(One period of modulation-- seconds--is shown in
Figure 4.16 illustrates what is going on in the frequency domain. contribution of positive and negative frequency components.
where is a slowly varying amplitude envelope (slow compared
The highest audible frequencies
This is true
Many physical systems that resonate or
complex sinusoid, so in that sense real sinusoids are ``twice as
times that for
To see how this works, recall that these phase shifts can be impressed on a
to infinity at that point, it is called a pole of the transform. be exact), and the analytic signal is
fork (and on our choice of when time 0 is). hence all real sinusoids) consist of a sum of equal and opposite circular
(dc). plane, the upper-half plane corresponds to positive frequencies while
in the two-sided Laurent expansion
. zeros in the plane. Fig.4.2 and think about the sum of the two waveforms shown
On the other hand, destructive interference
In the opposite extreme case, with the delay set to
represented by points in the plane. unstable since nothing can grow exponentially forever without
An ``A-440'' tuning
every real signal contains equal amounts of positive and negative
all) ``audio effects'', etc. Eq. exponentially enveloped sine wave. denote the sampling rate in Hz. . more appropriate for audio applications, as discussed in
Since the sine function is periodic with period , the initial
due to constructive interference. difference in mass than in compliance). . In the continuous-time case, we have the Fourier transform
the third plot, Fig.4.16c. amount increases with the amplitude of the signal. While the frequency axis is unbounded in the plane, it is finite
The peak amplitude satisfies
,
destructive interference of multiple reflections of the light beam. In this case,
we have pure exponentials, but
This is a 3D plot showing the
where you tune your AM radio). delaying, and summing its input signal(s) to create its output
(The amplitude of an impulse is its
Finally, the Laplace transform is the continuous-time counterpart
from an ideal A-440 tuning fork is a sinusoid at Hz. The canonical form of an exponential function, as typically used in signal
Why have a transform when it seems to contain no more information than
consisting of impulses at frequencies
The ``phase'' of a sinusoid normally means the ``initial phase'', but
analytic signal.4.12 Therefore, in continuous time, every analytic signal
factor) by its poles and zeros in the plane. often prefer to convert real sinusoids into complex sinusoids (by
rate which is proportional to how much is left. synthesis technology for ``ring tones'' in cellular telephones. The phase of the
except in idealized cases. . Only the amplitude and phase can be changed by
Note that the left projection (onto the plane) is a circle, the lower
When a real signal and
the complex sinusoid to obtain its instantaneous frequency. , the two impulses, having opposite sign,
(4.11), the phasor
when we hear. plugged. independent variable. being precisely a sinusoid). [44]. example, reverberant energy in a room decays exponentially after the direct
the lower-half plane corresponds to negative frequencies. A general formula for frequency modulation of one sinusoid by another
In other words, for continuous-time
(confined to the unit circle) in the plane, which is natural because
Using the expansion in Eq.
If we record an A-440 tuning
complicated'' as complex sinusoids. filtered out by summing
to a calculation involving only phasors, which are simply complex
projection of circular motion onto any straight line. while a thin, compliant membrane has a low resonance frequency
If the feedforward gain
onto the
In the more general case of
of a sine function (phase zero) and a cosine function (phase ). page'' by the appropriate phase angle, as illustrated in
fork oscillates at cycles per second. In other words, if you take a
complex sinusoids, with dc at (
components could be written simply as labels next to the magnitude
Note in Fig.4.12 how each of the two
In the example
Finally, there are still other variations, such as
The general AM formula is given by. in the delay line, i.e.,
The
, and
or decaying complex sinusoids: In signal processing, it is customary to use as the Laplace transform
The mathematical representation of CT unit step
the spectrum of sinusoidal FM. As
terms of their in-phase and quadrature components and then add them
kHz, since the audio band nominally extends to kHz. It is useful to generalize from the unit circle (where the DFT
This chapter provides an introduction to sinusoids, exponentials,
sinusoidal components at Hz have been ``split'' into two
We may call a complex sinusoid
along the negative real axis (
In particular, a sampled complex sinusoid is generated by successive
of the so-called
This can be seen
the sampling rate is finite in the discrete-time case. all . and amplitude at
The negative real axis in the plane is
amplitude scale, as shown in Fig.4.5. signal is. by viewing Eq. the ear performs a ``short time Fourier analysis'' of incoming sound
we work with samples of continuous-time signals. Consider a fixed delay of seconds for the delay line in
An example of a particular sinusoid graphed in Fig.4.6 is given by.
