For phase

*p* and frequency

*f* group delay GD is the negative of the derivative of the change in phase with respect to the change in frequency: GD = -d

*p*/d

*f*. In general phase is a function of frequency

*p(f)*; that is, phase changes with frequency. That's the math behind it.

A change in phase is equivalent to a time shift, for example a 180-degree phase shift is like shifting the time by one-half cycle of the signal. The derivative is a fancy expression for the slope of a line, so it is a measure of phase linearity. A straight (linear) line has the form

**y = mx+b** where every point

**x** is multiplied by the slope m and added to offset

**b** to produce a

**y** value. For a straight line,

**m** is constant (just a number, not a function of something else), and hopefully we remember this formula from school. Now replace

**m** with

**GD** so to get a straight line, that means the change in phase divided by the change in frequency must be constant, meaning group delay is a constant, and every frequency is delayed by the same amount of time. What goes in, comes out again, exactly as it was but just a little later in time.

Now, a pulse, or musical signal, includes many frequencies. If we send the signal through a component like an amplifier or speaker with constant group delay, then every frequency is delayed the same amount, and the output is just like the input except delayed in time. If the group delay is

*not *constant, that means different frequencies have different delays through the component, so at the other end the signal will be "smeared" in time with different frequencies arriving at different times. Things like transient attacks from drums or instruments will not be as clean. There are various studies discussing just how far off the delay can be at different frequencies before we notice it, some referenced in the Wikipedia article mentioned previously (

https://en.wikipedia.org/wiki/Group_delay_and_phase_delay).

HTH - Don