198
E. Gengel and A. Pikovsky
In the next section we describe a method to circumvent this problem by virtue
of iterated HT embeddings (IHTE) [12], illustrating the procedure with different
observables of the SL oscillator.
12.3.3 Iterated HT Embeddings
As discussed above, the HT embedding {X (t), ˆ
H [X (t)]} although does not provide
a closed looped line, allows one for an approximate determination of the protophase.
To accomplish this, one needs to define a variable monotonously growing along
the trajectory and gaining 2π at each approximate loop. A naive analytic-signalbased protophase arg(X + iY ) would work only for cosine-like waveforms like X 1,2 .
Therefore we employ another definition of the protophase, based on the trajectory
length [18]
L(t) =
t
0
˙
X 2 (τ ) + ˙
Y 2 (τ )dτ .
(12.14)
This length grows monotonously also in the case when the embedding has loops
(cf. Fig. 12.2), in which case the analytic-signal-based definition obviously fails.
Having calculated the length L(t), we can transform it to a protophase by interpolation. For this, we define in the signal features which we attribute to the zero (modulo 2π) protophase, and define the corresponding time instants t j . In the simplest
case, one can define a protophase θ(t) on the interval (t j , t j+1 ) as a linear function
of the length θ(t) = 2π j + 2π(L(t) − L(t j ))/(L(t j+1 ) − L(t j )). However, such a
protophase will be discontinuous in the first derivative. A better transformation is
achieved via splines: one constructs a spline approximation for the function θ(L),
provided one knows the values of this function at the signal features: θ(t j ) = 2π j at
L(t j ).
Constructed in this way, the protophase θ(t) is only approximate, because X (θ +
2π) = X (θ). Visually, on the plane {X, ˆ
H [X ]} one observes a band instead of a
single loop (see Fig. 12.2). Also, when X is plotted versus θ, one observes not a
single-valued function, but a band (see Fig. 12.1a).
Recently, in Ref. [12], we proposed to use iterative Hilbert transform embeddings
(IHTE) to improve the quality of the protophase definition above. Our idea is to perform subsequent Hilbert transforms based on the previously calculated protophases
θ n (t), where n denotes the step of iteration (see Fig. 12.3). Intuitively, the advantage
of iterations can be understood as follows: The widely used first iteration already
presents an approximation to the protophase, although not a perfect one. This means,
that the function X (θ 1 ) still has modulation, but less than X (t). Now, if we take θ 1
as a new time and again perform a demodulation by virtue of the Hilbert transform
embedding, we expect θ 2 (t) to be better than θ 1 (t), etc. A detailed analysis performed
in Ref. [12] shows that this procedure indeed converges to perfect demodulation.
In terms of iterations, the protophase θ(t) discussed above is the first iteration
θ 1 (t), while the time variable can be considered as the “zero” iteration θ 0 (t). At each
E. Gengel and A. Pikovsky
In the next section we describe a method to circumvent this problem by virtue
of iterated HT embeddings (IHTE) [12], illustrating the procedure with different
observables of the SL oscillator.
12.3.3 Iterated HT Embeddings
As discussed above, the HT embedding {X (t), ˆ
H [X (t)]} although does not provide
a closed looped line, allows one for an approximate determination of the protophase.
To accomplish this, one needs to define a variable monotonously growing along
the trajectory and gaining 2π at each approximate loop. A naive analytic-signalbased protophase arg(X + iY ) would work only for cosine-like waveforms like X 1,2 .
Therefore we employ another definition of the protophase, based on the trajectory
length [18]
L(t) =
t
0
˙
X 2 (τ ) + ˙
Y 2 (τ )dτ .
(12.14)
This length grows monotonously also in the case when the embedding has loops
(cf. Fig. 12.2), in which case the analytic-signal-based definition obviously fails.
Having calculated the length L(t), we can transform it to a protophase by interpolation. For this, we define in the signal features which we attribute to the zero (modulo 2π) protophase, and define the corresponding time instants t j . In the simplest
case, one can define a protophase θ(t) on the interval (t j , t j+1 ) as a linear function
of the length θ(t) = 2π j + 2π(L(t) − L(t j ))/(L(t j+1 ) − L(t j )). However, such a
protophase will be discontinuous in the first derivative. A better transformation is
achieved via splines: one constructs a spline approximation for the function θ(L),
provided one knows the values of this function at the signal features: θ(t j ) = 2π j at
L(t j ).
Constructed in this way, the protophase θ(t) is only approximate, because X (θ +
2π) = X (θ). Visually, on the plane {X, ˆ
H [X ]} one observes a band instead of a
single loop (see Fig. 12.2). Also, when X is plotted versus θ, one observes not a
single-valued function, but a band (see Fig. 12.1a).
Recently, in Ref. [12], we proposed to use iterative Hilbert transform embeddings
(IHTE) to improve the quality of the protophase definition above. Our idea is to perform subsequent Hilbert transforms based on the previously calculated protophases
θ n (t), where n denotes the step of iteration (see Fig. 12.3). Intuitively, the advantage
of iterations can be understood as follows: The widely used first iteration already
presents an approximation to the protophase, although not a perfect one. This means,
that the function X (θ 1 ) still has modulation, but less than X (t). Now, if we take θ 1
as a new time and again perform a demodulation by virtue of the Hilbert transform
embedding, we expect θ 2 (t) to be better than θ 1 (t), etc. A detailed analysis performed
in Ref. [12] shows that this procedure indeed converges to perfect demodulation.
In terms of iterations, the protophase θ(t) discussed above is the first iteration
θ 1 (t), while the time variable can be considered as the “zero” iteration θ 0 (t). At each