð1 þ aÞÞÞuð2Þ . k 2 Since the series ðuk ; k∈ZÞ∈l ðZÞ, we must have both series 2 ðr k ; k∈ZÞ and ðsk ; k∈ZÞ in l ðZÞ. We have arrived at Eq. (5.10). We now turn to the proof of sufficiency. It suffices to check that a function ρ∈L2 ðRÞ having the form (5.10) satisfies Eq. (3.8) by Lemma 3.6. Applying the Fourier transform to both sides of Eq. (5.10) and noting that the Fourier transform of the generalized sampling function given by Eq. (2.9) we get that ^ ¼ H a ðξÞðMðξÞi sgnðξÞGðξÞÞ ρðξÞ
262
Q. Chen et al. / Signal Processing 94 (2014) 255–263
with the two 1-periodic functions pffiffiffiffiffiffi pffiffiffiffiffiffi 2π 2π ð1 þ aÞ ∑ r k ei2kπξ ; GðξÞ ¼ ð1 þ aÞ ∑ sk ei2kπξ : M ðξÞ ¼ 2 2 k∈Z k∈Z Observing the identity H a ð þ nÞ ¼ an H a ðÞ for n≥0 and the 1-periodicity of M and G lead to that, for ξ∈Rþ , ^ þ jnjÞ ¼ H a ðξ þ jnjÞðMðξ þ jnjÞiGðξ þ jnjÞÞ ρðξ ^ ¼ ajnj H a ðξÞðMðξÞiGðξÞÞ ¼ ajnj ρðξÞ: The case with negative ξ can be shown similarly. This completes the proof of the lemma. □ Theorem 5.5. If ρ∈S a , then both ρ and Hρ are continuous. Proof. By Proposition 3.5, Hρ∈S a , thus it suffices to show ρ∈S a is continuous. Indeed, if ρ∈S a , then ρ has the representation (5.10) by Lemma 5.4. Since both sinca and H sinca are in L2 ðRÞ by Proposition 2.1 and Corollary 5.3, consequently by the Cauchy–Schwarz inequality, the two series on the right-hand side of Eq. (5.10) converge uniformly, hence the limiting function ρ is continuous. □ Lemma 5.6. If f ∈S a , then for k∈Z, 〈f ; sinca ð2kπÞ〉 ¼ f ð2kπÞ;
and
〈f ; Hsinca ð2kπÞ〉 ¼ Hf ð2kπÞ:
Proof. The 2πperiodicity of sin θa yields Z
〈f ; sinca ð2kπ Þ〉 ¼
f ðt Þ R
sin θa ðt2kπÞ dt ¼ Hðf sin θa Þð2kπ Þ: t2kπ
Therefor by the assumption on f, we have when k∈Z, Hðf sin θa Þð2kπÞ ¼ f ð2kπÞ cos θa ð2kπÞ ¼ f ð2kπÞ because cos θa ð2kπÞ ¼ cos θa ð0Þ ¼ 1 by Eq. (2.7). The second equality follows by noting the fact that Hf ∈S a due to Proposition 3.5 and the Hilbert transformer is an anti-self adjoint operator thus 〈f ; H sinca ð2kπ〉 ¼ 〈Hf ; sinca ð2kπ〉 ¼ Hf ð2kπÞ:
&
Lemmas 5.4 and 5.6 and Theorem 2.2 immediately implies the following theorem. Theorem 5.7. Any real-valued function ρ∈S a if and only if ρðt Þ ¼
1a 1a ∑ ρð2kπ Þ sinca ðt2kπ Þ ∑ Hρð2kπ ÞH sinca ðt2kπ Þ: 1 þ a k∈Z 1 þ a k∈Z
Moreover, the sampling sequences 2 ðHρð2kπÞ; k∈ZÞ are in l ðZÞ.
ðρð2kπÞ; k∈ZÞ
and
From Theorems 5.7 and 2.2 we have the following corollary. Corollary 5.8. The system Φ defined in (2.13) is a complete orthogonal system of the subspace S a ⊂L2 ðRÞ. That is, the subspace equals to the closure of the span of the set Φ, or symbolically, S a ¼ spanΦ: Lastly, we state some facts for the case a ¼0. Note when a ¼0, sin θa ðtÞ becomes sin t, cos θa ðtÞ becomes cos t, sinca becomes sinc and H sinca becomes H sinc which is given at t∈R by Eq. (5.9). Therefore Theorems 4.1 and 5.7
and the well-known Shannon sampling theorem imply the following corollary. Corollary 5.9. The following statements are equivalent: 1. The real signal ρ∈L2 ðRÞ is bandlimited such that supp ρ^ D ½1; 1. 2. HðρðÞ cos ðÞÞðtÞ ¼ ρðtÞ sin t, t∈R. 3. HðρðÞei ÞðtÞ ¼ iρðtÞeit , t∈R. 4. ρðtÞ ¼ ∑k∈Z ρð2kπÞ sincðt2kπÞ∑k∈Z Hρð2kπÞ ð1 cos ðt2kπÞÞÞ=ðt2kπÞ, t∈R. 5. ρðtÞ ¼ ∑k∈Z ρðkπÞ sincðtkπÞ, t∈R.
6. Conclusions In signal analysis, researchers have been trying to understand, for a given signal, what are its instantaneous amplitude, instantaneous phase, and instantaneous frequency. The rigorous mathematical definitions for those notions, however, have not been well agreed by signal analysts. The divergent understandings, as a matter of fact, have created many controversies. We give in this paper rigorous mathematical characterizations of an instantaneous amplitude when the instantaneous frequency is periodic and described by the Poisson kernel. Our main result indicates that such kind of amplitudes can be perfectly reconstructed in terms of a sampling formula using the generalized sampling function. A novel characterization of the band-limited functions is also given. We believe that our theoretic study is useful for further understanding analytic signals. References [1] E. Bedrosian, A product theorem for Hilbert transform, Proceedings of IEEE 51 (1963) 868–869. [2] B. Boashash, Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals, Proceedings of IEEE 80 (1992) 417–430. [3] P.L. Butzer, W. Engels, S. Ries, The Shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines, SIAM Journal of Applied Mathematics 46 (1986) 299–323. [4] P. Cerejeiras, Q. Chen, U. Kähler, A necessary and sufficient condition for a Bedrosian identity, Mathematical Methods in Applied Sciences 33 (2010) 493–500. [5] Q. Chen, C.A. Micchelli, Y. Wang, Functions with spline spectra and their applications, International Journal of Wavelets, Multiresolution and Information Processing 8 (2) (2010) 197–223. [6] Q. Chen, T. Qian, Sampling theorem and multi-scale spectrum based on non-linear Fourier atoms, Applicable Analysis 88 (6) (2009) 903–919. [7] Q. Chen, Y.B. Wang, Y. Wang, A sampling theorem for nonbandlimited signals using generalized sinc functions, Computational and Applied Mathematics 56 (2008) 1650–1661. [8] L. Cohen, Time–Frequency Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1995. [9] M.I. Doroslovacki, On nontrivial analytic signals with positive instantaneous frequency, Signal Processing 83 (2003) 655–658. [10] S.L. Hahn, On the uniqueness of the definition of the amplitude and phase of the analytic signal, Signal Processing 83 (2003) 1815–1820. [11] S.L. Hahn, The history of applications of analytic signals in electrical and radio engineering, in: International Conference on “Computer as a Tool”, EOROCONF2007, 2007, pp. 2627–2631. [12] N.E. Huang, et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proceedings of the Royal Society London Series A 454 (1998) 903–995.
