## Complex Sinusoids

Recall Euler's Identity,

*complex sinusoid*:

*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

### Circular Motion

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

*counter-clockwise*circular motion along the unit circle in the complex plane as increases, while

*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.

### 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

*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*):

### 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

*carrier wave*, is called the

*modulation index*(or

*AM index*), and 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:

In the case of

*sinusoidal*AM, we have

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 :

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.

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 (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].

#### 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 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.

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 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

The last expression can be interpreted as the Fourier superposition of the sinusoidal harmonics of ,

*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 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 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),

^{4.11}

*i.e.*, expressing a real-valued FM signal as the real part of a more analytically tractable complex-valued FM signal, we obtain

re | |||

re | |||

re | |||

re | |||

(4.9) |

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

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'':

^{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

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 *demodulation*^{4.14}is 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*:

*complex*, and further defined as

When , we obtain

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

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 the complex sinusoid's*phasor*, and its

*carrier wave*.

For a *real* sinusoid,

*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
.

#### 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

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 projection*^{4.16} 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

*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

*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:

*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.

### 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