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barcode lib ssrs when A(ejw) > 0 when A(eJY)c 0 in Software
when A(ejw) > 0 when A(eJY)c 0 Code 128C Decoder In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Making Code128 In None Using Barcode generation for Software Control to generate, create Code 128B image in Software applications. Similarly, a filter is said to have generalized linear phase if the frequency response has the form ~ ( ~ j= ~(~iw)~iWB) " ) Thus, filters with linear phase or generalized linear phase have a constant group delay. Scan Code 128A In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. Code 128 Code Set A Encoder In Visual C#.NET Using Barcode drawer for Visual Studio .NET Control to generate, create ANSI/AIM Code 128 image in .NET applications. 'system Analysis by Digital Computer. F. F. Kuo and J. F. Kaiser, Eds.. John Wiley and Sons, New York. 1966 Code 128C Printer In Visual Studio .NET Using Barcode maker for ASP.NET Control to generate, create Code 128 Code Set B image in ASP.NET applications. Making Code 128C In Visual Studio .NET Using Barcode maker for .NET Control to generate, create Code128 image in .NET applications. FOURIER ANALYSIS
Code 128C Creator In VB.NET Using Barcode drawer for Visual Studio .NET Control to generate, create Code 128 Code Set C image in .NET applications. UPCA Creator In None Using Barcode creation for Software Control to generate, create UPC Code image in Software applications. All pass
EAN13 Maker In None Using Barcode creation for Software Control to generate, create EAN13 image in Software applications. Code 128 Code Set A Creation In None Using Barcode creation for Software Control to generate, create Code 128 Code Set C image in Software applications. 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 Making Bar Code In None Using Barcode generator for Software Control to generate, create barcode image in Software applications. Code 3 Of 9 Printer In None Using Barcode maker for Software Control to generate, create ANSI/AIM Code 39 image in Software applications. where ct is any real number with l 1 < I. The unit sample response of this allpass filter is a h(n) = a8(n) Creating Postnet In None Using Barcode creator for Software Control to generate, create USPS POSTal Numeric Encoding Technique Barcode image in Software applications. UPC Code Generation In ObjectiveC Using Barcode encoder for iPad Control to generate, create UPCA Supplement 5 image in iPad applications. Frequency Selective Filters
Bar Code Creation In None Using Barcode printer for Font Control to generate, create barcode image in Font applications. GS1128 Maker In .NET Using Barcode maker for VS .NET Control to generate, create EAN128 image in .NET applications. + (1  ct2)an'u(n  1) EAN13 Drawer In None Using Barcode creation for Online Control to generate, create GS1  13 image in Online applications. Painting Data Matrix ECC200 In Java Using Barcode maker for Java Control to generate, create Data Matrix 2d barcode image in Java applications. Many of the filters that are important in applications have piecewise constant frequency response magnitudes. These include the lowpass, highpass, bandpass, and bandstop filters that are illustrated in Fig. 21. 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. Bar Code Scanner In .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET framework applications. Create Data Matrix 2d Barcode In Java Using Barcode printer for BIRT reports Control to generate, create Data Matrix image in BIRT applications. 1 H(eiW)l
lH(eiW)l
xw 2kw
o r ( a ) Ideal lowpassW C filter.
(b) Ideal highpass filter.
0.q
w=an  (c )IIdeal bandpass filrer. W WI 02
Fig. 21. (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 shiftinvariant systems is shown in the figure below. A cascade is equivalent to a single linear shiftinvariant 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 shiftinvariant systems is shown in the figure below.
A parallel network is equivalent to a single linear shiftinvariant system with a unit sample response Therefore, the frequency response of the parallel network is
EXAMPLE 2.4.1 The cascade of a lowpass filter with a highpass 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 lowpass filter with acutoff frequency y with a highpass filter that has a cutoff frequency w,. Similarly, the bandstop filter shown in Fig. 2l(d)may be realized with a with parallel connection of a lowpass filter with cutoff frequency w 1and a highpass 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 DISCRETETIME FOURIER TRANSFORM
The frequency response of a linear shiftinvariant system is found by multiplying h(n) by a complex exponential, eJn", and summing over n. The discretetime Fourier transform (DTFT) of a sequence, x(n), is defined in the same way, Thus, the frequency response of a linear shiftinvariant 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  aejw
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  aIejw
1 1  ae1" 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 ztransforms. 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 21 Some Common DTFT Pairs
Sequence 6(n) S(n  no) 1 eJ"wO
DiscreteTime 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)

