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Sinusoids

A sinusoid is any function having the following form:

$\displaystyle x(t) = A \sin(\omega t + \phi) $

where $ t$ is the independent (real) variable, and the fixed parameters $ A$, $ \omega$, and $ \phi$ are all real constants. In audio applications we typically have
\begin{eqnarray*} A &=& \mbox{peak amplitude (nonnegative)} \\ \omega &=& \mbox... ...s)}\\ \omega t + \phi &=& \mbox{instantaneous phase (radians).} \end{eqnarray*}
An example is plotted in Fig.4.1. The term ``peak amplitude'' is often shortened to ``amplitude,'' e.g., ``the amplitude of the tone was measured to be 5 Pascals.'' Strictly speaking, however, the amplitude of a signal $ x$ is its instantaneous value $ x(t)$ at any time $ t$. The peak amplitude $ A$ satisfies $ \left\vert x(t)\right\vert\leq A$. The ``instantaneous magnitude'' or simply ``magnitude'' of a signal $ x(t)$ is given by $ \vert x(t)\vert$, and the peak magnitude is the same thing as the peak amplitude. The ``phase'' of a sinusoid normally means the ``initial phase'', but in some contexts it might mean ``instantaneous phase'', so be careful. Another term for initial phase is phase offset. Note that Hz is an abbreviation for Hertz which physically means cycles per second. You might also encounter the notation cps (or ``c.p.s.'') for cycles per second (still in use by physicists and formerly used by engineers as well). Since the sine function is periodic with period $ 2\pi $, the initial phase $ \phi \pm 2\pi$ is indistinguishable from $ \phi$. As a result, we may restrict the range of $ \phi$ to any length $ 2\pi $ interval. When needed, we will choose

$\displaystyle -\pi \leq \phi < \pi, $

i.e., $ \phi\in[-\pi,\pi)$. You may also encounter the convention $ \phi\in[0,2\pi)$. Note that the radian frequency $ \omega$ is equal to the time derivative of the instantaneous phase of the sinusoid:

$\displaystyle \frac{d}{dt} (\omega t + \phi) = \omega $

This is also how the instantaneous frequency is defined when the phase is time varying. Let

$\displaystyle \theta(t) \isdef \omega t + \phi(t) $

denote the instantaneous phase of a sinusoid with a time-varying phase-offset $ \phi(t)$. Then the instantaneous frequency is again given by the time derivative of the instantaneous phase:

$\displaystyle \frac{d}{dt} [\omega t + \phi(t)] = \omega + \frac{d}{dt} \phi(t) $

Example Sinusoids

Figure 4.1 plots the sinusoid $ A \sin(2\pi f t + \phi)$, for $ A=10$, $ f=2.5$, $ \phi=\pi/4$, and $ t\in[0,1]$. Study the plot to make sure you understand the effect of changing each parameter (amplitude, frequency, phase), and also note the definitions of ``peak-to-peak amplitude'' and ``zero crossings.''
Figure 4.1: An example sinusoid.
\includegraphics[width=\twidth]{eps/sine}
A ``tuning fork'' vibrates approximately sinusoidally. An ``A-440'' tuning fork oscillates at $ 440$ cycles per second. As a result, a tone recorded from an ideal A-440 tuning fork is a sinusoid at $ f=440$ Hz. The amplitude $ A$ determines how loud it is and depends on how hard we strike the tuning fork. The phase $ \phi$ is set by exactly when we strike the tuning fork (and on our choice of when time 0 is). If we record an A-440 tuning fork on an analog tape recorder, the electrical signal recorded on tape is of the form

$\displaystyle x(t) = A \sin(2\pi 440 t + \phi). $

As another example, the sinusoid at amplitude $ 1$ and phase $ \pi/2$ (90 degrees) is simply

$\displaystyle x(t) = \sin(\omega t + \pi/2) = \cos(\omega t). $

Thus, $ \cos(\omega t)$ is a sinusoid at phase 90-degrees, while $ \sin(\omega t)$ is a sinusoid at zero phase. Note, however, that we could just as well have defined $ \cos(\omega t)$ to be the zero-phase sinusoid rather than $ \sin(\omega t)$. It really doesn't matter, except to be consistent in any given usage. The concept of a ``sinusoidal signal'' is simply that it is equal to a sine or cosine function at some amplitude, frequency, and phase. It does not matter whether we choose $ \sin()$ or $ \cos()$ in the ``official'' definition of a sinusoid. You may encounter both definitions. Using $ \sin()$ is nice since ``sinusoid'' naturally generalizes $ \sin()$. However, using $ \cos()$ is nicer when defining a sinusoid to be the real part of a complex sinusoid (which we'll talk about in §4.3.11).

