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

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

Contents of the Lecture Entire Semester:  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 Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Slide VI-2

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 Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Slide VI-3

Hilbert Transform Frequency and Impulse Response of a “Hilbert Transformer” – Part 1 Frequency-domain definition: 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: Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Slide VI-4

Hilbert Transform Frequency and Impulse Response of a “Hilbert Transformer” – Part 2 Frequency-domain definition (continued): 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). Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Slide VI-5

Hilbert Transform Frequency and Impulse Response of a “Hilbert Transformer” – Part 3 Frequency-domain definition (continued): By applying an inverse Fourier transform we obtain the impulse response of the filter: … derivation on the blackboard … Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Slide VI-6

Hilbert Transform Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 1 Definitions: 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! Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Slide VI-7

Hilbert Transform Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 2 Definitions (continued): For the spectrum of an analytic signal we get: … periodically repeated … The analytic signal has a one-sided spectrum ! This is very useful for a variety of applications. Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Slide VI-8

Hilbert Transform Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 3 Definitions (continued): 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: Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Slide VI-9

Hilbert Transform Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 4 Definitions (continued): 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. … 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) ... Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Slide VI-10

Hilbert Transform Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 5 Definitions (continued): 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 of the spectrum the analytic signal is uniquely defined by the real input sequence . Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Slide VI-11

Hilbert Transform Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 6 Example: 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 Loudspeaker signal is played back via one or more Amplifier loudspeakers. This allows for a better signal-to-noise ratio for Speaking person the listeners. However, since Feed- the loudspeaker signals might back paths couple back into the microphone a closed electro-acoustic loop is generated. Microphone Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Slide VI-12

Hilbert Transform Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 7 Example: 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“) Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Slide VI-13

Hilbert Transform Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 8 Example (continued): 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 Loudspeaker (about 3 to 10 Hz, realized by means Amplifier of a modulation) and finally compute the shifted output signal by means Speaking person of a second Hilbert transform. Feed- back paths Equalization and Microphone frequency shift Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Slide VI-14

Hilbert Transform Hilbert Transform, One-sided Spectra, and Analytic Signals – Part 9 Example (continued): Source: „The Big Bang Theory“, YouTube Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transfrom Digital Signal Processing and System Theory| Advanced Signals and Systems| Hilbert Transform Slide VI-15