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Complex Sinusoids

Recall Euler's Identity,

$\displaystyle e^{j\theta} = \cos(\theta) + j\sin(\theta). $

Multiplying this equation by $ A \geq 0$ and setting $ \theta = \omega t + \phi$, where $ t$ is time in seconds, $ \omega$ is radian frequency, and $ \phi$ is a phase offset, we obtain what we call the complex sinusoid:

$\displaystyle s(t) \isdef A e^{j(\omega t+\phi)} = A \cos(\omega t+\phi) + jA\sin(\omega t+\phi) $

Thus, a complex sinusoid consists of an ``in-phase'' component for its real part, and a ``phase-quadrature'' component for its imaginary part. Since $ \sin^2(\theta) + \cos^2(\theta) = 1$, we have

$\displaystyle \left\vert s(t)\right\vert \isdef \sqrt{\mbox{re}^2\left\{s(t)\right\} + \mbox{im}^2\left\{s(t)\right\}} \equiv A. $

That is, the complex sinusoid has a constant modulus (i.e., a constant complex magnitude). (The symbol ``$ \equiv$'' means ``identically equal to,'' i.e., for all $ t$.) The instantaneous phase of the complex sinusoid is

$\displaystyle \angle s(t) = \omega t+\phi. $

The derivative of the instantaneous phase of the complex sinusoid gives its instantaneous frequency

$\displaystyle \frac{d}{dt}\angle s(t) = \omega = 2\pi f. $

Circular Motion

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

$\displaystyle x(t) = e^{j\omega t} $

traces out counter-clockwise circular motion along the unit circle in the complex plane as $ t$ increases, while

$\displaystyle \overline{x(t)} = e^{-j\omega t} $

gives clockwise circular motion.

We may call a complex sinusoid $ e^{j\omega t}$ a positive-frequency sinusoid when $ \omega>0$. Similarly, we may define a complex sinusoid of the form $ e^{-j\omega t}$, with $ \omega>0$, 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,

\begin{eqnarray*} \mbox{re}\left\{e^{j\omega t}\right\} &=& \cos(\omega t) \\ \mbox{im}\left\{e^{j\omega t}\right\} &=& \sin(\omega t), \end{eqnarray*}

in the complex plane, we see that sinusoidal motion is the projection of circular motion onto any straight line. Thus, the sinusoidal motion $ \cos(\omega t)$ is the projection of the circular motion $ e^{j\omega t}$ onto the $ x$ (real-part) axis, while $ \sin(\omega t)$ is the projection of $ e^{j\omega t}$ onto the $ y$ (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 $ z$-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}

Note that the left projection (onto the $ z$ 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 $ z$ plane, and sinusoidal motion on the two other planes.

Positive and Negative Frequencies

In §2.9, we used Euler's Identity to show

\begin{eqnarray*} \cos(\theta) &= \frac{\displaystyle e^{j \theta} + e^{-j \thet... ...\theta) &= \frac{\displaystyle e^{j \theta} - e^{-j \theta}}{2j} \end{eqnarray*}

Setting $ \theta = \omega t + \phi$, 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 $ X(\omega)$ denotes the spectrum of the real signal $ x(t)$, we will always have $ \vert X(-\omega)\vert = \vert X(\omega)\vert$.

Note that, mathematically, the complex sinusoid $ Ae^{j(\omega t + \phi)}$ is really simpler and more basic than the real sinusoid $ A\sin(\omega t + \phi)$ because $ e^{j\omega t}$ consists of one frequency $ \omega$ while $ \sin(\omega t)$ really consists of two frequencies $ \omega$ and $ -\omega$. 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

$\displaystyle x(t) = A_x e^{j\omega_x t} $

as a positive-frequency sinusoid when $ \omega_x>0$. 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 $ A_x$ at the point $ \omega=\omega_x$, 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] \includegraphics{eps/csplot} More generally, however, a complex sinusoid has both an amplitude and a phase (or, equivalently, a complex amplitude):

$\displaystyle x(t) = \left(A_x e^{j\theta_x}\right)e^{j\omega_x t} $

To accommodate the phase angle $ \theta_x$ in spectral plots, the plotted vector may be rotated by the angle $ \theta_x$ in the plane orthogonal to the frequency axis passing through $ \omega_x$, as done in Fig.4.16b below (p. [*]) for phase angles $ \theta_x=\pm \pi/2$.

