152
Chapter 4
Convolution of Pulsars with Samples Pulsar synthesis can be harnessed as a method of sound transformation through convolution. Convolution is fundamental to the physics of waves (Rabiner and Gold 1975). It ``crosses'' two signals, creating a new signal that combines the time structures and spectra of both inputs. Many transformations emerge from convolution, including exotic ®lters, spatializers, models of excitation/ resonance, and a gamut of temporal transformations (echoes, reverberation, attack-smoothing, rhythm-mapping). Pure convolution, however, has no control parameters, that is, the type of e¨ect achieved depends entirely on the nature of the input signals. See Roads (1992b, 1993a, 1997) for applications of convolution in musical sound transformation. Sophisticated transformations involving rhythm- and spatial-mapping can be achieved through convolution. It is well known that any series of impulses convolved with a brief sound maps that sound into the time pattern of the impulses. These impulses can be emitted by a pulsar generator. If the pulsar train frequency is in the infrasonic range, then each pulsar is replaced by a copy of the sampled sound object, creating a rhythmic pattern. The convolution of a rhythmic pattern with a sound object causes each impulse to be replaced by a ®ltered copy of the sound object. Each instance of the sampled object is projected in space according to the spatial location of a speci®c pulsar's position. In convolution, each pulsar represents the impulse response of a ®lter. Thus timbral variations can derive from two factors: (1) ®ltering e¨ects imposed by the time-varying pulsar train, and (2) overlapping e¨ects caused by convolution with pulsar trains whose fundamental period is shorter than the duration of the sampled sound. Figure 4.19 shows the temporal and ®ltering e¨ects of convolution in the form of sonograms. The input signal (b) is the Italian word qui (pronounced ``kwee''). It convolves with the pulsar train (a) with a variable infrasonic fundamental frequency and a variable audio formant frequency. The resulting convolution (c) combines the time structure and the spectra of the two signals. The composer can stockpile a database of sampled sound objects for crossing with trains selected from the pulsar database. If the goal of the synthesis is to retain the time structure of the pulsar train (e.g., to maintain a speci®c rhythm), the sampled sound objects should be of short duration (less than the fundamental period of the pulsar train) and have a sharp attack (a rise time of less than 100 ms). These constraints minimize the time-smearing e¨ects of convolution (Roads 1992b, 1993a, 1997). A good starting point for a sound database