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[13] F. King, Hilbert Transform, vol. 1, Cambridge University Press, Cambridge, 2009. [14] P.Y. Ktonas, N. Papp, Instantaneous envelope and phase extraction from real signal: theory, implementation, and an application to EEG analysis, Signal Processing 2 (1980) 373–385. [15] R. Kumaresan, A. Rao, Model-based approach to envelope and positive instantaneous frequency estimation of signals with speech applications, Journal of the Acoustical Society of America 105 (3) (1999) 1912–1924. [16] A.H. Nuttall, On the quadrature approximation to the Hilbert transform of modulated signals, Proceedings of IEEE Letters (1966) 1458– 1459.
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[17] B. Picinbono, On instantaneous amplitude and phase of signals, IEEE Transactions on Signal Processing 45 (March (3)) (1997) 552–560. [18] T. Qian, Q.H. Chen, L.Q. Li, Analytic unit quadrature signals with nonlinear phase, Physica D: Nonlinear Phenomena 203 (1–2) (2005) 80–87. [19] C. Shannon, A mathematical theory of communication, Bell System Technical Journal 27 (1948) 379–423. [20] C. Shannon, Communication in the presence of noise, Proceedings of IRE 37 (1949) 10–21. [21] X.G. Xia, L. Cohen, On analytic signals with nonnegative instantaneous frequency, in: Proceedings of the ICASSP-99, Phoenix, March 1999, Paper 1483.
Contents lists available at ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Amplitudes of mono-component signals and the generalized sampling functions Qiuhui Chen a,1, Luoqing Li b,2, Yi Wang c,n a b c
Cisco School of Informatics, Guangdong University of Foreign Studies, Guangzhou, China Faculty of Mathematics and Computer Science, Hubei University, Wuhan 430062, China Department of Mathematics, Auburn University at Montgomery, P.O. Box 244023, Montgomery, AL 36124-4023, USA
a r t i c l e i n f o
abstract
Article history: Received 20 December 2012 Received in revised form 25 June 2013 Accepted 26 June 2013 Available online 6 July 2013
There is a recent trend to use mono-components to represent nonlinear and non-stationary signals rather than the usual Fourier basis with linear phase, such as the intrinsic mode functions used in Norden Huang's empirical mode decomposition [12]. A mono-component is a real-valued signal of finite energy that has non-negative instantaneous frequencies, which may be defined as the derivative of the phase function of the given real-valued signal through the approach of canonical amplitude-phase modulation. We study in this paper how the amplitude is determined by its phase for a class of signals, of which the instantaneous frequency is periodic and described by the Poisson kernel. Our finding is that such an amplitude can be perfectly represented by a sampling formula using the so-called generalized sampling functions that are related to the phase. The regularity of such an amplitude is identified to be at least continuous. Such characterization of mono-components provides the theory to adaptively decompose non-stationary signals. Meanwhile, we also make a very interesting and new characterization of the band-limited functions. & 2013 Elsevier B.V. All rights reserved.
Keywords: Mono-component Generalized sampling function Analytic signal Nonlinear phase Amplitude-phase modulation Hilbert transform Blaschke product Poisson kernel
1. Introduction Any real-valued non-stationary signal f of finite energy, that is, f is in the space L2 ðRÞ of square integrable functions on the set R of real numbers, may be represented as an amplitude-phase modulation with a time-varying amplitude ρ and a time-varying phase ϕ where phase ϕ is, in general, nonlinear [14]. Specifically, the value of f at t∈R may be represented as f ðtÞ ¼ ρðtÞ cos ϕðtÞ:
ð1:1Þ
Unfortunately, this type of representation is not unique because the modulation is obtained through a complex signal that can have various choices of the imaginary part.
However, one can determine a unique such factorization (1.1) by using the approach of analytic signals [10]. Indeed, let Aðf Þ be the analytic signal associated with f with the characteristic property ( 2f^ ðωÞ if ω≥0 ∧ ðAðf ÞÞ ðωÞ ¼ ð1:2Þ 0 if ω o0; where for any signal g∈L2 ðRÞ, g^ ¼ F g is the Fourier transform of g defined at ξ∈R by the equation Z 1 g^ ðξÞ ¼ ðF g ÞðξÞ≔ pffiffiffiffiffiffi g ðt Þeiξt dt: ð1:3Þ 2π R Eq. (1.2) is equivalent to for t∈R, Aðf ÞðtÞ ¼ f ðtÞ þ iHf ðtÞ;
n
Corresponding author. Tel.: +1 3342443318. E-mail addresses: [email protected] (Q. Chen), [email protected] (L. Li), [email protected] (Y. Wang). 1 The author is partially supported by the Natural Science Foundation of Guangdong Province under Grant no. S2011010004986. 2 The author is partially supported by the National Natural Science Foundation of China under Grant no. 11071058. 0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.06.034
where the operator H : L2 ðRÞ-L2 ðRÞ stands for the Hilbert transform, and for f ∈L2 ðRÞ, Hf at t∈R is defined through the principal value integral Z Z 1 f ðxÞ f ðxÞ dx ¼ lim dx: Hf ðtÞ≔p:v: π R tx ϵ-0 jxtj 4 ϵ tx
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Q. Chen et al. / Signal Processing 94 (2014) 255–263
The value Aðf ÞðtÞ at t∈R is complex which can be written into the quadrature form Aðf ÞðtÞ ¼ ρðtÞeiϕðtÞ : Under the conditions that the derivative value ϕ′ðtÞ is nonnegative for all t∈R, the quantities ρðtÞ and ϕ′ ðtÞ are called the instantaneous amplitude and instantaneous frequency at t∈R, of the real signal f, respectively. The corresponding modulation (1.1) is then called the canonical amplitude-phase modulation, or canonical modulation for short [8]. The signal f with such defined non-negative instantaneous frequencies is thus called a mono-component. A large body of literature addresses this problem, see for example, [1,2,16,17,9,21]. The notions of instantaneous amplitude and frequency are fundamental in many applications involving modulations of signals that appear especially in communications or information processing. Thus constructing the canonical pair ðρ; ϕÞ of the instantaneous amplitude and phase is important in the theories of analytic signals and in order to facilitate the modulation and demodulation techniques such as processing speech signals [15] or signals in electrical and radio engineering [11]. It is equivalent to the problem of seeking the function pair ðρ; ϕÞ such that for t∈R, the following equation holds true: HðρðÞ cos ϕðÞÞðtÞ ¼ ρðtÞ sin ϕðtÞ:
ð1:4Þ
We remark that (1.4) can be apparently considered as a special case of the Bedrosian identity
We shall in this paper characterize the amplitude function ρ when the phase function ϕ is chosen at t∈R by Z ϕðtÞ ¼ θa ðtÞ ¼ pa ðxÞ dx ½0;t
such that Eq. (1.4) is satisfied. Our main result indicates that such kind of amplitude can be perfectly reconstructed in terms of a sampling formula using the generalized sampling function whose value at t∈R is given by sinca ðt Þ≔
sin θa ðtÞ : t
ð1:8Þ
In Section 2, we review the construction of the generalized sampling function and discuss some properties pertaining to it. In Section 3, we introduce the concept of Bedrosian subspace of the Hilbert transform and investigate some properties of functions in this space. In Section 4, we make an important observation when a linear phase is chosen, the amplitude function must be bandlimited in order to satisfy equation (1.4). In Section 5, we present our main result in Theorem 5.7.
