Hilbert Transform Gerhard Schmidt Christian-Albrechts-Universitt zu - - PowerPoint PPT Presentation

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Sep 07, 2023 123 likes •371 views

Advanced Signals and Systems Hilbert Transform Gerhard Schmidt Christian-Albrechts-Universitt zu Kiel Faculty of Engineering Institute of Electrical and Information Engineering Digital Signal Processing and System Theory Digital Signal

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Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom

Gerhard Schmidt

Christian-Albrechts-Universität zu Kiel Faculty of Engineering Institute of Electrical and Information Engineering Digital Signal Processing and System Theory

Advanced Signals and Systems – Hilbert Transform

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Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform

Entire Semester:

Contents of the Lecture

 Introduction  Discrete signals and random processes  Spectra  Discrete systems  Idealized linear, shift-invariant systems  Hilbert transform  State-space description and system realizations  Generalizations for signals, systems, and spectra

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Contents of this Part

Hilbert Transform

 Frequency and impulse response of a “Hilbert transformer”  Frequency-domain definition  Impulse response  Hilbert transform, one-sided spectra, and analytic signals  Definitions  Example  Instantaneous amplitude, phase, and frequency  One-sided signals and causality

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Hilbert Transform

Frequency and Impulse Response of a “Hilbert Transformer” – Part 1

A Hilbert transformer is a special case of an ideal, linear-phase system. Such a system is used in several applications (e.g. modulation theory). The filter is defined by its frequency response: with Diagram:

Frequency-domain definition:

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Hilbert Transform

Frequency and Impulse Response of a “Hilbert Transformer” – Part 2

We obtain for the magnitude of the frequency response of a Hilbert filter: We can see that describes a nearly linear-phase all-pass filter. However, due to the „jumps“ at the filter is not free of distortions. The filter achieves a constant phase shift of ±90° with phase jumps at by 180° (in addition to a constant delay of samples).

Frequency-domain definition (continued):

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Hilbert Transform

Frequency and Impulse Response of a “Hilbert Transformer” – Part 3

By applying an inverse Fourier transform we obtain the impulse response of the filter:

Frequency-domain definition (continued):

… derivation on the blackboard …

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The meaning of the filter becomes more obvious if we look at the output of such a filter (please have a look also on the next slide): If we add now the input signal and the filtered input signal (after compensation for the filter‘s delay) we obtain the so-called analytic signal: Please note, that instead of using a „negative“ delay for the filtered signal a „positive“ delay can be applied to the input signal!

Hilbert Transform

Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 1

Definitions:

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For the spectrum of an analytic signal we get: The analytic signal has a one-sided spectrum! This is very useful for a variety of applications.

Hilbert Transform

Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 2

Definitions (continued):

… periodically repeated …

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Hilbert Transform

Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 3

The signal that is required for the one-sided spectral compensation is called Hilbert transform . For this transformation the delay is usually set to zero ( ) and a filter with a non-causal impulse response is used: This relation can be inverted easily (see next slide) and we obtain:

Definitions (continued):

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Hilbert Transform

Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 4

To understand the inversion of the Hilbert transform we start with the application of the Hilbert filter in the frequency domain. The following lines are restricted to the frequency range – outside this range periodical expansion is assumed.

Definitions (continued):

… exchanging both sides and dividing by the term in brackets with the sign function … … expanding the numerator with 1 ... … truncating the negative imaginary unit „-j“ ... … exploiting that a multiplication with a sign function is equal to a division with it (except at 0) ...

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Hilbert Transform

Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 5

Except for the sign we obtain the same relation as the original Hilbert transform. Thus, applying a Hilbert transform twice leads to the original signal multiplied with -1. Remarks:

 If we compute the analytic signal of a real input , we will obtain a complex

sequence . The sequences and are called a pair of Hilbert signals.