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), $

where $ (A_c,\omega_c,\phi_c)$ are parameters of the sinusoidal carrier wave, $ \alpha\in[0,1]$ is called the modulation index (or AM index), and $ a_m(t)\in[-1,1]$ is the amplitude modulation signal. In AM radio broadcasts, $ a_m(t)$ is the audio signal being transmitted (usually bandlimited to less than 10 kHz), and $ \omega_c$ is the channel center frequency that one dials up on a radio receiver. The modulated signal $ x_\alpha(t)$ can be written as the sum of the unmodulated carrier wave plus the product of the carrier wave and the modulating wave:

$\displaystyle x_\alpha(t) = x_0(t) + \alpha \cdot a_m(t) \cdot A_c\sin(\omega_c t + \phi_c) \protect$ (4.1)

In the case of sinusoidal AM, we have

$\displaystyle a_m(t) = \sin(\omega_m t + \phi_m). \protect$ (4.2)

Periodic amplitude modulation of this nature is often called the tremolo effect when $ \omega_m<20\pi$ or so ($ <10$ Hz).

Let's analyze the second term of Eq.$ \,$(4.1) for the case of sinusoidal AM with $ \alpha =1$ and $ \phi_m=\phi_c=0$:

$\displaystyle x_m(t) \isdef \sin(\omega_m t)\sin(\omega_c t) \protect$ (4.3)

An example waveform is shown in Fig.4.11 for $ f_c=100$ Hz and $ f_m=10$ 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}

When $ \omega_m$ is small (say less than $ 20\pi$ radians per second, or 10 Hz), the signal $ x_m(t)$ is heard as a ``beating sine wave'' with $ \omega_m/\pi=2f_m$ 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 $ x_m(t)$. (One period of modulation--$ 1/f_m$ 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:

$\displaystyle \cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B) $

Subtracting this from $ \cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B)$ leads to the identity

$\displaystyle \sin(A)\sin(B) = \frac{\cos(A-B) - \cos(A+B)}{2}. $

Setting $ A=\omega_m t$ and $ B=\omega_c t$ gives us an alternate form for our ``ring-modulator output signal'':

$\displaystyle x_m(t) \isdef \sin(\omega_m t)\sin(\omega_c t) = \frac{\cos[(\omega_m-\omega_c)t] - \cos[(\omega_m+\omega_c)t]}{2} \protect$ (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 $ \omega_c$ (since typically $ \omega_c\gg\omega_m>0$).

Equation (4.3) expresses $ x_m(t)$ as a ``beating sinusoid'', while Eq.$ \,$(4.4) expresses as it two unmodulated sinusoids at frequencies $ \omega_c\pm\omega_m$. Which case do we hear?

It turns out we hear $ x_m(t)$ 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 $ x_m(t)$ by inspection, as shown in Fig.4.12. In the example of Fig.4.12, we have $ f_c=100$ Hz and $ f_m=20$ Hz, where, as always, $ \omega=2\pi f$. For comparison, the spectral magnitude of an unmodulated $ 100$ Hz sinusoid is shown in Fig.4.6. Note in Fig.4.12 how each of the two sinusoidal components at $ \pm100$ Hz have been ``split'' into two ``side bands'', one $ 20$ Hz higher and the other $ 20$ Hz lower, that is, $ \pm100\pm20=\{-120,-80,80,120\}$. Note also how the amplitude of the split component is divided equally among its two side bands.

figure[htbp] \includegraphics{eps/sineamfd}

Recall that $ x_m(t)$ 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 $ a_m(t)$ given by Eq.$ \,$(4.2), the magnitude of the spectral representation appears as shown in Fig.4.13.

figure[htbp] \includegraphics{eps/sineamgfd}

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

$\displaystyle x(t) = A_c\cos[\omega_c t + \phi_c + A_m\sin(\omega_m t + \phi_m)], \protect$ (4.5)

where the parameters $ (A_c,\omega_c,\phi_c)$ describe the carrier sinusoid, while the parameters $ (A_m,\omega_m,\phi_m)$ 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] \includegraphics{eps/fmug}

