An analytical signal is a complex signal theory time signal whose imaginary part is the Hilbert transform of the real part. The term analytical expresses that the function is differentiable in the complex (see the analytical function). It follows that in the spectrum of the analytical signal as opposed to a real signal no negative frequencies occur. Analytical signal represents a special case of the group of signals monogenic
Applications of analytic signals lie in the signal processing in the range of modulation techniques, such as VSB modulation. This is a form of single-sideband modulation with respect to the digital signal processing.
X ( t) is a real time signal, with its Fourier transform X ( j? ), Then it can be obtained a spectrum of Xa ( j? ) With purely positive frequencies by multiplying by the step function occurs σ ( ω ).
Here ω denotes the angular frequency and j is the imaginary unit. The inverse Fourier transformation is performed according to the convention of the single-ended normalization. Furthermore, in the formation rule shows, for the analytical time signal xa (t) from the time signal x (t).
Thereby provide * the convolution operation, and the Hilbert transform dar. In an analytical signal of the imaginary part carries with respect to the real part of any additional information.
The real time signal x (t), consisting of a cosine wave, having the analytical signal xa (t). The Fourier transformation of the Euler's identity is found that xa (t) has a one-sided spectrum with no negative frequencies.
Like any complex number, the analytical signal can be expressed in complex polar representation.
In this case, γ (t) is the complex envelope of the A ( t) than the Betragseinhüllende and φ ( t) as the instantaneous phase.
Of importance, this representation is in communications technology, as is to allow a control a polar modulator, whereas the real and imaginary parts suitable for driving a Cartesian modulator (IQ ) modulator. Through the interaction with a suitably constructed power amplifier (English Power Amplifier abbreviated PA), the former 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.
By multiplication of the carrier frequency can be applied to a complex signal impressing, whereby the complex -modulated signal is produced. Demodulation is carried out by multiplication by a complex phasor which rotates in the opposite direction.
The real part and the imaginary part each require its own transmission path. In practice, however, as something like this proves unfeasible, since only one signal may indicate a transmission frequency. By calculation shows that there are m (t ) is an analytic signal when modulated signal, ie the imaginary part redundantly present to the real part, which only needs to be entrusted by both.
Provided with x (t) is a band-limited signal whose frequency components have an amplitude greater than zero, then the following is true Hilbert transform:
The imaginary part can be regenerated by the receiver side, the Hilbert transform.
In addition to the method shown there are other options for creating and dissolving the same modulated signal.