Complex Sinusoids
Recall Euler's Identity,













Circular Motion
Since the modulus of the complex sinusoid is constant, it must lie on a circle in the complex plane. For example,



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.
Projection of Circular Motion
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.)
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




figure[htbp]
More generally, however, a complex sinusoid has both an amplitude and
a phase (or, equivalently, a complex amplitude):




![[*]](../icons/crossref.png)

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
![$\displaystyle x_\alpha(t) = [1+\alpha \cdot a_m(t)]\cdot A_c\sin(\omega_c t + \phi_c),
$](http://www.dsprelated.com/josimages_new/mdft/img480.png)

![$ \alpha\in[0,1]$](http://www.dsprelated.com/josimages_new/mdft/img482.png)
![$ a_m(t)\in[-1,1]$](http://www.dsprelated.com/josimages_new/mdft/img483.png)



In the case of sinusoidal AM, we have
Periodic amplitude modulation of this nature is often called the tremolo effect when


Let's analyze the second term of Eq.(4.1) for the case of sinusoidal
AM with
and
:
An example waveform is shown in Fig.4.11 for


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:





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


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].
Example AM Spectra
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.
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.
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
where the parameters


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.
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
where






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)



Note that
is real when
is real. This can be seen
by viewing Eq.
(4.6) as the product of the series expansion for
times that for
(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.
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
where we have changed the modulator amplitude


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re![]() |
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re![]() |
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re![]() |
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(4.9) |
where we used the fact that






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




Any real sinusoid
may be converted to a
positive-frequency complex sinusoid
by simply generating a phase-quadrature component
to serve as the ``imaginary part'':

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


The analytic signal is then





![]() |
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.
Generalized Complex Sinusoids
We have defined sinusoids and extended the definition to include complex sinusoids. We now extend one more step by allowing for exponential amplitude envelopes:




When , we obtain




![\begin{eqnarray*}
y(t) &\isdef & {\cal A}e^{st} \\
&\isdef & A e^{j\phi} e^{(\...
... t} \left[\cos(\omega t + \phi) + j\sin(\omega t + \phi)\right].
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img604.png)
Defining
, we see that the generalized complex sinusoid
is just the complex sinusoid we had before with an exponential envelope:

Sampled Sinusoids
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
![\begin{eqnarray*}
y(nT) &\isdef & \left.{\cal A}\,e^{st}\right\vert _{t=nT}\\
...
...
\left[\cos(\omega nT + \phi) + j\sin(\omega nT + \phi)\right].
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img613.png)
Thus, the sampled case consists of a sampled complex sinusoid
multiplied by a sampled exponential envelope
.
Powers of z
Choose any two complex numbers and
, and form the sequence
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.
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.
Phasor and Carrier Components of Sinusoids
If we restrict in Eq.
(4.10) to have unit modulus, then
and we obtain a discrete-time complex sinusoid.
where we have defined

Phasor
It is common terminology to call

For a real 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
.
Why Phasors are Important
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






![\begin{eqnarray*}
y(n) &=& \sum_{i=1}^N g_i e^{j[\omega (n-d_i)T]}
= \sum_{i=1}...
...e^{-j \omega d_i T}
= x(n) \sum_{i=1}^N g_i e^{-j \omega d_i T}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img636.png)
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).
Importance of Generalized Complex Sinusoids
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


If, more generally,
(a sampled complex sinusoid with
exponential growth or decay), then the inner product becomes






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:






Comparing Analog and Digital Complex Planes
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 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
(
).
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
, with special cases being sampled complex
sinusoids
, sampled real exponentials
,
and the constant sequence
(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 Problems
Previous Section:
Exponentials