*
Introduction to
Signal Processing
*

*
*

The book's copyrights were transferred from Prentice Hall to Sophocles J. Orfanidis in 2009. A new version of the book, that includes corrections of all the typos, is now freely available in PDF format, and in a 2-up form. A solutions manual is available. A printed version is also available as a size-6x9 paperback.

Copyright (c) 2010 by Sophocles J. Orfanidis, All Rights Reserved.

Links to the book's web page, http://www.ece. rutgers.edu/~orfanidi/intro2sp/ , may be placed on any web site. Any part of this book may be downloaded and printed for personal or educational use only, as long as the printed or photocopied pages are not altered in any way from the original PDF file posted on the book's web page.

No part of this book may be reproduced, altered in any way, or transmitted in any form for commercial, profit, sale, or marketing purposes.

- Preface
- Table of Contents
- Highlights
- C and MATLAB Functions
- Errata and Feedback
- Publication Data
- Typesetting Notes
- About the Cover
- About the Author

Digital signal processing is everywhere. Today's college students hear "DSP" all the time in their everyday life - from their CD players, to their electronic music synthesizers, to the sound cards in their PCs. They hear all about "DSP chips", "oversampling digital filters", "1-bit A/D and D/A converters", "wavetable sound synthesis", "audio effects processors", "all-digital audio studios". By the time they reach their junior year, they are already very eager to learn more about DSP.

The book teaches by example and takes a hands-on
practical approach that emphasizes the * algorithmic, computational, and
programming * aspects of DSP. It contains a large number of worked examples,
computer simulations and applications, and
includes several C and MATLAB functions
for implementing various DSP operations. The practical slant of the book
makes the concepts more concrete.

A solutions manual, which also contains the results of the computer experiments,
is available from the publisher. The C and MATLAB functions may be obtained
via anonymous FTP from the Internet site `ece.rutgers.edu`
in the directory `/pub/sjo` or by pointing a Web browser to
the book's WWW home page on
`ftp://ece.rutgers.edu/pub/sjo/intro2sp.html`.

Chapter 2 discusses the *quantization process* and some practical
implementations of A/D and D/A converters, such as the conversion algorithm
for bipolar two's complement successive approximation converters. The standard
model of quantization noise is presented, as well as the techniques of
*oversampling, noise shaping, and dithering*. The tradeoff between
oversampling ratio and savings in bits is derived. This material is continued
in Section 12.7 where the implementation and operation of
delta-sigma noise shaping quantizers is considered.

Chapter 3 serves as a review of basic *discrete-time systems*
concepts, such as
linearity, time-invariance, impulse response, convolution, FIR and IIR
filters, causality, and stability. It can be covered quickly as most
of this material is assumed known from a prerequisite linear systems course.

Chapter 4 focuses on FIR filters and its purpose is to
introduce two basic signal processing methods:
*block-by-block* processing and *sample-by-sample* processing.
In the block processing part, we discuss convolution
and several ways of thinking about it,
transient and steady-state behavior, and real-time processing
on a block-by-block basis using the overlap-add method and its software
implementation. This is further discussed in Section 9.9 using the FFT.

In the sample processing part, we introduce the basic building blocks of
filters: adders, multipliers, and delays.
We discuss *block diagrams* for FIR filters
and their time-domain operation on a sample by sample basis.
We put a lot of emphasis on the concept of *sample processing
algorithm*, which is the repetitive series of computations that must be carried
out on each input sample.

We discuss the concept of *circular buffers* and their use in implementing
delays and FIR filters. We present a systematic treatment of the subject
and carry it on to the remainder of the book.
The use of circular delay-line
buffers is old, dating back at least 25 years with its application
to computer music. However, it has not been treated
systematically in DSP texts. It has acquired a new relevance because all
modern DSP chips use it to minimize the number of hardware instructions.

Chapter 5 covers the basics of z-transforms. We emphasize the z-domain view of causality, stability, and frequency spectrum. Much of this material may be known from an earlier linear system course.

Chapter 6 shows the equivalence of various ways of characterizing a linear filter and illustrates their relevance by example. It discusses also topics such as, sinusoidal and steady-state responses, time constants of filters, simple pole/zero designs of first and second order filters as well as comb and notch filters. The issues of inverse filtering and causality are also considered.