2. Generalized sampling functions Not very surprisingly the function sinca has many properties that are similar to the classic sinc defined at t∈R by the equation sin t : t
HðfgÞ ¼ f HðgÞ:
sincðt Þ≔
In [1], the author proved that, if both f ; g belong to L2 ðRÞ, f is of lower frequency, g is of higher frequency and f ; g have no overlapping frequency, then HðfgÞ ¼ f HðgÞ. This classic result of Bedrosian is not useful for constructing a mono-component. The reason lies in that the requirement of both f and g in L2 ðRÞ is invalid. Recently, an important phase function that renders monocomponents was given in [18]. The phase function is defined through the boundary values of a Blaschke product on a unit disk Δ≔fz : z∈C; jzj≤1g, where C indicates the set of complex numbers. Specifically, for a∈ð1; 1Þ, the Blaschke product at z∈C\f1=ag is given by
ð1:7Þ
Those properties include cardinality, orthogonality, decaying rate, among others. In the special case a ¼0, the function sinca reduces to the classic sinc, which will become clear later. Let us first review the approach to obtain an explicit form of sinca . The classic sinc function is fundamentally significant in digital signal processing due to the Shannon sampling theorem [19,20,3]. The Shannon sampling theorem enables to reconstruct a bandlimited signal from shifts of sinc functions weighted by the uniformly spaced samples of that signal. Recently efforts have been made to extend the classic sinc to generalized sampling functions, for example, in [5–7]. Intuitively, the spectrum of the sinc function is just the indicator function of a symmetric interval of finite measure. Hence, authors in [5] are inspired to consider functions with piecewise polynomial spectra to replace the usual sinc function for the purpose of sampling non-bandlimited signals. One kind of generalized sampling functions given in [5], denoted by sinca that is related to a constant a∈ð1; 1Þ, is defined as the inverse Fourier transform of a so-called symmetric cascade filter, denoted by H a . Specifically, rffiffiffi π ð1 þ aÞF 1 H a : sinca ≔ ð2:1Þ 2
then by taking the derivative of both sides of Eq. (1.6), we find that the phase θa is an anti-derivative of pa, and its derivative is always positive, that is,
Let N be the set of natural numbers, Z be the set of integers, and Zþ ≔f0g∪N. Let X be a subset of R, and for q∈N, we say a function f is in Lq ðXÞ if and only if the Lq ðXÞ norm
d θa ðt Þ ¼ pa ðt Þ 40: dt
∥f ∥q;X ≔
Ba ðzÞ ¼
za : 1az
ð1:5Þ
Subsequently the non-linear phase function, denoted by θa , is defined at t∈R by the equation eiθa ðtÞ ≔Ba ðeit Þ:
ð1:6Þ
If we recall that the periodic Poisson kernel pa whose value at t∈R is given by pa ðt Þ≔
1a2 ; 12a cos t þ a2
Z
1=q jf ðtÞjq dt X
o1:
Q. Chen et al. / Signal Processing 94 (2014) 255–263
Similarly, let Z be a subset of Z, a sequence y≔ðyk : k∈ZÞ is q q said to be in l ðZÞ if and only if the l ðZÞ norm !1=q ∥y∥q;Z ≔ ∑ jyk jq
and ð1 þ a2 Þ cos t2a : ð2:7Þ 12a cos t þ a2 We next list some properties of the function sinca .
