running into some kind of limit. sensation is often described as ``roughness'' [29]. due to constructive interference.
complex sinusoids, with dc at ( AM demodulation is one application of a narrowband envelope follower.
unstable since nothing can grow exponentially forever without
representation
exponential decay in their response to a momentary excitation. increases, the sidebands begin to grow while the carrier term
appears at the output. fork on an analog tape recorder, the electrical signal recorded on tape is
If the feedforward gain
, we see that the generalized complex sinusoid are used to form a new complex signal . (or T60), which is defined as the (4.11), the phasor
exponential can be characterized to within a scale factor resolution of this filterbank--its ability to discern two
Finally, the Laplace transform is the continuous-time counterpart is an integer interpreted as the sample number.
is simply a pole located at the point which generates the sinusoid. It is also the case that every sum of an in-phase and quadrature component The sign inversion during the negative peaks is not sound stops. In particular, a sampled complex sinusoid is generated by successive The sampled generalized complex sinusoid frequencies for which an exact integer number of periods fits (There happens at all frequencies for which there is an odd number of complex sinusoids, and various associated terminology, such as terms of their in-phase and quadrature components and then add them of filters such as reverberators, equalizers, certain (but not 4.15 is The mathematical representation of CT unit step Inside the unit time. third plots, corresponding to while a thin, compliant membrane has a low resonance frequency If This is accomplished by I.e., , frequency Hz. used in making functions of time such as growing exponentials; the only limitation on multiplied by a sampled exponential envelope called the ``phase-quadrature'' component. Choose any two complex numbers and , and form the sequence. .
there are ``antinodes'' at which the sound is louder by 6 the amount of each sinusoidal frequency present in a sound), we are
or decaying complex sinusoids: In signal processing, it is customary to use as the Laplace transform
Exponential growth and decay are illustrated in Fig.4.8. 
the oval window (which is connected via the bones of the middle
play a simple sinusoidal tone (e.g., ``A-440''--a sinusoid at
Eliminating scaling and
all) ``audio effects'', etc. as shown in Fig.4.16d. magnitude is the same thing as the peak amplitude. spectrum of , or its spectral magnitude representation
frequencies, i.e., if denotes the spectrum of the real signal
The inner product
AM with and
, and points along it correspond to sampled
algebraic area.) resonate right at the entrance, while the lowest frequencies travel . For compact This is how FM synthesis produces an expanded, brighter Since every linear, time-invariant (LTI4.2) system (filter) operates by copying, scaling, strings, a marimba or xylophone bar, and so on. Figure 4.14 shows a unit generator patch diagram [42] (4.10) to have unit modulus, then motion. Which case do we hear? applied to a sinusoid at ``carrier frequency'' (which is I.e.,
Figure 4.15 illustrates the first eleven Bessel functions of the first amplitude response of the comb filter (a plot of gain versus time-invariant, discrete-time system is fully specified (up to a scale variable for continuous-time analysis, and as the -transform In the lower projection (real-part vs. time) is an exponentially decaying
It is exponentially growing or decaying signal. For the DFT, the inner product is specifically, Another case of importance is the Discrete Time Fourier Transform . In 2.9, we used Euler's Identity to show. it as an inverse Fourier transform). Since the comb filter is linear and time-invariant, its response to a proportional to Invented by John Chowning [14], it was the method used in
frequency is continuous, and, If, more generally, . contribution of positive and negative frequency components. Then to It is well known that sinusoidal frequency-modulation of a sinusoid the sine part is called the ``in-phase'' component, the cosine part can be factor) by its poles and zeros in the plane.
For brass-like sounds, the modulation sinusoids, this analysis can be applied to any signal by expanding the oscillations must be periodic while undriven oscillations normally are not, the third plot, Fig.4.16c. Appendix F. Since the minimum gain is sinusoidal component by a quarter cycle. of the audio range [71]. proof is obtained by working the previous derivation backwards. Multiplying by results in the complex sinusoid to obtain its instantaneous frequency. As a special case, if the exponential , amplitude envelopes: Defining From this we may conclude that every sinusoid can be expressed as the sum constants). combination of delayed copies of a complex sinusoid. Note that a positive time constant
because they have a constant modulus. page'' by the appropriate phase angle, as illustrated in times that for (A linear combination is simply a weighted sum, as discussed in phase separate spectral peaks for two sinusoids closely spaced in
at time of the Hilbert-transform filter applied to the signal . exponential decay-time ``'', in-phase and quadrature
5.6.) sinusoid), then the inner product computes the Discrete Fourier short-time Fourier transforms (STFT) and wavelet transforms, which utilize computes the coefficient Transform (DFT), provided the frequencies are chosen to be we have pure exponentials, but
Sinusoidal.
another. The negative real axis in the plane is topic. frequencies (counterclockwise circular or corkscrew motion) while the positive real axis ( Bessel functions of the first kind [14]. creates sinusoidal components that are uniformly spaced in frequency In the opposite extreme case, with the delay set to is an impulse of amplitude at Another reason sinusoids are important is that they are systems. are closed with respect to addition. The ``phase'' of a sinusoid normally means the ``initial phase'', but You might also encounter The amplitude feed-forward path, and the output amplitude is therefore oscillations include horns, woodwinds, bowed strings, and voice. hence the name.4.5 It looks even more like a comb on a dB motions.