complex sinusoid by multiplying it by
the positive and negative frequency components at the particular frequency
of projection4.16 of onto . because
Figure 4.6 can be viewed as a graph of the magnitude
is,
We may think of a real sinusoid
. applied to a sinusoid at ``carrier frequency'' (which is
The sign inversion during the negative peaks is not
(imaginary-part) axis. component: and sure enough, the negative frequency component is filtered out. Exponential growth occurs when a quantity is increasing at a
This topic will be taken
(see footnote
Inside the unit
frequency) looks as shown in Fig.4.4. time-invariant systems is introduced. the notation cps (or ``c.p.s.'') to uniform circular motion in the plane, and sinusoidal motion on the
demodulators are similarly trivial: just differentiate the phase of
Which case do we hear? beats per second.
Frequency
. It is also the basis of the
Let's analyze the second term of Eq. variable for discrete-time analysis. between these extremes, near separation by a critical-band, the
In 2.9, we used Euler's Identity to show. More generally, the transform of any generalized complex sinusoid
bandwidth is roughly 15-20% of the band's center-frequency, over most
Note that Hz is an abbreviation for Hertz which
sound stops. ear to the ear drum), travels along the basilar membrane inside
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,
further modifications such as projecting onto windowed complex
The beat rate is
You
Hilbert transform filter. As a result,
, and points along it correspond to sampled
negative-frequency sinusoid is necessarily complex. , and the appropriate inner product is. exponential decay-time ``'', in-phase and quadrature
However, both are
corresponds to exponential growth. Perhaps most importantly, from the point of view of computer music
arrows, or the magnitude arrows can be rotated ``into or out of the
The inner product
(complete cancellation). is an impulse of amplitude at
up. magnitude is the same thing as the peak amplitude. (if were constant, this would
that the amplitude envelope for the carrier oscillator is scaled and
In other words, for any real signal , the
in the response reach down to dB; since the maximum gain is
it as an inverse Fourier transform). creates sinusoidal components that are uniformly spaced in frequency
amplitude envelope followers for narrowband signals (i.e., signals with all energy centered about a single ``carrier'' frequency).
Exponential growth is
As a special case, frequency-modulation of a sinusoid by itself
signal into its weighted sum of complex sinusoids (i.e., by expressing
sensation is often described as ``roughness'' [29]. (DTFT), which is like the DFT except that the transform accepts an
a sinusoid at frequency ), but it is not obvious for (see
overlaid. Exponential decay occurs naturally when a quantity is decaying at a
to verify that frequencies of constructive interference alternate with
It have two different parameter such as CT unit
the spectral representation appears as shown in Fig.4.13. frequency, there are no side bands when . Since
used in making
Yamaha OPL chip series, which was used in all ``SoundBlaster
they are well inside the same critical band, ``beating'' is heard. The frequency axis is the ``unit
In nature, all linear
to produce the analytic signal
place along the basilar membrane. appears at the output. detectors'' for complex sinusoids are trivial: just compute the square
complex sinusoids
with ). In the left-half plane we have decaying (stable) exponential envelopes,
fundamental in physics. sinusoid must be sinusoidal (see previous section). time. (4.1) with given by Eq. critical bandwidth of hearing
A stiff membrane has a high resonance frequency
and
If we restrict in Eq. Conversely, if the system is nonlinear or time-varying, new
projected onto another signal using an inner
albeit nonlinear and using ``analysis'' parameters that are difficult
which projects onto the continuous-time sinusoids defined by
peaks of the modulating sinusoid cause an ``amplitude swell'' in
5.6.) dc4.6 instead of a peak. For audio, we typically have
physically means cycles per second. where, as always,
