vili Contents 6.4 Quantization Effects in the Computation of the DFT 486 6.4.1 Quantization Errors in the Direct Computation of the DFT.487 6.4.2 Quantization Errors in FFT Algorithms.489 6.5 Summary and References 493 Problems 494 7 IMPLEMENTATION OF DISCRETE-TIME SYSTEMS 500 7.1 Structures for the Realization of Discrete-Time Systems 500 7.2 Structures for FIR Systems 502 7.2.1 Direct-Form Structure,503 7.2.2 Cascade-Form Structures.504 7.2.3 Frequency-Sampling Structures'.506 7.2.4 Lattice Structure.511 7.3 Structures for IIR Systems 519 7.3.1 Direct-Form Structures.519 7.3.2 Signal Flow Graphs and Transposed Structures.521 7.3.3 Cascade-Form Structures,526 7.3.4 Parallel-Form Structures.529 73.5 Lattice and Lattice-Ladder Structures for IIR Systems.531 7.4 State-Space System Analysis and Structures 539 7.4.1 State-Space Descriptions of Systems Characterized by Diflerence Equations.540) 7.4.2 Solution of the State-Space Equations.543 7.43 Relationships Between Input-Output and State-Space Descriptions.545 7.4.4 State-Space Analysis in the z-Domain.550 7.4.5 Additional State-Space Structures.554 7.5 Representation of Numbers 556 7.5.1 Fixed-Point Representation of Numbers.557 7.5.2 Binary Floating-Point Representation of Numbers.561 7.5.3 Errors Resulting from Rounding and Truncation.564 7.6 Quantization of Filter Coefficients 569 7.6.1 Analysis of Sensitivity to Quantization of Filter Coefficients.569 7.6.2 Quantization of Coefficients in FIR Filters.578 7.7 Round-Off Effects in Digital Filters 582 7.7.1 Limit-Cycle Oscillations in Recursive Systems.583 7.72 Scaling to Prevent Overfow.588 7.7.3 Statistical Characterization of Quantization Effects in Fixed-Point Realizations of Digital Filters,590 7.8 Summary and References 598 Problems 600