Terminology in Digital Signal Processin
versus discrete time are often made. Although they are often used interchangeably, different meanings should be attributed to the two sets of terms. 2) The term analog generallydescribesawaveform t h a t iscontinuous in time (or any other appropriate independent variable) and that belongs to a class that LAWRENCE R. RABINER, JAMES W. COOLEY, can take on a continuous range of amplitude values. HOWARD D. HELMS, LELAND B. JACKSON, JAMES F. KAISER,CHARLES M. RADER, Examples of analog wuveJorms or anulog signals are thosc RONALDW.SCHAFER, KENNETH STEIGLITZ, derived from acoustic sources. Such signals are repreand CLIFFORD J. WEINSTEIN sentedrnathematicallyasfunctions of acontinuous variable. The functions sin (ut) and the step function au-l(t) are examplesof common mathematical functions that coulddescribe“analogsignals.” T h e use of the Absfract-Thecommittee on DigitalSignalProcessing of the term “anaIog” in this context appears to stem fronl the IEEEGroup on Audio andElectroacousticshasundertakenthe field of analog computation, where a current or voltage project of recommending terminologg foruse in papers and texts on digital signal processing. The reasons for this project are twofold. waveform serves as a physical analog of some variable First, the meanings of many terms that are commonly used differ in a differential equation. from one author to another. Second, there are many terms that 3 ) The term continuous t i m e implies that only the one would like to have defined forwhich no standard term currently independent variable necessarily takes on a continuous exists. It is the purpose of this paper to propose terminology whichwe range of values. I n theory the amplitude may, butneed feel is self-consistent, and which is in reasonably good agreement with current practices. An alphabetic index of terms is included at not, be restricted to a finite or countable infinite set of the end of the paper. values (i.e., the amplitude may be quantized). Therefore, analog waveforms are continuous-time waveforms with continuous amplitude. In practice, however, “continuous-time waveforms” and “analog waveforms” are Introduction equivalent. Since most signal processing problems have As an aid to classifying the different types of terms nothing to do with analogs as such, the us11 of the term t o be defined, we have placed each term in one of the analogwaveform is often ambiguous at the least and following groups: may in fact be misleading. T ~ L Ithe S , term continuous1) Introductory Terms-GeneralDefinitions time waveform is preferable. 2) Discrete Systems-Block DiagramTerminology4) Discrete time implies that time (the independent 3) Relations Between Discrete and Continuous variable) is quantized. T h a t is, discrete-time signals are Signals defined only for discrete values of the independent vari4) Theory and Design of Digital Filters able.Suchsignalsarerepresentedrnathcmaticallyas 5) FiniteWordLength Effects-A/D, D/AConsequences of numbers. Those discrete-time signals that version take on a continuum of values are referredt o as sum$led6) Discrete Fourier Transforms and the FFT data signals. 7) Discrete Convolution and Spectrum Analysis. 5 ) The term digital implies that lmth timc and mlpli-In the above mentioned sectionsof this paper we will be tude are quantized. Thus d aigital system i s one i n w h i c h discussing terminology related to the processing of one- signals. are represented a s sequences o f n u m h - r s which dimensionalsignals.Forconvenience, we will assume take on only a finite set of values. Thlls onc uses d i g i t d that this dimension i s time-although thedefinitions whendiscussing actual phJ-sical realizations (as hardapply equally well to any single dimension. ware or programs) o f discrete-time signal processing systems, whereas the term discrete t i m e is a h;tt-er modifier when consideringmathematicalabstractions of such 1. introductory Terms-General Definitions systems in which the effects of amplitud- quantization 1) In discussingwaveformprocessingproblems, the are ignored, A digital signal or digital ~ ( ~ v e j o sisma scdistinctions analog versus digital and continuous time quence produced, for example,b y digital circuitry orby an analog-to-digital converter which is sampling a COIF tinuous-timewaveform. In digitalsignalprocessing Manuscript received August 1, 1972. L. R. Rabiner, J. F. Kaiser, and R. W. Schafer are with Bell these terms are commonly shortened to signal 01-waveTelephone Laboratories, Inc., Murray Hill, N . J . 