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Similarly, a filter is said to have generalized linear phase if the frequency response has the form ~ ( ~ j= ~(~iw)~-iW-B) " ) Thus, filters with linear phase or generalized linear phase have a constant group delay.
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'system Analysis by Digital Computer. F. F. Kuo and J. F. Kaiser, Eds.. John Wiley and Sons, New York. 1966
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A system is said to be allpass filter if the frequency response magnitude is constant: 1H(eJ")l = c An example of an allpass filter is the system that has a frequency response
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where ct is any real number with l 1 < I. The unit sample response of this allpass filter is a h(n) = -a8(n)
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+ (1 - ct2)an-'u(n - 1)
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Many of the filters that are important in applications have piecewise constant frequency response magnitudes. These include the low-pass, high-pass, bandpass, and bandstop filters that are illustrated in Fig. 2-1. The intervals over which the frequency response magnitude is equal to 1 are called the passbands, and the intervals over which it is equal to 0 are called the stopbands. The frequencies that mark the edges of the passbands and stopbands are the cutofffrequencies.
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1 H(eiW)l
lH(eiW)l
xw -2kw
-o r ( a ) Ideal low-passW C filter.
(b) Ideal high-pass filter.
-0.q
w-=a-n -
(-c )IIdeal bandpass filrer. W WI 02
Fig. 2-1.
(d) Ideal bandstop filter.
Ideal filters.
2.4 INTERCONNECTION OF SYSTEMS
Filters are often interconnected to create systems that have desirable properties. Two common types of connections are series (cascade) and parallel. A cascade of two linear shift-invariant systems is shown in the figure below.
A cascade is equivalent to a single linear shift-invariant system with a unit sample response
and a frequency response
FOURIER ANALYSIS
[CHAP. 2
H (el") = H l ( e j W ) ~ 2 ( e J W )
Note that the log magnitude of the cascade is the sum of the log magnitudes of the individual systems,
and the phase and group delay are additive,
A parallel connection of two linear shift-invariant systems is shown in the figure below.
A parallel network is equivalent to a single linear shift-invariant system with a unit sample response
Therefore, the frequency response of the parallel network is
EXAMPLE 2.4.1 The cascade of a low-pass filter with a high-pass filter may be used to implement a bandpass filter. For example, the ideal bandpass filter shown inFig. 2- I(c) may be realized by cascading a low-pass filter with acutoff frequency y with a high-pass filter that has a cutoff frequency w,. Similarly, the bandstop filter shown in Fig. 2-l(d)may be realized with a with parallel connection of a low-pass filter with cutoff frequency w 1and a high-pass filter with a cutoff frequency y, y > wl .
Another interconnection of systems that is commonly found in control applications is the feedback network shown in the figure below.
CHAP. 21
FOURIER ANALYSIS
we may use the Fourier analysis techniques described in the following section to show that the frequency response of this system, if it exists, is2
2.5 THE DISCRETE-TIME FOURIER TRANSFORM
The frequency response of a linear shift-invariant system is found by multiplying h(n) by a complex exponential, e-Jn", and summing over n. The discrete-time Fourier transform (DTFT) of a sequence, x(n), is defined in the same way,
Thus, the frequency response of a linear shift-invariant system, ~ ( e j " ) , the DTFT of the unit sample response, is h(n). In order for the DTFT of a sequence to exist, the summation in Eq. (2.3) must converge. This, in turn, requires that x(n) be absolutely summable:
EXAMPLE 2.5.1
The DTFT of the sequence
Using the geometric series, this sum is
Xl(e )
1 - ae-jw
provided la( < 1. Similarly, for the sequence
the DTFT is
Changing the limits on the sum, we have
If la1 > 1, this sum is
X2(e1") = -
1 + I = 1 - a-Iejw
1 1 - ae-1"
Therefore, x,(n) = anu(n)and x2(n) = -anu(-n - I) both have the same DTFT.
2 ~ is t
possible that g ( n ) will make the system unstable, in which case the DTFT of h(n) will not exist. Feedback systems are typically analyzed using z-transforms.
FOURIER ANALYSIS
[CHAP. 2
Given X ( e J w ) , sequence x ( n ) may be recovered using the inverse DTFT, the
The inverse DTFT may be viewed as adecomposition of x ( n ) into alinear combination of all complex exponentials that have frequencies in the range -17 i 5 IT. Table 2- 1 contains a list of some useful DTFT pairs. w
Table 2-1 Some Common DTFT Pairs
Sequence 6(n) S(n - no) 1
eJ"wO
Discrete-Time Fourier Transform
anu(n), la1 < I -anu(-n - I ) , (n la1 > 1
+ I)anu(n), la1 < 1
EXAMPLE 2.5.2
Suppose X(eJ")consists of an impulse at frequency w = wo: X(eJ")= 6(w - wO)
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