Hilbert transformation

In functional analysis , a branch of mathematics , the Hilbert transformation is a linear integral transformation . It is named after David Hilbert , who formulated it at the beginning of the 20th century while working on the Riemann-Hilbert problem for holomorphic functions .

It is used in the area of Fourier transformation and Fourier analysis . Further areas of application are in the area of signal processing , in which it is used to create an analytical signal or a monogenic signal from a real signal . The general phase shift of the imaginary part compared to the real part by π / 2 or 90 ° is characteristic.

Blue: Signal curve
Red: Hilbert transformation of the blue signal

definition

The Hilbert transformation is for real variables and and for real or complex valued functions and is defined as: ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle f}$ ${\displaystyle g}$

${\displaystyle g(y)={\mathcal {H}}\left\{{f}\right\}\left({y}\right)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {f(x)}{y-x}}\mathrm {d} x}$

This integral has the form of a convolution integral , so that the Hilbert transformation with the convolution operator can also be written in the following form: ${\displaystyle \ast }$

${\displaystyle g(y)={\mathcal {H}}\left\{{f}\right\}\left({y}\right)=f(y)\ast {\frac {1}{\pi y}}}$

This transformation is reversible. The inverse Hilbert transformation is given by:

${\displaystyle {\mathcal {H}}^{-1}\left\{{g}\right\}=-{\mathcal {H}}\left\{{g}\right\}=f}$

properties

Some essential properties of the Hilbert transformation for real variables and for real or complex functions or are: ${\displaystyle t}$ ${\displaystyle x}$ ${\displaystyle y}$

Linearity

${\displaystyle {\mathcal {H}}\left\{a\cdot x+b\cdot y\right\}=a\cdot {\mathcal {H}}\left\{x\right\}+b\cdot {\mathcal {H}}\left\{y\right\}}$

Filtering

${\displaystyle {\mathcal {H}}\left\{x\ast y\right\}={\mathcal {H}}\left\{x\right\}\ast y=x\ast {\mathcal {H}}\left\{y\right\}}$

Relationship to the Fourier transform

The reference to the Fourier transformation plays an important role, especially in communications engineering and its signal processing . For this, the transformation pairs are of interest in both directions. In the following, the notation common in engineering is used for the imaginary unit . In mathematics, the notation is common for the imaginary unit . It applies to the characteristic identity . ${\displaystyle \mathrm {j} }$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {j} }$ ${\displaystyle {\mathrm {j} }^{2}=-1}$

asymmetrical normalization		Transformation with frequency
${\displaystyle {\mathcal {F}}\left\{{\frac {1}{\pi t}}\right\}=-\mathrm {j} \cdot \operatorname {sgn} (\omega )=e^{-\mathrm {j} {\frac {\pi }{2}}\operatorname {sgn} (\omega )}}$ ${\displaystyle {\mathcal {F}}^{-1}\left\{{\frac {1}{\pi \omega }}\right\}=\mathrm {j} \cdot {\frac {1}{2\pi }}\cdot \operatorname {sgn} (t)}$		${\displaystyle {\mathcal {F}}\left\{{\frac {1}{\pi t}}\right\}=-\mathrm {j} \cdot \operatorname {sgn} (f)=e^{-\mathrm {j} {\frac {\pi }{2}}\operatorname {sgn} (f)}}$ ${\displaystyle {\mathcal {F}}^{-1}\left\{{\frac {1}{\pi f}}\right\}=\mathrm {j} \cdot \operatorname {sgn} (t)}$

Hilbert transformation as a transfer function in the frequency domain

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

${\displaystyle {\hat {x}}(t)={\mathcal {H}}\left\{x(t)\right\}=x(t)\ast {\frac {1}{\pi t}}}$

${\displaystyle {\hat {X}}(\mathrm {j} \omega )=X(\mathrm {j} \omega )\cdot (-\mathrm {j} \cdot \operatorname {sgn} (\omega ))}$

This leads to the transfer function

${\displaystyle H_{H}(\mathrm {j} \omega )=-\mathrm {j} \cdot \operatorname {sgn} (\omega )\,}$.

