Sinusoids and Exponentials
This chapter provides an introduction to
sinusoids,
exponentials,
complex sinusoids, and various associated terminology, such as
exponential decay-time ``

'', in-phase and
quadrature
sinusoidal components,
analytic signals, positive and
negative
frequencies, and constructive and
destructive interference. The
fundamental importance of sinusoids in the analysis of
linear
time-invariant systems is introduced. We also look at circular motion
expressed as the vector sum of in-phase and quadrature sinusoidal
motions. Both continuous and discrete-time sinusoids are considered.
In particular, a sampled complex sinusoid is generated by successive
powers of any
complex number 
.

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.
Exponentials
The canonical form of an exponential function, as typically used in
signal
processing, is
where

is called the
time constant of the exponential.

is
the peak amplitude, as before. The time constant is the time it takes to decay
by

,
i.e.,
A normalized
exponential decay is depicted in Fig.
4.7.
Figure 4.7:
The decaying exponential
, normalized to unit amplitude (
).
![\includegraphics[width=0.8 \twidth]{eps/exponential}](http://www.dsprelated.com/josimages_new/mdft/img441.png) |
Exponential decay occurs naturally when a quantity is decaying at a
rate which is proportional to how much is left. In nature, all
linear
resonators, such as musical instrument strings and
woodwind bores, exhibit
exponential decay in their response to a momentary excitation. As another
example, reverberant energy in a room decays exponentially after the direct
sound stops. Essentially all
undriven oscillations decay
exponentially (provided they are
linear and time-invariant). Undriven
means there is no ongoing source of driving energy. Examples of undriven
oscillations include the
vibrations of a
tuning fork, struck or
plucked
strings, a marimba or xylophone bar, and so on. Examples of driven
oscillations include horns, woodwinds,
bowed strings, and voice. Driven
oscillations must be
periodic while undriven oscillations normally are not,
except in idealized cases.
Exponential growth occurs when a quantity is increasing at a
rate proportional to the current amount. Exponential growth is
unstable since nothing can grow exponentially forever without
running into some kind of limit. Note that a positive
time constant
corresponds to
exponential decay, while a negative time constant
corresponds to exponential growth. In
signal processing, we almost
always deal exclusively with exponential decay (positive time
constants).
Exponential growth and decay are illustrated in Fig.
4.8.
Figure 4.8:
Growing and decaying exponentials.
![\includegraphics[width=0.8\twidth]{eps/decaygrowth}](http://www.dsprelated.com/josimages_new/mdft/img442.png) |
In audio, a decay by

(one
time-constant) is not enough to become inaudible, unless
the starting amplitude was extremely small.
In
architectural acoustics (which includes the design of
concert halls [
4]), a more commonly used measure of decay is ``

''
(or
T60), which is defined as the
time to decay by
dB.
4.7That is,

is obtained by solving the equation
Using the definition of the
exponential

, we find
Thus,

is about seven time constants. See where

is marked
on Fig.
4.7 compared with

.
Recall
Euler's Identity,
Multiplying this equation by

and setting

, where

is time in seconds,

is radian frequency, and

is a phase offset, we obtain what we call the
complex sinusoid:
Thus, a complex
sinusoid consists of an ``in-phase'' component for its
real part, and a ``
phase-quadrature'' component for its imaginary
part. Since

, we have
That is, the complex sinusoid has a
constant modulus (
i.e.,
a constant complex magnitude). (The symbol
``

