Read an Excerpt
Preface
Digitalsignal processing (DSP) has become a core body of material in undergraduateelectrical engineering programs. Several threads branch from this core toenable related disciplines, such as communication systems, source coding,multimedia entertainment, radar, sonar, medical and laboratory instruments, andothers. Multirate signal processing isone of these major threads. Multirate signal processing is the body of materialthat deals with concepts, algorithms, and architectures that embed sample ratechanges at one or more sites in the signal flow path.
There are two reasons to include multirate signal processing in thesolution of a particular signal-processing task. The first is reduction in costof the implementation. The second is enhanced performance of theimplementation. We might also include a third, personal incentive, which is,that it is fun to apply clever concepts to solve problems. We can hardlycomplete a multirate DSP design without a smile and the accompanying thought,“Boy, this is neat!”
Traditional concepts developed in the DSP world are the same asthose developed in the analog-processing world. In both domains we learn anduse concepts such as convolution, Fourier transforms, transfer functions, polesand zeros, and others. When required to distinguish the two approaches we usethe qualifier “discrete” when discussing the DSP version of thesefundamental concepts. The reason the two approaches are so similar is that theyboth emphasize linear time invariantxe"linear time invariant" (LTIxe"LTI" \t "See linear timeinvariant") systems for which the toolsofanalysis and synthesis are well developed.
Multirate signal processing brings to the designer an importanttool not available to the traditional DSP designer, who to the first orderapplies DSP techniques to emulate analog systems. We note that the interfacebetween the two versions of the world, continuous and discrete, is the samplingprocess. In the traditional DSP perspective, the sample rate is selected tosatisfy the Nyquist criterion but is otherwise incidental to the problem. Inmultirate signal processing, selection and modification of the sample rate areprimary considerations and options in the signal processing chain. The optionto change the sample rate is the additional tool offered to the DSP designer.Discrete systems with embedded sample rate changes are characterized as lineartime varyingxe "linear time varying" (LTV)xe"LTV" \t "See linear timevarying" or as periodically time varying.xe "periodically time varying" (PTVxe "PTV" \t "See periodically time varying") Most of us have very littleexperience in the continuous world with LTV filters, thus their uniqueproperties come as a pleasant surprise as we learn how to design and use them.
The ability to change sample rate within the processing streampresents a remarkable list of processing tricks and performance enhancements. Aconsistent theme in this book is the presentation of perspectives that accessthese processing tricks. The first perspective we present is that a processingtask should always be performed at the lowest rate commensurate with the signalbandwidth. This is the Nyquist rate of the signal component of interest. Wenote that a common processing task is to reduce the bandwidth of a signal byfiltering and then reduce the sample rate to match the reduced bandwidth. Ourfirst processing trick interchanges the order of filtering and sample ratechange so that the processing proceeds at the reduced output sample rate ratherthan at the high input sample rate. The condition under which this interchangeis permitted is known as the noble identityxe"noble identity". Reducing the sample rate prior to reducing the bandwidth causesaliasing of the input spectrum. Multirate signal processing permits, and infact, supports this intentional aliasing, which can be unwrapped by subsequentprocessing. In fact, most of the tricks and enhancements associated withmultirate signal processing are related to spectral aliasing due to the samplerate change. It seems counterintuitive to use intentional aliasing as part ofthe signal processing scheme particularly when we have been told over and overnot to alias the signal in the data collection process. It also seems a bitsuspicious to claim that aliasing can be reversed. But in fact it can be whenthe aliasing occurs with a specified structure that we ensure in the multirateprocessing scheme. We can also use the change in sample rate to intentionallyalias a signal at one center frequency to another. This option includesaliasing a signal from an intermediate center frequency to baseband by reducingthe sample rate, as well as aliasing a signal from baseband to an intermediatecenter frequency by increasing the sample rate.
