Implementing IIR/FIR Filters

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Gain at w(n) is: G w ( θ) H w e jθ ( )H w e jθ – = ( ) α ⎛1 -- – β⎞ ⎝2 ⎠ 2 sin 2 1 θ ⎛--+ β⎞ ⎝2⎠ 2 ( cosθ – cosθ 0 ) 2 = ------------------------------------------------------------------------------------------------------------- + 1 where: γ = ( - + β) cos θ0 has been used. 2 d Peak Gain is: ------ G dθ w ( θ ) = 0 θ = θm Frequency of peak gain is: ⎛1 -- + β⎞ ⎝2⎠ where: ξ 2 ≡ -------------------- 2β H w ( z) Wz ( ) α------------- Xz ( ) 1 -- γ z 2 1 – – β z 2 – = = --------------------------------------- + g 0 ≡G w ( θ m ) cosθ m = ξcosθ 0 =1 forξcosθ 0 ≤ 1 otherwise α ⎛1 -- – β⎞ γ 1 ⎝2 ⎠ 2 = ----------------------------------- forξcosθ 0 ≤ 1 – ------ 2β α = ----------------------------------------------- ⎛1 -- + β⎞( 1 – cosθ0) ⎝2⎠ EXAMPLE: Maximum Internal Node Gain for Bandpass Filter g 0 = 1 ----------------- 2sinθp otherwise forξcosθ 0 ≤ 1 where γ 2 = 2β 2 cos 2 θ p has been used and α= (1/2-β)/2 for a bandpass filter. If sinθ p > 1/2, then g 0 1) may occur at the internal node, w(n), unless the input is scaled down by 1/g 0 . Figure 3-26 Internal Node Gain Analysis of Second-Order Canonic Form MOTOROLA 3-9
MPY MAC MACR MAC MACR Difference Equation yn ( ) 2 1 -- wn ( ) 2 μ --wn ( – 1) 2 σ ⎧ ⎫ = ⎨ + + -- wn ( – 2) ⎩ 2 ⎬ ⎭ wn ( ) = 2{ αxn ( ) + γwn ( – 1) – βw( n – 2) } X:(R0) X0,Y1,A X0,Y0,A X1,Y0,A X0,Y0,A X1,Y0,A w(n-1) w(n-2) X:(R0)+,X0 X:(R0),X1 X0,X:(R0)- A,X:(R0) A,X0 Data Structures DSP56001 Code Y:(R4)+,Y0 Y:(R4)+,Y0 Y:(R4)+,Y0 Y:(R4)+,Y0 Y:(R4)+,Y0 Y:(R4) Total Instruction Cycles 5 Icyc @ 20 MHz = 500ns Figure 3-27 Internal Node Gain Analysis of Second-Order Canonic Form 3-10 MOTOROLA γ −β μ/2 σ/2 α ;Y1=x(n) (Input) ;X0=α ;A=αx(n) ;A=A+γw(n-1) ;A=A-βw(n-2) ;A=1/2w(n)+μ/2w(n-1) ;A=A+σ/2w(n-2). X0=2A ;(assumes scaling mode ;is set).X0 is Output.
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Motorola’s High-Performance DSP T
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© Motorola Inc. 1993 Motorola rese
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SECTION 6 Filter Design and Analysi
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Figure 2-3 Figure 2-4 Figure 2-5 Fi
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Figure 4-4 Figure 4-5 Figure 5-1 Fi
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Figure 7-9 Figure 7-10 Figure 7-11
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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 and 48: MPY MAC MAC MAC MAC Difference Equa
Page 49 and 50: Network Diagram x(n) Transfer Funct
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: Since this result does not depend o
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 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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