Chapter 7 develops the standard *filter realizations*
of canonical, direct, and
cascade forms, and their implementation with circular buffers.
Quantization effects are briefly discussed.

Chapter 8 presents three DSP application areas. The first is on digital
*waveform generation*, with particular emphasis on wavetable generators.
The second is on *digital audio effects*, such as flanging, chorusing,
reverberation, multitap delays, and dynamics processors, such as compressors and
expanders.
These two areas were chosen because of their appeal
to undergraduates and because they provide concrete illustrations of the use
of delays, circular buffers, and filtering concepts in the context of audio
signal processing.

The third area is on *noise reduction/signal enhancement*, which
is one of the most important applications of DSP and is of interest to
practicing engineers and scientists who remove noise from data on a
routine basis. Here, we develop the basic principles for designing noise
reduction and signal enhancement filters both in the frequency and time
domains.
We discuss the design and circular buffer implementation
of *notch and comb* filters for removing periodic interference, enhancing
periodic signals, signal averaging, and for separating the luminance and
chrominance components in digital color TV systems.
We also discuss *Savitzky-Golay* filters for data smoothing
and differentiation.

Chapter 9 covers *DFT/FFT algorithms*.
The first part emphasizes the issues of spectral analysis,
frequency resolution, windowing, and leakage.
The second part discusses the
computational aspects of the DFT and some of its pitfalls, the difference
between physical and computational frequency resolution,
the FFT, and fast convolution.

Chapter 10 covers *FIR filter design* using the window method, with particular
emphasis on the Kaiser window. We also discuss the use of the Kaiser window
in spectral analysis.

Chapter 11 discusses *IIR filter design* using the
bilinear transformation
based on Butterworth and Chebyshev filters.
By way of introducing the bilinear
transformation, we show how to design practical 2nd order
digital audio *parametric equalizer* filters having
prescribed widths, center frequencies, and gains. We also discuss
the design of periodic notch and comb filters with prescribed widths.

In these two filter design chapters, we have chosen to present only a few design methods that are simple enough for our intended level of presentation and effective enough to be of practical use.

Chapter 12 discusses *interpolation, decimation, oversampling DSP systems,
sample rate converters, and delta-sigma quantizers*. We discuss the use
of oversampling for alleviating the need for high quality analog prefilters and
postfilters. We present several practical
design examples of interpolation filters, including *polyphase and
multistage designs*. We consider the design of sample rate converters and
study the operation of oversampled
delta-sigma quantizers by simulation. This material is too advanced for
juniors but not for seniors. All undergraduates, however, have a strong interest in
it because of its use in digital audio systems such as CD and DAT
players.

The Appendix has four parts: (a) a review section on
*random signals*; (b) a discussion of
random number generators, including uniform, gaussian, low frequency,
and 1/f noise generators used in the simulations;
(c) C functions for performing the complex arithmetic in the
DFT routines; (d) listings of MATLAB functions.

Finally, I would like to thank my wife Monica and son John for their love, patience, encouragement, and support.

* Sophocles J. Orfanidis *

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**Sampling and Reconstruction**- 1.1 Introduction
- 1.2 Review of Analog Signals
- 1.3 Sampling Theorem
- 1.3.1 Sampling Theorem
- 1.3.2 Antialiasing Prefilters
- 1.3.3 Hardware Limits
- 1.4 Sampling of Sinusoids
- 1.4.1 Analog Reconstruction and Aliasing
- 1.4.2 Rotational Motion
- 1.4.3 DSP Frequency Units
- 1.5 Spectra of Sampled Signals*
- 1.5.1 Discrete-Time Fourier Transform
- 1.5.2 Spectrum Replication
- 1.5.3 Practical Antialiasing Prefilters
- 1.6 Analog Reconstructors*
- 1.6.1 Ideal Reconstructors
- 1.6.2 Staircase Reconstructors
- 1.6.3 Anti-Image Postfilters
- 1.7 Basic Components of DSP Systems
- 1.8 Problems

**Quantization**- 2.1 Quantization Process
- 2.2 Oversampling and Noise Shaping*
- 2.3 D/A Converters
- 2.4 A/D Converters
- 2.5 Analog and Digital Dither*
- 2.6 Problems