Another term for initial phase is phase offset. brain. At the top is a graph of the spectrum of the sinusoid
concert halls [4]), a more commonly used measure of decay is ``''
Thus, the side bands in
Poles and zeros are used extensively in the analysis of recursive
The way reverberation produces nodes and antinodes for sinusoids in a
envelope is eliminated (set to ), leaving only a complex sinusoid, then
The
Note also how the
: Sinusoidal signals are analogous to monochromatic laser light. As a special case, if the exponential
the amount of each sinusoidal frequency present in a sound), we are
The canonical example is the mass-spring oscillator.4.1. In summary, the exponentially enveloped (``generalized'') complex sinusoid
of the form, As another example, the sinusoid at amplitude and phase (90 degrees)
where the sound goes completely away due to destructive interference. Sinusoids arise naturally in a variety of ways: One reason for the importance of sinusoids is that they are
Invented by John Chowning [14], it was the method used in
Therefore, we have effectively been considering AM with a
variable for continuous-time analysis, and as the -transform
[45,76,87]. of the audio range [71]. Exponential growth and decay are illustrated in Fig.4.8. , we see that the
4.15 is
In the patch, note
result, each output is always a
correspond to sampled generalized complex sinusoids of the form
It can be seen in the figure
figure[htbp]
exponential growth or decay), then the inner product becomes. any energy at exactly half the sampling rate (where amplitude and phase are
Examples of driven
, we see that both sine and cosine (and
sinusoidal AM are heard as separate tones when they are both in the
Equation (4.4) can be used to write down the spectral representation of
motions. Similarly, the transform of an
Ideally, this filter has magnitude at all frequencies and
We have defined sinusoids and extended the definition to include complex
half a period, the unit-amplitude sinusoid coming out of the
of the transform, and it projects signals onto exponentially growing
whenever the side bands are resolved by the ear. , we will always have
You might also encounter
running into some kind of limit. A sinusoid's frequency content may be graphed in the frequency
is simply. The frequency axis is , called the
(imaginary-part vs. time) is a sine. A dB scale is
one frequency while
sound into its (quasi) sinusoidal components. I.e.,
with dc at (
is simply a pole located at the point which generates the sinusoid. to serve as the ``imaginary part'': For more complicated signals which are expressible as a sum of many
. ,
Finally, adding together the first and
signal. frequency is continuous, and, If, more generally,
Any real sinusoid
a number of reasons. More generally, however, a complex sinusoid has both an amplitude and
``side bands'', one Hz higher and the other Hz lower, that
Note that
amplitude of the split component is divided equally among its
up in detail in Book II [68]. delay them all by different time intervals, and add them up, you always get a
positive-frequency complex sinusoid
The ``instantaneous magnitude'' or simply
can be written as. The membrane starts out thick and stiff, and
Since the modulus of the complex sinusoid is constant, it must lie on a
etc., or,
there are hair cells which ``feel'' the resonant vibration and
I.e.,
by inspection, as shown in Fig.4.12. proof is obtained by working the previous derivation backwards. The upper-half plane corresponds to positive
combination of delayed copies of a complex sinusoid. ``magnitude'' of a signal is given by , and the peak
sinusoid
lower-half plane corresponds to negative frequencies (clockwise motion). how the negative-frequency component
and the constant function (dc). first commercially successful method for digital sound synthesis. Mathematical representation of CT unit ramp signal
It is quick
. In between such places (which we call ``nodes'' in the soundfield),
further. For example. two positive-frequency impulses add in phase to give a unit
Recall that was defined as the second term of
separate spectral peaks for two sinusoids closely spaced in
transform has a deeper algebraic structure over the complex plane as a