07974. J . W. Cooleyiswith the I B M T. J. \&’atson Research Center, f o r m . Sometimes the term signal is restricted to being a Vorktown Heights, 73. Y. 10598. desirable componentof a sequence instead of being used H. D. Helms is with Bell Telephone Laboratories, Inc., Whipinterchangeably with waveform. Noise is either defined pany, N . J . 07981. SystemsCorporation,West I,. B. Jackson is withRockland as a) an undesirable component of a sequence, or b) a Nyack, N.1’. 10994. C. M. Rader and C. 1. Weinstein are with M.I.T. Lincoln Lab- sequence of random variables. oratory, Lexington, Mas:. 02173. (Operated with support from the 6 ) (Digital)simulatiofz i s theexact or approxinlate U. S. Air Force.) or continK. Steiglitz is with Princeton University, Princeton,pi. J . 08540. representation o f a givensystem(discrete
IaBIirTm
et al.: DIGITAL
323
SIGKAL PROCESSING TERMIXOLOGY
uous) called source the system, by a (digital) system y ( n ) = x ( n - m), m >0 (2) called the object system. m 7) Next-state simulation is a method of digital simu(3) lation whereby the values of the digital system signals are represented by nodes ina block diagram representaIn ( 3 ) , X ( z ) appears multiplied by z-%. This result for tion. Usually, there isa close correlation between blocks m'= 1 accounts for the fact that 2-l is often termed the in the object system and elementsof the source system. unit delay operator, since a delay of the sequence by one Themethodentailsorderingthecalculations in the sample is equivalent to multiplication of the z transform digital system so that all the inputs to each block a t a by z - l . ( Similarly, z is often called the unit advance given sample time are computed before the output is operator.) computed. 2) In many cases,sequencesaredefinedoverboth 8) A real-time process is one for which, on the average, positive and negative valuesof In such cases, a somethe computing associated with each sampling interval what more general pointof view is called for. In general, can be completed in a time less than or equal to the the z transform is written as sampling interval. ,4 program running in 100 times real m timerequires 100 timesaslongto process the Sam(: X ( Z > = x(n)z-a. (4) number of samples; i.e., it is 100 times too slow for real n=--m time operation. A program ten times as fast as it needs t o be could be said to run in 1/10 real time. Obviously, I t should be noted that a common usage is t o call (1) the extra speed can onlybe used if other computing can simply thez transform, and(4)the two-sided z transform. be done in the interstices, or if the complete sequences Since (4) is most general, it would seem preferable to t o be processed have been stored beforehand. refer t o (4)as the z transform, and the special case, ( l ) , 9) Throughp.ut rate is the total rate at which digital as the one-sided z transform. information is processed byadiscrete-timesystem, 3 ) I t is possible to think of the z transform as simply measured in bits per second or samplesper second. In a a formal series whose properties can be tabulated, and multiplexed system, where several signals are processed, which never need be summed. However, it is generally we may refer to thethroughput rate per signal, measured preferable to realize that if certain convergence condiin bits per second per signal or samples per second per tions are met, both (1) and (4)are Laurent series in the signal. Thus, a multiplexed system which processes 10 complex variable z. As such, all the properties of the signals, each at 1000 bits/s, has a throughput rate of Laurent series apply. For example, if the series in (1) 10 000 bits/s,and a throughputrate/signal of 1000 converges, it must convergein a region zI > R+. If the bits/s/signal. series of (4) converges, it must converge in an annular 10) A multivatesystem is a discrete-timesystem in region R+ < zI (58) sequences with zeros so that they have the same length Wl=O N , which is at least as great as one less than the sum of
I
c
+ c
+
c
+
336
IEEE TRAKSACTIONS ON AUDIO AND ELECTROACOUSTICS, DECEMBEK
1972
