## Fixed-Point Signal ProcessingMore than 90 percent of signal-processing systems use finite-precision (fixed-point) arithmetic. This is because fixed-point hardware is lower cost and lower power, and often higher speed than floating-point hardware. The advantage of fixed-point hardware in a digital signal processing (DSP) microprocessor ( P) is largely due to the reduced data word size, since fixed-point is practical with 16 bit data, while floating-point usually requires 32 bit data. Field programmable gate arrays (FPGAs) gain similar advantages from fixed-point data word length and have the additional advantage of being customizable to virtually any desired word length. Unfortunately, most academic coursework ignores the unique challenges of fixed-point algorithm design. This text is intended to fill that gap by providing a solid introduction to fixed-point algorithm design. Readers will see both the theory and the application of the theory to useful algorithms. |

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### Contents

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3 | |

4 | |

22 ZTRANSFORM DOMAIN SYSTEMS ANALYSIS | 5 |

221 POLES AND ZEROS | 7 |

23 BLOCK DIAGRAMS AND FILTER IMPLEMENTATION | 9 |

231 TRANSPOSE FILTERS | 10 |

45 FLOATINGPOINT | 49 |

46 BLOCK FLOATINGPOINT | 50 |

Quantization Effects Data and Coefficients | 53 |

52 DATA QUANTIZATION | 54 |

522 RANGES | 57 |

523 QUANTIZATION NOISE POWER | 59 |

524 SIGNALTONOISE RATIO | 60 |

525 SATURATION AND OVERFLOW | 62 |

232 CASCADED SECONDORDER SECTIONS | 11 |

24 FREQUENCY RESPONSE | 13 |

241 FREQUENCY RESPONSE FROM THE ZTRANSFORM | 14 |

Random Processes and Noise | 19 |

311 EXPECTATIONS AND MOMENTS | 21 |

312 STATIONARY AND ERGODIC PROCESSES | 23 |

313 DEFINITIONS AND PROPERTIES | 24 |

32 RANDOM PROCESSES AND FOURIER ANALYSIS | 25 |

322 POWER SPECTRAL DENSITY | 26 |

323 FILTERING A RANDOM SEQUENCE | 27 |

325 PERIODOGRAMS | 28 |

Fixed Point Numbers | 31 |

412 ADDITION | 34 |

414 MULTIPLICATION | 35 |

416 SIGNED BINARY REPRESENTATION | 36 |

42 QFORMAT | 41 |

43 FIXEDPOINT ARITHMETIC | 43 |

431 MULTIPLICATION | 44 |

433 ROUNDING | 45 |

441 QUANTIZATION EXAMPLE COMPUTING Y0 | 46 |

442 QUANTIZATION EXAMPLE COMPUTING Y1 | 47 |

444 MATLAB EXAMPLE | 48 |

53 COEFFICIENT QUANTIZATION | 67 |

531 SIGNIFICANT FACTORS IN COEFFICIENT QUANTIZATION | 68 |

532 2NDORDER COUPLED FORM STRUCTURE | 69 |

533 DIRECT FORM IIR FILTERS COEFFICIENT QUANTIZATION PROBLEMS | 70 |

534 CASCADED SECONDORDER SECTION FILTERS | 73 |

Quantization Effects RoundOff Noise and Overﬂow | 83 |

611 CALCULATION EXAMPLE | 87 |

62 OVERFLOW AND SCALING | 90 |

622 SCALING | 91 |

624 L1t SCALING | 92 |

625 Lω SCALING | 94 |

626 L2 NORM SCALING | 95 |

628 CASCADED SECOND ORDER SECTIONS | 96 |

63 DESIGN EXAMPLE | 101 |

632 OUTPUT NOISE CALCULATIONS | 105 |

633 NOISE SPECTRUM CALCULATIONS | 107 |

635 COMPARING DIFFERENT CHOICES | 109 |

Limit Cycles | 113 |

Glossary | 117 |

119 | |

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### Common terms and phrases

adder algorithm amplitude analysis arithmetic attenuation autocorrelation avoid overﬂow binary numbers cascaded coefﬁcient quantization coefﬁcient values compute Convert crest factor deﬁned Deﬁnition difﬁcult digits Direct Form double precision effect example Figure Filter ﬁnal ﬁnd ﬁnite ﬁrst ﬁxed ﬁxed–point ﬂoating point ﬂoating–point fraction bits frequency response gain Gaussian hardware IIR ﬁlter implementation impulse response input signal integer L2 norm least signiﬁcant bit limit cycles LTI system magnitude Matlab multiply node noise power noise sources notation operation order ﬁlters output peak poles and zeros polynomials prevent overﬂow probability of overﬂow produce Q format quantization error quantization noise quantization noise power quantization step radix point random process random variable range reduce represent representation result round-off noise rounding samples saturation scale value second-order sections shown signal power signal processing sinusoid speciﬁc transfer function truncation two’s complement unit circle unquantized unsigned white noise wordlength z-transform