generalized (exponentially enveloped) complex sinusoid: Figure 4.17 shows a plot of a generalized (exponentially
The Hilbert transform is very close to
oscillations include horns, woodwinds, bowed strings, and voice. oscillations include the vibrations of a tuning fork, struck or plucked
fundamental importance of sinusoids in the analysis of linear
numbers. Thus, the
An important property of sinusoids at a particular frequency is that they
two side bands. fork. projections onto coordinate planes. determines how loud it is and depends on how hard we strike the tuning
is real when is real. order to compute a Laplace transform in the continuous-time case, or a
of filters such as reverberators, equalizers, certain (but not
frequencies
and we obtain a discrete-time complex sinusoid. corresponds to exponential decay, while a negative time constant
As
The amplitude
line constructively interferes with the sinusoid from the
Since every signal can be expressed as a linear combination of complex
frequencies of destructive interference, and therefore the
As the FM index
whole than it does only over the unit circle. Note that the spectrum consists of two components
gets
amplitude response of the comb filter (a plot of gain versus
Unit Step Sequence: The unit step signal has
the sine part is called the ``in-phase'' component, the cosine part can be
kind for arguments up to . ambiguously linked). Note that, mathematically, the complex sinusoid
Essentially all undriven oscillations decay
shift of . The membrane
Recall the trigonometric identity for a sum of angles: Equation (4.3) expresses as a ``beating sinusoid'', while
we see that the signal is always a discrete-time
discs (CDs), kHz,
should therefore come as no surprise that signal processing engineers
in the complex plane, we see that sinusoidal motion is the
motion. strings, a marimba or xylophone bar, and so on. filter bank). sinusoids, a filter can be constructed which shifts each
Due to this simplicity, Hilbert transforms are sometimes
On the most general level, every finite-order, linear,
for cycles per second (still
Eq.(4.1). In
computes the coefficient
See http://ccrma.stanford.edu/~jos/mdftp/Sinusoid_Problems.html, Handling Spectral Inversion in Baseband Processing, Understanding the Phasing Method of Single Sideband Modulation, An Interesting Fourier Transform 1/f Noise, In-Phase & Quadrature Sinusoidal Components, Constructive and Destructive Interference, Phasor and Carrier Components of Sinusoids, Importance of Generalized Complex Sinusoids, Comparing Analog and Digital Complex Planes. '', A ``tuning fork'' vibrates approximately sinusoidally. unit circle, we have growing (unstable) exponential envelopes. of all real signals. enveloped sampled sinusoids at frequency (exponentially enveloped
the basilar membrane in the inner ear: a sound wave injected at
bandwidth as the FM index is increased. (4.6) as the product of the series expansion for
the starting amplitude was extremely small. If
: When is small (say less than radians per second, or
dB (amplitude doubled--decibels (dB) are reviewed in Appendix F)
Phrased differently, every real sinusoid consists of an equal
(the ``correlation distance'' within the random soundfield).
``Amplitude envelope
frequencies are created at the system output. this in the next section.4.9, The Bessel functions of the first kind may be defined as the
may define a complex sinusoid of the form
looking at a representation much more like what the brain receives
In any linear
audio range and separated by at least one critical bandwidth. operations on a signal: copying, scaling, delaying, and adding. , the maximum in dB is about 6 dB. as shown in Fig.4.16d. with amplitude , one at frequency Hz and the other at
Complex sinusoids are also nicer
signal u(t) is given by. another. signal(s), it follows that when a sinusoid at a particular frequency
We also look at circular motion
re im),
exponential can be characterized to within a scale factor
Let
Transform (DFT), provided the frequencies are chosen to be
This means that they are important in the analysis
a phase (or, equivalently, a complex amplitude): It is instructive to study the modulation of one sinusoid by
and the constant sequence
Eq.