the lengthsof the two sequences, cyclic convolution can 9) Windows are useful in determining the coefficients be made to yield the same result as ordinary convolu- of a finite impulse response digitalfilter. In thiscase, the originalsequenceconsists of samples of the impulse tion. 5) This use of the fast Fourier transform to compute response corresponding t o a transfer function which is discrete convolutions is sometimes calledfast convolution approximated by the Fourier transformof the sequence or FFT convolution. Thistechniquecan be easily of pairwise products; the product sequence used is as the adapted for computing acyclic (i.e., nonperiodic) correcoefficients of the finite impulse response digital filter. lation functions. In this form, it is called fast correlation 10) Windows are used also in the indirect method of or FFT correlation. computing a power spectrum. In this method, the se6) If one of the two sequences is much shorter than quence consistingof samples of the autocorrelation functheother,the longersequence canbe sectioned into tion is multipliedbythe window. T h e D F T of the pieceswhose discreteconvolutionscanbecomputed resulting sequenceis an estimateof the power spectrum. separately.Thesediscreteconvolutionscanbecom11) Windowscanbe usedalso in estimatingcross bined t o produce the discrete convolution of the whole spectra where the estimates are obtained by multiplying sequence. (Sectioning is used because i t reduces the re- theproducts of theDFT’s of twoormoredistinct quired amounts of computation and memory.) One of sequences. these sectioning techniques (overlap-save or select-save) 12) The determination of afiniteimpulseresponse involves computing the inverse DFT of the product of described by an ordinary convolution is called deconthe DFT’s of a) N samples of the input sequence, and volution or FIR identification. b)theshorter sequenceaugmentedwith a sufficient number of zeros so that its sequence contains N samples. Acknowledgment (Usually N is at least twice as large as the lengthof the The authorswould like to acknowledge the comments shorter sequence.) Some of the members of the sequence resultingfromtheinverse DFT are members of the and criticismsof this paper provided byG. D. Bergland, sequence formed by the desired acyclic (i.e., nonperi- C. H. Coker, D. L. Favin, B. Gold, 0. Hermann, S. odic) convolution. (This numberof members equals one Lerman, A. V. Oppenheim, H. 0. Pollak,and H. F. Silverman. more than the number of zeros originally augmenting the shorter sequence.) T h e longer original sequence is advancedbythisnumber of members.Iteratingthis processgives the whole convolution. The overlap-add Alphabetic Index of Terms techniqueforsectioning uses a similar technique but A additionally requires adding shifted sequences of partial A/D Conversion Noise: V-2 convolutions. Aliasing: 111-4 7) A window is afinitesequence,eachelement of Amplitude Response: 11-9 1-2 which multiplies a corresponding element of the main Analog: Analog Signals: 1-2 sequence. (This is called windowing.) The sequence of Analog-to-Digital Converter: V - Z Analog Waveforms: 1-2 products formed by this element-by-element multiplicaAnalytical Design Technique: IV-32 tion is often more useful than the main sequence. T h e Fouriertransform of a typicalwindow(sometimes B called the spectral window) consists of a mainlobe, which Base: V-4 R:VI-13 usually contains a large percentageof the energy in the Base Bilinear Transformation: IV-31 window, and sidelobes which containtheremaining Binary Representation: V-4 Bit Reversal: VI-12 energy in the window. Block-Diagram: 11-11 8) Windows can be used in estimating power spectra. Block Floating-point h:umbers: V-5 VI-15 In the direct method, the power spectrum is estimated by Butterfly: Rutterworth Filter: TV-27 computing the square of the absolute value of the DFT of the windowed sequence. T h e D F T of the windowed C sequence is the convolution of the DFT’s of the window Canonic Form: IV-17 Cascade Canonic Form: IV-17 and the originalsequence. This convolution smoothes Causal: IV-22 the input power spectrum, consequently values of the Causality: IV-21 Filters: 1V-27 power spectrum at frequencies separated by less than Chebyshev Coefficient Quantization Error: V-13 Comb Filter: IV-11 the width of the rnainlobe of the spectral window canContinuous Time: 1-3 not be resolved. In addition to this limit onresolution, the Cooley-Tukey : VI-9 estimate of the power spectrum may contain significant Cyclic or Circular Discrete Conyolutigrl: V I 1 4 leakage, Le., erroneous contributions from components D of the power spectrum at frequencies possibly distant DC Point: 111-2 from the frequency of interest because of the nonzero Deadband Effect: V-9 energy in the spectral xy-indow sidelohes, Dgrimationvin-Frequency:VI-8