 Since for real sequences all information is in the „left“ as well as in the „right“ part

f the spectrum

the analytic signal is uniquely defined by the real input sequence .

Definitions (continued):

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Hilbert Transform

Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 6

In order to show some applications of the Hilbert transform we will mention now so-called public address systems as a first example. In such systems the signal of a speaking person is recorded by means of a

microphone. After amplification the

signal is played back via one or more

loudspeakers. This allows for a

better signal-to-noise ratio for the listeners. However, since the loudspeaker signals might couple back into the microphone a closed electro-acoustic loop is generated.

Example:

Amplifier Loudspeaker Microphone Feed- back paths Speaking person

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Hilbert Transform

Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 7

The transmission from the loudspeaker over the room to the microphone determines the maximum gain that can be used without generating a non-stable system (howling). Countermeasures and improvements:

 Equalization filters  Frequency shift filters

(„Hilbert transformer“)

Example:

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Hilbert Transform

Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 8

By means of an appropriate equalization usually a few decibels more gain can be achieved. However, also a frequency shift is able to increase the maximum gain. To realize the frequency shift, we can compute first the analytic signal, second shift this signal by a few Hertz (about 3 to 10 Hz, realized by means

f a modulation) and finally compute

the shifted output signal by means

f a second Hilbert transform.

Example (continued):

Amplifier Loudspeaker Microphone Feed- back paths Speaking person Equalization and frequency shift

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Hilbert Transform

Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 9

Example (continued):

Source: „The Big Bang Theory“, YouTube

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Hilbert Transform

Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 10

Example (continued):

Results: A gain increase of about 1 to 6 dB can be achieved (dependent on the type and size of the room). If larger shift frequencies would be applied also larger gains can be

achieved. However, the signal sounds

not very good any more …

0 Hz 5 Hz 10 Hz 15 Hz 20 Hz 25 Hz 30 Hz 35 Hz

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Hilbert Transform

Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 11

By means of a Hilbert transform we can change a real input signal to a complex analytic signal (assuming a delay-free Hilbert filter) and we obtain: In addition to that we know a well-known form for describing complex quantities: We can obtain the two quantities from above and compute . This term is called the instantaneous amplitude or the envelope of : In a similar way we can compute the instantaneous phase

Instantaneous amplitude, instantaneous phase, and instantaneous frequency

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Hilbert Transform

Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 12

Similar to the continuous case, where is termed the instantaneous frequency we define the instantaneous frequency of a discrete signal as

Instantaneous amplitude, instantaneous phase, and instantaneous frequency (continued)

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Hilbert Transform

One-sided Signals and Causality – Part 1

In the following we will assume a so-called right-hand sided signal , meaning that we have Such a signal can be decomposed into an even and an odd part:

Inverse relations

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Hilbert Transform

One-sided Signals and Causality – Part 2

Please add in the following diagram the even and the

dd part of the signal!

As we can see, we get the following relation between the even and the odd signal component (under our assumptions):

Inverse relations (continued)

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Hilbert Transform

One-sided Signals and Causality – Part 3

As a result we can conclude that either or describe the signal completely (except for the single value ). As a result also either the spectrum

fully describe the spectrum . If we transform the relation into the frequency (Fourier) domain, we obtain

Inverse relations (continued)

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Hilbert Transform

One-sided Signals and Causality – Part 4

This integral is also denoted as a Hilbert transform (even if slightly differently defined in comparison to its time-domain counterpart): In addition, we get

Inverse relations (continued)

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Hilbert Transform

One-sided Signals and Causality – Part 5

If the right-hand sided signal is real (in addition) we obtain the following relations for the spectra Thus, we obtain for the spectra of real, right-hand sided signals:

Inverse relations (continued)

The real and the imaginary part of the spectrum

f real, right-hand sided signals are a “pair” of Hilbert transforms

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Hilbert Transform

Contents of the Part on Hilbert Transforms

This part:

Next part:

 State-space description and system realizations