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.

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

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

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 initial 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:

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