Adding two sinusoids of the same
frequency
CMPT 368: Lecture 3
Additive Synthesis
Tamara Smyth, tamaras@cs.sfu.ca
School of Computing Science,
Simon Fraser University
• Adding two sinusoids of the same frequency but with
possibly different amplitudes and phases,
produces another sinusoid at that frequency.
January 16, 2008
3
sin(4πt)
2sin(4πt + π/4)
sum
Amplitude
2
1
0
−1
−2
−3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Figure 1: Adding two sinusoids of the same frequency.
1
CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
Mathematical Proof
Spectrum
• Recalling the expression for sinusoidal motion and
trigonometric identities1 , we may see that
• The spectrum of a signal is a graphical
representation of the frequency components it
contains and their complex amplitudes.
x = A sin(ω0t + φ) = A sin(φ + ω0t)
= [A sin φ] cos ω0t + [A cos φ] sin ω0t
= B cos ω0t + C sin ω0t,
• Signals may therefore be represented as the sum of N
sinusoids of arbitrary amplitudes, phases, AND
frequencies:
where the amplitude A is given by
√
A = B 2 + C 2,
and the phase φ is given by
φ = tan
−1
x(t) =
C
.
B
N
X
Ak cos(ωk t + φk )
k=0
• We may therefore, synthesize a sound by setting up a
bank of oscillators, each set to the appropriate
amplitude, phase and frequency.
Note: The derivation of A and φ will be given in a
subsequent lecture.
• The output of each oscillator is added together to
produce a synthesized sound, and thus the synthesis
technique is called additive synthesis.
Every sinusoid can be expressed as the sum
of a sine and cosine function, or equivalently,
an “in-phase” and “phase-quadrature” component.
1
2
For the homework you will need the identity cos(A + B) = cos(A) cos(B) − sin(A) sin(B)
CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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Additive Synthesis
Additive Synthesis Caveat
• Additive synthesis provides the maximum flexibility in
the types of sound that can be synthesized.
• In certain cases, it can realize tones that are
“indistinguishable from real tones.”
• Drawback: it often requires many oscillators to
produce good quality sounds, and can be very
computationally demanding.
• Also, many functions are useful only for a limited
range of pitch and loudness.
• Signal analysis is often a prerequisite for additive
synthesis of specific sounds. This will allow you to
determine the amplitude, phase and frequency
functions.
– For example, the timbre of a piano played at A4 is
different from one played at A2;
– similarly, the timbre of a trumpet played loudly is
quite different from one played softly at the same
pitch.
• Therefore, this technique is also sometimes called
Fourier recomposition.
• It is possible however, to use some knowledge of
acoustics to determine functions.
– For example, in specifying amplitude envelopes for
each of the oscillators, it is useful to know that in
many acoustic instruments, the higher harmonics
attack last and decay first.
CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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Creating Different Periodic Waveforms
Amplitude
1
• Different “standard” periodic waveforms can be
created using additive synthesis.
0.5
0
−0.5
−1
Table 1: Other Simple Waveforms Synthesized by Adding Cosine Functions
Type
Harmonics
Amplitude
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
Time (s)
Phase (cos) Phase (sin)
square
Odd, n = [1, 3, 5, ..., N]
1/n
−π/2
0
triangle
Odd, n = [1, 3, 5, ..., N]
1/n2
0
π/2
Even and Odd, n = [1, 2, 3, ..., N]
1/n
−π/2
0
sawtooth
Amplitude
2
1
0
−1
−2
0
0.1
0.2
0.3
0.4
0.5
Time (s)
Amplitude
2
1
0
−1
−2
0
0.1
0.2
0.3
0.4
0.5
Time (s)
Figure 2: Summing sinusoids to produce other simple waveforms
CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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Harmonics and Pitch
Square Wave Spectrum
Magnitude
1
0.8
0.6
• Notice that even though these new waveforms contain
more than one frequency component, they are still
periodic.
