
A
sinusoid is any function having the following form:
where

is the independent (real) variable, and the fixed parameters

,

, and

are all real constants. In audio
applications we typically have
An example is plotted in Fig.
4.1.
The term ``peak amplitude'' is often shortened to ``amplitude,''
e.g.,
``the amplitude of the tone was measured to be 5
Pascals.'' Strictly
speaking, however, the amplitude of a
signal 
is its instantaneous
value

at any time

. The peak amplitude

satisfies

. The ``instantaneous magnitude'' or simply
``magnitude'' of a signal

is given by

, and the peak
magnitude is the same thing as the peak amplitude.
The ``phase'' of a sinusoid normally means the ``initial phase'', but
in some contexts it might mean ``instantaneous phase'', so be careful.
Another term for initial phase is
phase offset.
Note that
Hz is an abbreviation for
Hertz which
physically means
cycles per second. You might also encounter
the notation
cps (or ``c.p.s.'') for cycles per second (still
in use by physicists and formerly used by engineers as well).
Since the sine function is
periodic with
period 
, the initial
phase

is indistinguishable from

. As a result,
we may restrict the range of

to any length

interval.
When needed, we will choose
i.e.,

. You may also encounter the convention

.
Note that the
radian frequency 
is equal to the time
derivative of the
instantaneous phase of the sinusoid:
This is also how the instantaneous frequency is defined when the
phase is
time varying. Let
denote the instantaneous phase of a sinusoid with a time-varying
phase-offset

. Then the instantaneous frequency is again
given by the time derivative of the instantaneous phase:
Figure
4.1 plots the
sinusoid

, for

,

,

, and
![$ t\in[0,1]$](http://www.dsprelated.com/josimages_new/mdft/img385.png)
. Study the plot to make sure you understand the effect of
changing each parameter (amplitude, frequency, phase), and also note the
definitions of ``peak-to-peak amplitude'' and ``zero crossings.''
A ``
tuning fork''
vibrates approximately sinusoidally. An ``A-440'' tuning
fork oscillates at

cycles per second. As a result, a tone recorded
from an ideal A-440 tuning fork is a sinusoid at

Hz. The amplitude

determines how loud it is and depends on how hard we strike the tuning
fork. The phase

is set by exactly
when we strike the tuning
fork (and on our choice of when time 0 is). If we record an A-440 tuning
fork on an analog tape recorder, the electrical
signal recorded on tape is
of the form
As another example, the sinusoid at amplitude

and phase

(90 degrees)
is simply
Thus,

is a sinusoid at phase 90-degrees, while

is a sinusoid at
zero phase. Note, however, that we could
just as well have defined

to be the zero-phase sinusoid
rather than

. It really doesn't matter, except to be
consistent in any given usage. The concept of a ``
sinusoidal signal''
is simply that it is equal to a sine or cosine function at some amplitude,
frequency, and phase. It does not matter whether we choose

or

in the ``official'' definition of a sinusoid. You may
encounter both definitions. Using