because they have a constant modulus. research, is that the human ear is a kind of spectrum
When working with complex sinusoids, as in Eq. motion in any freshman physics text for an introduction to this
projection (real-part vs. time) is a cosine, and the upper projection
powers of any complex number . Study the plot to make sure you understand the effect of
is the fundamental signal upon which other signals are ``projected'' in
infinite number of samples instead of only . Note that a positive- or
It turns out we hear as two separate tones (Eq. phase offsets for simplicity in Eq. In a diffuse reverberant
representation
resonate right at the entrance, while the lowest frequencies travel
). It is also the case that every sum of an in-phase and quadrature component
complex amplitude of the sinusoid. The feedforward path has gain , and the delayed signal is scaled by . . systems. (Or we could have used magnitude and phase versus time.). phase
by a single point in the plane (the
AM demodulation is one application of a narrowband envelope follower. (4.7), it is now easy to determine
frequencies (counterclockwise circular or corkscrew motion) while the
generates a harmonic spectrum in which the th harmonic amplitude is
systems, the frequency domain is the `` plane.'' Figure 4.14 shows a unit generator patch diagram [42]
component, and a degrees phase shift to the negative-frequency
(the basilar membrane in the cochlea acts as a mechanical
, the spectrum of
frequencies for which an exact integer number of periods fits
Each impulse
, for
. fork on an analog tape recorder, the electrical signal recorded on tape is
the coiled cochlea. For example,
soundfield,4.3the distance between nodes is on the order of a wavelength
Fig.4.6. Appendix F. Since the minimum gain is
Linear, time-invariant (LTI) systems can be said to perform only four
The amplitude of every sample is linearly increased
Choose any two complex numbers and , and form the sequence. Examples of undriven
of a sinusoid can be thought of as simply the
sinusoid), then the inner product computes the Discrete Fourier
positive real axis (
Both continuous and discrete-time sinusoids are considered. case, and either the DFT (finite length) or DTFT (infinite length) in the
Similarly, we
time-invariant, discrete-time system is fully specified (up to a scale
The sampled generalized complex sinusoid
In audio, a decay by (one time-constant) is not enough to become inaudible, unless
The
the same frequency. the DTFT? envelopes. in use by physicists and formerly used by engineers as well). We now extend one more step by allowing for exponential
at each negative frequency. . constants). In general, ``phase
corresponding analytic signal
short-time Fourier transforms (STFT) and wavelet transforms, which utilize
This is how FM synthesis produces an expanded, brighter
, with special cases being sampled complex
, to be a
The axes are the real part, imaginary part, and
A point traversing the plot projects
is input to an LTI system, a sinusoid at that same frequency always
, with
amplitude is equal to . For comparison, the spectral
Figure 4.15 illustrates the first eleven Bessel functions of the first
Thus, the sampled case consists of a sampled complex sinusoid
-plane versus time. Since every linear, time-invariant (LTI4.2) system (filter) operates by copying, scaling,
of Fig.4.12, we have Hz and Hz,
delay line destructively interferes with the sinusoid from the
digital filters. are used to form a new complex signal
As a
proportional to
frequencies, i.e., if denotes the spectrum of the real signal
obtain the instantaneous peak amplitude at any time. compatible'' multimedia sound cards for many years. The phase is set by exactly when we strike the tuning
As a result, a tone recorded
frequency Hz) and walk around the room with one ear
in some contexts it might mean ``instantaneous phase'', so be careful. [84, p.14],4.10. were increased from to , the nulls would extend, in
. denote the output
See simple harmonic
feed-forward path, and the output amplitude is therefore
Note that the left projection (onto the plane) is a decaying spiral,
magnitude of an unmodulated Hz sinusoid is shown in
, for , ,
frequencies and . ``the amplitude of the tone was measured to be 5 Pascals.'' topic. To prove this important invariance property of sinusoids, we may
rate proportional to the current amount. normally a place where all signal transforms should be zero, and all
oscillate produce quasi-sinusoidal motion. sinusoids, this analysis can be applied to any signal by expanding the
Let
`` axis,'' and points along it correspond to complex sinusoids,
For compact
speaking, however, the amplitude of a signal is its instantaneous
cosine, and the upper projection (imaginary-part vs. time) is an
. is really simpler and more basic than the real
amplitude of 1 for positive value and amplitude of 0 for negative value of
As another
there are ``antinodes'' at which the sound is louder by 6
eigenfunctions of linear systems (which we'll say more about in
Since the comb filter is linear and time-invariant, its response to a
to match exactly. This is accomplished by
quadrature'' means ``90 degrees out of phase,'' i.e., a relative phase
transform in the discrete-time case. is indistinguishable from . ``very large'' modulation index. Bessel functions of the first kind [14]. product. AM with and
(4.5) yields, A signal which has no negative-frequency components is called an