'' means ``identically equal to,''
i.e., for all

.) The
instantaneous phase of the complex sinusoid is
The derivative of the instantaneous phase of the complex sinusoid
gives its instantaneous frequency
Since the modulus of the complex
sinusoid is constant, it must lie on a
circle in the
complex plane. For example,
traces out
counter-clockwise circular motion along the unit
circle in the complex plane as

increases, while
gives
clockwise circular motion.
We may call a
complex sinusoid

a
positive-frequency sinusoid when

. Similarly, we
may define a complex sinusoid of the form

, with

, to be a
negative-frequency sinusoid. Note that a positive- or
negative-frequency sinusoid is necessarily complex.
Interpreting the real and imaginary parts of the complex
sinusoid,
in the
complex plane, we see that
sinusoidal motion is the
projection of circular motion onto any straight line. Thus, the
sinusoidal motion

is the projection of the circular
motion

onto the

(real-part) axis, while

is the projection of

onto the

(imaginary-part) axis.
Figure
4.9 shows a plot of a
complex sinusoid versus time, along with its
projections onto coordinate planes. This is a 3D plot showing the

-plane versus time. The axes are the real part, imaginary part, and
time. (Or we could have used magnitude and phase versus time.)
Figure 4.9:
A complex sinusoid and its projections.
![\includegraphics[scale=0.8]{eps/circle}](http://www.dsprelated.com/josimages_new/mdft/img464.png) |
Note that the left projection (onto the

plane) is a circle, the lower
projection (real-part vs. time) is a cosine, and the upper projection
(imaginary-part vs. time) is a sine. A point traversing the plot projects
to
uniform circular motion in the

plane, and sinusoidal motion on the
two other planes.
Positive and Negative Frequencies
In §
2.9, we used
Euler's Identity to show
Setting

, we see that both sine and cosine (and
hence all real
sinusoids) consist of a sum of equal and opposite circular
motion. Phrased differently, every real
sinusoid consists of an equal
contribution of positive and negative frequency components. This is true
of all real
signals. When we get to
spectrum analysis, we will find that
every real signal contains equal amounts of positive and negative
frequencies,
i.e., if

denotes the
spectrum of the real signal

, we will always have

.
Note that, mathematically, the
complex sinusoid

is really
simpler and
more basic than the real
sinusoid

because

consists of
one frequency

while

really consists of two
frequencies

and

. We may think of a real sinusoid
as being the sum of a positive-frequency and a negative-frequency
complex sinusoid, so in that sense real sinusoids are ``twice as
complicated'' as complex sinusoids. Complex sinusoids are also nicer
because they have a
constant modulus. ``Amplitude
envelope
detectors'' for complex sinusoids are trivial: just compute the square
root of the sum of the squares of the real and imaginary parts to
obtain the
instantaneous peak amplitude at any time. Frequency
demodulators are similarly trivial: just differentiate the phase of
the complex sinusoid to obtain its
instantaneous frequency. It
should therefore come as no surprise that signal processing engineers
often prefer to convert real sinusoids into complex sinusoids (by
filtering out the negative-frequency component) before processing them
further.
Plotting Complex Sinusoids versus Frequency
As discussed in the previous section, we regard the
signal
as a
positive-frequency sinusoid when

. In a
manner analogous to spectral magnitude plots (discussed in
§
4.1.6), we can plot this
complex sinusoid over a frequency
axis as a vertical line of length

at the point

, as shown in Fig.
4.10. Such a plot of
amplitude versus frequency may be called a
spectral plot, or
spectral representation [
44] of the (
zero-phase)
complex sinusoid.
figure[htbp]
More generally, however, a complex sinusoid has both an amplitude and
a
phase (or, equivalently, a
complex amplitude):
To accommodate the phase angle

in spectral plots, the
plotted vector may be rotated by the angle

in the plane
orthogonal to the frequency axis passing through

, as done
in Fig.
4.16b below (p.
![[*]](../icons/crossref.png)
)
for phase angles