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 $ k$th harmonic amplitude is proportional to $ J_k(\beta)$, where $ k$ is the order of the Bessel function and $ \beta $ 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 $ J_k(\beta)$ in the two-sided Laurent expansion of the so-called generating function [84, p. 14],4.10

$\displaystyle e^{\frac{1}{2}\beta\left(z-\frac{1}{z}\right)} = \sum_{k=-\infty}^\infty J_k(\beta) z^k \protect$ (4.6)

where $ k$ is the integer order of the Bessel function, and $ \beta $ is its argument (which can be complex, but we will only consider real $ \beta $). Setting $ z=e^{j\omega_mt}$, where $ \omega_m$ will interpreted as the FM modulation frequency and $ t$ as time in seconds, we obtain

$\displaystyle x_m(t)\isdef e^{j\beta\sin(\omega_m t)} = \sum_{k=-\infty}^\infty J_k(\beta) e^{jk\omega_m t}. \protect$ (4.7)

The last expression can be interpreted as the Fourier superposition of the sinusoidal harmonics of $ \exp[j\beta\sin(\omega_m t)]$, i.e., an inverse Fourier series sum. In other words, $ J_k(\beta)$ is the amplitude of the $ k$th harmonic in the Fourier-series expansion of the periodic signal $ x_m(t)$.

Note that $ J_k(\beta)$ is real when $ \beta $ is real. This can be seen by viewing Eq.$ \,$(4.6) as the product of the series expansion for $ \exp[(\beta/2) z]$ times that for $ \exp[-(\beta/2)/z]$ (see footnote pertaining to Eq.$ \,$(4.6)).

Figure 4.15 illustrates the first eleven Bessel functions of the first kind for arguments up to $ \beta=30$. It can be seen in the figure that when the FM index $ \beta $ is zero, $ J_0(0)=1$ and $ J_k(0)=0$ for all $ k>0$. Since $ J_0(\beta)$ is the amplitude of the carrier frequency, there are no side bands when $ \beta=0$. 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 $ k$ and argument $ \beta $.
\includegraphics[width=\twidth]{eps/bessel}

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

$\displaystyle x(t) = \cos[\omega_c t + \beta\sin(\omega_m t)], \protect$ (4.8)

where we have changed the modulator amplitude $ A_m$ to the more traditional symbol $ \beta $, 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
$\displaystyle x(t) \isdef \cos[\omega_c t + \beta\sin(\omega_m t)]$ $\displaystyle =$ re$\displaystyle \left\{e^{j[\omega_c t + \beta\sin(\omega_m t)]}\right\}$  
  $\displaystyle =$ re$\displaystyle \left\{e^{j\omega_c t} e^{j\beta\sin(\omega_m t)}\right\}$  
  $\displaystyle =$ re$\displaystyle \left\{e^{j\omega_c t} \sum_{k=-\infty}^\infty J_k(\beta) e^{jk\omega_m t}\right\}$  
  $\displaystyle =$ re$\displaystyle \left\{\sum_{k=-\infty}^\infty J_k(\beta) e^{j(\omega_c+k\omega_m) t}\right\}$  
  $\displaystyle =$ $\displaystyle \sum_{k=-\infty}^\infty J_k(\beta) \cos[(\omega_c+k\omega_m) t]$ (4.9)

where we used the fact that $ J_k(\beta)$ is real when $ \beta $ is real. We can now see clearly that the sinusoidal FM spectrum consists of an infinite number of side-bands about the carrier frequency $ \omega_c$ (when $ \beta\neq 0$). The side bands occur at multiples of the modulating frequency $ \omega_m$ away from the carrier frequency $ \omega_c$.

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 $ z(t)$ can be represented as

$\displaystyle z(t) = \frac{1}{2\pi}\int_0^{\infty} Z(\omega)e^{j\omega t}d\omega $

where $ Z(\omega)$ is the complex coefficient (setting the amplitude and phase) of the positive-frequency complex sinusoid $ \exp(j\omega t)$ at frequency $ \omega$.