Purpose of Book
The purpose of this book is to present aclear and intuitive description of the unique capabilities of multirate signalprocessing. This is accomplished by presenting the core material along withnumerous examples supported by the liberal use of figures to illustrate thetime and frequency representations of the multirate processing options. We alsopresent a number of useful perspectives to facilitate development of insight ofmultirate systems. One such insight is that when describing a multirate system,since the sample rate is changing, we don’t use the sample rate as ourreference as is done in conventional DSP. Rather, it is useful to use thesignal symbol rate or signal bandwidth as our reference since that is thesingle parameter that remains fixed in the process. The book includes manypractical applications of multirate systems to help the reader see novel waysthey can be applied to solve real problems. Commentary of traditional designtechniques and alternate improved options are sprinkled throughout the text.Some of the material presented in the book, by necessity, must mimic similarexpositions found in other texts on multirate processing, while other segmentsof the material reflect my unique perspective and experience. There arespecific segments of material that are covered quite extensively here that areonly lightly covered in other texts. In particular, the chapter on recursiveall-pass filters is a wealth of material deserving greater exposure thantraditionally allocated to this topic.
Much of the material presented in this textbook has been used in mygraduate course, “Multirate Signal Processing.” Significantsegments of this material have found their way into my undergraduate course,“Modem Design,” as well as a series of short courses andpresentations dealing with synchronization techniques in modern communicationsystems. A light sprinkling of multirate filters can even be found in myundergraduate course in “Digital Signal Processing” as well as ourundergraduate course in “Real-Time Digital Signal Processing.” This book can be used in an undergraduate course in advanced DSP concepts or in agraduate course in multirate signal processing. Segments can also be used insupport of various courses in communication system design and modem design, andcan be used as a source of real-world applications in a DSP programming lab.
One of the pleasures of being a faculty member proficient in DSPskills as well as knowledgeable in modern communication systems is the abilityto roam the halls of knowledge as well as the commercial centers of excellencethat feed the economic engines of our society. I have had the good fortune ofparticipating in the development of many systems that require the capabilitiesof high-performance, cost-effective DSP solutions. These systems includelaboratory instruments, cable modems, satellite modems, sonar systems, radarsystems, wireless systems, and consumer entertainment products. With a foot ineach camp, one in academia and one in commercial, I am exposed to a rich and variedset of questions of interest from residents of the two areas. Questions posedby commercial folks addressing focused problems are very different from thoseposed by those in academics.
Some of my most creative work has been spawned by questions posed,and challenges offered, by perceptive folks in the industrial arena. Theacademic environment provides me access to promising and talented students withwhom I can share the pleasure of learning and understanding an establishedknowledge base in multirate digital processing while developing and expandingthat same base. This text reflects much of that knowledge base tempered by theinsight gathered from problem-solving tasks in the commercial sector andnurtured by scholarly interaction with curious, motivated students.
Organizationof Book
This book is divided into 13 chapters.Chapter 1 is an introduction to multirate signal processing. The 1-to-4up-sampler of the common consumer CD player is shown as an example of aubiquitous application. Chapter 2 describes the process of sampling andresampling in time and frequency domains. Chapter 3 presents the relationshipbetween the specifications of a FIR filter and the number of taps, or thelength, of the filter. We also compare the window design and equal-ripple orRemez design techniques. The effects of in-band ripple and of constant levelstop band side-lobes are examined and various modifications to the designprocess are introduced to control stop band side-lobe levels. Chapter 4presents special filters such as square-root Nyquist filters and half-bandfilters. Discussions on standard and improved design techniques appropriate forthese specific filters are also presented.
Chapter 5 presents examples of systems that use multirate filtersand illustrates applications and demonstrates the wide range of applications.Chapter 6 presents resampling low pass and band-pass FIR filters for which thenoble identity has been applied to interchange the operations of resampling andfiltering. Up-sampling, down-sampling, and cascade up-down sampling filters areexamined. Chapter 7 describes polyphase interpolators and filters that performarbitrary sample rate change. We also examine Farrow filters as well as filtersthat interpolate while performing alias-based translation. Chapter 8 coversquadrature mirror filters and dyadic half-band FIR filters. Chapter 9 coversM-path modulators and demodulator channel banks. Also discussed aresimultaneous interpolation and channel bank formation.
Chapter 10 covers recursive all pass filters implemented asnonuniform phase and equiripple approximations to linear phase filters. Anumber of structures, including half-band, M-path filter banks, resamplingstructures and arbitrary bandwidth non-resampling structures, are presented andillustrated. Chapter 11 presents the CIC filter and its resampling version, theHogenauer filter. Chapter 12 describes cascades of low-order zero-packedup-sampled filters that exhibit periodic spectra with narrow transition band.Chapter 13 presents areas of applications in which multirate filters havesignificant presence.
