Sinusoids

A sinusoid is any function of time having the following form:

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

where all variables are real numbers, and

$$\begin{eqnarray*} A &=& \mbox{peak amplitude (nonnegative)} \\ \omega &=& \mbox... ...s)}\\ \omega t + \phi &=& \mbox{instantaneous phase (radians).} \end{eqnarray*}$$

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.

Note that Hz is an abbreviation for Hertz which physically means ``cycles per second.'' You might also encounter the older (and clearer) notation ``c.p.s.'' for cycles per second.

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

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

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:

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

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

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

denote the instantaneous phase of the sinusoid. Then the instantaneous frequency is given by the time derivative of the instantaneous phase:

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