Why Sinusoids are Important

Sinusoids arise naturally in a variety of ways: One reason for the importance of sinusoids is that they are fundamental in physics. Many physical systems that resonate or oscillate produce quasi-sinusoidal motion. See simple harmonic motion in any freshman physics text for an introduction to this topic. The canonical example is the mass-spring oscillator.4.1 Another reason sinusoids are important is that they are eigenfunctions of linear systems (which we'll say more about in §4.1.4). This means that they are important in the analysis of filters such as reverberators, equalizers, certain (but not all) ``audio effects'', etc. Perhaps most importantly, from the point of view of computer music research, is that the human ear is a kind of spectrum analyzer. That is, the cochlea of the inner ear physically splits sound into its (quasi) sinusoidal components. This is accomplished by the basilar membrane in the inner ear: a sound wave injected at the oval window (which is connected via the bones of the middle ear to the ear drum), travels along the basilar membrane inside the coiled cochlea. The membrane starts out thick and stiff, and gradually becomes thinner and more compliant toward its apex (the helicotrema). A stiff membrane has a high resonance frequency while a thin, compliant membrane has a low resonance frequency (assuming comparable mass per unit length, or at least less of a difference in mass than in compliance). Thus, as the sound wave travels, each frequency in the sound resonates at a particular place along the basilar membrane. The highest audible frequencies resonate right at the entrance, while the lowest frequencies travel the farthest and resonate near the helicotrema. The membrane resonance effectively ``shorts out'' the signal energy at the resonant frequency, and it travels no further. Along the basilar membrane there are hair cells which ``feel'' the resonant vibration and transmit an increased firing rate along the auditory nerve to the brain. Thus, the ear is very literally a Fourier analyzer for sound, albeit nonlinear and using ``analysis'' parameters that are difficult to match exactly. Nevertheless, by looking at spectra (which display the amount of each sinusoidal frequency present in a sound), we are looking at a representation much more like what the brain receives when we hear.

In-Phase & Quadrature Sinusoidal Components

From the trig identity $ \sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$, we have
\begin{eqnarray*} x(t) &\isdef & A \sin(\omega t + \phi) = A \sin(\phi + \omega ... ...omega t) \\ &\isdef & A_1 \cos(\omega t) + A_2 \sin(\omega t). \end{eqnarray*}
From this we may conclude that every sinusoid can be expressed as the sum of a sine function (phase zero) and a cosine function (phase $ \pi/2$). If the sine part is called the ``in-phase'' component, the cosine part can be called the ``phase-quadrature'' component. In general, ``phase quadrature'' means ``90 degrees out of phase,'' i.e., a relative phase shift of $ \pm\pi/2$. It is also the case that every sum of an in-phase and quadrature component can be expressed as a single sinusoid at some amplitude and phase. The proof is obtained by working the previous derivation backwards. Figure 4.2 illustrates in-phase and quadrature components overlaid. Note that they only differ by a relative $ 90$ degree phase shift.
Figure 4.2: In-phase and quadrature sinusoidal components.
\includegraphics{eps/quadrature}

Sinusoids at the Same Frequency

An important property of sinusoids at a particular frequency is that they are closed with respect to addition. In other words, if you take a sinusoid, make many copies of it, scale them all by different gains, delay them all by different time intervals, and add them up, you always get a sinusoid at the same original frequency. This is a nontrivial property. It obviously holds for any constant signal $ x(t)=c$ (which we may regard as a sinusoid at frequency $ f=0$), but it is not obvious for $ f\neq 0$ (see Fig.4.2 and think about the sum of the two waveforms shown being precisely a sinusoid). Since every linear, time-invariant (LTI4.2) system (filter) operates by copying, scaling, delaying, and summing its input signal(s) to create its output 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 appears at the output. Only the amplitude and phase can be changed by the system. We say that sinusoids are eigenfunctions of LTI systems. Conversely, if the system is nonlinear or time-varying, new frequencies are created at the system output. To prove this important invariance property of sinusoids, we may simply express all scaled and delayed sinusoids in the ``mix'' in terms of their in-phase and quadrature components and then add them up. Here are the details in the case of adding two sinusoids having the same frequency. Let $ x(t)$ be a general sinusoid at frequency $ \omega$:

$\displaystyle x(t) \isdef A\sin(\omega t+\phi) $

Now form $ y(t)$ as the sum of two copies of $ x(t)$ with arbitrary amplitudes and phase offsets:
\begin{eqnarray*} y(t) &\isdef & g_1 x(t-t_1) + g_2 x(t-t_2) \\ &=& g_1 A \sin[\omega (t-t_1) + \phi] + g_2 A \sin[\omega (t-t_2) + \phi] \end{eqnarray*}
Focusing on the first term, we have
\begin{eqnarray*} g_1 A \sin[\omega (t-t_1) + \phi] &=& g_1 A \sin[\omega t + (... ...omega t) \\ &\isdef & A_1 \cos(\omega t) + B_1 \sin(\omega t). \end{eqnarray*}
We similarly compute

$\displaystyle g_2 A \sin[\omega (t-t_2) + \phi] = A_2 \cos(\omega t) + B_2 \sin(\omega t) $

and add to obtain

$\displaystyle y(t) = (A_1+A_2) \cos(\omega t) + (B_1+B_2) \sin(\omega t). $

This result, consisting of one in-phase and one quadrature signal component, can now be converted to a single sinusoid at some amplitude and phase (and frequency $ \omega$), as discussed above.