is nice since
``sinusoid'' naturally generalizes

. However, using

is
nicer when defining a sinusoid to be the real part of a
complex sinusoid
(which we'll talk about in §
4.3.11).
Sinusoids arise naturally in a variety of ways:
One reason for the importance of sinusoids is that they are
fundamental in physics. Many physical systems that
resonate or
oscillate produce quasi-
sinusoidal motion. See
simple harmonic
motion in any
freshman physics text for an introduction to this
topic. The canonical example is the
mass-spring oscillator.
4.1
Another reason sinusoids are important is that they are
eigenfunctions of linear systems (which we'll say more about in
§
4.1.4). This means that they are important in the analysis
of
filters such as reverberators,
equalizers, certain (but not
all) ``audio effects'', etc.
Perhaps most importantly, from the point of view of computer music
research, is that the human
ear is a kind of
spectrum
analyzer. That is, the
cochlea of the
inner ear physically splits
sound into its (quasi) sinusoidal components. This is accomplished by
the
basilar membrane in the inner ear: a sound wave injected at
the
oval window (which is connected via the bones of the
middle
ear to the
ear drum), travels along the
basilar membrane inside
the coiled cochlea. The membrane starts out thick and stiff, and
gradually becomes thinner and more compliant toward its apex (the
helicotrema). A stiff membrane has a high resonance frequency
while a thin, compliant membrane has a low resonance frequency
(assuming comparable
mass per unit length, or at least less of a
difference in mass than in compliance). Thus, as the sound wave
travels, each frequency in the sound resonates at a particular
place along the basilar membrane. The highest audible frequencies
resonate right at the entrance, while the lowest frequencies travel
the farthest and resonate near the helicotrema. The membrane
resonance effectively ``shorts out'' the
signal energy at the resonant
frequency, and it travels no further. Along the basilar membrane
there are
hair cells which ``feel'' the resonant
vibration and
transmit an increased firing rate along the
auditory nerve to the
brain. Thus, the ear is very literally a Fourier analyzer for sound,
albeit
nonlinear and using ``analysis'' parameters that are difficult
to match exactly. Nevertheless, by looking at
spectra (which display
the amount of each sinusoidal frequency present in a sound), we are
looking at a representation much more like what the brain receives
when we hear.
From the trig identity

, we have
From this we may conclude that every
sinusoid can be expressed as the sum
of a sine function (phase zero) and a cosine function (phase

). If
the sine part is called the ``in-phase'' component, the cosine part can be
called the ``phase-quadrature'' component. In general, ``phase
quadrature'' means ``90 degrees out of phase,''
i.e., a relative phase
shift of

.
It is also the case that every sum of an in-phase and quadrature component
can be expressed as a single
sinusoid at some amplitude and phase. The
proof is obtained by working the previous derivation backwards.
Figure
4.2 illustrates in-phase and quadrature components
overlaid. Note that they only differ by a relative

degree phase
shift.
Figure 4.2:
In-phase and quadrature sinusoidal components.
 |
Sinusoids at the Same Frequency
An important property of
sinusoids at a particular frequency is that they
are
closed with respect to addition. In other words, if you take a
sinusoid, make many copies of it, scale them all by different gains,
delay them all by different time intervals, and add them up, you always get a
sinusoid at the same original frequency. This is a nontrivial property.
It obviously holds for any constant
signal 
(which we may regard as
a sinusoid at frequency

), but it is not obvious for

(see
Fig.
4.2 and think about the sum of the two waveforms shown
being precisely a sinusoid).
Since every linear, time-invariant (
LTI4.2) system (
filter) operates by copying, scaling,
delaying, and summing its input signal(s) to create its output
signal(s), it follows that when a sinusoid at a particular frequency
is input to an LTI system, a sinusoid at that same frequency always
appears at the output. Only the amplitude and phase can be changed by
the system. We say that sinusoids are
eigenfunctions of LTI
systems. Conversely, if the system is
nonlinear or time-varying, new
frequencies are created at the system output.
To prove this important invariance property of sinusoids, we may
simply express all scaled and delayed sinusoids in the ``mix'' in
terms of their in-phase and
quadrature components and then add them
up. Here are the details in the case of adding two sinusoids having
the same frequency. Let

be a general sinusoid at frequency

:
Now form

as the sum of two copies of

with arbitrary
amplitudes and phase offsets:
Focusing on the first term, we have
We similarly compute
and add to obtain
This result, consisting of one in-phase and one quadrature signal
component, can now be converted to a single sinusoid at some amplitude and
phase (and frequency

), as discussed above.
Sinusoidal signals are analogous to monochromatic laser light. You
might have seen ``speckle'' associated with laser light, caused by
destructive interference of multiple reflections of the light beam. In
a room, the same thing happens with sinusoidal sound. For example,
play a simple sinusoidal tone (
e.g., ``A-440''--a
sinusoid at
frequency