.
Sinusoidal Amplitude Modulation (AM)
It is instructive to study the
modulation of one
sinusoid by
another. In this section, we will look at sinusoidal
Amplitude
Modulation (AM). The general AM formula is given by
where

are parameters of the sinusoidal
carrier wave,
![$ \alpha\in[0,1]$](http://www.dsprelated.com/josimages_new/mdft/img482.png)
is called the
modulation index (or
AM index),
and
![$ a_m(t)\in[-1,1]$](http://www.dsprelated.com/josimages_new/mdft/img483.png)
is the
amplitude modulation signal. In
AM radio broadcasts,

is the audio signal being transmitted
(usually bandlimited to less than 10 kHz), and

is the channel
center frequency that one dials up on a radio receiver.
The modulated signal

can be written as the sum of the
unmodulated carrier wave plus the product of the carrier wave and the
modulating wave:
 |
(4.1) |
In the case of
sinusoidal AM, we have
 |
(4.2) |
Periodic amplitude modulation of this nature is often called the
tremolo effect when

or so (

Hz).
Let's analyze the second term of Eq.

(
4.1) for the case of sinusoidal
AM with

and

:
 |
(4.3) |
An example waveform is shown in Fig.
4.11 for

Hz and

Hz. Such a signal may be produced on an analog synthesizer
by feeding two differently tuned
sinusoids to a
ring modulator,
which is simply a ``four-quadrant multiplier'' for analog signals.
Figure:
Sinusoidal amplitude modulation as in Eq.
(4.3)--time
waveform.
![\includegraphics[width=3.5in]{eps/sineamtd}](http://www.dsprelated.com/josimages_new/mdft/img494.png) |
When

is small (say less than

radians per second, or
10 Hz), the signal

is heard as a ``beating
sine wave'' with

beats per second.
The beat rate is
twice the modulation frequency because both the positive and negative
peaks of the modulating sinusoid cause an ``amplitude swell'' in

. (One
period of modulation--

seconds--is shown in
Fig.
4.11.) The sign inversion during the negative peaks is not
normally audible.
Recall the trigonometric identity for a sum of angles:
Subtracting this from

leads to the identity
Setting

and

gives us an alternate form
for our ``ring-modulator output signal'':
![$\displaystyle x_m(t) \isdef \sin(\omega_m t)\sin(\omega_c t) = \frac{\cos[(\omega_m-\omega_c)t] - \cos[(\omega_m+\omega_c)t]}{2} \protect$](http://www.dsprelated.com/josimages_new/mdft/img505.png) |
(4.4) |
These two sinusoidal components at the
sum and difference
frequencies of the modulator and carrier are called
side bands
of the carrier wave at frequency

(since typically

).
Equation (
4.3) expresses

as a ``beating sinusoid'', while
Eq.

(
4.4) expresses as it two
unmodulated sinusoids at
frequencies

. Which case do we hear?
It turns out we hear

as two separate tones (Eq.

(
4.4))
whenever the side bands are
resolved by the ear. As
mentioned in §
4.1.2,
the ear performs a ``short time
Fourier analysis'' of incoming sound
(the
basilar membrane in the
cochlea acts as a mechanical
filter bank). The
resolution of this
filterbank--its ability to discern two
separate spectral peaks for two sinusoids closely spaced in
frequency--is determined by the
critical bandwidth of hearing
[
45,
76,
87]. A critical
bandwidth is roughly 15-20% of the band's center-frequency, over most
of the audio range [
71]. Thus, the side bands in
sinusoidal AM are heard as separate tones when they are both in the
audio range and separated by at least one critical bandwidth. When
they are well inside the same
critical band, ``beating'' is heard. In
between these extremes, near separation by a critical-band, the
sensation is often described as ``roughness'' [
29].
Equation (
4.4) can be used to write down the spectral representation of

by inspection, as shown in Fig.
4.12. In the example
of Fig.
4.12, we have

Hz and

Hz,
where, as always,

. For comparison, the spectral
magnitude of an
unmodulated 
Hz
sinusoid is shown in
Fig.
4.6. Note in Fig.
4.12 how each of the two
sinusoidal components at

Hz have been ``split'' into two
``side bands'', one

Hz higher and the other

Hz lower, that
is,

. Note also how the
amplitude of the split component is divided equally among its
two side bands.
figure[htbp]
Recall that

was defined as the
second term of
Eq.

