Thus, the slope of the phase versus radian frequency is . In general, the time delay in samples equals minus the slope of thelinear phase term.
If we express the original spectrum in polar form as
where and are the magnitude and phase of , respectively (both real), we can see that a linear phase term only modifies the spectral phase :
where . A positive time delay (waveform shift to the right) adds a negatively sloped linear phase to the original spectral phase. A negative time delay (waveform shift to the left) adds apositively sloped linear phase to the original spectral phase. If we seem to be belaboring this relationship, it is because it is one of the most useful in practice.
where is an indicator function which takes on the values or .