In general, for
Strictly speaking, we say
is linear phase if
Why is this important? A linear phase response gives the
same time delay for
ALL frequencies ! (Remember the shift theorem.) This is
very desirable in many applications, particularly when theappearance of the time-domain waveform is of interest, such as
in an oscilloscope. (see
)
Restrictions on h(n) to get linear phase
For linear phase, we require the right side of
to be
. For
, we thus require
Thus
is a
necessary condition for
the right side of
to be real valued, for
.
For
, or
, we require
Usually, one is interested in filters with
real-valued coefficients, or see
and
.
(Symmetric Filters).
.
(Anti-Symmetric Filters).
. Filter design techniques are usually slightly different for
each of these four different filter types. We will study themost common case, symmetric-odd length, in detail, and often
leave the others for homework or tests or for when oneencounters them in practice. Even-symmetric filters are often
used; the anti-symmetric filters are rarely used in practice,except for special classes of filters, like differentiators or
Hilbert transformers, in which the desired response isanti-symmetric.
So far, we have satisfied the condition that
where
is
real-valued . However, we have
not assured that
is
non-negative . In general,
this makes the design techniques much more difficult, so mostFIR filter design methods actually design filters with
Generalized Linear Phase :
, where
is
real-valued , but possible negative.
is called the
amplitude of the frequency
response .
usually goes negative only in the stopband, and the
stopband phase response is generally unimportant.
where
Lowpass filter
Desired |h()|
DesiredH()
The slope of each line is
.
Actual |h()|
goes negative.
ActualH()
phase jumps due to periodicity of phase.
phase jumps due to sign change in
. Time-delay introduces generalized linear phase.
For odd-length FIR filters, a linear-phase design
procedure is equivalent to a zero-phase design procedurefollowed by an
-sample delay of the
impulse response . For
even-length filters, the delay is non-integer, and thelinear phase must be incorporated directly in the desired
response!
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