(
4.1). The first term is simply the original unmodulated
signal. Therefore, we have effectively been considering AM with a
``very large'' modulation index. In the more general case of
Eq.

(
4.1) with

given by Eq.

(
4.2), the magnitude of
the spectral representation appears as shown in Fig.
4.13.
figure[htbp]
Sinusoidal Frequency Modulation (FM)
Frequency Modulation (FM) is well known as
the broadcast
signal format for FM radio. It is also the basis of the
first commercially successful method for
digital sound synthesis.
Invented by John Chowning [
14], it was the method used in
the the highly successful Yamaha
DX-7 synthesizer, and later the
Yamaha OPL chip series, which was used in all ``SoundBlaster
compatible'' multimedia sound cards for many years. At the time of
this writing, descendants of the OPL chips remain the dominant
synthesis technology for ``ring tones'' in cellular telephones.
A general formula for frequency modulation of one
sinusoid by another
can be written as
![$\displaystyle x(t) = A_c\cos[\omega_c t + \phi_c + A_m\sin(\omega_m t + \phi_m)], \protect$](http://www.dsprelated.com/josimages_new/mdft/img516.png) |
(4.5) |
where the parameters

describe the
carrier sinusoid, while the parameters

specify the
modulator sinusoid. Note that, strictly speaking,
it is not the frequency of the carrier that is modulated sinusoidally,
but rather the
instantaneous phase of the carrier. Therefore,
phase modulation would be a better term (which is in fact used).
Potential confusion aside, any modulation of phase implies a
modulation of frequency, and vice versa, since the instantaneous
frequency is always defined as the time-derivative of the
instantaneous phase. In this book, only phase modulation will be
considered, and we will call it FM, following common
practice.
4.8
Figure
4.14 shows a unit generator patch diagram [
42]
for brass-like FM synthesis. For brass-like sounds, the modulation
amount increases with the amplitude of the signal. In the patch, note
that the amplitude
envelope for the carrier
oscillator is scaled and
also used to control amplitude of the modulating oscillator.
figure[htbp]
It is well known that sinusoidal frequency-modulation of a sinusoid
creates sinusoidal components that are uniformly spaced in frequency
by multiples of the modulation frequency, with amplitudes given by the
Bessel functions of the first kind [
14].
As a special case, frequency-modulation of a sinusoid by itself
generates a
harmonic spectrum in which the

th
harmonic amplitude is
proportional to

, where

is the
order of the
Bessel function and

is the
FM index. We will derive
this in the next section.
4.9
Bessel Functions
The
Bessel functions of the first kind may be defined as the
coefficients

in the two-sided
Laurent expansion
of the so-called
generating function
[
84, p. 14],
4.10
 |
(4.6) |
where

is the integer
order
of the Bessel function, and

is its argument (which
can be complex, but we will only consider real

).
Setting

, where

will interpreted as the
FM modulation frequency and

as time in seconds, we obtain
 |
(4.7) |
The last expression can be interpreted as the Fourier superposition of the
sinusoidal harmonics of
![$ \exp[j\beta\sin(\omega_m t)]$](http://www.dsprelated.com/josimages_new/mdft/img528.png)
,
i.e., an
inverse Fourier series sum. In other words,

is
the amplitude of the

th
harmonic in the
Fourier-series expansion of
the
periodic signal 
.
Note that

is real when

is real. This can be seen
by viewing Eq.

(
4.6) as the product of the
series expansion for
![$ \exp[(\beta/2) z]$](http://www.dsprelated.com/josimages_new/mdft/img529.png)
times that for
![$ \exp[-(\beta/2)/z]$](http://www.dsprelated.com/josimages_new/mdft/img530.png)
(see footnote
pertaining to Eq.

