What is an analytic signal

Analytical signal

 

A analytical signal is a complex-valued function of time in signal theory, the imaginary part of which is the Hilbert transform of the real part. The designation analytically 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.

Is x(t) a real time signal with its Fourier transformX(jω), then a spectrum can be derived from it Xa(jω) can be obtained with purely positive frequencies by multiplying with the step function σ (ω).

Here, ω denotes the angular frequency and j the imaginary unit. The inverse Fourier transform takes place according to the convention of asymmetrical normalization. In addition, the formation specification for the analytical time signal is shown xa(t) from the time signal x(t).

Ask the represents the Hilbert transformation. In the case of an analytical signal, the imaginary part has no additional information content in relation to the real part.

example
 

The real time signal x(t), consisting of a cosine oscillation, has the analytical signal xa(t). The Fourier transform of Euler's identity shows that xa(t) has a one-sided spectrum with no negative frequencies.

Representations

A modulated signal m(t) in blue and the history of the amount A.(t) of the associated analytical signal

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

Thereby γ (t) as the complex envelopes, A.(t) as the Amount envelope and φ (t) as the Momentary phase designated.

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). Through the interaction with an appropriately designed power amplifier (EnglishPower amplifier abbreviated PA), the former system has a better efficiency.

A modulated signal m(t) is derived from the carrier frequency and the complex envelope is generated according to the following equation.

modulation

Modulation and demodulation of a complex signal

The carrier frequency can be obtained by multiplication a complex signal imprint, creating the complex modulated signal arises. The demodulation is done by multiplication with a complex pointer that rotates in the opposite direction.

The real part and the imaginary part each require their own transmission path. In practice, this is often not done. Calculation shows that this is the case with the modulated signal m(t) is an analytical signal, i.e. the imaginary part is redundant to the real part, which means that only one of the two has to be transmitted.

Provided with x(t) is a band-limited signal whose frequency components are above have an amplitude of zero, then the following Hilbert transformation applies:

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

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

Based on an article in: Wikipedia.de
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Date of the last change: Jena, 21.02. 2020