AudioBanter.com - View Single Post

(Mike Rivers) writes:

In article writes:

My point was two-fold:

a) phase is used differently in different contexts

That was my point, too. It's used correctly in some contexts and
incorrectly in others.

The two contexts I was alluding to are both "correct," just different.

and the
"derivative" in one context gives you a completely different
quantity than the derivative in the other.

I assume you're talking about the mathematical derivitive, the change
with respect to time.

Not all "mathematical" derivatives are with respect to time, but
they are all with respect to "something," i.e., some specific
variable.

I'm not sure of the significance of this on the
practical level. Enlighten me if you can.

In one context (the one we're primarily discussing here), "phase" is
the phase response of a system as a function of frequency f, which
I'll denote as phi(f). In this context, the derivative of phi(f) with
respect to f is related to the group delay G(f),

G(f) = -(1/(2*pi)) * (d phi(f) / df).

Note that the units of G(f) are seconds.

In another context, "phase" is the argument of a sinusoid as a
function of time, which I'll denote as theta(t). For example,
the sine wave

x(t) = sin(2*pi*f*t + a)

where f is the frequency of the sinusoid, t is time, and a is
a constant, has a time-varying phase of

theta(t) = 2*pi*f*t + a.

The derivative of theta(t) with respect to t is the frequency
of the sine wave,

d theta(t) / dt = 2*pi*f,

in radians per second.

Note that in both contexts I'm using radians as the unit of phase
rather than degrees.

b) stating that A is a derivative of a function B is, generally,
useless unless you state which variable the derivative is taken by.

Yup, just like stating that the phase shift is 90 degrees without
knowing what's 90 degrees different than what, and in the case of a
non-repetitive waveform, when.

The terminology is well-defined and the concepts well-understood
within the discipline of electrical engineering. The term "phase
shift" means the shift, or change, in phase a signal undergoes
when it is processed by a linear, time-invariant system H(f). It
is a function of frequency.

In linear system theory you learn that phase shift does not depend on
the signal but only the system, and in fact the phase shift is
precisely the phase response phi(f) of the system,

phi(f) = angle(H(f)),

where "angle(z)" denotes the angle, in radians, of the complex number
z, i.e., it is the "phi" in the polar form of the complex number z,

z = r * exp(j*phi).

So to say that a signal undergoes a 90 degree (or pi/2) phase
shift is to say that it passes through a system which has a phase
response of 90 degrees and unity magnitude at all frequencies, i.e.,
the transfer function H(f) = exp(sgn(f)*j*pi/2).
--
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
, 919-472-1124