**Discrete-Time Systems**- 3.1 Input/Output Rules
- 3.2 Linearity and Time Invariance
- 3.3 Impulse Response
- 3.4 FIR and IIR Filters
- 3.5 Causality and Stability
- 3.6 Problems

**FIR Filtering and Convolution**- 4.1 Block Processing Methods
- 4.1.1 Convolution
- 4.1.2 Direct Form
- 4.1.3 Convolution Table
- 4.1.4 LTI Form
- 4.1.5 Matrix Form
- 4.1.6 Flip-and-Slide Form
- 4.1.7 Transient and Steady-State Behavior
- 4.1.8 Convolution of Infinite Sequences
- 4.1.9 Programming Considerations
- 4.1.10 Overlap-Add Block Convolution Method
- 4.2 Sample Processing Methods
- 4.2.1 Pure Delays
- 4.2.2 FIR Filtering in Direct Form
- 4.2.3 Programming Considerations
- 4.2.4 Hardware Realizations and Circular Buffers
- 4.3 Problems

**z-Transforms**- 5.1 Basic Properties
- 5.2 Region of Convergence
- 5.3 Causality and Stability
- 5.4 Frequency Spectrum
- 5.5 Inverse z-Transforms
- 5.6 Problems

**Transfer Functions**- 6.1 Equivalent Descriptions of Digital Filters
- 6.2 Transfer Functions
- 6.3 Sinusoidal Response
- 6.3.1 Steady-State Response
- 6.3.2 Transient Response
- 6.4 Pole/Zero Designs
- 6.4.1 First-Order Filters
- 6.4.2 Parametric Resonators and Equalizers
- 6.4.3 Notch and Comb Filters
- 6.5 Deconvolution, Inverse Filters, and Stability
- 6.6 Problems

**Digital Filter Realizations**- 7.1 Direct Form
- 7.2 Canonical Form
- 7.3 Cascade Form
- 7.4 Cascade to Canonical
- 7.5 Hardware Realizations and Circular Buffers
- 7.6 Quantization Effects in Digital Filters
- 7.7 Problems

**Signal Processing Applications**- 8.1 Digital Waveform Generators
- 8.1.1 Sinusoidal Generators
- 8.1.2 Periodic Waveform Generators
- 8.1.3 Wavetable Generators
- 8.2 Digital Audio Effects
- 8.2.1 Delays, Echoes, and Comb Filters
- 8.2.2 Flanging, Chorusing, and Phasing
- 8.2.3 Digital Reverberation
- 8.2.4 Multitap Delays
- 8.2.5 Compressors, Limiters, Expanders, and Gates
- 8.3 Noise Reduction and Signal Enhancement
- 8.3.1 Noise Reduction Filters
- 8.3.2 Notch and Comb Filters
- 8.3.4 Line and Frame Combs for Digital TV
- 8.3.5 Signal Averaging
- 8.3.6 Savitzky-Golay Smoothing Filters*
- 8.4 Problems

**DFT/FFT Algorithms**- 9.1 Frequency Resolution and Windowing
- 9.2 DTFT Computation
- 9.2.1 DTFT at a Single Frequency
- 9.2.2 DTFT over a Frequency Range
- 9.2.3 DFT
- 9.2.4 Zero Padding
- 9.3 Physical versus Computational Resolution
- 9.4 Matrix Form of DFT
- 9.5 Modulo-N Reduction
- 9.6 Inverse DFT
- 9.7 Sampling of Periodic Signals and the DFT
- 9.8 FFT
- 9.9 Fast Convolution
- 9.9.1 Circular Convolution
- 9.9.2 Overlap-Add and Overlap-Save Methods
- 9.10 Problems

**FIR Digital Filter Design**- 10.1 Window Method
- 10.1.1 Ideal Filters
- 10.1.2 Rectangular Window
- 10.1.3 Hamming Window
- 10.2 Kaiser Window
- 10.2.1 Kaiser Window for Filter Design
- 10.2.2 Kaiser Window for Spectral Analysis
- 10.3 Frequency Sampling Method
- 10.4 Other FIR Design Methods
- 10.5 Problems

