Hω=∑h=0M−1hne−(iωn)=h0+h1e−(iω)+h2e−(i2ω)+…+hM−1e−(iω(M−1))=e−(iωM−12)(h0eiωM−12+…+hM−1e−(iωM−12))=e−(iωM−12)((h0+hM−1)cosM−12ω+(h1+hM−2)cosM−32ω+…+i(h0sinM−12ω+…))
H
ω
h
0
M
1
h
n
ω
n
h
0
h
1
ω
h
2
2
ω
…
h
M
1
ω
M
1
ω
M
1
2
h
0
ω
M
1
2
…
h
M
1
ω
M
1
2
ω
M
1
2
h
0
h
M
1
M
1
2
ω
h
1
h
M
2
M
3
2
ω
…
h
0
h
M
1
M
1
2
ω
…

(1)
For linear phase, we require the right side of

Equation 1 to be

e−(i
θ
0
)(real,positive function of ω)
θ
0
(real,positive function of ω)
. For

θ
0
=0
θ
0
0
, we thus require

h0+hM−1=real number
h
0
h
M
1
real number
h0−hM−1=pure imaginary number
h
0
h
M
1
pure imaginary number
h1+hM−2=pure real number
h
1
h
M
2
pure real number
h1−hM−2=pure imaginary number
h
1
h
M
2
pure imaginary number
⋮
⋮
Thus

hk=
h
*
M−1−k
h
k
h
*
M
1
k
is a

*necessary* condition for
the right side of

Equation 1 to be real valued, for

θ
0
=0
θ
0
0
.

For
θ
0
=π2
θ
0
2
, or
e−(i
θ
0
)=−i
θ
0
, we require
h0+hM−1=pure imaginary
h
0
h
M
1
pure imaginary
h0−hM−1=pure real number
h
0
h
M
1
pure real number
⋮
⋮
⇒
hk=−(
h
*
M−1−k)
⇒
h
k
h
*
M
1
k

Usually, one is interested in filters with
*real-valued* coefficients, or see Figure 2 and Figure 3.

Filter design techniques are usually slightly different for
each of these four different filter types. We will study the
most common case, symmetric-odd length, in detail, and often
leave the others for homework or tests or for when one
encounters 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 is
anti-symmetric.

So far, we have satisfied the condition that
Hω=Aωe−(i
θ
0
)e−(iωM−12)
H
ω
A
ω
θ
0
ω
M
1
2
where
Aω
A
ω
is *real-valued*. However, we have
*not* assured that
Aω
A
ω
is *non-negative*. In general,
this makes the design techniques much more difficult, so most
FIR filter design methods actually design filters with
Generalized Linear Phase:
Hω=Aωe−(iωM−12)
H
ω
A
ω
ω
M
1
2
, where
Aω
A
ω
is *real-valued*, but possible negative.
Aω
A
ω
is called the amplitude of the frequency
response.

Aω
A
ω
usually goes negative only in the stopband, and the
stopband phase response is generally unimportant.

|Hω|=±Aω=Aωe−(iπ12(1−signAω))
H
ω
±
A
ω
A
ω
1
2
1
sign
A
ω
where
signx={1 if x>0-1 if x<0
sign
x
1
x
0
-1
x
0

Time-delay introduces generalized linear phase.

For odd-length FIR filters, a linear-phase design
procedure is equivalent to a zero-phase design procedure
followed by an

M−12
M
1
2
-sample delay of the

impulse response. For
even-length filters, the delay is non-integer, and the
linear phase must be incorporated directly in the desired
response!