At the end of each chapter are a number of problems designed tohighlight key concepts presented in the chapter. These problems also serve totest the reader’s knowledge and understanding of the material. Followingeach problem set is a list of references to guide the reader to related areasfor further study.
Acknowledgements
Weare able to see further than those who came before us because we stand on theirshoulders. In addition, each day presents a fresh horizon because of the workwe have performed as well as that performed by those around us. Interactionwith our peers, our students, and our patrons enriches us by moving boundariesand lifting veils through the exchange of ideas. I cherish a group of peoplewith whom I have had the great pleasure of sharing work and developingknowledge and with whom I have established long-standing friendships. I wouldlike to cite some of them with the understanding that their ordering is a randomprocess involving fleeting pleasant remembrances.
Tony Constantinides of Imperial College in London has had a specialinfluence on my work. I met Tony at a conference in Brisbane where we started alifelong friendship and working relationship. It was through Tony that I gainedan appreciation of the recursive polyphase half-band filter. I have yet to putdown this valuable filter structure. It was Markku Renfors of Tempre Universityin Finland who first touched me through a classic paper on M-path recursivepolyphase filters. That paper brought forth a torrent of ideas, papers, studentprojects, and commercial products. Many years later, meeting Markku at aconference in Santorini I quickly warmed to him as a fellow traveler and Icontinue to enjoy our too infrequent academic and personal encounters. In asimilar vein, Ron Crochiere through the classic book that he and Larry Rabinerauthored, ‘Multirate Digital Signal Processing’ helped form many ofthe ideas I nurtured into my concepts and perspectives on multirate filters. Ithank Ron for reviewing the manuscript of this book and I’m relieved thatit passed muster. I am always pleased to see and spend time with Ron. From theother side of the playing field I take great pleasure in my nearly daily interactionwith Bernie Sklar formerly with Aerospace Corporation and now withCommunications Engineering Services. I hone my applied DSP skills on thecommunication system questions and problems that Bernie brings to me.Interacting with Bernie is like dancing with a skilled partner with bothparticipants exhilarated and charmed by the encounter. Another fellow dancerwho sways to the music of applied multirate DSP is Chis Dick of Xilinx. Chisand I met at a conference in Adelaide while he was at LaTrobe University. Ourcommon interest and passion have been the core of a special friendship and ourwives tell us that we are boring company when we climb into our common DSPshell to do our DSP magic. Michael Rice of BYU is another special person withwhom I have been sharpening my applied DSP skills.
The list of people with whom I initially interacted throughprofessional contacts that have warmed to friendships mingled with professionalrespect and admiration hold another special place for me. These include DonSteinbrecher, a very creative individual with a piece of his mind firmlyplanted in our future. Itzhak Gurantz, formerly of Comstream and Rockwell andnow with Entropic, is a very smart person who many times has posed fascinatingquestions that guided me to places I would never have thought to tread. RalphHudson with whom I shared an office at Hughes Aircraft while I was on a leaveof absence from SDSU, and David Lynch, who enticed me to visit Hughes, areclever individuals with enormous vision and ability to see beyond the view ofmost folks. I have not seen either of these fellows for many years but theyflooded me with great problems from which I was able to develop insight I couldnot have had without them. Others with whom I have worked and from whom I have learnedare Pranish Sinha, Ragavan Sreen, Anton Monk, and Ron Porat.
The third group of people I feel compelled to acknowledge are thosewith whom I initially interacted through a student-mentor relationship and havecontinued to interact as friend and colleague. I will limit this list tocurrent and former students who have worked with me in the area of multiratesignal processing. Maximilien d’Oreye de Lantramange and Michael Orchardare the best of the best. Dragan Vuletic and Wade Lowdermilk with whom Iinteract daily on applied problems are rising stars following other exceptionalformer students such as Bob Bernardi, and Christian Bettwieser and the linecontinues with Benjamin Egg, a new graduate student at its head. In thirty-fiveyears of teaching I have touched the minds of many students whom I hope haveenjoyed the encounters as much as I have. I know that many names are missingfrom my acknowledgment list and if you feel offended by this, don’t be.Please attribute it to a state of absentmindedness rather than to one ofintentional omission.
fredric j harris
San Diego, California
