Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

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Two-Dimensional IIR Filters 23-39If linear-phase subfilters are to be used, the equalities in Equation 23.99 must be satisfied. This impliesthat the subfilters must have constant group delays. If zero-phase subfilters are employed, where f i (z 1 )and f i (z1 1 ),and g i (z 2 ) and g i (z2 1 )contribute equally to the amplitude response of the 2-D filter. The designcan be accomplished by assuming that the desired amplitude responses for subfilters F i , G i are f 1=2 i , g 1=2 i ,for i ¼ 1, 2, k, ..., K, respectively.Error Compensation Procedure. When the main section and correction sections are designed by anoptimization procedure as described above, approximation errors inevitably occur that will accumulateand manifest themselves as the overall error. The accumulation of error can be reduced by the followingcompensation procedure.When filters F 1 and G 1 are designed, the approximation error matrix E 1 can be calculated asE 1 ¼ A f 1 (e jpm l )g1 (e jpnm ) (23:106)and then perform SVD on E 1 to obtainE 1 ¼ S 22 f 22 g T 22 þþS r2f r2 g T r2(23:107)Data f 22 and g 22 can be used to deduce filters f 2 (z 1 ) and g 2 (z 2 ). Thus, the first correction section can bedesigned. Next, form the error matrix E 2 asand then perform SVD on E 2 to obtainE 2 ¼ E 1 S 22 f 2 (e jpm l )g2 (e jpvm ) (23:108)E 2 ¼ S 33 f 33 g T 33 þþS r3 f r3 g T r3(23:109)and use data f 33 and g 33 to design the second correction section. The procedure is continued until thenorm of the error matrix becomes sufficiently small that a satisfactory approximation to the desiredamplitude response is reached.Design of 1-D filters by using optimization can sometimes yield unstable filters. This problem can beeliminated by replacing poles outside the unit circle of the z plane by their reciprocals and simultaneouslyadjusting the multiplier constant to compensate for the change in gain [19].Example 23.7 [26]Design a circularly symmetric, zero-phase 2-D filter specified by(jHðv 1 , v 2 Þj ¼ 1 for v2 1 þ 1=2 v2 2 < 0:35p0 for v 2 1 þ 1=22 v2 0:65 passuming that v s1 ¼ v s2 ¼ 2p.Solution1. Construct a sampled amplitude response matrix. By taking L ¼ M ¼ 21 and assuming that theamplitude response varies linearly with the radius in the transition band, the amplitude responsematrix can be obtained as
23-40 Passive, Active, and Digital FilterswhereA ¼ A 1 0 0 021211 1 1 1 1 1 1 1 1 0:75 0:5 0:251 1 1 1 1 1 1 1 0:75 0:5 0:25 01 1 1 1 1 1 1 1 0:75 0:5 0:25 01 1 1 1 1 1 1 0:75 0:5 0:25 0 01 1 1 1 1 1 1 0:75 0:5 0:25 0 01 1 1 1 1 1 1 0:75 0:5 0:25 0 0A ¼1 1 1 1 1 0:75 0:5 0:25 0 0 0 01 1 1 0:75 0:75 0:5 0:25 0 0 0 0 01 0:75 0:75 0:5 0:5 0:25 0 0 0 0 0 00:75 0:5 0:5 0:25 0:25 0 0 0 0 0 0 00:5 0:25 0:25 0 0 0 0 0 0 0 0 0 0:25 0 0 0 0 0 0 0 0 0 0 0 The ideal amplitude response of the filter is illustrated in Figure 23.20a.2. Perform SVD to matrix A to obtain the amplitude response of the main section of the 2-D filter. Itis worth noting that when a circularly symmetric 2-D filter is required, matrix A is symmetric and,therefore, Equation 23.95 becomes110.80.80.60.60.40.40.20.201(a)0.50–0.5–1 –1–0.500.5101(b)0.50–0.5–1 –1–0.500.51110.80.80.60.60.40.40.20.2(c)010.50–0.5–1 –1–0.500.5101(d)0.50–0.5–1 –1–0.500.51FIGURE 23.20 Amplitude responses of (a) the ideal circularly symmetric 2D filter; (b) the main section; (c) themain section plus the first correction; (d) the main section plus the first and second correction sections.
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Passive, Active,and DigitalFilters
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The Circuits and Filters HandbookTh
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ContentsPreface ...................
