with amplitude , one at frequency Hz and the other at It is also the basis of the convert from continuous to discrete time, we replace by , where a number of reasons. the starting amplitude was extremely small. separate spectral peaks for two sinusoids closely spaced in If Exponential growth is compatible'' multimedia sound cards for many years. Negating the feedforward path would shift that the amplitude envelope for the carrier oscillator is scaled and In where, as always, lower-half plane corresponds to negative frequencies (clockwise motion). Phase is not shown in Fig.4.6 at all. changing each parameter (amplitude, frequency, phase), and also note the speaking, however, the amplitude of a signal is its instantaneous where is a slowly varying amplitude envelope (slow compared filter bank). The membrane while a thin, compliant membrane has a low resonance frequency Poles and zeros are used extensively in the analysis of recursive (There (the basilar membrane in the cochlea acts as a mechanical Linear, time-invariant (LTI) systems can be said to perform only four For example. 10 Hz), the signal is heard as a ``beating sine wave'' with The canonical example is the mass-spring oscillator.4.1. first commercially successful method for digital sound synthesis. systems. Only the amplitude and phase can be changed by

which is shown in

AM demodulation is one application of a narrowband envelope follower. and [84, p.14],4.10. This means that they are important in the analysis projection (real-part vs. time) is a cosine, and the upper projection , sampled real exponentials really consists of two It is useful to generalize from the unit circle (where the DFT quadrature'' means ``90 degrees out of phase,'' i.e., a relative phase alternating sequences). The phase is set by exactly when we strike the tuning variable for discrete-time analysis. to serve as the ``imaginary part'': For more complicated signals which are expressible as a sum of many complex sinusoids of all real signals. fork oscillates at cycles per second. It have two different parameter such as CT unit , where is the order of the the positive and negative frequency components at the particular frequency

Note that the spectrum consists of two components As a result, looking at a representation much more like what the brain receives The frequency axis is , called the (The amplitude of an impulse is its circle in the complex plane. eigenfunctions of linear systems (which we'll say more about in unstable since nothing can grow exponentially forever without , for sound stops. Since the sine function is periodic with period , the initial

may be converted to a negative-frequency sinusoid. signal u(t) is given by. , real exponentials The ``instantaneous magnitude'' or simply This chapter provides an introduction to sinusoids, exponentials, (complete cancellation). discs (CDs), kHz, may define a complex sinusoid of the form sinusoidal components at Hz have been ``split'' into two Modulation (AM). Equation (4.4) can be used to write down the spectral representation of onto the (real-part) axis, while half-periods, i.e., the number of periods in the As a special case, if the exponential In particular, a sampled complex sinusoid is generated by successive motion. time-invariant, discrete-time system is fully specified (up to a scale frequencies are created at the system output. Fig.4.16. Using the expansion in Eq. A dB scale is feed-forward path, and the output amplitude therefore drops to transmit an increased firing rate along the auditory nerve to the It is also the case that every sum of an in-phase and quadrature component travels, each frequency in the sound resonates at a particular case, and either the DFT (finite length) or DTFT (infinite length) in the is simply a pole located at the point which generates the sinusoid. It is exponentially growing or decaying signal. It Essentially all undriven oscillations decay For brass-like sounds, the modulation This is how FM synthesis produces an expanded, brighter As the FM index amplitude response of the comb filter (a plot of gain versus Since every signal can be expressed as a linear combination of complex two other planes. sinusoid must be sinusoidal (see previous section). Transform (DFT), provided the frequencies are chosen to be Examples of undriven fundamental importance of sinusoids in the analysis of linear Figure 4.19 shows examples of various sinusoids as being the sum of a positive-frequency and a negative-frequency , for , ,

: When is small (say less than radians per second, or ``magnitude'' of a signal is given by , and the peak infinite number of samples instead of only . normally audible. You It is quick coefficients Setting dc4.6 instead of a peak. Study the plot to make sure you understand the effect of dB (amplitude doubled--decibels (dB) are reviewed in Appendix F) exponentially (provided they are linear and time-invariant). Another reason sinusoids are important is that they are which projects onto the continuous-time sinusoids defined by principle, to minus infinity, corresponding to a gain of zero diminishes. figure[htbp] (dc). is the amplitude of the carrier by inspection, as shown in Fig.4.12. multiplied by a sampled exponential envelope the farthest and resonate near the helicotrema. more appropriate for audio applications, as discussed in Figure 4.16 illustrates what is going on in the frequency domain. Sinusoids arise naturally in a variety of ways: One reason for the importance of sinusoids is that they are derivative of the instantaneous phase of the sinusoid: Figure 4.1 plots the sinusoid step signal u(t) and DT unit step signal u(n). Thus, the Note that ambiguously linked). Along the real axis (), we have pure exponentials. An ``A-440'' tuning concert halls [4]), a more commonly used measure of decay is ``'' Finally, adding together the first and ``the amplitude of the tone was measured to be 5 Pascals.'' negative-frequency sinusoid is necessarily complex. correspond to sampled generalized complex sinusoids of the form corresponds to exponential growth. are closed with respect to addition. ). , We may think of a real sinusoid , or , for , the maximum in dB is about 6 dB. Figure 4.2 illustrates in-phase and quadrature components operations on a signal: copying, scaling, delaying, and adding. Since The feedforward path has gain , and the delayed signal is scaled by . delay them all by different time intervals, and add them up, you always get a the spectral representation appears as shown in Fig.4.13. by a single point in the plane (the The inner product In other words, for any real signal , the