Any real sinusoid $ A\cos(\omega t + \phi)$ may be converted to a positive-frequency complex sinusoid $ A\exp[j(\omega t + \phi)]$ by simply generating a phase-quadrature component $ A\sin(\omega t + \phi)$ to serve as the ``imaginary part'':

$\displaystyle A e^{j(\omega t + \phi)} = A\cos(\omega t + \phi) + j A\sin(\omega t + \phi) $

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 $ {\cal H}_t\{x\}$ denote the output at time $ t$ of the Hilbert-transform filter applied to the signal $ x$. Ideally, this filter has magnitude $ 1$ at all frequencies and introduces a phase shift of $ -\pi/2$ at each positive frequency and $ +\pi/2$ at each negative frequency. When a real signal $ x(t)$ and its Hilbert transform $ y(t) = {\cal H}_t\{x\}$ are used to form a new complex signal $ z(t) = x(t) + j y(t)$, the signal $ z(t)$ is the (complex) analytic signal corresponding to the real signal $ x(t)$. In other words, for any real signal $ x(t)$, the corresponding analytic signal $ z(t)=x(t) + j {\cal H}_t\{x\}$ has the property that all ``negative frequencies'' of $ x(t)$ 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 $ \exp(\pm j\pi/2) = \pm j$. Consider the positive and negative frequency components at the particular frequency $ \omega_0$:

\begin{eqnarray*} x_+(t) &\isdef & e^{j\omega_0 t} \\ x_-(t) &\isdef & e^{-j\omega_0 t} \end{eqnarray*}

Now let's apply a $ -90$ degrees phase shift to the positive-frequency component, and a $ +90$ degrees phase shift to the negative-frequency component:

\begin{eqnarray*} y_+(t) &=& e^{-j\pi/2} e^{j\omega_0 t} = -j e^{j\omega_0 t} \\ y_-(t) &=& e^{j\pi/2} e^{-j\omega_0 t} = j e^{-j\omega_0 t} \end{eqnarray*}

Adding them together gives

\begin{eqnarray*} z_+(t) &\isdef & x_+(t) + j y_+(t) = e^{j\omega_0 t} - j^2 e^{... ... x_-(t) + j y_-(t) = e^{-j\omega_0 t} + j^2 e^{-j\omega_0 t} = 0 \end{eqnarray*}

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

$\displaystyle x(t)=2\cos(\omega_0 t) = e^{j\omega_0 t} + e^{-j\omega_0 t}. $

Applying the ideal phase shifts, the Hilbert transform is

\begin{eqnarray*} y(t) &=& e^{j\omega_0 t-j\pi/2} + e^{-j\omega_0 t + j\pi/2}\\ &=& -je^{j\omega_0 t} + je^{-j\omega_0 t} = 2\sin(\omega_0 t). \end{eqnarray*}

The analytic signal is then

$\displaystyle z(t) = x(t) + j y(t) = 2\cos(\omega_0 t) + j 2\sin(\omega_0 t) = 2 e^{j\omega_0 t}, $

by Euler's identity. Thus, in the sum $ x(t) + j y(t)$, the negative-frequency components of $ x(t)$ and $ jy(t)$ cancel out, leaving only the positive-frequency component. This happens for any real signal $ x(t)$, not just for sinusoids as in our example.

Figure 4.16: Creation of the analytic signal $ z(t)=e^{j\omega _0 t}$ from the real sinusoid $ x(t) = \cos(\omega_0 t)$ and the derived phase-quadrature sinusoid $ y(t) = \sin(\omega_0 t)$, viewed in the frequency domain. a) Spectrum of $ x$. b) Spectrum of $ y$. c) Spectrum of $ j y$. d) Spectrum of $ z = x + jy$.
\includegraphics[width=2.8in]{eps/sineFD}