cos θa ðt Þ ¼
o1:
k∈Z
Proposition 2.1. Let the generalized sampling function sinca be defined by Eq. (2.4) or Eq. (1.8). Then the following statements hold:
The symmetric cascade filter Ha is a piecewise constant function whose value at ξ∈R is given by H a ðξÞ≔ ∑ an χ In ðξÞ;
ð2:2Þ
n∈Zþ
1. For t∈R,
where χ I is the indicator function of the set I, and the interval In, n∈Zþ , is the union of two symmetric intervals given by the equation
1
ð1a2 Þ sin t ¼ pa ðt Þ sincðt Þ: 12a cos t þ a2 t
ð2:8Þ
F sinca ¼
rffiffiffi π ð1 þ aÞH a : 2
ð2:9Þ
2
Of course, we have that H a ∈L ðRÞ∩L ðRÞ because the 1 2 sequence b≔ðan : n∈Zþ Þ∈l ðZþ Þ∩l ðZþ Þ, and hence sinca ∈L2 ðRÞ since the Fourier operator is closed in L2 ðRÞ and sinca is continuous because H a ∈L1 ðRÞ. The symmetric cascade filter H a can be associated with an analytic function F on the open unit disk Δ defined at z∈Δ by FðzÞ≔ ∑ an zn :
3. sinca ðnπÞ ¼ ðð1 þ aÞ=ð1aÞÞδn;0 , where δn;0 ¼ 1 if n ¼0 and δn;0 ¼ 0 if n∈Z\f0g. 4. sinca is even, bounded, infinitely differentiable. 5. jsinca ðtÞj≤ðð1 þ jajÞ=ð1jajÞÞ2=ð1 þ jtjÞ for t∈R, and sinca ∈L2 ðRÞ. 6. The set fsinca ðnπÞ : n∈Zg is an orthogonal set, that is
ð2:3Þ
n∈Zþ
〈sinca ; sinca ðnπ Þ〉 ¼
Thus, by substituting Eq. (2.2) into Eq. (2.1) and making use of Eq. (2.3) an alternative form of sinca ðtÞ, t∈R in terms of F can be found as t RefFðeit Þeð1=2Þit g; ð2:4Þ sinca ðt Þ ¼ ð1 þ aÞ sinc 2
Ba ðzÞ1 : z1
Proof. Eq. (2.8) directly follows by substituting Eq. (2.6) into Eq. (1.8). Eq. (2.9) is obtained by taking the inverse Fourier transform of both sides of Eq. (2.1). The third and fourth statements directly follow from Eq. (2.8). The fifth statement follows from Eq. (2.8) and noticing sincðtÞ≤ 2=ð1 þ jtjÞ and pa ðtÞ≤ð1 þ jajÞ=ð1jajÞ for any t∈R. The last statement is a special case of Corollary 3.2 of [5]. For the convenience of readers, we provide a direct proof here. By Parseval's theorem and Eq. (2.9) we have Z Z π sinca ðt Þ sinca ðtnπ Þ dt ¼ ð1 þ aÞ2 H 2a ðxÞeinπx dx 2 R R Z π 2 2k inπx ¼ ð1 þ aÞ ∑ a χ Ik ðxÞe dx 2 R k∈Zþ Z π 1þa πδn;0 ; ¼ ð1 þ aÞ2 ∑ a2k einπx dx ¼ 2 1a Ik k∈Zþ
ð2:5Þ
Plugging the formula (2.5) into Eq. (2.4) we readily obtain the explicit expression of sinca given earlier in Eq. (1.8). The next two formulas shall be used later. Expanding the left-hand side of Eq. (1.6) using Euler's formula and separating the real part from the imaginary part of the right-hand side, one obtains that for t∈R ð1a2 Þ sin t ¼ pa ðt Þ sin t 12a cos t þ a2
ð1 þ aÞ πδn;0 ; 1a
R where 〈u; v〉 ¼ R uðtÞvn ðtÞ dt denotes the usual inner product of the two functions u; v∈L2 ðRÞ, and vn is the complex conjugate of v.
where ReðzÞ is the real part of a complex number z. A very interesting fact, as discovered in the paper [5], is that the function F is linked to the Blashke Product Ba by the equation
sin θa ðt Þ ¼
sinca ðt Þ ¼ 2.
I n ≔ððn þ 1Þ; n∪½n; ðn þ 1ÞÞ:
ð1 þ aÞF ðzÞ≔
257
ð2:6Þ
3
2 1.8
sinca(t) sinc(t)
2.5
1.6
2
1.4 1.2
1.5
1 1
0.8 0.6
0.5
0.4
0
0.2 0 −20
−15
−10
−5
0
5
10
Fig. 1. Left: the graph of ð
15
20
−0.5 −20
−15
−10
−5
0
5
10
pffiffiffiffiffiffiffiffi π=2Þð1 þ aÞHa , a ¼0.5. Right: the graph of sinca , a¼ 0.5.
15
20
258
Q. Chen et al. / Signal Processing 94 (2014) 255–263
where, in the last equality we have used the orthogonality R identity Ik einπx dx ¼ 2δn;0 , k∈Zþ . The interchange of the integral operator and the infinite sum is guaranteed by the absolute convergence of the series. □ In Fig. 1, we show pffiffiffiffiffiffiffiffiffi ffi an example of the Fourier transform pair sinca and ðπ=2Þð1 þ aÞH a with a ¼0.5. In the plot of sinca , the graph of the standard sinc is also shown, which corresponds to the case a ¼0. Next we review several basic properties of the Hilbert transform which we will need frequently later. These properties can be found, for example, in the book [13]. First the Hilbert transform is an anti-involution, that is, H2 ¼ I
ð2:10Þ
where I is the identity operator. Second the operator H is anti-self adjoint, that is, 〈Hu; v〉 ¼ 〈u; Hv〉:
ð2:11Þ
Third, for any f ∈L2 ðRÞ and t∈R, the composition of the Fourier transform and the Hilbert transform is given by F ðHf ÞðtÞ ¼ i sgnðtÞF f ðtÞ;
ð2:12Þ
where sgnðÞ is the signum function having values defined by sgnðxÞ ¼ 1 if x∈Rþ ≔ft∈R : t 4 0g, sgnðxÞ ¼ 1 if x∈R ≔ ft∈R : t o0g, and sgnð0Þ ¼ 0. Theorem 2.2. The system Φ≔fsinca ð2kπÞ; H sinca ð2kπÞ : k∈Zg
ð2:13Þ
is an orthogonal system in L2 ðRÞ. Proof. By the third statement of Proposition 2.1, we have that 〈sinca ; sinca ð2kπ Þ〉 ¼
1þa δ : 1a 0k
〈H sinca ; H sinca ð2kπ Þ〉 ¼ 〈sinca ; H2 sinca ð2kπ Þ〉 1þa δ : ¼ 〈sinca ; sinca ð2kπ Þ〉 ¼ 1a 0k By Parseval's theorem and Eq. (2.9) we have
R
sinca ðt ÞH sinca ðt2kπ Þ dt ¼
ð1 þ aÞ2 πi 2
Z R
Invoking the expression (2.2) of Ha, Eq. (2.14) becomes Z sinca ðt ÞH sinca ðt2kπ Þ dt 2
¼ ¼
ð1 þ aÞ πi 2
Z ∑ a2n χ In ðt Þ sgnðt Þei2kπt dt
R n∈Zþ
ð1 þ aÞ2 πi ∑ a2n 2 n∈Zþ
ð1 þ aÞ2 πi ∑ a2n ¼ 2 n∈Zþ
Z Z i2kπt
e ½n;nþ1Þ
□
3. Bedrosian subspace of the Hilbert transform In this section, we pay attention to the set S a ≔ff : f ∈L2 ðRÞ; Hðf cos θa ðÞÞ ¼ f sin θa ðÞg:
ð3:1Þ
2
It is clear that the set S a is a subspace of L ðRÞ due to the linearity of the Hilbert transform. The subspace S a shall be called the Bedrosian subspace of the Hilbert transform. We first make a simple observation that we shall need frequently later. Lemma 3.1. Eq. (1.4) is true if and only if for t∈R, HðρðÞeiϕ ÞðtÞ ¼ iρðtÞeiϕt :
ð3:2Þ
Proof. Applying the Hilbert transform H to both sides of Eq. (1.4) and utilizing Eq. (2.10) yields HðρðÞ sin ðϕÞÞðtÞ ¼ ρðtÞ cos ðϕtÞ:
ð3:3Þ
Combining Eqs. (1.4) and (3.3) produces Eq. (3.2). Thus Eq. (1.4) is equivalent to Eq. (3.2). □ Lemmas 3.2–3.4 are several technical lemmas we need in the sequel. Lemma 3.2 concerns the Fourier transform of the product of a L2 ðRÞ function and a periodic function. It is an immediate result of the assumption of the absolute convergence of the series ∑k∈Z ck and the definition (1.3) of the Fourier transform. We state it below by omitting its proof.