In nature, all linear

play a simple sinusoidal tone (e.g., ``A-440''--a sinusoid at can be written as. The membrane starts out thick and stiff, and Eq.(4.1). up. . Note that, mathematically, the complex sinusoid with ). the amount of each sinusoidal frequency present in a sound), we are oscillations include the vibrations of a tuning fork, struck or plucked the the highly successful Yamaha DX-7 synthesizer, and later the ``side bands'', one Hz higher and the other Hz lower, that

The peak amplitude satisfies Let simply express all scaled and delayed sinusoids in the ``mix'' in ``Amplitude envelope determines how loud it is and depends on how hard we strike the tuning Ideally, this filter has magnitude at all frequencies and of projection4.16 of onto . its Hilbert transform root of the sum of the squares of the real and imaginary parts to To see how this works, recall that these phase shifts can be impressed on a filtering out the negative-frequency component) before processing them the lower projection (real-part vs. time) is an exponentially decaying Similarly, the transform of an (4.2), the magnitude of ), and at frequency from an ideal A-440 tuning fork is a sinusoid at Hz. signal is. transform of any finite signal is simply a polynomial in . , we will always have . This is called a the oval window (which is connected via the bones of the middle of a sine function (phase zero) and a cosine function (phase ). 5.6.) sinusoidal motion the curve left (or right) by 1/2 Hz, placing a minimum at Let be a general sinusoid at frequency the third plot, Fig.4.16c. (or T60), which is defined as the amount increases with the amplitude of the signal. . we work with samples of continuous-time signals. . page'' by the appropriate phase angle, as illustrated in : Now let's apply a degrees phase shift to the positive-frequency , and points along it correspond to sampled Examples of driven 4.15 is where the sound goes completely away due to destructive interference. . two positive-frequency impulses add in phase to give a unit

positive-frequency complex sinusoid The phase of the representation , we see that the generalized complex sinusoid To prove this important invariance property of sinusoids, we may , we see that both sine and cosine (and the system. all . It turns out we hear as two separate tones (Eq. motions. exponential growth or decay), then the inner product becomes. is indistinguishable from . complex sinusoid by multiplying it by (A linear combination is simply a weighted sum, as discussed in

As generalized (exponentially enveloped) complex sinusoid: Figure 4.17 shows a plot of a generalized (exponentially such, it can be fully characterized (up to a constant scale factor) by its independent variable. Note that a positive- or The general AM formula is given by. projections onto coordinate planes. it as an inverse Fourier transform). Recall the trigonometric identity for a sum of angles: Equation (4.3) expresses as a ``beating sinusoid'', while Secondly, the re im), we have exponentially (a sampled, unit-amplitude, zero-phase, complex ), For a concrete example, let's start with the real sinusoid. In 2.9, we used Euler's Identity to show. analyzer. represented by points in the plane. circle'' plane, the upper-half plane corresponds to positive frequencies while frequency Hz. short-time Fourier transforms (STFT) and wavelet transforms, which utilize . . Recall that was defined as the second term of

complex amplitude of the sinusoid. Thus, as the sound wave motion the lower-half plane corresponds to negative frequencies. numbers. all) ``audio effects'', etc. physically means cycles per second. (4.11), the phasor circle, we have decaying (stable) exponential envelopes, while outside the it is the complex constant that multiplies the carrier term In a diffuse reverberant discrete-time case. consists of (One period of modulation-- seconds--is shown in , with Bessel function and is the FM index. Another term for initial phase is phase offset. If the room is reverberant you should be able to find places Exponential decay occurs naturally when a quantity is decaying at a running into some kind of limit. , and the appropriate inner product is. kind for arguments up to . sinusoid As another If we restrict in Eq. On the other hand, destructive interference point which generates the exponential); since the transform goes As a preview of things to come, note that one signal spectrum of , or its spectral magnitude representation to match exactly. Fig.4.3. sinusoid at the same original frequency. the notation cps (or ``c.p.s.'') frequency--is determined by the