Figure 4.16 illustrates what is going on in the frequency domain. At the top is a graph of the spectrum of the sinusoid $ \cos(\omega_0 t)$ consisting of impulses at frequencies $ \omega=\pm\omega_0$ and zero at all other frequencies (since $ \cos(\omega_0 t) = (1/2)\exp(j\omega_0 t) + (1/2)\exp(-j\omega_0 t)$). Each impulse amplitude is equal to $ 1/2$. (The amplitude of an impulse is its algebraic area.) Similarly, since $ \sin(\omega_0 t) = (1/2j)\exp(j\omega_0 t) - (1/2j)\exp(-j\omega_0 t) = -(j/2) \exp(j\omega_0 t) + (j/2)\exp(-j\omega_0 t)$, the spectrum of $ \sin(\omega_0 t)$ is an impulse of amplitude $ -j/2$ at $ \omega=\omega_0$ and amplitude $ +j/2$ at $ \omega=-\omega_0$. Multiplying $ y(t)$ by $ j$ results in $ j\sin(\omega_0 t) = (1/2)\exp(j\omega_0 t) - (1/2)\exp(-j\omega_0 t)$ which is shown in the third plot, Fig.4.16c. Finally, adding together the first and third plots, corresponding to $ z(t) = x(t) + j y(t)$, we see that the two positive-frequency impulses add in phase to give a unit impulse (corresponding to $ \exp(j\omega_0 t)$), and at frequency $ -\omega_0$, the two impulses, having opposite sign, cancel in the sum, thus creating an analytic signal $ z$, as shown in Fig.4.16d. This sequence of operations illustrates how the negative-frequency component $ \exp(-j\omega_0 t)$ gets filtered out by summing $ \cos(\omega_0 t)$ with $ j\sin(\omega_0 t)$ to produce the analytic signal $ \exp(j\omega_0 t)$ corresponding to the real signal $ \cos(\omega_0 t)$.

As a final example (and application), let $ x(t) = A(t)\cos(\omega t)$, where $ A(t)$ is a slowly varying amplitude envelope (slow compared with $ \omega$). This is an example of amplitude modulation applied to a sinusoid at ``carrier frequency'' $ \omega$ (which is where you tune your AM radio). The Hilbert transform is very close to $ y(t)\approx A(t)\sin(\omega t)$ (if $ A(t)$ were constant, this would be exact), and the analytic signal is $ z(t)\approx A(t)e^{j\omega t}$. Note that AM demodulation4.14is now nothing more than the absolute value. I.e., $ A(t) = \left\vert z(t)\right\vert$. 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:

$\displaystyle y(t) \isdef {\cal A}e^{st} $

where $ {\cal A}$ and $ s$ are complex, and further defined as

\begin{eqnarray*} {\cal A}&=& Ae^{j\phi} \\ s &=& \sigma + j\omega. \end{eqnarray*}

When $ \sigma=0$, we obtain

$\displaystyle y(t) \isdef {\cal A}e^{j\omega t} = A e^{j\phi} e^{j\omega t} = A e^{j(\omega t + \phi)} $

which is the complex sinusoid at amplitude $ A$, frequency $ \omega$, and phase $ \phi$. More generally, we have

\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*}

Defining $ \tau = -1/\sigma$, we see that the generalized complex sinusoid is just the complex sinusoid we had before with an exponential envelope:

\begin{eqnarray*} \mbox{re}\left\{y(t)\right\} &=& A e^{- t/\tau} \cos(\omega t ... ...{im}\left\{y(t)\right\} &=& A e^{- t/\tau} \sin(\omega t + \phi) \end{eqnarray*}

Sampled Sinusoids

In discrete-time audio processing, such as we normally do on a computer, we work with samples of continuous-time signals. Let $ f_s$ denote the sampling rate in Hz. For audio, we typically have $ f_s>40$ kHz, since the audio band nominally extends to $ 20$ kHz. For compact discs (CDs), $ f_s= 44.1$ kHz, while for digital audio tape (DAT), $ f_s= 48$ kHz.

Let $ T\isdef 1/f_s$ denote the sampling interval in seconds. Then to convert from continuous to discrete time, we replace $ t$ by $ nT$, where $ n$ 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*}

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}$.

Powers of z

Choose any two complex numbers $ z_0$ and $ z_1$, and form the sequence

$\displaystyle x(n) \isdef z_0 z_1^n, \quad n=0,1,2,3,\ldots\,. \protect$ (4.10)

What are the properties of this signal? Writing the complex numbers as

\begin{eqnarray*} z_0 &=& A e^{j\phi} \\ z_1 &=& e^{sT} = e^{(\sigma + j\omega)T}, \end{eqnarray*}

we see that the signal $ x(n)$ is always a discrete-time generalized (exponentially enveloped) complex sinusoid:

$\displaystyle x(n) = A e^{\sigma n T} e^{j(\omega n T + \phi)} $

Figure 4.17 shows a plot of a generalized (exponentially decaying, $ \sigma<0$) complex sinusoid versus time.