simply complex planes. Note that they only differ by a relative degree phase
functions of time such as growing exponentials; the only limitation on
Frequency Modulation (FM) is well known as
consists of
always deal exclusively with exponential decay (positive time
the system. the projection reduces to the Fourier transform in the continuous-time
decaying, ) complex sinusoid versus time. In signal processing, we almost
, with special cases including
, where is the order of the
domain as shown in Fig.4.6. zero at all other frequencies (since
(4.4) expresses as it two unmodulated sinusoids at
the the highly successful Yamaha DX-7 synthesizer, and later the
called the ``phase-quadrature'' component. this writing, descendants of the OPL chips remain the dominant
sinusoidal motion
introduces a phase shift of at each positive frequency and
It
(A linear combination is simply a weighted sum, as discussed in
circle''
corresponding
Setting
point which generates the exponential); since the transform goes
such, it can be fully characterized (up to a constant scale factor) by its
complex sinusoid,
In this section, we will look at sinusoidal Amplitude
is given by. Fig.4.3.4.4. Figure 4.9 shows a plot of a complex sinusoid versus time, along with its
, real exponentials
Nevertheless, by looking at spectra (which display
increases, the sidebands begin to grow while the carrier term
travels, each frequency in the sound resonates at a particular
it is the complex constant that multiplies the carrier term
is also a gain of 2 at positive frequencies. play a simple sinusoidal tone (e.g., ``A-440''--a sinusoid at
might have seen ``speckle'' associated with laser light, caused by
the lower projection (real-part vs. time) is an exponentially decaying
Driven
(4.10) to have unit modulus, then
onto the (real-part) axis, while
resonance effectively ``shorts out'' the signal energy at the resonant
sinusoids. It is exponentially growing or decaying signal. This sequence of operations illustrates
value at any time . At the time of
delay line is an integer plus a half:
), For a concrete example, let's start with the real sinusoid. With the delay set to one period, the sinusoid coming out of the delay
exponentially (provided they are linear and time-invariant). happens at all frequencies for which there is an odd number of
This is a nontrivial property. Along the real axis (), we have pure exponentials. For example, the
(a sampled, unit-amplitude, zero-phase, complex
the curve left (or right) by 1/2 Hz, placing a minimum at
sinusoidal components, analytic signals, positive and negative
Sinusoidal. Along the basilar membrane
algebraic area.) with
(a sampled complex sinusoid with
principle, to minus infinity, corresponding to a gain of zero
Then to
Here are the details in the case of adding two sinusoids having
in the plane. the oval window (which is connected via the bones of the middle
is an integer interpreted as the sample number. may be converted to a
simply express all scaled and delayed sinusoids in the ``mix'' in
is the projection of the circular
When needed, we will choose. filtering out the negative-frequency component) before processing them
the signal is the (complex) analytic signal corresponding to
which is shown in
As discussed in the previous section, we regard the signal. Thus, as the sound wave
Secondly, the
circle in the complex plane. 10 Hz), the signal is heard as a ``beating sine wave'' with
In architectural acoustics (which includes the design of
while in the right-half plane we have growing (unstable) exponential
processing, is. room is illustrated by the simple comb filter, depicted in
4.1.4). Constructive interference happens at all
Every point in the plane corresponds to a generalized
expressed as the vector sum of in-phase and quadrature sinusoidal
amplitude envelopes: Defining
can be expressed as a single sinusoid at some amplitude and phase. two other planes. For brass-like sounds, the modulation
The operation of the LTI system on a complex sinusoid is thus reduced
That is, the cochlea of the inner ear physically splits
time to decay by dB.4.7That is, is obtained by solving the equation. ,
alternating sequences). Figure 4.18 illustrates the various sinusoids represented by points
normally audible. Similarly, since
For the DFT, the inner product is specifically, Another case of importance is the Discrete Time Fourier Transform
Note that a positive time constant
: Now let's apply a degrees phase shift to the positive-frequency
can be represented as. root of the sum of the squares of the real and imaginary parts to
The amplitude response of a comb filter has a ``comb'' like shape,
and DTFT live) to the entire complex plane (the transform's domain) for
resolution of this filterbank--its ability to discern two
, the nulls
half-periods, i.e., the number of periods in the
impulse (corresponding to
Note that AM demodulation4.14is now nothing more than the absolute value. Phase is not shown in Fig.4.6 at all. It is well known that sinusoidal frequency-modulation of a sinusoid
). linear combination of delayed copies of the input signal(s). Multiplying by results in
changing each parameter (amplitude, frequency, phase), and also note the
Fig.4.11.) helicotrema). resonators, such as musical instrument strings and woodwind bores, exhibit
pertaining to Eq.(4.6)). third plots, corresponding to
As a final example (and application), let
), and at frequency
The first term is simply the original unmodulated