Hilbert transform

The Hilbert transform is in the functional analysis, a branch of mathematics, a linear integral transformation. It is named after David Hilbert, which they put in work on the Riemann - Hilbert problem for holomorphic functions in the early 20th century.

It is applied in the range of the Fourier transform and the Fourier analysis. Other areas of application are in the range of signal processing in which it serves to form an analytic signal or a monogenic signal of a real signal.

  • 4.1 Relationship to the Kramers- Kronig relations

Definition

The Hilbert transform is defined for real variables x and y and for real - or complex-valued functions f and g as:

This integral has the form of a convolution integral, so that the Hilbert transform can be written using the convolution operator * in the following form:

This transformation is reversible. Inverse Hilbert transform is given by:

Properties

Some essential properties of the Hilbert transform on real variable t and for real or complex functions are x and y, respectively:

Relation to the Fourier transform

In particular in communications technology, and the signal processing with respect to the Fourier transform plays a crucial role. This transformation pairs in both directions of interest.

Considered is now the convolution operation in the time domain, which corresponds to the multiplication in the frequency domain.

This leads to the transfer function

The Hilbert transform may be regarded in this context as a phase shift of π / 2 (or 90 degrees) and negative frequencies to - π / 2 (or -90 ° ) for positive frequencies. Nachrichtentechnische applications are in the range of modulation techniques, and in particular the single-sideband component as an analytical signal. The technical implementation is done approximately in the form of special all-pass filters, which are also known as Hilbert transformers.

Discrete Hilbert transform

A band-limited signal g (t) limited the Hilbert transform of g (t ) in the same bandwidth. When the band limiting maximum half the sampling frequency, according to the Nyquist -Shannon sampling theorem can be used without loss of information, a time- discrete sequence g [ k] are formed with K a positive integer. The discrete Hilbert transform is given as:

With the impulse response h [k] of the time-discrete Hilbert transform:

The time-discrete Hilbert transformation is not causal - for practical implementations in the context of digital signal processing where it plays a role in the form, H [k] is implemented by means of an approximately finite length. Note that the time-discrete pulse response h [k ] is not equal to the sampled continuous impulse response h (t).

Causality condition in the frequency range

By the impulse response of a system can be fully described. If the condition of causality are met, then the impulse response for the time before the excitation must have the value zero. Abstract this can be expressed by a multiplication with the step function.

Fourier transformation in the frequency domain can be determined from the impulse response of the transfer function corresponding to H. This eventually leads to a convolution integral, which corresponds to the Hilbert transform.

From this follow the causality conditions for an arbitrary transfer function:

And

Correspondences

Some important correspondence from the Hilbert transform are: ( Note: The requirements as valid value range or domain have been omitted for clarity. )

Implementation

For practical implementations, the discrete Hilbert transform of a real number sequence of length N using the discrete Fourier transform (DFT) can be approximately achieved: Firstly, the Fourier transform of the input sequence is calculated, then, in the calculated spectrum of all spectral components for negative frequency components are set to 0. Finally ( IDFT ) is computed, the output sequence by means of the inverse Fourier transform.

  • Value of 1 for the index i = 1 ( N / 2) 1
  • 2 value for the index i = 2, 3, ..., ( N / 2)
  • A value of 0 for the index i = ( N / 2) 2, ..., N

Alternatively, even order can be realized in the form of an all-pass FIR filters with the Hilbert transform in approximation, as shown in the illustration for a Hilbert transform filter the 6th order. Be seen here that Hilbert transform filters are always the odd Filterkoeffizenten value of 0, and the remaining straight filter coefficients α0, α2, α4 ... can be summarized in pairs with inverted sign due to symmetry reasons. The output signal yI [k ] (I- component) is the filter time only delayed by the filtered signal yQ [k ] ( Q component ) to be in phase. The combination thus formed

Is referred to as an analytical signal of the real-valued input signal x [k].

Functional Analysis

The Hilbert transformation has as an operator between function spaces of some importance. It is a non-trivial fact that the Hilbert transform a limited operator for defined.

The Hilbert transform is unitary and satisfies the equation, where is the identity mapping.

The Hilbert transform is restricted to non, however weak.

Relationship with the Kramers- Kronig relations

The Kramers- Kronig relations in physics is obtained with the formal identity (see Distribution (mathematics) )

Wherein the first portion of the integration over the Cauchy principal value of x CH of (1 / x) and the second part results in the π times the δ - function.

The Hilbert transform then is used when a real function of the real axis to the overlying half-space holomorphic complex function should be continued.

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