In this context, the Hilbert transformation can be understood as a phase shift by (or + 90 °) for negative frequencies and by (or −90 °) for positive frequencies. Telecommunications applications are in the area of modulation methods , in particular single sideband modulation as a component of an analytical signal . The technical implementation takes place approximately in the form of special all-pass filters , which are also referred to as Hilbert transformers . ${\displaystyle {\tfrac {\pi }{2}}}$ ${\displaystyle -{\tfrac {\pi }{2}}}$

Discrete Hilbert Transformation

A band-limited signal also limits the Hilbert transform of to the same bandwidth. If the band limit is a maximum of half the sampling frequency , a time-discrete sequence with positive and whole numbers can be formed according to the Nyquist-Shannon sampling theorem without loss of information . The discrete Hilbert transformation is then given as: ${\displaystyle g\left(t\right)}$ ${\displaystyle g\left(t\right)}$ ${\displaystyle g\left[k\right]}$ ${\displaystyle k}$

${\displaystyle {\mathcal {H}}\left\{g[k]\right\}=h[k]\ast g[k]}$

with the impulse response of the time-discrete Hilbert transformation: ${\displaystyle h\left[k\right]}$

${\displaystyle h[k]={\frac {1-\mathrm {cos} (\pi k)}{\pi k}}={\begin{cases}0,&k{\text{ gerade}},\\{\frac {2}{\pi k}}&k{\text{ ungerade}}\end{cases}}}$

The time-discrete Hilbert transformation is not causal; for practical implementations in the context of digital signal processing where this form plays a role, the implementation is approximately finite length. It should be noted that the time-discrete impulse response does not correspond to the sampled, continuous impulse response . ${\displaystyle h\left[k\right]}$ ${\displaystyle h\left[k\right]}$ ${\displaystyle h\left(t\right)}$

Causality condition in the frequency domain

A system can be fully described by the impulse response. If the causality condition is to be fulfilled, then the impulse response for the time before the excitation must have the value zero. This can be expressed abstractly by multiplying with the step function .

${\displaystyle h(t)=h(t)\cdot \sigma _{\frac {1}{2}}(t)}$

The corresponding transfer function in the frequency domain can be determined from the impulse response by means of Fourier transformation . This ultimately leads to a convolution integral that corresponds to the Hilbert transformation. ${\displaystyle H}$

${\displaystyle H(\mathrm {j} \omega )=-\mathrm {j} {\mathcal {H}}\left\{H(f)\right\}=-\mathrm {j} \cdot \left[\Re \left({\mathcal {H}}\left\{H(f)\right\}\right)+\mathrm {j} \Im \left({\mathcal {H}}\left\{H(f)\right\}\right)\right]=\Re \left(H(f)\right)+\mathrm {j} \Im \left(H(f)\right)}$

The causality conditions for any transfer function follow from this:

${\displaystyle \Re \left(H(f)\right)={\mathcal {H}}\left\{\Im \left(H(f)\right)\right\}}$

and

${\displaystyle \Im \left(H(f)\right)=-{\mathcal {H}}\left\{\Re \left(H(f)\right)\right\}}$

Correspondence

Some important correspondences of the Hilbert transformation are: (Note: The requirements such as valid value range or definition range have been omitted for the sake of clarity.)

signal ${\displaystyle x(t)\,}$	Hilbert transform ${\displaystyle {\mathcal {H}}\{x(t)\}}$
${\displaystyle \sin(t)\,}$	${\displaystyle -\cos(t)\,}$
${\displaystyle \cos(t)\,}$	${\displaystyle \sin(t)\,}$
${\displaystyle {\frac {1}{t^{2}+1}}}$	${\displaystyle {\frac {t}{t^{2}+1}}}$
${\displaystyle {\frac {\sin(t)}{t}}}$ Sinc function	${\displaystyle {\frac {1-\cos(t)}{t}}}$
${\displaystyle \sqcap (t)}$ Rectangle function	${\displaystyle {1 \over \pi }\ln \left|{\frac {t+{\frac {1}{2}}}{t-{\frac {1}{2}}}}\right|}$
${\displaystyle \delta (t)}$ Dirac delta function	${\displaystyle {\frac {1}{\pi t}}}$
${\displaystyle e^{-t^{2}}}$	${\displaystyle e^{-t^{2}}\operatorname {erfi} (t)}$ Imaginary error function erfi

implementation

Calculation via Fourier transformation

For practical implementations, the discrete Hilbert transformation of a real number sequence of length can be approximated using the discrete Fourier transformation : First, the Fourier transform of the input sequence is calculated, then all spectral components that represent negative frequency components are included in the calculated spectrum 0 set. Finally, the inverse Fourier transformation is used to calculate the output sequence. ${\displaystyle N}$