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PrefaceAs circuit complexity contin
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ContributorsPhillip E. AllenSchool
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IPassive FiltersWai-Kai ChenUnivers
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1General Characteristicsof FiltersA
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General Characteristics of Filters
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1-12 Passive, Active, and Digital F
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2-2 Passive, Active, and Digital Fi
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Approximation 2-9K(ω)Large nSmall
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Approximation 2-132π21φ = cos -1
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Approximation 2-15This result is us
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Approximation 2-19Butterworth; put
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Approximation 2-21This however impl
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Approximation 2-23TABLE 2.3 Bessel
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Approximation 2-25R mn (ω)b + εbb
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Approximation 2-27is a measure of t
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Approximation 2-29Recall, now, the
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Approximation 2-31TABLE 2.4 Zeros o
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Approximation 2-33References1. A. M
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4Sensitivityand SelectivityIgor M.
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Sensitivity and Selectivity 4-34.3
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Sensitivity and Selectivity 4-5Beca
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Sensitivity and Selectivity 4-7Simi
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Sensitivity and Selectivity 4-9comp
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Sensitivity and Selectivity 4-11In
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Sensitivity and Selectivity 4-13and
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Sensitivity and Selectivity 4-154.8
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Sensitivity and Selectivity 4-17cha
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Sensitivity and Selectivity 4-19Tak
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Sensitivity and Selectivity 4-21I p
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Sensitivity and Selectivity 4-23Att
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Sensitivity and Selectivity 4-25 In
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Sensitivity and Selectivity 4-27the
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Sensitivity and Selectivity 4-29is
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Sensitivity and Selectivity 4-311.
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5Passive Immittancesand Positive-Re
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Passive Immittances and Positive-Re
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6Passive CascadeSynthesisWai-Kai Ch
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Passive Cascade Synthesis 6-3not gr
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Passive Cascade Synthesis 6-5degree
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Passive Cascade Synthesis 6-7M < 0L
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Passive Cascade Synthesis 6-9LN AFI
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Passive Cascade Synthesis 6-11ζCN
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Passive Cascade Synthesis 6-13the d
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Passive Cascade Synthesis 6-15The e
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7Synthesis of LCM andRC One-Port Ne
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Synthesis of LCM and RC One-Port Ne
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V g-V g-9-8 Passive, Active, and Di
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10Design of BroadbandMatching Netwo
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Design of Broadband Matching Networ
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IIActive FiltersWai-Kai ChenUnivers
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11Low-Gain Active FiltersPhillip E.
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Low-Gain Active Filters 11-3The dam
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Low-Gain Active Filters 11-5at a fr
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Low-Gain Active Filters 11-7+RK++CK
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Low-Gain Active Filters 11-9R 2+R 1
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Low-Gain Active Filters 11-11For th
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Low-Gain Active Filters 11-13The se
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Low-Gain Active Filters 11-15Y 1H22
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Low-Gain Active Filters 11-17R 2+R
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Low-Gain Active Filters 11-19To con
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Low-Gain Active Filters 11-21+10k 1
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Low-Gain Active Filters 11-23TABLE
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Low-Gain Active Filters 11-25The ph
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Low-Gain Active Filters 11-27TABLE
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Low-Gain Active Filters 11-29I n1E
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Low-Gain Active Filters 11-3150 nVS
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12Single-AmplifierMultiple-Feedback
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Single-Amplifier Multiple-Feedback
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13-10 Passive, Active, and Digital
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14The CurrentGeneralizedImmittance
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The Current Generalized Immittance
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15High-Order FiltersRolf SchaumannP
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High-Order Filters 15-3V inV o2V oi
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High-Order Filters 15-5Choosing the
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High-Order Filters 15-7where we def
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High-Order Filters 15-9where we int
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High-Order Filters 15-11where s is
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High-Order Filters 15-1315.4.1 Sign
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High-Order Filters 15-15Ri2R 4V oR
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High-Order Filters 15-17Notice that
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High-Order Filters 15-19v i g i(α
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High-Order Filters 15-21TABLE 15.1
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High-Order Filters 15-23R a1 ¼ ^C
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High-Order Filters 15-25I 1 1V 1 sk
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High-Order Filters 15-27664154015.2
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16Continuous-TimeIntegrated Filters
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Continuous-Time Integrated Filters
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17Switched-CapacitorFiltersJose Sil
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Switched-Capacitor Filters 17-3C SC
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Switched-Capacitor Filters 17-5For
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Switched-Capacitor Filters 17-7φ 2
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Switched-Capacitor Filters 17-9A 3
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Switched-Capacitor Filters 17-11whe
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Switched-Capacitor Filters 17-1317.
