In working with phase delay, it is often necessary to ``unwrap''
the phase response
. Phase unwrapping ensures that
all appropriate multiples of have been included in
. We defined
simply as the complex
angle of the frequency response
, and this is not sufficient
for obtaining a phase response which can be converted to true time
delay. If multiples of are discarded, as is done in the
definition of complex angle, the phase delay is modified by multiples
of the sinusoidal period. Since LTI filter analysis is based on
sinusoids without beginning or end, one cannot in principle
distinguish between ``true'' phase delay and a phase delay with
discarded sinusoidal periods when looking at a sinusoidal output at
any given frequency. Nevertheless, it is often useful to define the
filter phase response as a continuous function of frequency
with the property that
or (for real filters). This
specifies how to unwrap the phase response at all frequencies
where the amplitude response is finite and nonzero. When the
amplitude response goes to zero or infinity at some frequency, we can
try to take a limit from below and above that frequency.

Matlab and Octave have a function called unwrap() which
implements a numerical algorithm for phase unwrapping.
Figures 7.4(a) and 7.4(b) show the effect of the
unwrap function on the phase response of the example elliptic
lowpass filter of §7.5.2, modified to contract the zeros from
the unit circle to a circle of radius in the plane:

[B,A] = ellip(4,1,20,0.5); % design lowpass filter
B = B .* (0.95).^[1:length(B)]; % contract zeros by 0.95
[H,w] = freqz(B,A); % frequency response
theta = angle(H); % phase response
thetauw = unwrap(theta); % unwrapped phase response

In Fig.7.4(a), the phase-response minimum has ``wrapped
around'' to the top of the plot. In Fig.7.4(b), the phase
response is continuous. We have contracted the zeros away from the
unit circle in this example, because the phase response really does
switch discontinuously by radians when frequency passes through
a point where the phases crosses zero along the unit circle (see
Fig.7.3(b)). The unwrap function need not modify
these discontinuities, but it is free to add or subtract any integer
multiple of in order to obtain the ``best looking''
discontinuity. Typically, for best results, such discontinuities
should alternate between and , making the phase
response resemble a distorted ``square wave'', as in Fig.7.3(b).
A more precise example appears in Fig.11.2.

Figure 7.4:
Phase response of a modified order 4 elliptic
function lowpass filter cutting off at .