Analytical signal

In signal theory, an analytic signal is a complex-valued function of time, the imaginary part of which is the Hilbert transform of the real part. The term analytical expresses that the function is differentiable in the complex (see analytical function ). This means that in contrast to a real signal, no negative frequencies occur in the spectrum of an analytical signal . The analytical signal represents a special case from the group of monogenic signals .

Applications of analytical signals in signal processing are in the area of single sideband modulation .

definition

Transfer function by which an analytical signal is formed.

If x ( t ) is a real time signal with its Fourier transform X (jω), then a spectrum X _a (jω) with purely positive frequencies can be obtained therefrom by multiplying with the step function σ (ω).

${\displaystyle {\begin{aligned}X_{a}(\mathrm {j} \omega )&=X(\mathrm {j} \omega )\cdot 2\sigma _{\frac {1}{2}}(\omega )\\&=X(\mathrm {j} \omega )+X(\mathrm {j} \omega )\cdot \operatorname {sgn}(\omega )\end{aligned}}}$

Here, ω denotes the angular frequency and j the imaginary unit . The inverse Fourier transformation follows the convention of asymmetrical normalization. Furthermore, the formation rule for the analytical time signal x _a ( t ) from the time signal x ( t ) is shown. ${\displaystyle {\mathcal {F}}^{-1}}$

${\displaystyle {\begin{aligned}x_{a}(t)&=x(t)+\mathrm {j} \,x(t)\ast {\frac {1}{\pi t}}\\&=x(t)+\mathrm {j} \,{\mathcal {H}}\{x(t)\}\\&=x(t)+\mathrm {j} \,{\hat {x}}(t)\\\end{aligned}}}$

Thereby make the convolution operation and the Hilbert transform . In an analytical signal the imaginary bears with respect to the real part no additional information content . ${\displaystyle \ast }$ ${\displaystyle {\mathcal {H}}}$

example

The real time signal x ( t ), consisting of a cosine oscillation , has the analytical signal x _a ( t ). The Fourier transformation of Euler's identity shows that x _a ( t ) has a one-sided spectrum without negative frequencies .

${\displaystyle {\begin{aligned}&x(t)=\cos(\omega _{0}t)&\Rightarrow \quad &x_{\mathrm {a} }(t)=\cos(\omega _{0}t)+\mathrm {j} \,\sin(\omega _{0}t)=\mathrm {e} ^{\mathrm {j} \omega _{0}t}\\&\Downarrow &&\Downarrow \\&X(\mathrm {j} \omega )={\frac {\delta (\omega +\omega _{0})}{2}}+{\frac {\delta (\omega -\omega _{0})}{2}}&&X_{a}(\mathrm {j} \omega )=\delta (\omega -\omega _{0})\\\end{aligned}}}$

Representations

A modulated signal m ( t ) in blue and the magnitude curve A ( t ) of the associated analytical signal

Like any complex number, the analytic signal can also be expressed in a complex polar representation .

${\displaystyle x_{\mathrm {a} }(t)={\underline {\gamma }}(t)=A(t)\mathrm {e} ^{\mathrm {j} \varphi (t)}\,}$

Here, γ ( t ) is referred to as the complex envelope , A ( t ) as the magnitude envelope and φ ( t ) as the instantaneous phase.

${\displaystyle A(t)=|x_{\mathrm {a} }(t)|={\sqrt {x^{2}(t)+{\hat {x}}^{2}(t)}}}$

${\displaystyle \varphi (t)=\arg \left\{x_{\mathrm {a} }(t)\right\}}$

System with polar modulator

This representation is important in communications engineering, since it can be used to control a polar modulator, whereas the real and imaginary part is suitable for controlling a Cartesian modulator ( IQ modulator ). Due to the interaction with a suitably designed power amplifier ( English Power Amplifier abbreviated PA ), the first-mentioned system has a better efficiency.

A modulated signal m ( t ) is generated from the carrier frequency and the complex envelope according to the following equation. ${\displaystyle \Omega _{T}}$

${\displaystyle m(t)=\Re \left({\underline {\gamma }}(t)\cdot \mathrm {e} ^{j\Omega _{T}t}\right)=A(t)\cdot \cos(\Omega _{T}t+\varphi (t))}$

modulation

Modulation and demodulation of a complex signal

A complex signal can be impressed on the carrier frequency through multiplication , which creates the complex modulated signal . The demodulation is done by multiplication with a complex pointer that rotates in the opposite direction. ${\displaystyle \Omega _{T}}$ ${\displaystyle {\underline {g}}(t)}$ ${\displaystyle {\underline {m}}(t)}$

${\displaystyle {\underline {m}}(t)={\underline {g}}(t)\cdot \mathrm {e} ^{\mathrm {j} \Omega _{T}t}}$

${\displaystyle {\underline {g}}(t)={\underline {m}}(t)\cdot \mathrm {e} ^{-\mathrm {j} \Omega _{T}t}}$

${\displaystyle \mathrm {e} ^{\mathrm {j} \Omega _{T}t}\cdot \mathrm {e} ^{-\mathrm {j} \Omega _{T}t}=1}$

The real part and the imaginary part each require their own transmission path. In practice this is often not done. Calculation shows that the modulated signal is an analytical signal, i.e. the imaginary part is redundant to the real part, so that only one of the two has to be transmitted. ${\displaystyle {\underline {m}}(t)}$

${\displaystyle {\underline {m}}(t)=\left[\Re ({\underline {g}}(t))+\mathrm {j} \Im ({\underline {g}}(t))\right]\ \cdot \ \left[\cos(\Omega _{T}t)+\mathrm {j} \sin(\Omega _{T}t)\right]}$

Assuming that x ( t ) is a band-limited signal, the frequency components of which have an amplitude of zero above , then the following Hilbert transformation applies: ${\displaystyle \omega _{0}}$

${\displaystyle {\mathcal {H}}\{x(t)\cdot \cos(\omega _{0}t)\}=x(t)\cdot \sin(\omega _{0}t)}$

${\displaystyle {\underline {m}}(t)=m(t)+\mathrm {j} {\mathcal {H}}\{m(t)\}}$

The imaginary part can be regenerated on the receiver side by the Hilbert transformation.

${\displaystyle m(t)=\Re \left({\underline {g}}(t)\right)\cdot \cos(\Omega _{T}t)-\Im \left({\underline {g}}(t)\right)\cdot \sin(\Omega _{T}t)}$

${\displaystyle {\underline {g}}(t)=\left[m(t)+\mathrm {j} {\mathcal {H}}\{m(t)\}\right]\cdot \mathrm {e} ^{-\mathrm {j} \Omega _{T}t}}$

In addition to the method shown, there are other options for generating and resolving the same modulated signal.

literature

• Karl-Dirk Kammeyer, Kristian Kroschel: Digital signal processing . 6th edition. Teubner, 2006, ISBN 3-8351-0072-6 .

Web links

• Analytic signal and Hilbert transform filter (Engl.)