INTRODUCTIONSwept phase-shifting devices have become a widely-used and effective means of modifying the tonal and spatial characteristics of musical signals. In anticipation of all-digital musical and audio-processing systems in the near future, the phase-shifting effect has been implemented using the Digital Filter Module of the Bell Labs Digital Synthesis System. The digital method of "phase-shifting" has its own set of peculiarities, advantages, and disadvantages, and these will be discussed. Digital filter models and appropriate coefficients are presented for use in other digital processing systems.ANALOG PHASE-SHIFTERThe analog phase-shift system is already well known in the electronic music field, and will be summarized here as a background for the digital implementation. For a more detailed treatment of the mathematical aspects of analog phase-shifters, see Reference 1.Originally used as a substitute for a variable time-delay in musical vibrato and "rotating sound" systems, the variable phase-shift element is a unity-gain amplifier which provides a phase-shift of -360 degrees at low frequencies, approaching -180 degrees at higher frequencies. A popular implementation of the single phase-shift stage, its transfer functions and phase-frequency response are shown in Figure I (See also Reference 2). These phase-shift stages are cascaded (usually in multiples of 2) and the final output is summed with the input signal, resulting in cancellations in the frequency response. For six phase-shift stages with identical time-constants the frequency response and system diagram is shown in Figure 2.
In the present case, it was decided to simulate the characteristics of the six-stage analog phase shifter, using the Digital Filter Module available on the Bell Labs Digital Synthesis System. Digital filter stages approximating the transfer function of the phase-shift stage (Figure 1) are cascaded and connected in the same configuration as described in Figure 3. using digital filter sections and envelope-multipliers available on the Bell Labs Digital Synthesizer. The filter system is a time-multiplexed real-time processor which provides thirty-two second-order filter stages operating at a 30 kHz sampling rate. The module interfaces to an LSI-II computer by means of a IK-word address space in the LSI-II memory. By means of this interface, the module is configured in terms of the filter coefficients, interconnection of filter stages, and other control signals. A simplified diagram of the basic filter stage is shown in Figure 5. Overload protect, precision rectification, full-scale limiting, and specialized interconnect functions are available; a more complete specification may be found in Reference 3. Each of the filter sections is completely configured by a 72-bit program. The program specifies the filter coefficients (a0, a1, a2, B1, B2) and a number of other processing options. Thirty-two of these programs are available, and are used here for specifying phase-shift stages with different time-constants. In order to provide a filter with dynamically variable properties, two The individual filter stages may be interconnected and cascaded in a wide variety of configurations. Used as a sub-section of the Bell Labs Digital Synthesizer, the filter programs and interconnections are specified by means of a special operating system designed for the synthesizer by Doug Bayer (See Reference 4). The operating system enables the user to specify all programs and configurations in a high-level language (Bell Labs "C" compiler) with minimum attention to hardware-oriented details, and integrates the filter module into the totality of the Digital Synthesis System. This configuration uses only one-half of the available processing in filter section , so another digital filter simulating two analog stages was constructed (Figure 8):
The envelope-generators ENVI and ENV2 are each composed of a program-controllable digital multiplier (with coefficients between zero and 1) and a summing junction (the multiplier coefficient is 1) for the input signal and the phased signal, and can also function as a switch or mixer for the phased signal. ENV2 is controlled by a front-panel slider or a program variable, and controls the amount of positive feedback around the phasing loop. The 30kHz sampling rate of the digital system, and the associated rolloff of the anti-aliasing filters in the 14-bit ADC system presently used, provide a slightly compromised frequency-response characteristic. Rolloff begins at about 13 kHz, and thus the extreme high-end of the audible frequency response is attenuated. Analog systems can currently provide better high-frequency performance. The dynamic range of the present system, which uses 16-bit binary arithmetic, is also slightly inferior to the best available analog systems. The theoretical dynamic range is about 90 dB using internally generated 16-bit signals, or 78 dB using signals from the 14-bit ADC in the Bell Labs Synthesizer. Two other factors further limit these values: the use of positive feedback around the phasing loop will reduce the effective dynamic range in proportion to the overall feedback gain, and the results of rounding errors in the arithmetic will further reduce the signal-to-noise ratio. The results of the arithmetic (rounding) errors increase at low input signal frequencies, and also for the lower values of frequency response. In these cases, the ratio of change in the input values to the limits of the arithmetic resolution becomes significant, and instability can result. Therefore the dynamic range becomes quite compressed under these conditions. The rounding and scaling errors are proportionally worse for the two-stage filter sections than for the one-stage filters, since second order calculations are involved in the former case. MUSICAL APPLICATIONS, AND DIRECTIONS FOR FURTHER RESEARCHThe conventional notion of the Phase-shifting effect is that of a periodic or "rotating" effect which lends interest and motion to musical signals. Less commonly used variations are envelope or pedal-controlled sweep of the filters to provide multi-pole dynamic filter effects, or multiple channel phasing to provide spatial vibrato effects.The accuracy and versatility of the digital system, especially in the context of a programmable multi-channel synthesizer, introduces additional possibilities. Among these are: 1). Multi-pole filter effects in which the filters emphasize definite harmonics of the musical signal 2). Precisely controllable spatial vibrato and quadraphonic rotation effects 3). Stepped or "arpeggiated" sweep patterns to produce overtone "harmonics" within individual notes or chords. The additional programmable control of the mixing and feedback functions provides a facility for dynamic variations in the intensity or prominence of the effect. Many combinations and extensions of these techniques should provide materials for creative new uses of "phase-shifting." In order to make the digital approach more viable in terms of frequency response and dynamic range, two hardware modifications are indicated: 1). Sampling rate could be increased to 40 kHz or more, at the expense of the number of available filter stages. At present, a quadraphonic system of four six-stage filters uses at most 24 of the 32 available filter sections on the Bell Labs Module. 2). Internal arithmetic could be expanded beyond the present 16-bit architecture. To provide excellent 16-bit "terminal" characteristics, an architecture of 24 bits binary or 18 bits floating point is indicated. ACKNOWLEDGMENTSThis research was made possible by the generous assistance and guidance of the following people: H. G. Alles, Greg Sims, Doug Bayer and J. F. Kaiser of Bell Telephone Laboratories at Murray Hill, N. J.; and R. L. Kosut of Singer Company, Binghamton, N. Y.REFERENCESW. M. Hartmann, "Flanging and Phaser's" J. Audio Eng. Soc., vol. 26, pp 439-443, (June 1978) Jerald G. Graeme, Applications of Operational Amplifiers pp 102-103. McGraw-Hill Book Co. c 1973 H. G. Alles and W. C. Fischer, "A New General Purpose Digital Filter" October 7, 1976, Bell Telephone Laboratories Douglas L. Bayer, "Real-Time Software for a Digital Music Synthesizer" Computer Music Journal Volume I, Number 4 |

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