Implementing IIR/FIR Filters

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LHS = h 0 sinθτ g + h 1 sinθτ ( g – 1) + h 2 sinθτ ( g – 2) + … +h N – 3 so that if: Eqn. 7-24 then for every positive argument there will be a corresponding negative argument. For example, the argument for the h 1 term becomes: which is the negative of the argument for the h N-2 term, i.e., so that if: τ g sinθτ ( g – N+ 3) + h N– 2 sinθτ ( g – N+ 2) + h N– 1 sinθτ ( g – N+ 1) ( N – 1) = ----------------- 2 N 1 θ – ⎛------------ – 1⎞ N 3 θ ⎝ 2 ⎠ – = ⎛------------ ⎞ ⎝ 2 ⎠ N 1 θ – ⎛------------ – N + 2⎞ – θ ⎝ 2 ⎠ N 3 – = ⎛------------ ⎞ ⎝ 2 ⎠ hi () = hN ( – 1– i) for 0 ≤ i ≤ N – 1 Eqn. 7-25 then Eqn. 7-9 and Eqn. 7-25 represent the solution to . By substituting Eqn. 7-9 into Eqn. 7-20, the phase of a linear-phase FIR filter is given by: φθ ( ) N – 1 = – ⎛------------ ⎞θ ⎝ 2 ⎠ Eqn. 7-26 MOTOROLA 7-7
Therefore, a nonrecursive filter (FIR), unlike a recursive filter (IIR), can have a constant time delay for all frequencies over the entire range (from 0 to fs /2). It is only necessary that the coefficients (and therefore impulse response) be symmetrical about the midpoint between samples (N-2)/2 and N/2 for even N or about sample (N-1)/2 for odd N (see Eqn. 7-25). When this symmetry exists, τg for the filter will be ⎛N – 1 ------------ ⎞ T seconds. ⎝ 2 ⎠ 7.20 Linear-Phase FIR Filter Design Using the Frequency Sampling Method Implementing a FIR filter in DSP hardware such as the DSP56001 is a relatively simple task. Determining a set of coefficients that describe a given impulse response of a filter is also a straightforward procedure. However, deriving the optimal coefficients necessary to obtain a particular response in the frequency domain is not always as easy. The following paragraphs introduce a simple method to determine a set of coefficients based on a desired arbitrary frequency response (often referred to as the frequency sampling method). This method has the distinct advantage of being done in real time. However, the most efficient determination of coefficients is best done by utilizing a software filter 7-8 MOTOROLA
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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
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is taken to calculating the filter
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solving the differential equation i
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Gain Phase 1.5 1.0 0.5 0 -π/2 -π
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1.3 Analog Highpass Filter The pass
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Gain Phase 1.5 1.0 0.5 π π/2 0 0
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V 0 R ----- = ---------------------
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V 0 R ----- = ---------------------
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1-18 MOTOROLA
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In the digital domain, the continuo
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One mapping or transformation from
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Although this transformation was de
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property in particular as follows:
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Figure 2-10 are similar to the anal
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case but also in the code for the h
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Lowpass Gain Lowpass Phase 1.5 1.0
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Network Diagram Transfer Function G
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MPY MAC MAC MAC MAC Difference Equa
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Network Diagram x(n) Transfer Funct
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BANDSTOP GAIN BANDSTOP PHASE 1.5 1.
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Network Diagram x(n) Transfer Funct
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BANDPASS GAIN BANDPASS PHASE 1.5 1.
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Pole Equation of H(z) Zp = rcosθp
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x(n) b 0 + + Figure 3-24 The Second
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The last step uses the definition f
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where: θ p is the angle (see Figur
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Since this result does not depend o
Page 67 and 68: MPY MAC MACR MAC MACR Difference Eq
Page 69 and 70: 3-12 MOTOROLA
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
Page 117: ut is twice the width of the origin
Page 121 and 122: sides of by e j2πkm/N and summing
Page 123 and 124: hi () 1 j πr πkN 1 --- Ak ( )e N
Page 125 and 126: given with AN ( ⁄ 2) = 0 and r= 1
Page 127 and 128: Likewise, the y component (imaginar
Page 129 and 130: h(n) 0.2 0.1 0 -0.1 0 16 31 Figure
Page 131 and 132: The continuous frequency gain and p
Page 133 and 134: Filter Gain, G(θ), (dB) Filter Gai
Page 135 and 136: 7-24 MOTOROLA Figure 7-40 FDAS Outp
Page 137 and 138: If it is assumed that linear phase
Page 139 and 140: Reference 5). Basically, a guess is
Page 141 and 142: Figure 7-41 shows an example of an
Page 143 and 144: R0 x(n) b 0 X:(R0) x(n) x(n-1) x(n-
Page 147: 16. McClellan, J. H. and T. W. Park
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Implementing IIR/FIR Filters

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