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Figure 6.1 Block diagram symbols. (a) Addition of two sequences. (b) Multiplication of a sequence by a constant. (c) Unit delay.
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Figure 6.2 Example of a block diagram representation of a difference equation.
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Figure 6.3 Block diagram representation for a general Nth-order difference equation.
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Figure 6.4 Rearrangement of block diagram of Figure 6.3. We assume for convenience that N = M. If N ≠ M, some of the coefficients will be zero.
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Figure 6.5 Combination of delays in Figure 6.4.
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Figure 6.6 Direct form I implementation of Eq. (6.16).
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Figure 6.7 Direct form II implementation of Eq. (6.16).
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Figure 6.8 Example of nodes and branches in a signal flow graph.
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Figure 6.9 Example of a signal flow graph showing source and sink nodes.
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Figure 6.10 (a) Block diagram representation of a 1st-order digital filter. (b) Structure of the signal flow graph corresponding to the block diagram in (a).
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Figure 6.11 Signal flow graph of Figure 6.10(b) with the delay branch indicated by z−1.
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Figure 6.12 Flow graph not in standard direct form.
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Figure 6.13 Direct form I equivalent of Figure 6.12.
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Figure 6.14 Signal flow graph of direct form I structure for an Nth-order system.
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Figure 6.15 Signal flow graph of direct form II structure for an Nth-order system.
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Figure 6.16 Direct form I structure for Example 6.4.
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Figure 6.17 Direct form II structure for Example 6.4.
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Figure 6.18 Cascade structure for a 6th-order system with a direct form II realization of each 2nd-order subsystem.
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Figure 6.19 Cascade structures for Example 6.5. (a) Direct form I subsections. (b) Direct form II subsections.
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Figure 6.20 Parallel form structure for 6th-order system (M = N = 6) with the real and complex poles grouped in pairs.
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Figure 6.21 Parallel form structure for Example 6.6 using a 2nd-order system.
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Figure 6.22 Parallel form structure for Example 6.6 using 1st-order systems.
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Figure 6.23 (a) System with feedback loop. (b) FIR system with feedback loop. (c) Noncomputable system.
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Figure 6.24 (a) Flow graph of simple 1st-order system. (b) Transposed form of (a). (c) Structure of (b) redrawn with input on left.
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Figure 6.25 Direct form II structure for Example 6.8.
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Figure 6.26 Transposed direct form II structure for Example 6.8.
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Figure 6.27 General flow graph resulting from applying the transposition theorem to the direct form I structure of Figure 6.14.
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Figure 6.28 General flow graph resulting from applying the transposition theorem to the direct form II structure of Figure 6.15.
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Figure 6.29 Direct form realization of an FIR system.
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Figure 6.30 Transposition of the network of Figure 6.29.
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Figure 6.31 Cascade form realization of an FIR system.
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Figure 6.32 Direct form structure for an FIR linear-phase system when M is an even integer.
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Figure 6.33 Direct form structure for an FIR linear-phase system when M is an odd integer.
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Figure 6.34 Symmetry of zeros for a linear-phase FIR filter.
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Figure 6.35 One section of the lattice structure for FIR lattice filters. (a) Block diagram representation of a two-port building block (b) Equivalent flow graph.
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Figure 6.36 Cascade connection of M basic building block sections.
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Figure 6.37 Lattice flow graph for an FIR system based on a cascade of M two-port building blocks of Figure 6.35(b).
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Figure 6.38 Algorithm for converting from k-parameters to FIR filter coefficients.
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Figure 6.39 Algorithm for converting from FIR filter coefficients to k-parameters.
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Figure 6.40 Flow graphs for example. (a) Direct form. (b) Lattice form (coefficients rounded).
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Figure 6.41 One stage of computation for an all-pole lattice system.
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Figure 6.42 All-pole lattice system.
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Figure 6.43 Signal flow graph of IIR filter; (a) direct form, (b) lattice form.
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Figure 6.44 Nonlinear relationships representing two’s-complement (a) rounding and (b) truncation for B = 2.
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Figure 6.45 Two’s-complement rounding. (a) Natural overflow. (b) Saturation.
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Figure 6.46 Implementation of discrete-time filtering of an analog signal. (a) Ideal system. (b) Nonlinear model. (c) Linearized model.
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Table 6.1 UNQUANTIZED DIRECT-FORM COEFFICIENTS FOR A 12TH-ORDER ELLIPTIC FILTER
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Figure 6.47 IIR coefficient quantization example. (a) Log magnitude for unquantized elliptic bandpass filter. (b) Magnitude in passband for unquantized (solid line) and 16-bit quantized cascade form (dashed line).
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Table 6.2 ZEROS AND POLES OF UNQUANTIZED 12TH-ORDER ELLIPTIC FILTER.
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Figure 6.48 IIR coefficient quantization example. (a) Poles and zeros of H(z) for unquantized coefficients. (b) Poles and zeros for 16-bit quantization of the direct form coefficients.
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Table 6.3 UNQUANTIZED CASCADE-FORM COEFFICIENTS FOR A 12TH-ORDER ELLIPTIC FILTER
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Table 6.4 SIXTEEN-BIT QUANTIZED CASCADE-FORM COEFFICIENTS FOR A 12TH-ORDER ELLIPTIC FILTER
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Figure 6.49 Direct form implementation of a complex-conjugate pole pair.