Figure 4.17: Exponentially decaying complex sinusoid and projections.
\includegraphics[scale=0.8]{eps/circledecaying}

Note that the left projection (onto the $ z$ 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 $ z_1$ in Eq.$ \,$(4.10) to have unit modulus, then $ \sigma=0$ and we obtain a discrete-time complex sinusoid.

$\displaystyle x(n) \isdef z_0 z_1^n = \left(Ae^{j\phi}\right) e^{j\omega n T} = A e^{j(\omega n T+\phi)}, \quad n=0,1,2,3,\ldots \protect$ (4.11)

where we have defined

\begin{eqnarray*} z_0 &\isdef & Ae^{j\phi}, \quad \hbox{and}\\ z_1 &\isdef & e^{j\omega T}. \end{eqnarray*}

Phasor

It is common terminology to call $ z_0 = Ae^{j\phi}$ the complex sinusoid's phasor, and $ z_1^n = e^{j\omega n T}$ its carrier wave.

For a real sinusoid,

$\displaystyle x_r(n) \isdef A \cos(\omega n T+\phi), $

the phasor is again defined as $ z_0 = Ae^{j\phi}$ and the carrier is $ z_1^n = e^{j\omega n T}$. However, in this case, the real sinusoid is recovered from its complex-sinusoid counterpart by taking the real part:

$\displaystyle x_r(n) =$   re$\displaystyle \left\{z_0z_1^n\right\} $

The phasor magnitude $ \left\vert z_0\right\vert=A$ is the amplitude of the sinusoid. The phasor angle $ \angle{z_0}=\phi$ is the phase of the sinusoid.

When working with complex sinusoids, as in Eq.$ \,$(4.11), the phasor representation $ Ae^{j\phi}$ 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 $ e^{j\omega nT}$.

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

$\displaystyle y(n) = \sum_{i=1}^N g_i x(n-d_i) $

where $ g_i$ is a weighting factor, $ d_i$ is the $ i$th delay, and

$\displaystyle x(n)\isdef e^{j\omega nT} $

is a complex sinusoid, the ``carrier term'' $ e^{j\omega nT}$ can be ``factored out'' of the linear combination:

\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*}

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 $ y(\cdot)$4.15 is projected onto another signal $ x(\cdot)$ using an inner product. The inner product $ \left<y,x\right>$ computes the coefficient of projection4.16 of $ y$ onto $ x$. If $ x_k(n) = e^{j\omega_k n T}, n=0,1,2,\ldots,N-1$ (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 $ \omega_k = 2\pi k f_s/N$. For the DFT, the inner product is specifically

$\displaystyle \left<y,x_k\right> \isdef \sum_{n=0}^{N-1}y(n)\overline{x_k(n)} ... ...^{N-1}y(n)e^{-j 2\pi n k/N} \isdef \hbox{\sc DFT}_k(y) \isdef Y(\omega_k). $

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 $ N$. In this case, frequency is continuous, and

$\displaystyle \left<y,x_\omega\right> = \sum_{n=0}^\infty y(n) e^{-j \omega n T} \isdef \hbox{\sc DTFT}_\omega(y). $

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 $ -\infty$. The DTFT is discussed further in §B.1.

If, more generally, $ x(n) = z^n$ (a sampled complex sinusoid with exponential growth or decay), then the inner product becomes

$\displaystyle \left<y,x\right> = \sum_{n=0}^\infty y(n) z^{-n} $

and this is the definition of the $ z$ transform. It is a generalization of the DTFT: The DTFT equals the $ z$ transform evaluated on the unit circle in the $ z$ plane. In principle, the $ z$ transform can also be recovered from the DTFT by means of ``analytic continuation'' from the unit circle to the entire $ z$ plane (subject to mathematical disclaimers which are unnecessary in practical applications since they are always finite).