RABINER
et
d.: DIGITAL SIGNAL PROCESSING TERMINOLOGY
Decimation-in-Time: VI-9 Deconvolution: VII-12 Digital: o - - - -1-5 -- -Digital Aliasing: 111-5 Digital Filter: IV-1 Digital Filter Realizations: IV-17 Digital ImDulse: 11-5 Digital Siinal: 1-5 Digital Simulation: 1-6 Digital System: 1-5 Digital-to-Analog (D/A) Converter: V-1 Digital Waveform: 1-5 Direct Form 1: 1V-17 Direct Form 2: 1V-17 Direct Method: VII-8 Discrete Convolution: VII-3 Discrete Filters: IV-1 Iliscrete Fourier Transform (DFT): V I - 1 Iliscrete Fourier Transformation: Vi-1 Discrete Time: 1-4, 1-5 Discrete-Time Convolution: 11-8 Discrete-Time Impulse: 11-5 Discrete-Time Impulse a t k = k o : 11-5 Discrete-Time Linear Filter: 11-7 Tliscrete-Time Linear System: 11-7 Dither: V-9 Dynamic Range: V-1 1
E Elliptic (Cauer) Filter: IV-27 Equiripple (Optimal) Filter: IV-14 Exponent: V-4 Extraripple Filter: IV-13
F Fast Convolution: VII-5 Fast Correlation: V I I J :VI4 Fast Fourier Transform (FFT) FFT Convolution: VI14 FFT Correlation: VII-5 Filter Bandwidth: IV-25 Finite Impulse Response (FIR): IV-6 F I R Identification: VII-12 First-Order Section: IV-17 Fixed-point Number: V-3 Floating-point &'umber: V-4 Flow Graph: VI-16 Folding Frequency: 111-3 Fourier Transform: 111-2 Frequency Response: 11-9 Frequency Samples: IV-12 Frequency-Sampling Filter: IV-12 Frequency-Sampling Realization: IV-15 Frequency-Scale Factor: IV-24
G Gain of a Discrete Filter: IV-23 Guard Filter: IV-31
H Hexadecimal Representation: V-4 1
Impulse: 11-5 Impulse Invariance: IV-31 Impulse Response: 11-5 In-Band Ripple: IV-30 Indirect Method: VII-10 Infinite Impulse Response (IIR): IV-7 In-Place: VI-12
337
Inverse Discrete Fourier 'Transformation (IDFT): VI-3 Inverse z Transform: 11-3 Iterative Optimization Technique: IV-32
K Kalman Filter (Discrete Time): IV-16
L Leakage: VII-8 Limit Cycle: V-9
M Mainlobe: VII-7 Mantissa: V-4 Matched z Transform: IV-31 Minimum Stopband Attenuation: IV-30 Mixed Radix: VI-13 Multiplexed Filter: IV-3 Multirate System: 1-10
N Negative Frequency: VII-1 Next-State Simulation: 1-7 Noise: 1-5 Nonrecursive Filter: IV-5 Nth-Order Systems: 11-10 Nyquist Frequency: 111-3 Nyquist Interval: 111-3 Nyquist Rate: 111-3
0 Object System: 1-6 Octal Representation: V-4 Ones Complement: V-6 One-sided z Transform: 11-1, 11-2 Optimization Technique: IV-32 Ordering: IV-20 Out-of-Band Ripple: IV-30 Overflow: V-10 Overflow Oscillations: V-10 Overlap-Add: VII-6 Overlap-Save: VI1-6
P Pairing: IV-20 Parallel Canonic Form: IV-17 Parameter Quantization Error: V-13 Passband Ripple: IV-29 Periodic Discrete Convolution: VII-3 Phase Factors: VI-7 Phase Response: 11-9 Positive Frequency: VII-1 Postfilter: V-1 Prime Factor Algorithm:VI-11 Principle Root of Unity: VI-14
Resolution: VII-8 Resolved : VI 1-8 Ripple: IV-29 Rotation Factors: VI-7 Rounding: V-8 Roundoff Error: V-9 Roundoff Noise: V-9 5 Sample Value: 11-6 Sampled Data: 1-4 Sampled-Data Filter: IV-1 Sampling Frequency: 111-3 Sampling Interval : 111-3 Sampling Rate: 111-3 Sande-Tukey: VI-8 Saturation Arithmetic: V-10 Second-Order Section: IV-17 Sectioned: VII-6 Sectioning: VII-6 Select-Save: VII-6 Sequences: 1-4 Series Form: IV-17 Sidelobes: VII-7 Sign and Magnitude: V-6 Signal: 1-5 Signal-to-Noise Ratio: V- 12 Source System: 1-6 Spectral Window: VII-7 Stability: IV-21 Stopband Ripple: IV-30 Switched Filter: IV-2 System Function: 11-9
T Tapped Delay Line: IV-10 Throughput Rate: 1-9 Throughput Rate per Signal: 1-9 Transition Band: IV-28 Transition Ratio: IV-28 Transpose Configurations: IV-19 Transversal Filter: IV-10 Truncation: V-8 Truncation Error: V-9 Truncation Noise: V-9 Twiddle Factors: VI-7 Twos Complement: V-6 Two-sided z Transform: 11-2
U IJndersamnled : 111-4 . Unit Advcnce: 11-1 Unit Circle: 111-2 Unit Delay Operator: 11-1 Unit Pulse: 11-5 Unit Sample: 11-5 Unit Sample Response: 11-5
Q Quantizing: V-2 Quantizing Noise: V-2
R Radix R:VI-13 Real-Time Process: 1-8 Reconstruction Device: V-1 Reconstruction Filter: V-1 Recursive Filter: IV-4 Recursive Realization: 11-10
W Waveform: 1-5 Wiener Filter: IV-16 Window: VII-7 Windowing: VII-7 1
Plane: 11-4 z-1 Plane: 11-4 z Transform: 11-1 Zero-Input Limit Cycle: V-9 2;