**IIR Digital Filter Design**- 11.1 Bilinear Transformation
- 11.2 First Order Lowpass and Highpass Filters
- 11.3 Second Order Peaking and Notching Filters
- 11.4 Parametric Equalizer Filters
- 11.5 Comb Filters
- 11.6 Higher Order Filters
- 11.6.1 Analog Lowpass Butterworth Filters
- 11.6.2 Digital Lowpass Filters
- 11.6.3 Digital Highpass Filters
- 11.6.4 Digital Bandpass Filters
- 11.6.5 Digital Bandstop Filters
- 11.6.6 Chebyshev Filter Design*
- 11.7 Problems

**Interpolation, Decimation, and Oversampling**- 12.1 Interpolation and Oversampling
- 12.2 Interpolation Filter Design*
- 12.2.1 Direct Form
- 12.2.2 Polyphase Form
- 12.2.3 Frequency-Domain Characteristics
- 12.2.4 Kaiser Window Designs
- 12.2.5 Multistage Designs
- 12.3 Design Examples*
- 12.3.1 4-fold Interpolators
- 12.3.2 Multistage 4-fold Interpolators
- 12.3.3 DAC Equalization
- 12.3.4 Postfilter Design and Equalization
- 12.3.5 Multistage Equalization
- 12.4 Decimation and Oversampling*
- 12.5 Sampling Rate Converters*
- 12.6 Noise Shaping Quantizers*
- 12.8 Problems

**Appendix**- A Random Signals*
- A.1 Autocorrelation Functions and Power Spectra
- A.2 Filtering of Random Signals
- B Random Number Generators
- B.1 Uniform and Gaussian Generators
- B.2 Low-Frequency Noise Generators*
- B.3 1/f Noise Generators*
- B.4 Problems
- C Complex Arithmetic in C
- D MATLAB Functions

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- Sampling and reconstruction. Practical antialiasing prefilters and anti-image postfilters.
- Quantization. A/D & D/A converters. Noise shaping, oversampling DSP systems, dither.
- Block processing and sample processing methods. Convolution.
- Circular buffer implementations of delays, FIR, and IIR filters.
- Discrete-time systems. Z-transforms. Transfer functions. Digital filter realizations.
- Wavetable generators. Digital audio effects and dynamics processors.
- Noise reduction and signal enhancement principles.
- Notch filters for canceling periodic interference.
- Comb filters for periodic signal enhancement and digital TV.
- Signal averaging. Savitzky-Golay smoothing filters.
- DFT/FFT. Spectral analysis. Frequency resolution and windowing. Fast convolution.
- FIR filter design using the Kaiser window.
- IIR filter design using the bilinear transformation. Butterworth and Chebyshev designs.
- Parametric equalizer filter design for digital audio. Parametric comb filters.
- Interpolation, decimation, and oversampling. Multistage and polyphase designs.
- Sample rate converters. Noise shaping delta-sigma quantizers.
- Random signals. Random noise generators: uniform, gaussian, white, low-frequency, 1/f.

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To improve the readability of the C functions, we use the old K&R way of declaring function arguments. The functions may be easily edited to conform with ANSI C. For example, the K&R declaration:

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- blockcon.c - block convolution
- can.c - canonical realization
- can2.c - canonical realization
- can3.c - canonical realization
- cas2can.c - cascade to canonical
- cas.c - cascade realization
- ccan.c - circular-buffer canonical realization
- ccan2.c - circular-buffer canonical realization
- ccas.c - circular-buffer cascade realization
- ccas2.c - circular-buffer cascade realization
- cdelay.c - circular delay line
- cdelay2.c - circular delay line
- cfir.c - circular-buffer FIR filter
- cfir1.c - circular-buffer FIR filter
- cfir2.c - circular-buffer FIR filter
- conv.c - convolution
- csos.c - circular-buffer second-order section
- csos2.c - circular-buffer second-order section
- delay.c - delay line
- dir.c - direct form realization
- dir2.c - direct form realization
- fir.c - FIR filter in direct form
- fir2.c - FIR filter in direct form
- fir3.c - FIR filter in direct form
- sos.c - second-order section
- tap.c - circular delay-line tap outputs
- tap2.c - circular delay-line tap outputs
- wrap.c - circular-buffer pointer wrapping
- wrap2.c - circular-buffer index wrapping

