ï~~Multirate Additive Synthesis
Desmond PHILLIPS *
d.k.phillips @ Iboro.ac.uk
Alan PURVIS
alan.purvis @dur.ac.uk
Simon JOHNSON
simon.johnson @dur.ac.uk
Durham Music Technology *
School of Engineering, University of Durham, Science Laboratories,
South Road, Durham City, County Durham, DHl 3LE, UK.
Abstract
The advantages of musical signal modelling in frequency domain via sinusoidal additive synthesis are well
known, as is the prime disadvantage of high computational overhead. This paper proposes a reduction in
overhead by the application of standard multirate DSP techniques to optimise the sample rate of oscillators, with
a view to VLSI implementation. The optimisation works by adapting computation to oscillator frequency ranges,
which are often known a priori for partials in note-based music, and places few restrictions on functionality. The
cost of signal interpolation up to an industry-standard sample rate is minimised by grouping oscillators, spatially
related in the final sound image, to common synthesis QMF filterbanks which have a hierarchical sub-band
decomposition that balances interpolation cost with goodness-of-fit to expected oscillator frequency ranges.
Finally, an overview is provided of the mapping to a hypothetical "single chip" coprocessor.
1. Introduction
Additive Synthesis (AS) is a low-level transformation
that underpins all spectral modelling paradigms [Serra
& Smith, 1990]. The potential wide application and
simplicity of AS make VLSI an attractive solution.
Intriguingly, a problem is over-generality in the
context of musical signals which are computed with
substantial redundancy. However, optimising AS to
assumed signal properties (e.g. harmonicity) restricts
the application of AS to that particular signal class,
compromising generality and making VLSI less
attractive. To this end, a fresh solution is proposed -
Multirate Additive Synthesis (MAS) - that has a
graduated trade-off between generality and cost.
2. Background
AS computes the Inverse Fourier Transform (IFT) of
spectra comprising discrete lines, which is a property
of tonal sounds such as musical timbres. In eqn. (1)
y[n] is the sum of l<i<_N sinusoids, each modulated
by individual amplitude and frequency envelopes
A[n] and Fn]; n is a time index at a sample rate of
f>40kHz. By this means, the spectral evolution of
any tonal sound can be described over time.
N
y[n] =, Ai[n]sin(D1[n])
(i [n] = (Di[n-1]+21tFi[n]/fs
The prime disadvantage is that for N oscillators, 2N
envelope data streams are required at f. For example,
a 100-voice ensemble with 40 sinusoids per voice (to
ensure synthesis quality) at f,=44.lkHz requires a net
oscillator update rate of 176.4MHz and an envelope
data bandwidth of 352.8Mhz. Fortunately, envelope
data may be short-time averaged with little
perceptible loss of quality using a piecewise linear
approximation described by a compact breakpoint set.
This technique is reported as achieving data
compression ratios of 100:1 and is central to AS and
many spectral modelling paradigms [Serra & Smith,
1990].
However, for playback, uncompression is required in
real-time for AS via an oscillator bank. Control data
bandwidths of the magnitude discussed previously
recur and swamp the throughput of CPUs optimised
for the execution of high complexity instruction
streams. AS is a fine-grain data-parallel algorithm and
a traditional "form follows function" solution is to
map eqn. (1) into a direct-form dedicated coprocessor
exploiting deep pipelining, thus freeing CPU
throughput for high-level tasks more suited to its
design. A recently proposed alternative is to simulate
an oscillator bank in software via the Inverse Fast
Fourier Transform (IFFT) in an overlap-add scheme
[Rodet & Depalle, 1992]. The algorithm is
characterised by the data conditioning required to
generate smooth, linear inter-frame changes in A;[n],
Fi[n] [Goodwin & Kogon, 1995].
Durham Music Technology is a collaboration between the Schools of Engineering and Music. Desmond Phillips is now at
the Dept. of Electronic Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK.
Phillips et al.
496
ICMC Proceedings 1996