(
4.6)).
Figure
4.15 illustrates the first eleven Bessel functions of the first
kind for arguments up to

. It can be seen in the figure
that when the FM index

is zero,

and

for
all

. Since

is the amplitude of the carrier
frequency, there are no side bands when

. As the FM index
increases, the sidebands begin to grow while the carrier term
diminishes. This is how
FM synthesis produces an expanded, brighter
bandwidth as the FM index is increased.
Figure 4.15:
Bessel functions of the first kind
for a range of orders
and argument
.
![\includegraphics[width=\twidth]{eps/bessel}](http://www.dsprelated.com/josimages_new/mdft/img537.png) |
FM Spectra
Using the expansion in Eq.

(
4.7), it is now easy to determine
the
spectrum of
sinusoidal FM. Eliminating scaling and
phase offsets for simplicity in Eq.

(
4.5) yields
![$\displaystyle x(t) = \cos[\omega_c t + \beta\sin(\omega_m t)], \protect$](http://www.dsprelated.com/josimages_new/mdft/img538.png) |
(4.8) |
where we have changed the modulator amplitude

to the more
traditional symbol

, called the
FM index in FM sound
synthesis contexts. Using
phasor analysis (where
phasors
are defined below in §
4.3.11),
4.11i.e., expressing a real-valued FM
signal as the real part of a more
analytically tractable complex-valued FM signal, we obtain
where we used the fact that

is real when

is real.
We can now see clearly that the sinusoidal FM spectrum consists of an
infinite number of side-bands about the carrier frequency

(when

). The side bands occur at multiples of the
modulating frequency

away from the carrier frequency

.
Analytic Signals and Hilbert Transform Filters
A signal which has no
negative-frequency components is called an
analytic signal.
4.12 Therefore, in continuous time, every analytic signal

can be represented as
where

is the complex coefficient (setting the amplitude and
phase) of the positive-frequency complex
sinusoid

at
frequency

.
Any real
sinusoid

may be converted to a
positive-frequency
complex sinusoid
![$ A\exp[j(\omega t +
\phi)]$](http://www.dsprelated.com/josimages_new/mdft/img554.png)
by simply generating a
phase-quadrature component

to serve as the ``imaginary part'':
The phase-
quadrature component can be generated from the
in-phase component
by a simple quarter-cycle time shift.
4.13
For more complicated signals which are expressible as a sum of many
sinusoids, a
filter can be constructed which shifts each
sinusoidal component by a quarter cycle. This is called a
Hilbert transform filter. Let

denote the output
at time

of the Hilbert-transform filter applied to the signal

.
Ideally, this filter has magnitude

at all frequencies and
introduces a phase shift of

at each positive frequency and

at each negative frequency. When a real signal

and
its Hilbert transform

are used to form a new complex signal

,
the signal

is the (complex)
analytic signal corresponding to
the real signal

. In other words, for any real signal

, the
corresponding analytic signal

has the property
that all ``
negative frequencies'' of

have been ``filtered out.''
To see how this works, recall that these phase shifts can be impressed on a
complex sinusoid by multiplying it by

. Consider
the positive and negative frequency components at the particular frequency

:
Now let's apply a

degrees phase shift to the positive-frequency
component, and a

degrees phase shift to the negative-frequency
component:
Adding them together gives
and sure enough, the negative frequency component is filtered out. (There
is also a gain of 2 at positive frequencies.)
For a concrete example, let's start with the real sinusoid
Applying the ideal phase shifts, the Hilbert transform is
The analytic signal is then
by
Euler's identity. Thus, in the sum

, the
negative-frequency components of

and

cancel out,
leaving only the positive-frequency component. This happens for any
real signal

, not just for sinusoids as in our example.
Figure 4.16:
Creation of the analytic signal
from the real sinusoid
and the derived phase-quadrature sinusoid
, viewed in the frequency domain. a) Spectrum of
. b) Spectrum
of
. c) Spectrum of
. d) Spectrum of
.
![\includegraphics[width=2.8in]{eps/sineFD}](http://www.dsprelated.com/josimages_new/mdft/img576.png) |
Figure
4.16 illustrates what is going on in the frequency domain.
At the top is a graph of the spectrum of the sinusoid

consisting of
impulses at frequencies

and
zero at all other frequencies (since

). Each impulse
amplitude is equal to

. (The amplitude of an impulse is its
algebraic area.) Similarly, since

, the spectrum of

is an impulse of amplitude

at

and amplitude

at

.
Multiplying

by

results in

which is shown in
the third plot, Fig.
4.16c. Finally, adding together the first and
third plots, corresponding to