- adc.c - A/D converter
- dac.c - D/A converter

- allpass.c - allpass reverberator
- lowpass.c - lowpass reverberator
- plain.c - plain reverberator
- tapi.c - interpolated circular delay-line tap outputs
- tapi2.c - interpolated circular delay-line tap outputs

- gdelay2.c - generalized circular delay
- sine.c - sinusoidal wavetable
- square.c - square wavetable
- trapez.c - trapezoidal wavetable
- wavgen.c - wavetable generator (truncation)
- wavgenr.c - wavetable generator (rounding)
- wavgeni.c - wavetable generator (interpolation)

- bitrev.c - bit reversed index
- complex.c - complex arithmetic in C
- cmplx.h - header file for complex.c
- dft.c - DFT
- dftmerge.c - DFT merging
- dtft.c - DTFT at single frequency
- dtftr.c - DTFT over frequency range
- fft.c - FFT
- ifft.c - inverse FFT
- modwrap.c - modulo-N reduction
- shuffle.c - shuffling in FFT
- swap.c - swapping in FFT

- gran.c - gaussian random number generator
- ran.c - uniform random number generator
- ran1f.c - 1/f noise generator
- ranh.c - low-frequency hold generator
- ranl.c - linearly interpolated generator

- cheby.c - Chebyshev polynomial evaluator
- corr.c - correlation
- delta.c - unit impulse
- dot.c - dot product
- I0.c - modified Bessel function
- u.c - unit step

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- cas.m - cascade realization
- cas2can.m - cascade to canonical
- cdelay2.m - delay (circular buffer)
- cfir2.m - FIR filter in direct form (circular buffer)
- delay.m - delay (linear buffer)
- fir.m - FIR filter in direct form (linear buffer)
- sos.m - second order section
- wrap2.m - circular delay-line wrapping

- dtft.m - DTFT computation

- dbp.m - ideal bandpass filter impulse response
- ddiff.m - ideal differentiator impulse response
- dhilb.m - ideal Hilbert transformer impulse response
- dlh.m - ideal lowpass/highpass filter impulse response
- I0.m - Modified Bessel function
- kbp.m - Kaiser bandpass design
- kdiff.m - Kaiser differentiator design
- khilb.m - Kaiser Hilbert transformer design
- klh.m - Kaiser lowpass/highpass design
- kparm2.m - Kaiser window parameters for spectral analysis
- kparm.m - Kaiser window parameters for filter design
- kwind.m - Kaiser window

- bpcheb2.m - bandpass Chebyshev type 2 design
- bpsbutt.m - bandpass/bandstop Butterworth design
- bscheb2.m - bandstop Chebyshev type 2 design
- lhbutt.m - lowpass/highpass Butterworth design
- lhcheb1.m - lowpass/highpass Chebyshev type 1 design
- lhcheb2.m - lowpass/highpass Chebyshev type 2 design

- combeq.m - parametric comb/notch equalizer design
- parmeq.m - parametric equalizer design
- peq.m - J. Audio Eng. Soc., vol.45, 444 (1997).

- sg.m - Savitzky-Golay filter design
- sgfilt.m - Savitzky-Golay filtering
- sigav.m - signal averaging
- ecg.m - simulated ECG waveform generator

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College Division

Prentice Hall,
Upper Saddle River, NJ 07458

ISBN: 0-13-209172-0

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The dvi previewers were Y&Y's dviwindo and emtex's dvidrv. Several LaTeX style files from the CTAN collection were used: equation.sty, jeep.sty, aip.sty, psfig.sty, alltt.sty, amslatex.sty. The table.tex macros from PCTeX and the ps2pk conversion utility were also used.

The data graphs were plotted by the Scientific Endeavors GraphiC package, exported to EPS postscript format, and inserted into the dvi file by psfig.sty. The illustrations were prepared by the author using CorelDraw and exported to EPS; they were also inserted with psfig.

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e-mail:
`orfanidi@ece.rutgers.edu`, or,
`orfanidi@rci.rutgers.edu`

Sophocles J. Orfanidis

Department of Electrical and Computer Engineering

Rutgers University

P. O. Box 909

Piscataway, NJ 08855-0909

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