0.4
0.2
0
0
5
10
15
Frequency (Hz)
Triangle Wave Spectrum
• Because each of these frequency components are
integer multiples of some fundamental frequency
f0, they are called harmonics.
Magnitude
1
0.8
0.6
• Signals with harmonic spectra have a fundamental
frequency and therefore have a periodic waveform
(the reverse is, of course, also true).
0.4
0.2
0
0
5
10
15
Frequency (Hz)
Sawtooth Wave Spectrum
• Pitch is our subjective response to the fundamental
frequency.
Magnitude
1
0.8
• The harmonics contribute to the timbre of a sound,
but do not necessarily alter the pitch.
0.6
0.4
0.2
0
0
5
10
15
Frequency (Hz)
Figure 3: Spectra of complex waveforms
CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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Relating to Acoustics
10
Pitch
• In acoustics, to excite a higher harmonic of an
instrument, your excitation mechanism, whether it be
blowing through a mouthpiece, plucking or bowing a
string, must contain a component at a frequency
matching that harmonic.
OPEN−OPEN
f
PRESSURE VARIATIONS
CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
2f
• There is a nonlinear relationship between pitch
perception and frequency.
• Listeners usually compare tones on the basis of the
musical interval separating them. For example, the
pitch interval of an octave corresponds to a frequency
ratio of 2:1.
• We will often encounter a pitch notation wich
designates a pitch with an octave: C4 is middle C.
3f
4f
• We often hear of the pitch A4 as “A440”, or A at 440
Hz. The pitch one octave below, A3 is therefore 220
Hz.
Figure 4: Vibrational modes of a tube open at both ends.
CLOSED−OPEN
• In equal tempered tuning, there are 12 evenly spaced
tones in an octave, called semi-tones.
PRESSURE VARIATIONS
f
3f
• How do you calculate the pitch 1 semitone above
A440?
5f
7f
Figure 5: Vibrational modes of a tube open at one end and closed at the other.
CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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• The resulting waveform shows that there is a periodic,
low frequency amplitude envelope superimposed on a
higher frequency sinusoid.
Beat Notes
• What happens when we add two frequencies that are
not harmonically related?
Beat Note Waveform (f0 = 220 Hz, f1 = 2 Hz)
1
0.8
• The Beat note comes about by adding two sinusoids
that are very close in frequency.
0.6
0.4
Spectrum of Beat Note
0.2
Amplitude
1
0.9
0.8
0
−0.2
0.7
−0.4
Magnitude
0.6
0.5
−0.6
0.4
−0.8
0.3
−1
0
0.2
0.4
0.6
0.2
0.8
1
1.2
1.4
1.6
1.8
2
Time (s)
0.1
Figure 7: Waveform of a Beat Note.
0
205
210
215
220
225
230
235
Frequency (Hz)
• What is going on?
Figure 6: Beat Note made by adding sinusoids at frequencies 218 Hz and 222 Hz.
CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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Multiplication of Sinusoids
• What happens when we multiply a low frequency
sinusoids with a higher frequency sinusoid? Begin by
using the inverse Euler formula:
x(t) = sin(2π(220)t) cos(2π(2)t)
j2π(220)t
j2π(2)t
e
− e−j2π(220)t
e
+ e−j2π(2)t
=
2j
2
i
1 h j2π(222)t
−j2π(222)t
j2π(218)t
e
−e
+e
− e−j2π(218)t
=
4j
1
= [sin(2π(222)t) + sin(2π(218)t)]
2
which is a sum of real sine functions.
• From this we can now see that there will be four
frequency components in the spectrum (including the
negative frequencies), none of which are the two
multiplied frequency components. Rather, the
spectrum has their sum and the difference.
• Sinusoidal multiplication can therefore be expressed
as an addition (which makes sense because all signals
can can be represented by the sum of sinusoids).
CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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CMPT 368: Computer Music Theory and Sound Synthesis: Lecture 3
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