We say that a complex τ-periodic signal g is circularly analytic if its Fourier expansion is the one-sided series gðtÞ ¼ ∑k∈Zþ ck eið2π=τÞkt ; t∈R. Lemma 3.3. Suppose that f ∈L2 ðRÞ is a real-valued function and the complex function g is τ-periodic and circularly analytic with real Fourier coefficients ck, k∈Zþ . Then HðfgÞðtÞ ¼ if ðtÞgðtÞ
sgnðt Þei2kπt dt In
for any k∈Z.
H 2a ðt Þ sgnðt Þei2kπt dt:
ð2:14Þ
R
〈sinca ; ðH sinca Þð2kπÞ〉 ¼ 0
Lemma 3.2. Suppose that f ∈L2 ðRÞ and g∈L2 ðτ=2; τ=2Þ is a τperiodic function having Fourier series gðtÞ ¼ ∑k∈Z ck R eikð2π=τÞt with ck ¼ ð1=τÞ ðτ=2;τ=2Þ gðtÞeiðð2πÞ=τÞkt dt. Moreover, the series ∑k∈Z ck is absolutely convergent. Then fg∈L2 ðRÞ and its Fourier transform at ξ∈R is given by 2π ðfgÞ∧ ðξÞ ¼ ∑ ck f^ ξ k : τ k∈Z
Invoking Eqs. (2.10) and (2.11) yields
Z
brackets is zero. Therefore we have
Z dt ððnþ1Þ;n
i2kπt
e
dt ;
where the interchange of the integral and the infinite sum in the second equality is guaranteed by the absolute convergence of the series. When k ¼0, the difference in the pair of brackets is zero, while when k∈Z\f0g, each integral inside the pair of
if and only if for a.e. ξ∈R , 2π ∑ ck f^ ξ k ¼ 0: τ k∈Zþ
ð3:4Þ
ð3:5Þ
Proof. Eq. (3.4) indicates that fg is an analytic signal. This is equivalent to say that fg has no negative frequency, that is, ðfgÞ∧ ðξÞ ¼ 0; ξ∈R . By Lemma 3.2 and the fact that g is circularly analytic, we obtain Eq. (3.5). □
Q. Chen et al. / Signal Processing 94 (2014) 255–263
259
Lemma 3.4. The 2π-periodic function g≔eiθa ðÞ is circularly analytic and the Fourier expansion of eiθa ðÞ at t∈R is given by
Lemma 3.6. The real-valued signal ρ∈S a if and only if for a.e. ξ∈R\f0g, n∈Zþ ,
eiθa ðtÞ ¼ a þ ∑ ð1a2 Þak1 eikt :
^ þ sgnðξÞnÞ ¼ an ρðξÞ: ^ ρðξ
ð3:6Þ
ð3:8Þ
k∈N
Proof. Recalling Eq. (1.6), the Fourier coefficients ck, k∈Z, of g is given by Z 1 eit a ikt ck ¼ e dt: 2π ðπ;πÞ 1aeit Let Rezðf ðzÞ; cÞ indicate the residue of the function f at the point c∈C. Denote the boundary of the unit disc Δ by ∂Δ. For k ¼0, we have c0 ¼ ð1=2πiÞ∮∂Δ ððzaÞ=ð1azÞÞdz=z ¼ ResððzaÞ=ð1azÞz; 0Þ ¼ a. In the case k is a negative integer, by using the Cauchy theorem we get that Z 1 eit a ijkjt 1 ðzaÞzjkj1 ck ¼ ∮∂Δ dz ¼ 0; e dt ¼ 2π ðπ;πÞ 1aeit 2πi 1az because the integrand is analytic in the unit disk Δ as jajo 1. We are left to consider the case of k∈N. Using the residue theorem and the formula for j∈N,
d za ¼ 1a2 aj1 j!ð1azÞj1 ; dzj 1az j
Proof. By Lemma 3.1 ρ∈S a is equivalent to for t∈R, HðρðÞeiθa ðÞ ÞðtÞ ¼ iρðtÞeiθa ðtÞ ;
which is equivalent to saying that the signal ρðÞeiθa ðÞ is an analytic signal. Applying Lemmas 3.3 and 3.4 to the signal ρðÞeiθa ðÞ yields that ρ∈S a holds if and only if for a.e. ξ∈R , ^ ¼ ð1a2 Þ ∑ ak1 ρðξkÞ: ^ aρðξÞ
In Eq. (3.10), separating the first term from the rest in the sum on the right-hand side, we have ! ^ ^ ^ ¼ ð1a2 Þ ρðξ1Þ þ ∑ ak1 ρðξkÞ aρðξÞ k∈1þN 2 ^ ^ ^ ¼ ρðξ1Þa ρðξ1Þ þ ð1a2 Þ ∑ ak1 ρðξkÞ:
On the other hand, multiplying both sides of Eq. (3.10) by a and replacing ξ by ξ1 we also have k∈N
Combining Eqs. (3.11) and (3.12) deduces that for a:e: ξ∈R ,
1 d za lim k ¼ 1a2 ak1 : k! z-0 dz 1az
^ ^ ρðξ1Þ ¼ aρðξÞ: By induction, we can get that for n∈Zþ and a:e: ξ∈R ,
□
Next proposition indicates that the subspace S a is invariant under the Hilbert transform. Proposition 3.5. Suppose that ρ∈S a , then Hρ∈S a . Proof. We first claim that the complex-valued signal ðρ þ iHρÞeiθa is an analytic signal. Indeed, by Lemma 3.4 the signal eiθa is circularly analytic, and we write eiθa ðtÞ ¼ ∑k∈Zþ g k eikt . Then by Lemma 3.2, we have for ξ∈R, F ððρ þ iHρÞeiθa ÞðξÞ ¼ ∑ g k F ðρ þ iHρÞðξkÞ:
ð3:7Þ
k∈Zþ
Note that ðρ þ iHρÞ is an analytic signal thus it has no negative spectrum, that is F ðρ þ iHρÞðξÞ ¼ 0 when ξ∈R . Consequently, when ξ∈R , ∑ g k F ðρ þ iHρÞðξkÞ ¼ 0;
^ ^ ρðξnÞ ¼ an ρðξÞ:
ð3:13Þ
Reversing the above calculations reveals that condition (3.13) is also sufficient for (3.10). Taking the conjugate of both sides of Eq. (3.10) and using ^ we obtain an equivalent the Hermitian property of ρ, equation ^ ¼ ð1a2 Þ ∑ ak1 ρðξ ^ þ kÞ aρðξÞ
ð3:14Þ
k∈N
for a.e. ξ∈R . A similar discussion to the case ξ∈R gives that Eq. (3.14) is equivalent to ^ þ nÞ ¼ an ρðξÞ ^ ρðξ
ð3:15Þ
for n∈Zþ and a.e. ξ∈R . Finally combining Eqs. (3.13) and (3.15) produces (3.8). □ Proposition 3.7.