, we see that the
two positive-frequency impulses
add in phase to give a unit
impulse (corresponding to

), and at frequency

, the two impulses, having opposite sign,
cancel in the sum, thus creating an analytic signal

,
as shown in Fig.
4.16d. This sequence of operations illustrates
how the negative-frequency component

gets
filtered out by summing

with

to produce the analytic signal

corresponding
to the real signal

.
As a final example (and application), let

,
where

is a slowly varying amplitude
envelope (slow compared
with

). This is an example of
amplitude modulation
applied to a sinusoid at ``carrier frequency''

(which is
where you tune your AM radio). The Hilbert transform is very close to

(if

were constant, this would
be exact), and the analytic signal is

.
Note that AM
demodulation4.14is now nothing more than the
absolute value.
I.e.,

. Due to this simplicity, Hilbert transforms are sometimes
used in making
amplitude envelope followers for narrowband signals (
i.e., signals with all energy centered about a single ``carrier'' frequency).
AM demodulation is one application of a narrowband envelope follower.
We have defined
sinusoids and extended the definition to include
complex
sinusoids. We now extend one more step by allowing for
exponential
amplitude envelopes:
where

and

are
complex, and further defined as
When

, we obtain
which is the complex
sinusoid at amplitude

, frequency

,
and phase

.
More generally, we have
Defining

, we see that the generalized complex sinusoid
is just the complex sinusoid we had before with an
exponential envelope:
In discrete-time audio processing, such as we normally do on a computer,
we work with
samples of continuous-time
signals. Let

denote the
sampling rate in Hz. For audio, we typically have

kHz, since the audio band nominally extends to

kHz. For compact
discs (CDs),

kHz,
while for digital audio tape (DAT),

kHz.
Let

denote the
sampling interval in seconds. Then to
convert from continuous to discrete time, we replace

by

, where

is an integer interpreted as the
sample number.
The sampled generalized complex
sinusoid
is then
Thus, the sampled case consists of a sampled
complex sinusoid
multiplied by a sampled
exponential envelope
![$ \left[e^{\sigma
T}\right]^n = e^{-nT/\tau}$](http://www.dsprelated.com/josimages_new/mdft/img614.png)
.
Choose any two
complex numbers 
and

, and form the sequence
 |
(4.10) |
What are the properties of this
signal?
Writing the complex numbers as
we see that the signal

is always a discrete-time
generalized (exponentially
enveloped) complex
sinusoid:
Figure
4.17 shows a plot of a generalized (exponentially
decaying,

)
complex sinusoid versus time.
Figure 4.17:
Exponentially decaying
complex sinusoid and projections.
![\includegraphics[scale=0.8]{eps/circledecaying}](http://www.dsprelated.com/josimages_new/mdft/img620.png) |
Note that the left projection (onto the

plane) is a decaying spiral,
the lower projection (real-part vs. time) is an exponentially decaying
cosine, and the upper projection (imaginary-part vs. time) is an
exponentially enveloped
sine wave.
If we restrict

in Eq.

(
4.10) to have unit modulus, then

and we obtain a discrete-time
complex sinusoid.
 |
(4.11) |
where we have defined
It is common terminology to call

the complex
sinusoid's
phasor, and

its
carrier wave.
For a
real sinusoid,
the phasor is again defined as

and the carrier is

. However, in this case, the real sinusoid
is recovered from its
complex-sinusoid counterpart by taking the real part:

re
The
phasor magnitude

is the
amplitude of the sinusoid.
The
phasor angle

is the
phase of the sinusoid.
When working with complex sinusoids, as in Eq.