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Switched-Capacitor Filters 17-15C i
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Switched-Capacitor Filters 17-1717.
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Switched-Capacitor Filters 17-19The
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Switched-Capacitor Filters 17-21Dur
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Switched-Capacitor Filters 17-23sig
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Switched-Capacitor Filters 17-25C 1
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Switched-Capacitor Filters 17-27φC
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Switched-Capacitor Filters 17-29for
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Switched-Capacitor Filters 17-31If
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Switched-Capacitor Filters 17-33FIG
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Switched-Capacitor Filters 17-35CH1
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IIIDigital FiltersRashid AnsariUniv
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18FIR FiltersM. H. ErNanyang Techno
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FIR Filters 18-3Consequently,tan (v
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FIR Filters 18-5TABLE 18.1Frequency
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FIR Filters 18-7H d (e jω )1To und
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FIR Filters 18-90-10-20Amplitude re
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FIR Filters 18-110-5-10Amplitude re
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FIR Filters 18-130-20Amplitude resp
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FIR Filters 18-15TABLE 18.2Spectral
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FIR Filters 18-17If it were possibl
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FIR Filters 18-19The alternation th
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FIR Filters 18-21respectively, and
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FIR Filters 18-23Rejection of super
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FIR Filters 18-25The selective step
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FIR Filters 18-27TABLE 18.4 Impulse
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FIR Filters 18-29selective step-by-
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FIR Filters 18-31Odd filter lengthM
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FIR Filters 18-33and1a 1 ¼ ~c 02 ~
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FIR Filters 18-35a useful property
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FIR Filters 18-37with the number of
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FIR Filters 18-39F(z L )z −LN F /
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FIR Filters 18-411F(Lω)G 1 (ω)0 0
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FIR Filters 18-43TABLE 18.11Algorit
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FIR Filters 18-451N ove ¼ N optL
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FIR Filters 18-47TABLE 18.13 Implem
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FIR Filters 18-494. If r ¼ R, then
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FIR Filters 18-51In this case, the
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FIR Filters 18-53Estimated orders11
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FIR Filters 18-550Amplitude (db)−
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FIR Filters 18-57In the above, the
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FIR Filters 18-59TABLE 18.15Data fo
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FIR Filters 18-614. Y. C. Lim and Y
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20Finite WordlengthEffectsBruce W.
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Finite Wordlength Effects 20-3the q
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Finite Wordlength Effects 20-5From
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Finite Wordlength Effects 20-7In ge
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Finite Wordlength Effects 20-9To ob
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Finite Wordlength Effects 20-11Howe
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Finite Wordlength Effects 20-13wher
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Finite Wordlength Effects 20-15 y(4
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Finite Wordlength Effects 20-17j1.0
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Finite Wordlength Effects 20-1921.
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Page 594 and 595: 23Two-DimensionalIIR FiltersA. G. C
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Page 642 and 643: 241-D MultirateFilter BanksNick G.
Page 644 and 645: 1-D Multirate Filter Banks 24-324.2
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1-D Multirate Filter Banks 24-41x0
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1-D Multirate Filter Banks 24-43pri
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1-D Multirate Filter Banks 24-45The
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1-D Multirate Filter Banks 24-47Two
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1-D Multirate Filter Banks 24-49" #
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1-D Multirate Filter Banks 24-51Tre
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1-D Multirate Filter Banks 24-537.
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ω 1πω 1π25-8 Passive, Active, a
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26Nonlinear FilteringUsing Statisti
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Nonlinear Filtering Using Statistic
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27Nonlinear Filtering forImage Deno
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Nonlinear Filtering for Image Denoi
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IN-2Indeximpulse invariance method,
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IN-4Indexprescribed specification,2
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IN-6Indexfilter rank ordering, 2-32
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IN-8Indexpassband and stopband ripp
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IN-10Indexstate-space filter realiz
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IN-12Indexnonessential singularitye
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IN-14Indexreal-valued weights andop
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IN-16Indexbaseband communication,26
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IN-18Indexarbitrary specificationCh
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IN-20IndexVoltage-controlled curren
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