1. Calculation of the Fourier transform of the input sequence with the length . The Fast Fourier Transformation (FFT) is used for reasons of efficiency . ${\displaystyle X\left[k\right]}$ ${\displaystyle N}$
2. Creation of a vector of length that only has the values ​​0, 1 and 2 according to the following rule: ${\displaystyle H\left[k\right]}$ ${\displaystyle N}$
   • ${\displaystyle H[k]=1}$ For ${\displaystyle k=1,\,{\tfrac {N}{2}}+1}$
   • ${\displaystyle H[k]=2}$ For ${\displaystyle k=2,\,3,\ldots ,\,{\tfrac {N}{2}}}$
   • ${\displaystyle H[k]=0}$ For ${\displaystyle k={\tfrac {N}{2}}+2,\ldots ,\,N}$
3. Formation of the element-wise products ${\displaystyle Y\left[k\right]=H\left[k\right]\cdot X\left[k\right]}$
4. Compute the inverse Fourier transform of to determine the output sequence. ${\displaystyle Y\left[k\right]}$

Calculation with FIR filter

Hilbert transform filter (FIR) with 6th order

Alternatively, the Hilbert transformation can also be implemented approximately with FIR filters of an even order in the form of an all-pass filter , as shown in the figure on the right for a Hilbert transformation filter of the 6th order. It can be seen here that with Hilbert transform filters the odd filter coefficients always have the value 0, and the remaining even filter coefficients (for even n) can be combined in pairs with inverted signs due to reasons of symmetry. The output signal (I component) is only delayed in time in the filter in order to be in phase with the filtered signal (Q component). The combination so formed ${\displaystyle \alpha _{0},\,\alpha _{2},\,\alpha _{4},\ldots ,\alpha _{n}={\tfrac {2}{\pi (n+1)}}}$ ${\displaystyle y_{I}\left[k\right]}$ ${\displaystyle y_{Q}\left[k\right]}$

${\displaystyle y_{I}[k]+\mathrm {j} \cdot y_{Q}[k]}$

is called the analytic signal of the real-valued input signal . ${\displaystyle x\left[k\right]}$

Functional analysis

The Hilbert transformation has some meaning as an operator between function spaces. It is a non-trivial fact that the Hilbert transform defines a bounded operator for . ${\displaystyle {\mathcal {H}}\colon L^{p}(\mathbb {R} ){\stackrel {\mathcal {H}}{\to }}L^{p}(\mathbb {R} )}$ ${\displaystyle 1<p<\infty }$

The Hilbert transform is an isometric isomorphism (for a unitary operator ) and satisfies the equation , where is the identical mapping . ${\displaystyle p=2}$ ${\displaystyle {\mathcal {H}}^{2}=-\mathbb {I} }$ ${\displaystyle \mathbb {I} }$

The Hilbert transformation is not weakly restricted for for , but for already. ${\displaystyle {\mathcal {H}}\colon L^{p}(\mathbb {R} ){\stackrel {\mathcal {H}}{\to }}L^{p}(\mathbb {R} )}$ ${\displaystyle p\in (1,\infty ]}$ ${\displaystyle p=1}$

Relationship to the Kramers-Kronig relations

The Kramers-Kronig relations of physics are obtained with the formal identity (see distribution (mathematics) )

${\displaystyle {\frac {1}{x}}=\lim \limits _{\varepsilon \to 0}{\frac {1}{x+\mathrm {i} \varepsilon }}=\lim \limits _{\varepsilon \to 0}{\frac {x}{x^{2}+\varepsilon ^{2}}}-\mathrm {i} \cdot \lim \limits _{\varepsilon \to 0}\,{\frac {\varepsilon }{x^{2}+\varepsilon ^{2}}}\,,}$

where the first part results in the integration via the Cauchy principal value CH of and the second part results in -fold the Dirac distribution . ${\displaystyle x}$ ${\displaystyle {\tfrac {1}{x}}}$ ${\displaystyle \pi }$ ${\displaystyle \delta }$

The Hilbert transformation is used when a real function is to be continued from the real axis ${\displaystyle \mathbb {R} }$ to a holomorphic function in the complex half-plane above it .

literature

• Karl Dirk Kammeyer: MATLAB in telecommunications . J. Schlembach Fachverlag, 2001, ISBN 3-935340-05-2 .
• Bernd Girod, Rudolf Rabenstein, Alexander KE Stenger: Introduction to systems theory: signals and systems in electrical engineering and information technology . 4th edition. Teubner Verlag, Wiesbaden 2007, ISBN 978-3-8351-0176-0 .

Web links

• Julius O. Smith III Analytic Signals and Hilbert Transform Filters , Stanford University (engl.)
• Python SciPy.org: Hilbert transformation with example for envelope calculation (scipy.signal.hilbert)

Individual evidence

1. ^ S. Lawrence Marple: Computing the discrete-time analytic signal via FFT , IEEE Transactions on Signal Processing, Issue 47, No. 9, September 1999, pp. 2600-2603.