k∈Zþ
which implies by Lemma 3.3 that iθa
k∈1þN
ð3:12Þ
1 za za ∮∂Δ dz ¼ Res ;0 kþ1 kþ1 2πi ð1azÞz ð1azÞz
The proof of this lemma is completed.
ð3:11Þ
k∈1þN
a2 ρðξ1Þ ^ ¼ ð1a2 Þ ∑ ak ρðξ1kÞ ^ ¼ ð1a2 Þ ∑ ak1 ρðξkÞ: ^
k
¼
ð3:10Þ
k∈N
leads to when k∈N, ck ¼
ð3:9Þ
sinca ∈S a
and
H sinca ∈S a :
iθa
H½ðρ þ iHρÞe ¼ iðρ þ iHρÞe : Simplifying both sides leads to H½ðHρÞeiθa ¼ i½ðHρÞeiθa , which indicates that Hρ∈S a . □ The next lemma indicates that any real-valued function ρ∈S a is completely determined by its spectrum ρχ ^ ð1;1Þ on the interval ð1; 1Þ.
Proof. By Lemma 3.5, it suffices to show sinca ∈S a . Observing the identity H a ð þ nÞ ¼ an H a ðÞ for a.e. ξ∈R\f0g, n∈Zþ and recall Eq. (2.9) we thus have F ðsinca Þðξ þ jnjsgnðξÞÞ ¼ ajnj F ðsinca ÞðξÞ: Hence by Lemma 3.6, we conclude that sinca ∈S a .
□
260
Q. Chen et al. / Signal Processing 94 (2014) 255–263
4. An observation – how a linear phase determines the amplitude In this section, we specifically consider the case when the phase ϕ in Eq. (1.4) is a linear phase and investigate the representation of the corresponding amplitude. The result is given in the following theorem. We use the notation suppðf Þ for the set of real numbers on which the values f(x) at x∈R are nonzero. Theorem 4.1. Suppose that γ is a positive real number and ρ is a non-zero real signal in L2 ðRÞ. Then the following equation: HðρðÞ cos ðγÞÞðtÞ ¼ ρðtÞ sin ðγtÞ
ð4:1Þ
^ holds if and only if ρ is bandlimited with suppðρÞ⊂½γ; γ.
and 1 1 1eiξ ∧ ðH : a ðÞÞ ðξÞ ¼ pffiffiffiffiffiffi 2π 1aeiξ iξ Proof. We start with the observation k k1 Hþ χ ½0;kÞ ðtÞ: a ðtÞ ¼ ∑ a χ ½k;kþ1Þ ðtÞ ¼ ð1aÞ ∑ a k∈Zþ
1a eikξ 1 ; ¼ pffiffiffiffiffiffi ∑ ak1 iξ 2π k∈N
Proof. Lemma 3.1 implies that Eq. (4.1) is equivalent to HðρðÞeiγ ÞðtÞ ¼ iρðtÞeiγt ;
ð4:2Þ
or equivalently, the complex signal ρðtÞeiγt is an analytic signal. Note that the Fourier transform of the function on ^ the left-hand side of Eq. (4.2) is given by i sgnðξÞρðξγÞ and the Fourier transform of the right-hand side of ^ Eq. (4.2) is iρðξγÞ. This leads to the equivalent equation of (4.1) in the frequency domain ðsgnðξÞ1ÞρðξγÞ ^ ¼ 0;
ð4:3Þ
i.e. ρðξÞ ^ ¼0
for
ξ∈ð1; γ:
ð4:4Þ
Thus if ρ is bandlimited with supp ρ⊂½γ; ^ γ, then clearly Eq. (4.4) is true, hence Eq. (4.1) is true. On the other hand, assuming that Eq. (4.1) is true, we obtain that ρðξÞ ^ ¼ 0 for ξ∈ð1; γ. Since ρ is real-valued, by the Hermitian ^ ¼0 property of the Fourier transform of a real signal, ρðξÞ for ξ∈½γ; þ 1Þ. Consequently we conclude that ρ is a bandlimited function with its support in the frequency domain belongs to ½γ; γ. □ Remark. From this theorem we know that the amplitude function ρ can be represented by shifts of sinc function if and only if Eq. (4.1) is true.