Hz) and walk around the room with one ear
plugged. If the room is reverberant you should be able to find places
where the sound goes completely away due to destructive interference.
In between such places (which we call ``
nodes'' in the soundfield),
there are ``
antinodes'' at which the sound is louder by 6
dB (amplitude doubled--
decibels (
dB) are reviewed in Appendix
F)
due to constructive interference. In a
diffuse reverberant
soundfield,
4.3the distance between nodes is on the order of a
wavelength
(the ``
correlation distance'' within the random soundfield).
figure[htbp]
The way
reverberation produces nodes and antinodes for
sinusoids in a
room is illustrated by the simple
comb filter, depicted in
Fig.
4.3.
4.4
Since the
comb filter is
linear and time-invariant, its response to a
sinusoid must be sinusoidal (see previous section).
The feedforward path has gain

, and the delayed signal is scaled by

.
With the delay set to one
period, the sinusoid coming out of the
delay
line constructively interferes with the sinusoid from the
feed-forward path, and the output amplitude is therefore

.
In the opposite extreme case, with the delay set to
half a period, the unit-amplitude sinusoid coming out of the
delay line destructively interferes with the sinusoid from the
feed-forward path, and the output amplitude therefore drops to

.
Consider a fixed delay of

seconds for the delay line in
Fig.
4.3. Constructive interference happens at all
frequencies for which an
exact integer number of periods fits
in the delay line,
i.e.,

, or

, for

. On the other hand, destructive interference
happens at all frequencies for which there is an
odd number of
half-periods,
i.e., the number of periods in the
delay line is an integer plus a half:

etc., or,

, for

. It is quick
to verify that frequencies of constructive interference alternate with
frequencies of destructive interference, and therefore the
amplitude response of the comb
filter (a plot of gain versus
frequency) looks as shown in Fig.
4.4.
Figure 4.4:
Comb filter amplitude response when delay
sec.
![\includegraphics[width=4in,height=2.0in]{eps/combfilterFR}](http://www.dsprelated.com/josimages_new/mdft/img424.png) |
The amplitude response of a comb filter has a ``comb'' like shape,
hence the name.
4.5 It looks even more like a comb on a
dB
amplitude scale, as shown in Fig.
4.5. A
dB scale is
more appropriate for audio applications, as discussed in
Appendix
F. Since the minimum gain is

, the nulls
in the response reach down to

dB; since the maximum gain is

, the maximum in dB is about 6 dB. If the feedforward gain
were increased from

to

, the nulls would extend, in
principle, to minus infinity, corresponding to a gain of zero
(complete cancellation). Negating the feedforward path would shift
the curve left (or right) by 1/2 Hz, placing a minimum at
dc4.6 instead of a peak.
Figure 4.5:
Comb filter amplitude
response in dB.
![\includegraphics[width=4in,height=2.0in]{eps/combfilterFRDB}](http://www.dsprelated.com/josimages_new/mdft/img428.png) |
A
sinusoid's frequency content may be graphed in the
frequency
domain as shown in Fig.
4.6.
figure[htbp]
An example of a particular sinusoid graphed in Fig.
4.6 is given by
where
That is, this sinusoid has amplitude 1, frequency 100 Hz, and phase
zero (or

, if

is defined as the
zero-phase
case).
Figure
4.6 can be viewed as a graph of the
magnitude
spectrum of

, or its
spectral magnitude representation
[
44]. Note that the
spectrum consists of two components
with amplitude

, one at frequency

Hz and the other at
frequency

Hz.
Phase is not shown in Fig.
4.6 at all. The phase of the
components could be written simply as labels next to the magnitude
arrows, or the magnitude arrows can be rotated ``into or out of the
page'' by the appropriate phase angle, as illustrated in
Fig.
4.16.
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