(
4.11), the phasor
representation

of a sinusoid can be thought of as simply the
complex amplitude of the sinusoid.
I.e.,
it is the complex constant that multiplies the carrier term

.
Linear, time-invariant (
LTI) systems can be said to perform only four
operations on a
signal: copying, scaling, delaying, and adding. As a
result, each output is always a
linear combination of delayed copies of the input signal(s).
(A
linear combination is simply a weighted sum, as discussed in
§
5.6.) In any linear
combination of delayed copies of a complex
sinusoid
where

is a weighting factor,

is the

th delay, and
is a
complex sinusoid, the ``carrier term''

can be ``factored out'' of the linear combination:
The operation of the LTI system on a complex
sinusoid is thus reduced
to a calculation involving only
phasors, which are simply
complex
numbers.
Since every signal can be expressed as a linear combination of complex
sinusoids, this analysis can be applied to any signal by expanding the
signal into its weighted sum of complex sinusoids (
i.e., by expressing
it as an inverse
Fourier transform).
As a preview of things to come, note that one
signal
4.15 is
projected onto another signal

using an
inner
product. The inner product

computes the
coefficient
of projection4.16 of

onto

. If

(a sampled, unit-amplitude,
zero-phase, complex
sinusoid), then the inner product computes the
Discrete Fourier
Transform (
DFT), provided the frequencies are chosen to be

. For the DFT, the inner product is specifically
Another case of importance is the
Discrete Time Fourier Transform
(
DTFT), which is like the DFT except that the transform accepts an
infinite number of samples instead of only

. In this case,
frequency is continuous, and
The DTFT is what you get in the limit as the number of samples in the
DFT approaches infinity. The lower limit of summation remains zero
because we are assuming all signals are zero for negative time (such
signals are said to be
causal). This means we are working with
unilateral Fourier transforms. There are also corresponding
bilateral transforms for which the lower summation limit is

. The DTFT is discussed further in
§
B.1.
If, more generally,

(a sampled
complex sinusoid with
exponential growth or decay), then the inner product becomes
and this is the definition of the
transform. It is a
generalization of the DTFT: The DTFT equals the

transform evaluated on
the
unit circle in the

plane. In principle, the

transform
can also be recovered from the DTFT by means of ``analytic continuation''
from the unit circle to the entire

plane (subject to mathematical
disclaimers which are unnecessary in practical applications since they are
always finite).
Why have a

transform when it seems to contain no more information than
the DTFT? It is useful to generalize from the unit circle (where the DFT
and DTFT live) to the entire
complex plane (the

transform's domain) for
a number of reasons. First, it allows transformation of
growing
functions of time such as growing
exponentials; the only limitation on
growth is that it cannot be faster than exponential. Secondly, the

transform has a deeper algebraic structure over the complex plane as a
whole than it does only over the unit circle. For example, the

transform of any finite signal is simply a
polynomial in

. As
such, it can be fully characterized (up to a constant scale factor) by its
zeros in the

plane. Similarly, the

transform of an
exponential can be characterized to within a scale factor
by a single point in the

plane (the
point which
generates the exponential); since the

transform goes
to infinity at that point, it is called a
pole of the transform.
More generally, the

transform of any
generalized complex sinusoid
is simply a
pole located at the point which generates the
sinusoid.
Poles and zeros are used extensively in the analysis of
recursive
digital filters. On the most general level, every
finite-order, linear,
time-invariant, discrete-time system is fully specified (up to a scale
factor) by its poles and zeros in the

plane. This topic will be taken
up in detail in Book II [
68].
In the
continuous-time case, we have the
Fourier transform
which projects

onto the continuous-time sinusoids defined by

, and the appropriate inner product is
Finally, the
Laplace transform is the continuous-time counterpart
of the

transform, and it projects signals onto exponentially growing
or decaying complex sinusoids:
The Fourier transform equals the Laplace transform evaluated along the
``