Why have a $ z$ 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 $ z$ 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 $ z$ transform has a deeper algebraic structure over the complex plane as a whole than it does only over the unit circle. For example, the $ z$ transform of any finite signal is simply a polynomial in $ z$. As such, it can be fully characterized (up to a constant scale factor) by its zeros in the $ z$ plane. Similarly, the $ z$ transform of an exponential can be characterized to within a scale factor by a single point in the $ z$ plane (the point which generates the exponential); since the $ z$ transform goes to infinity at that point, it is called a pole of the transform. More generally, the $ z$ 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 $ z$ 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 $ y$ onto the continuous-time sinusoids defined by $ x(t)=e^{j\omega t}$, and the appropriate inner product is

$\displaystyle \left<y,x\right> = \int_{0}^{\infty} y(t) e^{-j\omega t} dt \isdef Y(\omega). $

Finally, the Laplace transform is the continuous-time counterpart of the $ z$ transform, and it projects signals onto exponentially growing or decaying complex sinusoids:

$\displaystyle \left<y,x\right> = \int_{0}^{\infty} y(t) e^{-s t} dt \isdef Y(s) $

The Fourier transform equals the Laplace transform evaluated along the ``$ j\omega$ axis'' in the $ s$ plane, i.e., along the line $ s=j\omega$, for which $ \sigma=0$. 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 $ z$ transform outlined above.

Comparing Analog and Digital Complex Planes

In signal processing, it is customary to use $ s$ as the Laplace transform variable for continuous-time analysis, and $ z$ as the $ z$-transform variable for discrete-time analysis. In other words, for continuous-time systems, the frequency domain is the ``$ s$ plane'', while for discrete-time systems, the frequency domain is the ``$ z$ plane.'' However, both are simply complex planes.

Figure 4.18: Generalized complex sinusoids represented by points in the $ s$ plane.
\includegraphics[width=4.5in]{eps/splane}

Figure 4.19: Sampled generalized complex sinusoids represented by points in the $ z$ plane.
\includegraphics[width=\twidth]{eps/zplane}

Figure 4.18 illustrates the various sinusoids $ e^{st}$ represented by points in the $ s$ plane. The frequency axis is $ s=j\omega$, called the ``$ j\omega$ axis,'' and points along it correspond to complex sinusoids, with dc at $ s=0$ ( $ e^{j0t}\equiv 1$). The upper-half plane corresponds to positive frequencies (counterclockwise circular or corkscrew motion) while the lower-half plane corresponds to negative frequencies (clockwise motion). In the left-half plane we have decaying (stable) exponential envelopes, while in the right-half plane we have growing (unstable) exponential envelopes. Along the real axis ($ s=\sigma$), we have pure exponentials. Every point in the $ s$ plane corresponds to a generalized complex sinusoid, $ x(t) = {\cal A}e^{st}, t\geq 0$, with special cases including complex sinusoids $ {\cal A}e^{j\omega t}$, real exponentials $ A e^{\sigma t}$, and the constant function $ x(t)=1$ (dc).

Figure 4.19 shows examples of various sinusoids $ z^n=(e^{sT})^n$ represented by points in the $ z$ plane. The frequency axis is the ``unit circle'' $ z=e^{j\omega T}$, and points along it correspond to sampled complex sinusoids, with dc at $ z=1$ ( $ 1^n = [e^{j0T}]^n = 1$). While the frequency axis is unbounded in the $ s$ plane, it is finite (confined to the unit circle) in the $ z$ plane, which is natural because the sampling rate is finite in the discrete-time case. As in the $ s$ 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$ \left\{z\right\}>0, \;$   im$ \left\{z\right\}=0$), we have pure exponentials, but along the negative real axis ( re$ \left\{z\right\}<0, \;$   im$ \left\{z\right\}=0$), we have exponentially enveloped sampled sinusoids at frequency $ f_s/2$ (exponentially enveloped alternating sequences). The negative real axis in the $ z$ plane is normally a place where all signal $ z$ 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 $ z$ 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$, with special cases being sampled complex sinusoids $ {\cal A}e^{j\omega nT}$, sampled real exponentials $ A e^{\sigma nT}$, and the constant sequence $ x=[1,1,1,\ldots]$ (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 $ z$ transform in the discrete-time case. As a special case, if the exponential envelope is eliminated (set to $ 1$), 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.

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Sinusoid Problems
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Exponentials