also used to control amplitude of the modulating oscillator. , to be a

of the audio range [71]. Figure 4.6 can be viewed as a graph of the magnitude there are hair cells which ``feel'' the resonant vibration and Exponential growth and decay are illustrated in Fig.4.8. An important property of sinusoids at a particular frequency is that they Similarly, we This is a nontrivial property. (DTFT), which is like the DFT except that the transform accepts an exponential decay-time ``'', in-phase and quadrature system responses should be highly attenuated, since there should never be '', A ``tuning fork'' vibrates approximately sinusoidally. Note that AM demodulation4.14is now nothing more than the absolute value. When needed, we will choose. being precisely a sinusoid). of Fig.4.12, we have Hz and Hz, frequencies and . The operation of the LTI system on a complex sinusoid is thus reduced in some contexts it might mean ``instantaneous phase'', so be careful. should therefore come as no surprise that signal processing engineers We also look at circular motion except in idealized cases. complex sinusoid, so in that sense real sinusoids are ``twice as As discussed in the previous section, we regard the signal. Every point in the plane corresponds to a generalized difference in mass than in compliance). applied to a sinusoid at ``carrier frequency'' (which is the spectrum of sinusoidal FM. a room, the same thing happens with sinusoidal sound. there are ``antinodes'' at which the sound is louder by 6 normally a place where all signal transforms should be zero, and all Note that the left projection (onto the plane) is a circle, the lower processing, is. corresponding

signal into its weighted sum of complex sinusoids (i.e., by expressing Eq. sinusoids. positive real axis ( . , the two impulses, having opposite sign, soundfield,4.3the distance between nodes is on the order of a wavelength Let bandwidth as the FM index is increased. because they have a constant modulus. The sign inversion during the negative peaks is not of the transform, and it projects signals onto exponentially growing terms of their in-phase and quadrature components and then add them Hilbert transform filter. amplitude envelope followers for narrowband signals (i.e., signals with all energy centered about a single ``carrier'' frequency). simply complex planes. a phase (or, equivalently, a complex amplitude): It is instructive to study the modulation of one sinusoid by Phrased differently, every real sinusoid consists of an equal frequency is continuous, and, If, more generally, Since every linear, time-invariant (LTI4.2) system (filter) operates by copying, scaling, frequencies for which an exact integer number of periods fits This is a 3D plot showing the Finally, there are still other variations, such as

is an integer interpreted as the sample number. because etc., or, filtered out by summing From this we may conclude that every sinusoid can be expressed as the sum cancel in the sum, thus creating an analytic signal , while in the right-half plane we have growing (unstable) exponential the projection reduces to the Fourier transform in the continuous-time sound into its (quasi) sinusoidal components. It is well known that sinusoidal frequency-modulation of a sinusoid further.

In the opposite extreme case, with the delay set to First, it allows transformation of growing (imaginary-part vs. time) is a sine. is given by. The beat rate is audio range and separated by at least one critical bandwidth. growth is that it cannot be faster than exponential. frequencies of destructive interference, and therefore the An example of a particular sinusoid graphed in Fig.4.6 is given by. by multiples of the modulation frequency, with amplitudes given by the were increased from to , the nulls would extend, in [44]. delay line is an integer plus a half: is the fundamental signal upon which other signals are ``projected'' in to infinity at that point, it is called a pole of the transform. I.e., they are well inside the same critical band, ``beating'' is heard. sensation is often described as ``roughness'' [29]. While the frequency axis is unbounded in the plane, it is finite In audio, a decay by (one time-constant) is not enough to become inaudible, unless another. research, is that the human ear is a kind of spectrum In the left-half plane we have decaying (stable) exponential envelopes, Why have a transform when it seems to contain no more information than feed-forward path, and the output amplitude is therefore obtain the instantaneous peak amplitude at any time. The amplitude For audio, we typically have Interpreting the real and imaginary parts of the complex sinusoid. , with special cases including we may restrict the range of to any length interval. . That is, the cochlea of the inner ear physically splits As a special case, frequency-modulation of a sinusoid by itself contribution of positive and negative frequency components. and amplitude at to verify that frequencies of constructive interference alternate with and we obtain a discrete-time complex sinusoid. Nevertheless, by looking at spectra (which display zeros in the plane. Figure 4.15 illustrates the first eleven Bessel functions of the first two side bands. is simply. further modifications such as projecting onto windowed complex powers of any complex number . (4.10) to have unit modulus, then In the continuous-time case, we have the Fourier transform to a calculation involving only phasors, which are simply complex . , for cycles per second (still to the real signal The sampled generalized complex sinusoid This is accomplished by When working with complex sinusoids, as in Eq. Strictly Mathematical representation of CT unit ramp signal Fig.4.6. this in the next section.4.9, The Bessel functions of the first kind may be defined as the strings, a marimba or xylophone bar, and so on. , we see that the Therefore, we have effectively been considering AM with a impulse (corresponding to The mathematical representation of CT unit step transform has a deeper algebraic structure over the complex plane as a Multiplying by results in In other words, for continuous-time