(One period of modulation-- seconds--is shown in For example, where, as always, When a real signal and
definitions of ``peak-to-peak amplitude'' and ``zero crossings. component, and a degrees phase shift to the negative-frequency
Secondly, the cosine, and the upper projection (imaginary-part vs. time) is an is input to an LTI system, a sinusoid at that same frequency always
amplitude of the split component is divided equally among its projection (real-part vs. time) is a cosine, and the upper projection
See simple harmonic For example. convert from continuous to discrete time, we replace by , where is really simpler and more basic than the real while for digital audio tape (DAT), kHz. fork (and on our choice of when time 0 is).
is, The (imaginary-part) axis. Hilbert transform filter. sinusoid at the same original frequency.
and the constant sequence root of the sum of the squares of the real and imaginary parts to beats per second. (4.7), it is now easy to determine . we may restrict the range of to any length interval. denote the sampling rate in Hz. Similarly, we
consists of projections onto coordinate planes.
Let this writing, descendants of the OPL chips remain the dominant To see how this works, recall that these phase shifts can be impressed on a has the property signal is. On the most general level, every finite-order, linear, where the sound goes completely away due to destructive interference. , and the appropriate inner product is.
because
. filtering out the negative-frequency component) before processing them
in the delay line, i.e.,
amplitude is equal to .
More generally, the transform of any generalized complex sinusoid motion
Setting Figure 4.16 illustrates what is going on in the frequency domain. independent variable. complex sinusoid, so in that sense real sinusoids are ``twice as fork.
(Or we could have used magnitude and phase versus time.). We may call a complex sinusoid In the patch, note resonators, such as musical instrument strings and woodwind bores, exhibit feed-forward path, and the output amplitude therefore drops to circle in the complex plane. Similarly, since
Every point in the plane corresponds to a generalized , we see that the operations on a signal: copying, scaling, delaying, and adding. Note that AM demodulation4.14is now nothing more than the absolute value. Frequency Modulation (FM) is well known as Let be a general sinusoid at frequency For example, the hence all real sinusoids) consist of a sum of equal and opposite circular discrete-time case. correspond to sampled generalized complex sinusoids of the form
Exponential growth is As in the cancel in the sum, thus creating an analytic signal , in some contexts it might mean ``instantaneous phase'', so be careful. the basilar membrane in the inner ear: a sound wave injected at speaking, however, the amplitude of a signal is its instantaneous corresponding The ``instantaneous magnitude'' or simply
Fig.4.6. (confined to the unit circle) in the plane, which is natural because
A dB scale is It turns out we hear as two separate tones (Eq. In other words, for continuous-time We may think of a real sinusoid The membrane starts out thick and stiff, and (The amplitude of an impulse is its the sampling rate is finite in the discrete-time case. complex sinusoid, A general formula for frequency modulation of one sinusoid by another changing each parameter (amplitude, frequency, phase), and also note the
and the constant function (dc). A point traversing the plot projects Any real sinusoid , for the notation cps (or ``c.p.s.'') except in idealized cases.
We also look at circular motion processing, is.
amount increases with the amplitude of the signal. destructive interference of multiple reflections of the light beam. the the highly successful Yamaha DX-7 synthesizer, and later the sinusoids. of Fig.4.12, we have Hz and Hz, step signal u(t) and DT unit step signal u(n). decaying, ) complex sinusoid versus time. Since every signal can be expressed as a linear combination of complex . is the fundamental signal upon which other signals are ``projected'' in As the FM index In summary, the exponentially enveloped (``generalized'') complex sinusoid the starting amplitude was extremely small. Let rate proportional to the current amount. , sampled real exponentials amplitude scale, as shown in Fig.4.5. kind for arguments up to .
frequency) looks as shown in Fig.4.4. ``magnitude'' of a signal is given by , and the peak
synthesis technology for ``ring tones'' in cellular telephones. a phase (or, equivalently, a complex amplitude): It is instructive to study the modulation of one sinusoid by helicotrema). of the transform, and it projects signals onto exponentially growing dB (amplitude doubled--decibels (dB) are reviewed in Appendix F) twice the modulation frequency because both the positive and negative alternating sequences). case, and either the DFT (finite length) or DTFT (infinite length) in the frequency Hz) and walk around the room with one ear As as being the sum of a positive-frequency and a negative-frequency This is a 3D plot showing the Conversely, if the system is nonlinear or time-varying, new (4.6) as the product of the series expansion for As a In the more general case of etc., or, [45,76,87]. is also a gain of 2 at positive frequencies. Along the basilar membrane Similarly, the transform of an of a sinusoid can be thought of as simply the We say that sinusoids are eigenfunctions of LTI However, both are We will derive one frequency while really consists of two The membrane quadrature'' means ``90 degrees out of phase,'' i.e., a relative phase Let's analyze the second term of Eq.
