Hilbert-Transformation

The Hilbert-Transformation plays an important role both in system theory and in communications. In system theory it is helpful for the description of causal and analytic signals and their spectra. In communications for example, the Hilbert-Transformer is used in modulators and demodulators.

In contrast to the Fourier-Transformation the Hilbert-Transformation is a tansformation within one domain.

It is defined as:



The inverse Hilbert-Transformation is the negative Hilbert-Transformation:



The Hilbert-Transformation is a linear operation. It can be characterized by its real impulse response hH (t) respectively by its imaginary transfer function HH (jw ) :

(see textbook at page 383)

As you can see the Hilbert-Transformer is an allpass. It only leads to a constant phase shift of +90° for negative and -90° for positive frequencies.






Causal/Analytic Signals




Causal Signals

A signal v(t) with v(t) = 0 for t < 0 is called causal.

Causality is an important property in system theory, because every system that is realizable has to be causal, i.e. its impulse response h(t) = 0 for t < 0.

This time condition has two effects for the frequency domain:

  • The unsymmetry of the time signal v(t) leads to a complex spectrum V(jw)
    • .
  • The real and the imaginary part of the spectrum depend on each other.
    They are connected by the Hilbert-Transformation:

Re{V(jw)} =  H{Im(V(jw))}
 Im{V(jw)} = -H{Re(V(jw))}

(see textbook at page 379)

Analytic Signals

The properties of causal signals and their spectra can be transformed to right-sided spectra and their appertaining time signals. A signal v(t) that has a right-sided spectrum (V(jw) = 0 for w < 0) is called analytic.

This condition means for the time domain:

  • An analytic signal has to be complex
    • .
  • The real and the imaginary part of the time signal v(t) depend on each other:

Re{v(t)} = -H{Im(v(t))}
Im{v(t)} =  H{Re(v(t))}

(see textbook at page 382)


Although analytic signals are complex and therefore seem to be very academic, they are used for the description of systems in communications and signal processing (e.g. sampling of bandpass signals).






Single-Sideband Modulation (SSM)



The Single-Sideband Modulation is an example for an application of the Hilbert-Transformation in communications. The SSM is used in the ham-radio and analog telephony for instance.
In contrast to the Double-Sideband Amplitude Modulation (AM) only one sideband -the lower or the upper- of a real source signal is transmitted . The redundancy of the symmetric spectrum of a real signal makes transmission possible without loosing information.

Advantage: only half the band width is required.

A single-sideband signal can be generated by the phase shift method for instance. Here the input signal passes two pathes:
  • In the first one a cosinus-modulation with the modulation frequency wm takes place.
  • In the second path a phase shift of ± 90° (Hilbert-Transformer) is employed followed by a sinus-modulation. As a consequence the sign of the upper sideband of the spectrum is changed, whereas the lower one stays unchanged.
Therefore the addition or subtraction of these two signals causes the cancellation of one sideband and so a single-sideband signal is generated (see textbook at page 388 ff.).

After transmitting this signal the original signal can be recovered by a synchronous demodulation for instance. This contains another cosinus-modulation followed by a low pass filter with the amplification factor 2. The low pass suppresses the spectral parts at 2·wm. Of course the recovering of the source signal strongly depends on the channel properties. For instance a transfer function with a phase depending on the frequency leads to a distorted output signal.


Single-Sideband Modulation






Sampling of Bandpass Signals



Another application of the Hilbert-Transformer is the sampling of bandpass signals.

For sampling of a real bandpass signal with the band limits w1 and w2 a sampling rate ws of at least 2·(w2 - w1) is necessary to avoid aliasing. Besides that the condition w1 = n·(w2 - w1)  (n e N). must be fulfilled.

real (and even) bandpass signal and its symmetric spectrum


In contrary a complex bandpass signal can be critically sampled with half the sampling rate, namely w2 - w1. Compared to real bandpass signals, sampling of a complex signal is always possible, as this doesn´t depend on the position of the band limits.

(See textbook at page 280 ff.)


complex bandpass signal and its spectrum


Thus sampling of a complex bandpass signal has some advantages over sampling of a real one.

But how do you get a complex signal with a one-sided spectrum from a real signal x(t)? Based on the theory of analytic signals (see chapter "analytic Signals") you receive the appertaining complex signal with

x1(t) = x(t) + j·H{x(t)}

x1(t) can always be critically sampled with the Nyquist rate w2 - w1. The resulting signal x2(t) contains no aliasing, because possibly existing aliasing of its real and imaginary part would cancel each other out. To recover the input signal again the real and the imaginary part of x2(t) have to pass a complex band pass filter with the band limits w1 and w2 first. The combination of the resulting complex signals x3(t) and x4(t) leads to the analytic signal x1(t):

x1(t) = x3(t) + j·x4(t)

With x(t) = Re{x1(t)} = Re{x3(t)} + Re{j·x4(t)} = Re{x3(t)} - Im{x4(t)} you receive the original signal x(t) again.


Sampling of Bandpass Signals (see textbook at page 384)