k∈N
By the definition of the Fourier transform (1.3), we obtain that Z 1a ∧ ðH þ ∑ ak1 χ ½0;kÞ ðt Þeiξt dt a ðÞÞ ðξÞ ¼ pffiffiffiffiffiffi 2π R k∈N Z 1a ¼ pffiffiffiffiffiffi ∑ ak1 eiξt dt 2π k∈N ½0;kÞ
where, the interchange of the order of the summation and the integral is justified by the absolute convergence of the series. Continue by noting ∑k∈N ak1 eikξ ¼ eiξ =ð1aeiξ Þ we have 1 1aeiξ ð1aÞeiξ ∧ ðH þ a ðÞÞ ðξÞ ¼ pffiffiffiffiffiffi iξð1aeiξ Þ 2π 1 1 1eiξ : ¼ pffiffiffiffiffiffi 2π 1aeiξ iξ þ Using the identity H a ðÞ ¼ H a ðÞ, we obtain that
1 1 1eiξ þ ∧ ∧ : ðH a ðÞÞ ðξÞ ¼ ðH a ðÞÞ ðξÞ ¼ pffiffiffiffiffiffi 2π 1aeiξ iξ
□
Lemma 5.2. The quasi-Bedrosian type identity
1þa p ðt ÞH sincðt Þ H pa ðÞ sincðÞ ðt Þ ¼ 1a a
ð5:3Þ
is true for t∈R and 1 eit 1 1 ¼ ðsinca ðt Þ þ iH sinca ðt ÞÞ: 1þa 1aeit it
ð5:4Þ
Proof. Let rðtÞ≔ð1=ð1aeit ÞÞðeit 1Þ=it. Separating the real part from the imaginary part of r(t) produces 1a sin t 1þa 1 cos t þi t 12a cos t þ a2 t 12a cos t þ a2 1 1 p ðt Þ sincðt Þ þ i p ðt ÞH sincðt Þ; ¼ ð5:5Þ 1þa a 1a a
r ðt Þ ¼ 5. How does non-linear phase determine amplitude? In this section, we shall completely characterize a realvalued function ρ∈S a . We begin with introducing two onesided filters that are related to the two-sided symmetric cascade filter Ha: Hþ a ðtÞ≔H a ðtÞχ Rþ ðtÞ;
H a ðtÞ≔H a ðtÞχ R ðtÞ;
ð5:1Þ
where t∈R. The next lemma provides the Fourier trans form pairs of H þ a and H a , respectively. Lemma 5.1. The Fourier transform of Hþ a and H a are given, respectively, by
1 1 1eiξ ∧ ; ðH þ a ðÞÞ ðξÞ ¼ pffiffiffiffiffiffi iξ iξ 2π 1ae
ð5:2Þ
where we have used the identity H sincðt Þ ¼
1 cos t : t
ð5:6Þ
pffiffiffiffiffiffi ∧ On the other hand, observepthat rðtÞ ¼ 2π ðH þ a Þ ðtÞ, so ffiffiffiffiffiffi þ ^ the Fourier transform r ðtÞ ¼ 2π H a ðtÞ has zero negative spectrum. This implies r is an analytic signal. Therefore the imaginary part of r(t) equals to the Hilbert transform of its real part, that is, 1 1 pa ðÞ sincðÞ ðt Þ ¼ p ðt ÞH sincðt Þ: H 1þa 1a a After rearranging the above equation we obtain Eq. (5.3). Now we turn to show Eq. (5.4). Applying Eq. (5.3) to the
Q. Chen et al. / Signal Processing 94 (2014) 255–263
that for t∈R,
imaginary part of Eq. (5.5) we continue to have that r ðt Þ ¼ ¼
1 p ðt Þ sincðt Þ þ iH pa ðÞ sincðÞ ðt Þ 1þa a
ρðtÞ ¼ ∑ r k sinca ðt2kπÞ þ ∑ sk ðH sinca Þðt2kπÞ: k∈Z
1 ðsinca ðt Þ þ iH sinca ðt ÞÞ; 1þa
which is Eq. (5.4).
□
Corollary 5.3. Let cosinca ðtÞ≔ð1 cos θa ðtÞÞ=t and define cosinca ð0Þ ¼ limt-0 ð1 cos θa ðtÞÞ=t, then for t∈R, H sinca ðtÞ ¼ cosinca ðtÞ:
ð5:7Þ
Moreover, the function cosinca is odd, bounded, infinitely differentiable, cosinca ðtÞ≤ðð1 þ jajÞ=ð1jajÞÞð3=jtj þ 1Þ for t∈R, and cosinca ∈L2 ðRÞ. Proof. Eq. (5.7) is just a new form of Eq. (5.3). Indeed, the left-hand side of Eq. (5.3) is just H sinca ðtÞ. Utilizing Eqs. (1.7) and (5.9) then the right-hand side of Eq. (5.3) is simplified to cosinca ðtÞ. The second statement follows by rewriting cosinca as cosinca ðt Þ ¼
1þa 1 cos t p ðt Þ ; 1a a t
ð5:8Þ
1 cos t t
ð5:9Þ
is infinitely differentiable if we define H sincð0Þ ¼ limt-0 ð1 cos tÞ=t, and jð1 cos tÞ=tj≤ð3=1 þ jtjÞ for any t∈R. □ An example of the graph of cosinca with a¼0.5 is shown in Fig. 2. As a comparison, the graph of H sincðtÞ ¼ ð1 cos tÞ=t is also shown in Fig. 2. The following lemma appeared in [4]. However, a completely new proof by direct construction is given here for both sufficiency and necessity. The construction approach casts a new insight to understand the relation of the spectrum ρ^ of a function ρ∈S a to the symmetric cascade filter Ha. Lemma 5.4. A real signal ρ∈S a if and only if there are two 2 real sequences r ¼ fr k : k∈Zg and s ¼ fsk : k∈Zg in l ðZÞ such 2
Proof. We first show the necessity, that is, if ρ∈S a , then ρ is given by Eq. (5.10). From Lemma 3.6, we know that ρðξÞχ Rþ ðξÞ and ρðξÞχ R ðξÞ are completely determined by ρðξÞχ ð0;1Þ ðξÞ and ρðξÞχ ð1;0Þ ðξÞ, respectively. Expand ρðξÞχ ð0;1Þ ðξÞ and ρðξÞχ ð1;0Þ ðξÞ into 1-periodic functions γ 1 ðξÞ≔∑k∈Z uk ei2kπt and γ 2 ðξÞ≔∑k∈Z vk ei2kπt , where the two series 2 ðuk : k∈ZÞ and ðvk : k∈ZÞ are in l ðZÞ. By Lemma 3.6 we thus obtain that for a.e. ξ∈R, ρðξÞ ^ ¼ γ 1 ðξÞH þ a ðξÞ þ γ 2 ðξÞH a ðξÞ:
The Hermitian property of ρ^ implies that for ξ∈R\f0g, γ 2 ðξÞ ¼ ðγ 1 ðξÞÞn : For convenience let γ ¼ γ 1 . Consequently ρ^ has the representation n ρðξÞ ^ ¼ γðξÞH þ a ðξÞ þ ðγðξÞÞ H a ðξÞ;
for a.e. ξ∈R. Let ρA be the analytic signal associated with ρ, which may be defined by
or equivalently for t∈R, ρA ðtÞ ¼ ρðtÞ þ iHρðtÞ: Then we have ρðtÞ ¼ ReðρA ðtÞÞ. We need to investigate ρA . The inverse Fourier transform of ρ^ A yields Z 1 iξt ρA ðt Þ ¼ pffiffiffiffiffiffi 2γ ðξÞH þ dξ a ðξÞe 2π R Z 2 iξt ¼ pffiffiffiffiffiffi ∑ uk ei2kπξ Hþ dξ: a ðξÞe 2π R k∈Z Applying the Lebesgue dominated convergence theorem to interchange the order of the integral and the sum, and appealing to Eq. (5.2) we obtain that Z 2 iξðt2kπÞ ρA ðt Þ ¼ pffiffiffiffiffiffi ∑ uk H þ dξ a ðξÞe 2π k∈Z R 2 1 eiðt2kπÞ 1 : ¼ pffiffiffiffiffiffi ∑ uk iðt2kπÞ iðt2kπÞ 2π k∈Z 1ae Denote uk ¼ uð1Þ þ iuð2Þ . Recalling Eq. (5.4) we readily obtain k k that
cosinca(t) Hsinc(t)
1.5
2 ρðt Þ ¼ pffiffiffiffiffiffi 2π ð1 þ aÞ
1 0.5 0 −0.5 −1 −1.5 −2 −20
−15
−10
−5
ð5:10Þ
k∈Z
^ ρ^ A ðξÞ ¼ 2ρðξÞχ Rþ ðξÞ
and noting the function H sincðt Þ ¼
261
0
5
10
Fig. 2. The graph of cosinca , a ¼0.5.