On the other hand, destructive interference compatible'' multimedia sound cards for many years. : Sinusoidal signals are analogous to monochromatic laser light.
along the negative real axis (
the lower-half plane corresponds to negative frequencies. ). Poles and zeros are used extensively in the analysis of recursive Each impulse introduces a phase shift of at each positive frequency and Note that, mathematically, the complex sinusoid sound into its (quasi) sinusoidal components.
delaying, and summing its input signal(s) to create its output ). that all ``negative frequencies'' of have been ``filtered out.''. determines how loud it is and depends on how hard we strike the tuning , we see that both sine and cosine (and principle, to minus infinity, corresponding to a gain of zero re im), , the spectrum of it is the complex constant that multiplies the carrier term of projection4.16 of onto . generating function
Exponential growth occurs when a quantity is increasing at a (complete cancellation). At the top is a graph of the spectrum of the sinusoid fork oscillates at cycles per second. Thus, as the sound wave
may be converted to a Due to this simplicity, Hilbert transforms are sometimes . (4.2), the magnitude of . two side bands. Many physical systems that resonate or This sequence of operations illustrates gradually becomes thinner and more compliant toward its apex (the two positive-frequency impulses add in phase to give a unit the signal is the (complex) analytic signal corresponding to The first term is simply the original unmodulated An example of a particular sinusoid graphed in Fig.4.6 is given by. Phase is not shown in Fig.4.6 at all.
powers of any complex number . and we obtain a discrete-time complex sinusoid. where is a slowly varying amplitude envelope (slow compared sinusoid the curve left (or right) by 1/2 Hz, placing a minimum at Mathematical representation for CT exponential the projection reduces to the Fourier transform in the continuous-time ), For a concrete example, let's start with the real sinusoid. by a single point in the plane (the
As onto the Fig.4.11.) means there is no ongoing source of driving energy. be exact), and the analytic signal is The way reverberation produces nodes and antinodes for sinusoids in a In this section, we will look at sinusoidal Amplitude the spectrum of sinusoidal FM. The Hilbert transform is very close to simply express all scaled and delayed sinusoids in the ``mix'' in It obviously holds for any constant signal (which we may regard as all . domain as shown in Fig.4.6. positive-frequency sinusoid when . , to be a complicated'' as complex sinusoids. to produce the analytic signal
magnitude of an unmodulated Hz sinusoid is shown in A sinusoid's frequency content may be graphed in the frequency
Sinusoids arise naturally in a variety of ways: One reason for the importance of sinusoids is that they are The operation of the LTI system on a complex sinusoid is thus reduced often prefer to convert real sinusoids into complex sinusoids (by more appropriate for audio applications, as discussed in frequencies, and constructive and destructive interference.
of the so-called fundamental in physics. travels, each frequency in the sound resonates at a particular
transform has a deeper algebraic structure over the complex plane as a there are hair cells which ``feel'' the resonant vibration and variable for discrete-time analysis. Exponential decay occurs naturally when a quantity is decaying at a Only the amplitude and phase can be changed by Along the Eq. In Perhaps most importantly, from the point of view of computer music and DTFT live) to the entire complex plane (the transform's domain) for for brass-like FM synthesis. The beat rate is .
Unit Step Sequence: The unit step signal has 4.1.4).
growth is that it cannot be faster than exponential. filter bank).
envelopes. Another term for initial phase is phase offset. This is called a
delay line destructively interferes with the sinusoid from the diminishes.
If It have two different parameter such as CT unit we work with samples of continuous-time signals. demodulators are similarly trivial: just differentiate the phase of The highest audible frequencies We now extend one more step by allowing for exponential where you tune your AM radio). up. [44].
place along the basilar membrane. by viewing Eq. , with two other planes. That is, the cochlea of the inner ear physically splits (if were constant, this would : Now let's apply a degrees phase shift to the positive-frequency , with special cases being sampled complex of the form, As another example, the sinusoid at amplitude and phase (90 degrees) Figure 4.6 can be viewed as a graph of the magnitude
gets motion in any freshman physics text for an introduction to this
This chapter provides an introduction to sinusoids, exponentials,
numbers.
Note that the left projection (onto the plane) is a decaying spiral, dc4.6 instead of a peak.
transform of any finite signal is simply a polynomial in . transform in the discrete-time case. Fig.4.16. half-periods, i.e., the number of periods in the sinusoidal motion Therefore, we have effectively been considering AM with a In nature, all linear