axis'' in the

plane,
i.e., along the line

, for
which

. Also, the Laplace transform is obtainable from the
Fourier transform via analytic continuation. The usefulness of the Laplace
transform relative to the Fourier transform is exactly analogous to that of
the

transform outlined above.
In
signal processing, it is customary to use

as the
Laplace transform
variable for continuous-time analysis, and

as the

-transform
variable for discrete-time analysis. In other words, for continuous-time
systems, the
frequency domain is the ``

plane'', while for discrete-time
systems, the frequency domain is the ``

plane.'' However, both are
simply
complex planes.
Figure 4.18:
Generalized complex sinusoids
represented by points in the
plane.
![\includegraphics[width=4.5in]{eps/splane}](http://www.dsprelated.com/josimages_new/mdft/img651.png) |
Figure
4.18 illustrates the various
sinusoids 
represented by points
in the

plane. The frequency axis is

, called the
``

axis,'' and points along it correspond to
complex sinusoids,
with
dc at

(

).
The upper-half plane corresponds to positive
frequencies (counterclockwise circular or corkscrew motion) while the
lower-half plane corresponds to
negative frequencies (clockwise motion).
In the left-half plane we have decaying (stable)
exponential envelopes,
while in the right-half plane we have growing (unstable)
exponential
envelopes. Along the real axis (

), we have pure exponentials.
Every point in the

plane corresponds to a generalized
complex sinusoid,

, with special cases including
complex sinusoids

, real exponentials

,
and the constant function

(dc).
Figure
4.19 shows examples of various sinusoids

represented by points in the

plane. The frequency axis is the ``unit
circle''

, and points along it correspond to
sampled
complex sinusoids, with dc at

(
![$ 1^n = [e^{j0T}]^n = 1$](http://www.dsprelated.com/josimages_new/mdft/img664.png)
).
While the frequency axis is unbounded in the

plane, it is finite
(confined to the unit circle) in the

plane, which is natural because
the
sampling rate is finite in the discrete-time case.
As in the

plane, the upper-half plane corresponds to positive frequencies while
the lower-half plane corresponds to negative frequencies. Inside the unit
circle, we have decaying (stable) exponential envelopes, while outside the
unit circle, we have growing (unstable) exponential envelopes. Along the
positive real axis (
re

im

),
we have pure exponentials, but
along the negative real axis (
re

im

), we have exponentially
enveloped sampled sinusoids at frequency

(exponentially enveloped
alternating sequences). The negative real axis in the

plane is
normally a place where all signal

transforms should be zero, and all
system responses should be highly attenuated, since there should never be
any energy at exactly half the
sampling rate (where amplitude and phase are
ambiguously linked). Every point in the

plane can be said to
correspond to sampled generalized complex sinusoids of the form
![$ x(n) = {\cal A}z^n
= {\cal A}[e^{sT}]^n, n\geq 0$](http://www.dsprelated.com/josimages_new/mdft/img669.png)
, with special cases being sampled complex
sinusoids

, sampled real exponentials

,
and the constant sequence
![$ x=[1,1,1,\ldots]$](http://www.dsprelated.com/josimages_new/mdft/img672.png)
(dc).
In summary, the exponentially enveloped (``generalized'') complex sinusoid
is the fundamental signal upon which other signals are ``projected'' in
order to compute a Laplace transform in the continuous-time case, or a

transform in the discrete-time case. As a special case, if the exponential
envelope is eliminated (set to

), leaving only a complex sinusoid, then
the projection reduces to the
Fourier transform in the continuous-time
case, and either the
DFT (finite length) or
DTFT (infinite length) in the
discrete-time case. Finally, there are still other variations, such as
short-time Fourier transforms (
STFT) and wavelet transforms, which utilize
further modifications such as projecting onto
windowed complex
sinusoids.
See
http://ccrma.stanford.edu/~jos/mdftp/Sinusoid_Problems.html
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