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.
Next Section: Sinusoid ProblemsPrevious Section: Exponentials