Constructive and Destructive Interference

Sinusoidal signals are analogous to monochromatic laser light. You might have seen ``speckle'' associated with laser light, caused by destructive interference of multiple reflections of the light beam. In a room, the same thing happens with sinusoidal sound. For example, play a simple sinusoidal tone (e.g., ``A-440''--a sinusoid at frequency $ f=440$ Hz) and walk around the room with one ear plugged. If the room is reverberant you should be able to find places where the sound goes completely away due to destructive interference. In between such places (which we call ``nodes'' in the soundfield), there are ``antinodes'' at which the sound is louder by 6 dB (amplitude doubled--decibels (dB) are reviewed in Appendix F) due to constructive interference. In a diffuse reverberant soundfield,4.3the distance between nodes is on the order of a wavelength (the ``correlation distance'' within the random soundfield). figure[htbp] \includegraphics{eps/combfilter} The way reverberation produces nodes and antinodes for sinusoids in a room is illustrated by the simple comb filter, depicted in Fig.4.3.4.4 Since the comb filter is linear and time-invariant, its response to a sinusoid must be sinusoidal (see previous section). The feedforward path has gain $ 1$, and the delayed signal is scaled by $ 0.99$. With the delay set to one period, the sinusoid coming out of the delay line constructively interferes with the sinusoid from the feed-forward path, and the output amplitude is therefore $ 1+0.99=1.99$. In the opposite extreme case, with the delay set to half a period, the unit-amplitude sinusoid coming out of the delay line destructively interferes with the sinusoid from the feed-forward path, and the output amplitude therefore drops to $ \left\vert 1-0.99\right\vert=0.01$. Consider a fixed delay of $ \tau$ seconds for the delay line in Fig.4.3. Constructive interference happens at all frequencies for which an exact integer number of periods fits in the delay line, i.e., $ f\tau=0,1,2,3,\ldots\,$, or $ f=n/\tau$, for $ n=0,1,2,3,\ldots\,$. On the other hand, destructive interference happens at all frequencies for which there is an odd number of half-periods, i.e., the number of periods in the delay line is an integer plus a half: $ f\tau = 1.5, 2.5, 3.5,$ etc., or, $ f = (n+1/2)/\tau$, for $ n=0,1,2,3,\ldots\,$. It is quick to verify that frequencies of constructive interference alternate with frequencies of destructive interference, and therefore the amplitude response of the comb filter (a plot of gain versus frequency) looks as shown in Fig.4.4.
Figure 4.4: Comb filter amplitude response when delay $ \tau =1$ sec.
\includegraphics[width=4in,height=2.0in]{eps/combfilterFR}
The amplitude response of a comb filter has a ``comb'' like shape, hence the name.4.5 It looks even more like a comb on a dB amplitude scale, as shown in Fig.4.5. A dB scale is more appropriate for audio applications, as discussed in Appendix F. Since the minimum gain is $ 1-0.99=0.01$, the nulls in the response reach down to $ -40$ dB; since the maximum gain is $ 1+0.99 \approx 2$, the maximum in dB is about 6 dB. If the feedforward gain were increased from $ 0.99$ to $ 1$, the nulls would extend, in principle, to minus infinity, corresponding to a gain of zero (complete cancellation). Negating the feedforward path would shift the curve left (or right) by 1/2 Hz, placing a minimum at dc4.6 instead of a peak.
Figure 4.5: Comb filter amplitude response in dB.
\includegraphics[width=4in,height=2.0in]{eps/combfilterFRDB}

Sinusoid Magnitude Spectra

A sinusoid's frequency content may be graphed in the frequency domain as shown in Fig.4.6. figure[htbp] \includegraphics{eps/sinefd} An example of a particular sinusoid graphed in Fig.4.6 is given by

$\displaystyle x(t) = \cos(\omega_x t) = \frac{1}{2}e^{j\omega_x t} + \frac{1}{2}e^{-j\omega_x t} $

where

$\displaystyle \omega_x = 2\pi 100. $

That is, this sinusoid has amplitude 1, frequency 100 Hz, and phase zero (or $ \pi/2$, if $ \sin(\omega_x t)$ is defined as the zero-phase case). Figure 4.6 can be viewed as a graph of the magnitude spectrum of $ x(t)$, or its spectral magnitude representation [44]. Note that the spectrum consists of two components with amplitude $ 1/2$, one at frequency $ 100$ Hz and the other at frequency $ -100$ Hz. Phase is not shown in Fig.4.6 at all. The phase of the components could be written simply as labels next to the magnitude arrows, or the magnitude arrows can be rotated ``into or out of the page'' by the appropriate phase angle, as illustrated in Fig.4.16.
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