The Hilbert Transform Library

Extension

Introduction

The Hilbert Transform is offered in various forms:

• HilbertW - via Weaver's Third Method.
• HilbertH - via Weaver's Second Method, known as Hartley Phasing.
• HilbertPDN - via Hartley Phasing, expressed as a 12th-order Phase Differencing Network.

These pseudo-UGens return phase-quadrature outputs in a form suitable for use as an analytic signal. Each form offers separate pseudo-UGens for returning just the real or imaginary output independently by appending Re or Im, respectively, to the class name. E.g., HilbertWRe & HilbertWIm.

Additionally, each class includes further Hilbert related transforms and analyses:

• Phase Rotation
• Single-Sideband Modulation (SSB)
• Instantaneous Amplitude Analysis
• Instantaneous Phase Analysis

Further Reading

Ansari, R. “IIR Discrete-Time Hilbert Transformers.” Acoustics, Speech and Signal Processing, IEEE Transactions On, vol. 35, no. 8, 1987, pp. 1116–1119.

Hutchins, B. “Calculation of the Poles of 90° Phase Difference Networks [after Weaver]” in Musical Engineer's Handbook: Musical Engineering for Electronic Music. Ithaca, NY: Electronotes. 1975. http://electronotes.netfirms.com/MEHCh6aPart.PDF, accessed 2017-08-08.

Hutchins, B. “The Design of Wideband Analog 90° Phase Differencing Networks without Large Spread of Capacitor Values”, Electronotes, Special Issue G, No. 168, http://electronotes.netfirms.com/EN168-90degreePDN.PDF, accessed 2017-08-08.

Picinbono, B. “On Instantaneous Amplitude and Phase of Signals.” Signal Processing, IEEE Transactions On, vol. 45, no. 3, 1997, pp. 552–560.

Schüssler, H., and W. Steffen. “Halfband Filters and Hilbert Transformers.” Circuits, Systems and Signal Processing, vol. 17, no. 2, 1998, pp. 137–164.

Smith, J.O. “Analytic Signals and Hilbert Transform Filters”, in Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications, Second Edition, https://ccrma.stanford.edu/~jos/st/Analytic_Signals_Hilbert_Transform.html, online book, 2007 edition, accessed 2017-08-08.

Weaver, Donald. “A Third Method of Generation and Detection of Single-Sideband Signals.” Proceedings of the IRE, vol. 44, no. 12, 1956, pp. 1703–1705.

US Patent 1,666,206, Modulation System, April 17, 1928, United States Patent and Trademark Office.