A sinusoid is any function having the following form:
where
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
is its instantaneous
value
at any time
. The peak amplitude
satisfies
. The ``instantaneous magnitude'' or simply
``magnitude'' of a signal
is given by
, 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
, the initial
phase
is indistinguishable from
. As a result,
we may restrict the range of
to any length
interval.
When needed, we will choose
i.e.,
Note that the radian frequency
is equal to the time
derivative of the instantaneous phase of the sinusoid:
This is also how the instantaneous frequency is defined when the phase is time varying. Let
denote the instantaneous phase of a sinusoid with a time-varying phase-offset