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Figure 6.50 Pole-locations for the 2nd-order IIR direct form system of Figure 6.49. (a) Four-bit quantization of coefficients. (b) Seven-bit quantization.
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Figure 6.51 Coupled form implementation of a complex-conjugate pole pair.
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Figure 6.52 Pole locations for coupled form 2nd-order IIR system of Figure 6.51. (a) Four-bit quantization of coefficients. (b) Seven-bit quantization.
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Figure 6.53 Representation of coefficient quantization in FIR systems.
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Table 6.5 UNQUANTIZED AND QUANTIZED COEFFICIENTS FOR AN OPTIMUM FIR LOWPASS FILTER (M = 27)
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Figure 6.54 FIR quantization example. (a) Log magnitude for unquantized case. (b) Approximation error for unquantized case. (Error not defined in transition band.) (c) Approximation error for 16-bit quantization.
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Figure 6.54 (continued) (d) Approximation error for 14-bit quantization. (e) Approximation error for 13-bit quantization. (f) Approximation error for 8-bit quantization.
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Figure 6.55 Effect of impulse response quantization on zeros of H(z). (a) Unquantized. (b) 16-bit quantization. (c) 13-bit quantization. (d) 8-bit quantization.
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Figure 6.56 Subsystem to implement 4th-order factors in a linear-phase FIR system such that linearity of the phase is maintained independently of parameter quantization.
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Figure 6.57 Models for direct form I system. (a) Infinite-precision model. (b) Nonlinear quantized model. (c) Linear-noise model.
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Figure 6.58 Probability density function for quantization errors. (a) Rounding. (b) Truncation.
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Figure 6.59 Linear-noise model for direct form I with noise sources combined.
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Figure 6.60 1st-order linear noise model.
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Figure 6.61 Linear-noise models for direct form II. (a) Showing quantization of individual products. (b) With noise sources combined.
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Figure 6.62 Scaling of direct form systems. (a) Direct form I. (b) Direct form II.
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Figure 6.63 Scaled 1st-order system.
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Table 6.6 COEFFICIENTS FOR ELLIPTIC LOWPASS FILTER IN CASCADE FORM
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Figure 6.64 Models for 6th-order cascade system with transposed direct form II subsystems. (a) Infinite-precision model. (b) Linear-noise model for scaled system, showing quantization of individual multiplications. (c) Linear-noise model with noise sources combined.
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Figure 6.65 Pole–zero plot for 6th-order system of Figure 6.64, showing pairing of poles and zeros.
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Figure 6.66 Frequency-response functions for example system. (a) 20 log10|H1(ejω)|. (b) 20 log10|H2(ejω)|. (c) 20 log10 |H3(ejω)|.
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Figure 6.66 (continued) (d) 20 log10 |H’1(ejω)|. (e) 20 log10 |H’1(ejω)H’2 (ejω)|. (f) 20 log10 |H’1(ejω)H’2(ejω)H’3(ejω)| = 20 log10 |H’(ejω)|.
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Figure 6.67 Output noise power spectrum for 123 ordering (solid line) and 321 ordering (dashed line) of 2 nd-order sections.
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Figure 6.68 Direct form realization of an FIR system. (a) Infinite-precision model. (b) Linear-noise model.
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Figure 6.69 1st-order IIR system. (a) Infinite-precision linear system. (b) Nonlinear system due to quantization.
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Figure 6.70 Response of the 1st-order system of Figure 6.69 to an impulse. (a) a = ½. (b) a = −½.
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Figure P6.1
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Figure P6.2
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Figure P6.3
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Figure P6.3 (continued)
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Figure P6.5
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Figure P6.6
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Figure P6.8
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Figure P6.9
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Figure P6.10
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Figure P6.12
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Figure P6.16
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Figure P6.18
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Figure P6.23
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Figure P6.25
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Figure P6.27
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Figure P6.28-1
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Figure P6.28-2
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Figure P6.29
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Figure P6.31
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Figure P6.32
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Figure P6.33-1
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Figure P6.33-2
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Figure P6.34-1
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Figure P6.34-2
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Figure P6.35-1
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Figure P6.35-2 Polyphase structure of the system.
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Figure P6.35-3
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Figure P6.35-4
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Figure P6.36-1
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Figure P6.36-2
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Figure P6.36-3
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Figure P6.37
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Figure P6.40-1
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Figure P6.40-2
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Figure P6.41
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Figure P6.42
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Figure P6.43-1
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Figure P6.43-2
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Figure P6.45
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Figure P6.47
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Figure P6.48
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Figure P6.49
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Figure P6.52
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Figure P6.53
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Figure P6.54
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Figure P6.55
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Figure P6.56
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Discrete-Time Signal Processing, Third EditionAlan V. Oppenheim • Ronald W. Schafer
Figure P6.57
Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc.All rights reserved.
Discrete-Time Signal Processing, Third EditionAlan V. Oppenheim • Ronald W. Schafer
Figure P6.57
Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc.All rights reserved.
Discrete-Time Signal Processing, Third EditionAlan V. Oppenheim • Ronald W. Schafer
Figure P6.59