Implementing IIR/FIR Filters

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2.8 Digital Bandstop Filter The formulas and network diagram for the digital bandstop filter are presented in Figure 2-16. The DSP56001 code from Figure 2-17 is identical to that for the lowpass and highpass cases except for the coefficient data calculated from the equations of Figure 2-16. Scaling of this filter is not a problem for the singlesection case since the gain from the equation in Figure 2-16 never exceeds unity (as is true in the analog case as seen by the gain equation from Figure 1-5). Figure 2-18 is the calculated gain and phase of the digital filter, which should compare to the response curves of the equivalent analog filter plotted in Figure 1-6. 2.9 Digital Bandpass Filter Because there is one less coefficient in the bandpass network (see Figure 2-19), one instruction can be saved in the DSP56001 code implementation shown in Figure 2-20. Otherwise, the instructions are identical to those in the other three filter routines. Like the second-order bandstop network, the maximum response at the center frequency, θ 0 , is unity for any value of Q so that scaling need not be considered in the implementation of a single-section bandpass filter. This is true when the formulas for α, β, and γ (from Figure 2-19) are used in the direct-form implementation in Figure 2-20. Figure 2-21 is the calculated gain and phase of the digital filter, which should compare to the response curves of the equivalent analog filter plotted in Figure 1-8. MOTOROLA 2-19
Network Diagram x(n) Transfer Function Gain Phase Coefficients z -1 z -1 G( θ) α −2αcosθ 0 α Hz ( ) Σ α 1– 2cosθ0z 1 – z 2 – ( + ) 1/2 γz 1 – – βz 2 – = -------------------------------------------------------- + cosθ – cosθc ( dsinθsinθ0) 2 4( cosθ – cosθ0) 2 [ + ] 1/2 = -------------------------------------------------------------------------------------------------- φθ ( ) tan 1 – 2 θ θ ( cos – cos 0) = ---------------------------------------dsinθsinθ0 d 2tan( θ0 /2Q) = ------------------------------sinθ0 1 β ⎛--⎞ ⎝2⎠ 1 tan θ0 /2Q ( ) – = ------------------------------------- 1+ tan( θ0 /2Q) γ = ( 1/2 + β) cosθ0 α = ( 1/2 + β)/2 Figure 2-16 Direct-Form Implementation of Second-Order Bandstop <strong>IIR</strong> Filter and Analytical Formulas Relating Desired Response to Filter Coefficients 2-20 MOTOROLA 2 γ −β z -1 z -1 y(n)
Page 1 and 2: Motorola’s High-Performance DSP T
Page 3 and 4: © Motorola Inc. 1993 Motorola rese
Page 5 and 6: SECTION 6 Filter Design and Analysi
Page 7 and 8: Figure 2-3 Figure 2-4 Figure 2-5 Fi
Page 9 and 10: Figure 4-4 Figure 4-5 Figure 5-1 Fi
Page 11 and 12: Figure 7-9 Figure 7-10 Figure 7-11
Page 13 and 14: equivalent functions in the digital
Page 15 and 16: is taken to calculating the filter
Page 17 and 18: solving the differential equation i
Page 19 and 20: Gain Phase 1.5 1.0 0.5 0 -π/2 -π
Page 21 and 22: 1.3 Analog Highpass Filter The pass
Page 23 and 24: Gain Phase 1.5 1.0 0.5 π π/2 0 0
Page 25 and 26: V 0 R ----- = ---------------------
Page 27 and 28: V 0 R ----- = ---------------------
Page 29 and 30: 1-18 MOTOROLA
Page 31 and 32: In the digital domain, the continuo
Page 33 and 34: One mapping or transformation from
Page 35 and 36: Although this transformation was de
Page 37 and 38: property in particular as follows:
Page 39 and 40: Figure 2-10 are similar to the anal
Page 41 and 42: case but also in the code for the h
Page 43 and 44: Lowpass Gain Lowpass Phase 1.5 1.0
Page 45 and 46: Network Diagram Transfer Function G
Page 47: MPY MAC MAC MAC MAC Difference Equa
Page 51 and 52: BANDSTOP GAIN BANDSTOP PHASE 1.5 1.
Page 53 and 54: Network Diagram x(n) Transfer Funct
Page 55 and 56: BANDPASS GAIN BANDPASS PHASE 1.5 1.
Page 57 and 58: Pole Equation of H(z) Zp = rcosθp
Page 59 and 60: x(n) b 0 + + Figure 3-24 The Second
Page 61 and 62: The last step uses the definition f
Page 63 and 64: where: θ p is the angle (see Figur
Page 65 and 66: Since this result does not depend o
Page 67 and 68: MPY MAC MACR MAC MACR Difference Eq
Page 71 and 72: expression for the maximum gain at
Page 73 and 74: GAIN AT INTERNAL NODE, u(n) Transfe
Page 75 and 76: Gain Gain Gain 1.2 1.0 0.8 0.6 0.4
Page 79 and 80: Hs ( ) damping factors, dk, of each
Page 81 and 82: Using , Eqn. 5-7 becomes: Hs ( ) N/
Page 83 and 84: α k = β k = γ k = tan 2 ( θ c /
Page 85 and 86: pass Butterworth filter (three seco
Page 87 and 88: Network Diagram x(n) α1 z -1 z -1
Page 91 and 92: can be assembled by the DSP56001 as
Page 93 and 94: Phase (Radians) π π/2 0 -π/2 -π
Page 95 and 96: Seconds 6.3 E-03 4.7 E-03 3.1 E-03
Page 97 and 98: ZEROES OF TRANSFER FUNCTION Hd(z) R
Page 99 and 100:
1 COEFF ident 1,0 2 include 'head2.
Page 101 and 102:
77 P:0000 0C0040 jmp start ;jump to
Page 103 and 104:
6.2 Transpose Implementation (Direc
Page 105 and 106:
POLES OF TRANSFER FUNCTION Hd(z) Re
Page 107 and 108:
1 C0EFF ident 1,0 2 include 'head1.
Page 109 and 110:
75 76 P:0040 0003F8 ori #3,mr ;disa
Page 111 and 112:
6-22 MOTOROLA
Page 113 and 114:
7.19 Linear-Phase FIR Filter Struct
Page 115 and 116:
G( θ) hi ()e jiθ - N- 1 = i = 0 E
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ut is twice the width of the origin
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Therefore, a nonrecursive filter (F
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sides of by e j2πkm/N and summing
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hi () 1 j πr πkN 1 --- Ak ( )e N
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given with AN ( ⁄ 2) = 0 and r= 1
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Likewise, the y component (imaginar
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h(n) 0.2 0.1 0 -0.1 0 16 31 Figure
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The continuous frequency gain and p
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Filter Gain, G(θ), (dB) Filter Gai
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7-24 MOTOROLA Figure 7-40 FDAS Outp
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If it is assumed that linear phase
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Reference 5). Basically, a guess is
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Figure 7-41 shows an example of an
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R0 x(n) b 0 X:(R0) x(n) x(n-1) x(n-
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7-34 MOTOROLA
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16. McClellan, J. H. and T. W. Park
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Implementing IIR/FIR Filters

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