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CHAP. 51
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TRANSFORM ANALYSIS OF SYSTEMS
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Fig. 5-2. Evaluating the frequency response geometrically from the
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poles and zeros of the system function.
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where 0, is the angle subtended by the vector from the zero at z = ,Ek to the unit circle at z = eJw(see Fig. 5-2). Similarly, for each term in the denominator
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where Q2 is the angle of the vector from the pole at z = a!k to the unit circle at z = eJW.When a pole (zero) is close to the unit circle, the phase decreases (increases) rapidly as we move past the pole (zero). Because the group delay is the negative of the derivative of the phase, this implies that the group delay is large and positive close to a pole and large and negative when close to a zero.
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5.3 SYSTEMS WITH LINEAR PHASE
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A linear shift-invariant system is said to have linear phase if the frequency response has the form
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H (ejw)= I H (eiW)1e-jaw
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where a! is a real number. Thus, linear phase systems have a constant group delay,
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In some applications, one is interested in designing systems that have what is referred to as generalized linear phase. A system is said to have generalized linear phase if the frequency response has the form
where A(ejw)is a real-valued (possibly bipolar) function of w , and p is a constant. Often, the term linearphase is used to denote a system that has either linear or generalized linear phase.
EXAMPLE 5.3.1
Consider the FIR system with a unit sample response
h(n) =
n=0,1, ..., N
else
The frequency response is
TRANSFORM ANALYSIS OF SYSTEMS
[CHAP. 5
Therefore, this system has generalized linear phase, with a = N / 2 and
= 0.
In order for a causal system with a rational system function to have linear phase, the unit sample response must be finite in length. Therefore, IIR filters cannot have (generalized) linear phase. For an FIR filter with a real-valued unit sample response of length N I , a sufficient condition for this filter to have generalized linear phase is that the unit sample response be symmetric,
In this case, cr = N / 2 and B = 0 or YC. Another sufficient condition is that h(n) be antisymmetric,
which corresponds to the case in which cr = N / 2 and B = n / 2 or 3x12. Linear phase filters may be classified into four types, depending upon whether h(n) is symmetric or antisymmetric and whether N is even or odd. Each of these filters has specific constraints on the locations of the zeros in H ( z ) which, in turn, place constraints on the frequency response magnitude. For each of these types, which are described below, it is assumed that h(n) is real-valued, and that h(0) is the first nonzero value of h(n).
Q p e I Linear Phase Filters
A type I linear phase filter has a symmetric unit sample response,
and N is even. The center of symmetry is about the point cr = N/2, which is an integer, as illustrated in Fig. 5-3(a).
h(n)
It Center of symmetry
h(n)
Center of symmetry
.I. I I I
- -1
L s ;a
( 6 )Type 11 filter.
I I \ I
h(n)
I+ I I I I
c Center of symmetry
Center of symmetry
- -1
I t 3
(c)Type 111 filter.
(d)v p e IV filter.
Fig. 5-3. Symmetries in the unit sample response for generalized linear phase systems.
CHAP. 51
TRANSFORM ANALYSIS OF SYSTEMS
The frequency response of a type I linear phase filter may be expressed in the form
where
Type I1 Linear Phase Filters
A type I1 linear phase filter has a symmetric unit sample response, and N is odd. Therefore, the center of symmetry of h ( n ) occurs at the half-integer value a, = N / 2 , as illustrated in Fig. 5-3(b). The frequency response of a type I1 linear phase filter may be written as
where
Type 111 Linear Phase Filters
A type I11 linear phase filter has a unit sample response that is antisymmetric,
and N is even. Therefore, h ( n ) is antisymmetric about a = N / 2 , which is an integer, as illustrated in Fig. 5-3(c). The frequency response of a type 111 linear phase filter may be written as
where
'Qpe 1V Linear Phase Filters
A type IV linear phase filter has a unit sample response that is antisymmetric, and N is odd. Therefore, h ( n ) is antisymmetric about the half-integer value a = N / 2 , and the frequency response has the form
where
The z-Wansform of Linear Phase Systems
The symmetries in the unit sample response of a linear phase system impose constraints on the system function H ( z ) . For a type I or I1 filter, h ( n ) = h ( N - n ) , which implies that
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