The Analytic Function from the Hilbert Transform and End Effects Theory and Implementation

End Effects In both Hilbert Transform (HT) and EMD

End Effects End effects are problems for any data analysis method. Traditionally, window with tapered ends were used to alleviate the effects. As data are hard to collect, I proposed window frame: to extend the data beyond the available range. This approach involves predicting data based on available points.

End Effects in HHT There are two kind of end effects in HHT: the one in Hilbert Transform and the one in Empirical Mode Decomposition. The end effects in Hilbert Transform is relative easy to deal with, for it involves in Fourier Transform. The end effects in EMD is much harder to resolve, for it involves spline fittings.

End Effects in HT Hilbert Transform also suffers end effect, for it is equivalent to two Fourier Transforms.

End Effects in EMD (I) The end effects arise from the end points are not extrema. The simplest way to alleviate the effects is to extend the data beyond the available range to the next two extrema, using trigonometric functions with the amplitudes and periods equal the mean of the three oscillations next to the ends.

End Effects in EMD (II) The second way is to use linear prediction based on ARMA, which guarantees the same spectral shape of the extended part with the original data. The third way is to match the pattern of the end section of the data with some interior sections, and repeat the pattern at the ends. All prediction method are based on the stationary assumption, a luxury we do not have. But without it nothing works.

Observations In principle, the end effects of either HT or EMD are ill- posed: to make predications for nonstationary and nonlinear processes. The correction of end effects is based on rectifications empirically: by extending the data to the next zero with both ends having the same sign of slopes (for HT); to a couple of extrema (for EMD). In practice, however, the end effects of Hilbert Transform is relatively easy to fix, for HT involves Fourier Transform and applies only to IMFs. The end effects for EMD is relatively harder, for the first rectifying step involves nonstationary data, and applies to spline fitting.

Hilbert Transform : Definition

The Analytic function fits data at a point

Implementation

Implementation of Hilbert Transform : Analytic signal Leon Cohen : Time-frequency Analysis, Prentice Hall, 1995

An Example of Hilbert Transform x (t) = Sin ω t

Data : Sin ω t

FFT : Sin ω t

Hilbert Transform : Sin ω t

Because the implementation of Hilbert Transform is through FFT and IFFT, the End Effects are very important.

End Effects in Hilbert The First Example

Data Model Equation : Whole cycle

FFT Model Equation : Whole cycle

Positive FFT Model Equation : Whole cycle

Positive IFFT Model Equation : Whole cycle

Data Model Equation : Partial cycle

FFT Model Equation : Partial cycle

IFFT Model Equation : Partial cycle

Hilbert Transform : Whole vs Partial cycle

Model Equation : Hilbert Transform vs Quadrature

Data Model Equation : Ends

Hilbert Model Equation : Ends of same values and same slope

End Effects with Sine Waves The Second Example

Data

Hilbert Transform

Instantaneous Frequency

Rectifications The rectification process involves extending the data to the next zero at both ends with trigonometric functions. Make sure that the slope at the ends have the same sense.

Modified Hilbert Transform

Instantaneous Frequency : Modified

Instantaneous Frequency : Modified Details

Instantaneous Frequency

Problems for IF from Hilbert Transform Singularity points

Singularity points for Instantaneous Frequency

Singularity for two sinusoidal waves

Examples of IF from Analytic Functions FM and AM Frequencies a sin ω t + b sin φ t

This is simple example for Fourier analysis, but a more intriguing one for instantaneous frequency analysis. Depending on the relative magnitudes of a,b and ω, φ, we can have a variety of cases. Instantaneous Frequency through analytic function gives only the FM frequency. The AM frequency is defined by the envelope.

sin ω t sin 4 ω t

Hilbert : sin ω t sin 4 ω t

sin ω t sin 4 ω t

Hilbert : sin ω t sin 4 ω t

sin ω t + sin 4 ω t

Hilbert : sin ω t + sin 4 ω t

a sin ωt + b sin φt The data need to be sifted first. Whenever analytic function has a loop away from the original (negative maximum or positive minimum), there will be negative frequency. Whenever the analytic function pass through the original (both real and imaginary parts are zero), there will be a frequency singularity. The analytic function derived from the Hilbert Transform is local to a degree of 1/t.

IMF : sin ω t sin 4 ω

IMF : sin ω t sin 4 ω

IMF : sin ω t + sin 4 ω

Empirical Mode Decomposition method gives Instantaneous Frequency computation from an analytic function a new life

Limitation on IF computed from Analytic Functions Data need to be mono-component. Traditional applications using band-pass filter, which distorts the wave form. Bedrosian and Nuttall Theorems.

Bedrosian Theorem Let f(x) and g(x) denotes generally complex functions in L 2 (-∞, ∞) of the real variable x. If (1) the Fourier transform F(ω) of f(x) vanished for │ω│> a and the Fourier transform G(ω) of g(x) vanishes for │ω│< a, where a is an arbitrary positive constant, or (2) f(x) and g(x) are analytic (i. e., their real and imaginary parts are Hilbert pairs), then the Hilbert transform of the product of f(x) and g(x) is given H { f(x) g(x) } = f(x) H { g(x) }. Bedrosian, E., 1963: A Product theorem for Hilbert Transform, Proceedings of the IEEE, 51,

Nuttall Theorem For any function x(t), having a quadrature xq(t), and a Hilbert transform xh(t); then, where Fq(ω) is the spectrum of xq(t). Nuttall, A. H., 1966: On the quadrature approximation to the Hilbert Transform of modulated signal, Proc. IEEE, 54, 1458

Normalized HHT To overcome some of the limitations on compute IF from analytic functions, we proposed Normalized HHT. To overcome the end effects we introduced modified Hilbert Transform, in which data were extended beyond the end points by adding characteristic waves to make the extension always end at zero and with the same signs of slopes. The combination have improved the IF computation greatly.