15
20
! ∑ uð1Þ sinca ðt2kπÞ ∑ uð2Þ ðH sinca ðÞÞðt2kπÞ : k k
k∈Z
k∈Z
pffiffiffiffiffiffi pffiffiffiffiffiffi Let r k ¼ ð2=ð 2π ð1 þ aÞÞÞukð1Þ and sk ¼ ð2=ð 2π ð1 þ aÞÞÞuð2Þ . k 2 Since the series ðuk ; k∈ZÞ∈l ðZÞ, we must have both series 2 ðr k ; k∈ZÞ and ðsk ; k∈ZÞ in l ðZÞ. We have arrived at Eq. (5.10). We now turn to the proof of sufficiency. It suffices to check that a function ρ∈L2 ðRÞ having the form (5.10) satisfies Eq. (3.8) by Lemma 3.6. Applying the Fourier transform to both sides of Eq. (5.10) and noting that the Fourier transform of the generalized sampling function given by Eq. (2.9) we get that ^ ¼ H a ðξÞðMðξÞi sgnðξÞGðξÞÞ ρðξÞ
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with the two 1-periodic functions pffiffiffiffiffiffi pffiffiffiffiffiffi 2π 2π ð1 þ aÞ ∑ r k ei2kπξ ; GðξÞ ¼ ð1 þ aÞ ∑ sk ei2kπξ : M ðξÞ ¼ 2 2 k∈Z k∈Z Observing the identity H a ð þ nÞ ¼ an H a ðÞ for n≥0 and the 1-periodicity of M and G lead to that, for ξ∈Rþ , ^ þ jnjÞ ¼ H a ðξ þ jnjÞðMðξ þ jnjÞiGðξ þ jnjÞÞ ρðξ ^ ¼ ajnj H a ðξÞðMðξÞiGðξÞÞ ¼ ajnj ρðξÞ: The case with negative ξ can be shown similarly. This completes the proof of the lemma. □ Theorem 5.5. If ρ∈S a , then both ρ and Hρ are continuous. Proof. By Proposition 3.5, Hρ∈S a , thus it suffices to show ρ∈S a is continuous. Indeed, if ρ∈S a , then ρ has the representation (5.10) by Lemma 5.4. Since both sinca and H sinca are in L2 ðRÞ by Proposition 2.1 and Corollary 5.3, consequently by the Cauchy–Schwarz inequality, the two series on the right-hand side of Eq. (5.10) converge uniformly, hence the limiting function ρ is continuous. □ Lemma 5.6. If f ∈S a , then for k∈Z, 〈f ; sinca ð2kπÞ〉 ¼ f ð2kπÞ;
and
〈f ; Hsinca ð2kπÞ〉 ¼ Hf ð2kπÞ:
Proof. The 2πperiodicity of sin θa yields Z
〈f ; sinca ð2kπ Þ〉 ¼
f ðt Þ R
sin θa ðt2kπÞ dt ¼ Hðf sin θa Þð2kπ Þ: t2kπ
Therefor by the assumption on f, we have when k∈Z, Hðf sin θa Þð2kπÞ ¼ f ð2kπÞ cos θa ð2kπÞ ¼ f ð2kπÞ because cos θa ð2kπÞ ¼ cos θa ð0Þ ¼ 1 by Eq. (2.7). The second equality follows by noting the fact that Hf ∈S a due to Proposition 3.5 and the Hilbert transformer is an anti-self adjoint operator thus 〈f ; H sinca ð2kπ〉 ¼ 〈Hf ; sinca ð2kπ〉 ¼ Hf ð2kπÞ:
&
Lemmas 5.4 and 5.6 and Theorem 2.2 immediately implies the following theorem. Theorem 5.7. Any real-valued function ρ∈S a if and only if ρðt Þ ¼
1a 1a ∑ ρð2kπ Þ sinca ðt2kπ Þ ∑ Hρð2kπ ÞH sinca ðt2kπ Þ: 1 þ a k∈Z 1 þ a k∈Z
Moreover, the sampling sequences 2 ðHρð2kπÞ; k∈ZÞ are in l ðZÞ.
ðρð2kπÞ; k∈ZÞ
and
From Theorems 5.7 and 2.2 we have the following corollary. Corollary 5.8. The system Φ defined in (2.13) is a complete orthogonal system of the subspace S a ⊂L2 ðRÞ. That is, the subspace equals to the closure of the span of the set Φ, or symbolically, S a ¼ spanΦ: Lastly, we state some facts for the case a ¼0. Note when a ¼0, sin θa ðtÞ becomes sin t, cos θa ðtÞ becomes cos t, sinca becomes sinc and H sinca becomes H sinc which is given at t∈R by Eq. (5.9). Therefore Theorems 4.1 and 5.7
and the well-known Shannon sampling theorem imply the following corollary. Corollary 5.9. The following statements are equivalent: 1. The real signal ρ∈L2 ðRÞ is bandlimited such that supp ρ^ D ½1; 1. 2. HðρðÞ cos ðÞÞðtÞ ¼ ρðtÞ sin t, t∈R. 3. HðρðÞei ÞðtÞ ¼ iρðtÞeit , t∈R. 4. ρðtÞ ¼ ∑k∈Z ρð2kπÞ sincðt2kπÞ∑k∈Z Hρð2kπÞ ð1 cos ðt2kπÞÞÞ=ðt2kπÞ, t∈R. 5. ρðtÞ ¼ ∑k∈Z ρðkπÞ sincðtkπÞ, t∈R.
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