Recall Euler's Identity,
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):
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
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 . 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'' .
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 , 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  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 . 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
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.
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),4.11i.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
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
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 .
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
PhasorIt is common terminology to call the complex sinusoid's phasor, and its carrier wave.
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
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 .
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.