Methods for Tone and Signal Synthesis

1
Methods for Tone and Signal Synthesis

• R.C. Maher
• ECEN4002/5002 DSP Laboratory
• Spring 2002

2
Signal Synthesis

• So far in this course, we have considered signal
  processing input signal sent through LTI system
• For signal synthesis, we generally do not have an
  explicit input signal output is generated by an
  algorithm

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Applications of Synthesis

• Synthesis is useful in DSP systems for
• Communications signals (modulation, coherent
  detection, modem tones, etc.)
• Test signals (function generator, noise test
  signals, transfer function measurement)
• Warning and notification tones
• Music synthesis

4
General Synthesis Types

• Algorithmic synthesis
• Pseudo-random signals (noise with specified
  statistical properties)
• Periodic signals using a mathematical formula
  (transcendental functions, discrete summation)
• Look-up table synthesis
• Signal synthesized by reproducing samples from
  data in memory a wavetable

5
Random Signal Generation

• Want an algorithm to produce a sequence of
  numbers that appear to be chosen randomly
• Usually use a periodic sequence with a period
  sufficiently long that the repetition is not a
  problem sequence is pseudo-random
• Method applicable to DSP chips is the linear
  congruential algorithm

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Linear Congruential Method

• A cheap and not particularly rigorous method to
  make pseudo-random sequences
• Uses a recursive method and a modulo
• For numbers between 0 ? y ? (M-1), select y0
  (seed) and values for multiplier a,
  displacement c, and modulus M. Not all choices
  give maximal-length sequence!!
• yn(ayn-1 c)mod M

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Periodic Signal Algorithm

• Generate the signal by implementing a
  discrete-time formula
• Works well for square, pulse, triangle, sawtooth
  waveforms, but these are generally not
  bandlimited
• If sufficient processing power available, can
  approximate an arbitrary function (cosine,
  exponential, etc.)

8
Digital Sinusoid Generator

• Consider a filter with hnRn sin(w0n) u(n),
  which is a causal decaying sinusoid for 0ltRlt1,
  and a pure sinusoid for R1
• w02pf/fs
• Start with impulse input

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Stored Signal Synthesis

• Concept fill a delay line with one cycle of the
  desired waveform, then loop the signal over and
  over to produce the extended output signal
• Vary the frequency by decimation / interpolation
  of the stored samples
• Vary the amplitude by multiplying a slowly
  varying envelope function

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Wavetable Frequency Control

• Assume table is length N, holds one period of the
  signal, and the sample rate is fs
• Output frequency for entire table is fs/N

New sample every 1/fs Table frequency fs/N
Wavetable (N)
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Frequency Control (cont.)

• To get a different output frequency from the same
  stored wavetable, define a sample increment (SI)
  and a lookup accumulator (LUA).
• SI is typically a fixed-point number (integer
  part and fractional part) defined by

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Frequency Control (cont.)

• Lookup accumulator (LUA) has an integer part and
  a fractional part. The integer part is modulo-N
  counts zero to N-1, then rolls over to zero.
• At each sample time, add SI to LUA, then use
  integer part of LUA as lookup index into stored
  wavetable. Can use fractional part of LUA for
  inter-sample interpolation

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Wavetable Lookup Example

• fs48000 Hz, N1024, desired fout440 Hz
• SI1024?440 / 48000 9.386667

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Implementation on DSP

• Load a buffer with one cycle (length N) of the
  waveform
• Define SIinteger and SIfraction variables
• Define LUAinteger and LUAfraction variables
• For each sample
• Add SIfractional to LUAfractional
• Add with carry SIinteger to LUAinteger
• If LUA?N, subtract N from LUAinteger
• Fetch wavetable sample at offset LUAinteger

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Details to watch out for

• Wavetable lookup (resampling) with SIgt1 will
  cause aliasing if wavetable contains components
  above fs/(2SI)
• Choosing N to be a power-of-2 makes LUA modulo a
  simple masking operation
• Frequency resolution determined by number of
  fractional bits in LUA

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Linear Interpolation

• LUA contains desired location of next output
  sample. If LUAfraction ? 0, we need to
  interpolate the table.
• Ideally, need to do bandlimited (Shannon)
  interpolation, but often settle for cheap linear
  interpolation between adjacent table samples

yL1
yL
L
L1
L fract
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Basic Wavetable Synthesizer
Wavetable
Gain Multiplier
Sample Increment (SI)
Envelope
18
Musical Frequency Scale

• Most Western music uses a Well-Tempered Scale
  for the 12 notes of the diatonic scale
• Scale based on note A 440 Hz
• All other A note frequencies are multiples of 2n,
  where n is an integer (A110, 220, etc.)
• Other notes (B, B, C, etc.) related by 2k/12
• Factor 21/12 1.059463

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Musical Scale (cont.)

• Example for A4440 Hz, N1024, fs48000