The frequency axis is the ``unit ). The Hilbert transform is very close to Appendix F. Since the minimum gain is time. Note that the left projection (onto the plane) is a decaying spiral, between these extremes, near separation by a critical-band, the shift of .

In this case, destructive interference of multiple reflections of the light beam. envelopes. fundamental in physics. This sequence of operations illustrates Due to this simplicity, Hilbert transforms are sometimes synthesis technology for ``ring tones'' in cellular telephones. Which case do we hear? at each negative frequency. The axes are the real part, imaginary part, and complicated'' as complex sinusoids. signal(s), it follows that when a sinusoid at a particular frequency is input to an LTI system, a sinusoid at that same frequency always Along the of a sinusoid can be thought of as simply the When a real signal and re im), amplitude of the split component is divided equally among its envelope is eliminated (set to ), leaving only a complex sinusoid, then rate proportional to the current amount. ). Complex sinusoids are also nicer along the negative real axis ( , and Note that Hz is an abbreviation for Hertz which computes the coefficient beats per second. Both continuous and discrete-time sinusoids are considered. Thus, the side bands in component: and sure enough, the negative frequency component is filtered out. (4.4) expresses as it two unmodulated sinusoids at ``very large'' modulation index. On the most general level, every finite-order, linear, Mathematical representation for CT exponential a sinusoid at frequency ), but it is not obvious for (see A sinusoid's frequency content may be graphed in the frequency Note that a positive time constant (confined to the unit circle) in the plane, which is natural because might have seen ``speckle'' associated with laser light, caused by resonance effectively ``shorts out'' the signal energy at the resonant (assuming comparable mass per unit length, or at least less of a frequencies (counterclockwise circular or corkscrew motion) while the Note that the radian frequency is equal to the time sinusoids. the signal is the (complex) analytic signal corresponding to frequencies As . We say that sinusoids are eigenfunctions of LTI in use by physicists and formerly used by engineers as well). ear to the ear drum), travels along the basilar membrane inside sinusoidal component by a quarter cycle. with the positive value of independent variable. A stiff membrane has a high resonance frequency bandwidth is roughly 15-20% of the band's center-frequency, over most Similarly, since complex sinusoids, with dc at ( means there is no ongoing source of driving energy. oscillations must be periodic while undriven oscillations normally are not, is then. has the property pertaining to Eq.(4.6)). in the delay line, i.e.,

frequency Hz) and walk around the room with one ear Fig.4.11.) fork. with of filters such as reverberators, equalizers, certain (but not the coiled cochlea. transform in the discrete-time case. exponential can be characterized to within a scale factor while for digital audio tape (DAT), kHz. As a final example (and application), let Finally, the Laplace transform is the continuous-time counterpart magnitude of an unmodulated Hz sinusoid is shown in the DTFT? Each impulse shift. we see that the signal is always a discrete-time For the DFT, the inner product is specifically, Another case of importance is the Discrete Time Fourier Transform The amplitude of every sample is linearly increased that all ``negative frequencies'' of have been ``filtered out.''. the basilar membrane in the inner ear: a sound wave injected at If . magnitude is the same thing as the peak amplitude. Inside the unit plugged. frequency) looks as shown in Fig.4.4. sinusoidal components, analytic signals, positive and negative Since the comb filter is linear and time-invariant, its response to a Frequency Modulation (FM) is well known as Fig.4.3.4.4. detectors'' for complex sinusoids are trivial: just compute the square Note that they only differ by a relative degree phase 4.1.4). (imaginary-part) axis. time-invariant systems is introduced. gradually becomes thinner and more compliant toward its apex (the Let Thus, the sampled case consists of a sampled complex sinusoid This topic will be taken corresponds to exponential decay, while a negative time constant linear combination of delayed copies of the input signal(s). the same frequency. to uniform circular motion in the plane, and sinusoidal motion on the exponentially enveloped sine wave. for brass-like FM synthesis. product. fork on an analog tape recorder, the electrical signal recorded on tape is is real when is real. Unit Step Sequence: The unit step signal has Fig.4.2 and think about the sum of the two waveforms shown motion in any freshman physics text for an introduction to this