Digital Filters: Concepts, Classes, and Realizations, Study Guides, Projects, Research of Hispanic American Literature

Download Digital Filters: Concepts, Classes, and Realizations and more Study Guides, Projects, Research Hispanic American Literature in PDF only on Docsity! Terminology in Digital Signal Processin LAWRENCE R. RABINER, JAMES W. COOLEY, HOWARD D. HELMS, LELAND B. JACKSON, JAMES F. KAISER, CHARLES M. RADER, RONALD W. SCHAFER, KENNETH STEIGLITZ, and CLIFFORD J. WEINSTEIN Absfract-The committee on Digital Signal Processing of the IEEE Group on Audio and Electroacoustics has undertaken the project of recommending terminologg for use in papers and texts on digital signal processing. The reasons for this project are twofold. First, the meanings of many terms that are commonly used differ from one author to another. Second, there are many terms that one would like to have defined for which no standard term currently exists. It is the purpose of this paper to propose terminology which we feel is self-consistent, and which is in reasonably good agreement with current practices. An alphabetic index of terms is included at the end of the paper. Introduction As an aid to classifying the different types of terms to be defined, we have placed each term in one of the following groups: 1) Introductory Terms-General Definitions 2) Discrete Systems-Block Diagram Terminology- 3) Relations Between Discrete and Continuous 4) Theory and Design of Digital Filters 5) Finite Word Length Effects-A/D, D/A Con- 6) Discrete Fourier Transforms and the FFT 7) Discrete Convolution and Spectrum Analysis. Signals version In the above mentioned sections of this paper we will be discussing terminology related to the processing of one- dimensional signals. For convenience, we will assume that this dimension i s time-although the definitions apply equally well to any single dimension. 1. introductory Terms-General Definitions 1) In discussing waveform processing problems, the distinctions analog versus digital and continuous time Manuscript received August 1, 1972. L. R. Rabiner, J. F. Kaiser, and R. W. Schafer are with Bell J . W. Cooley is with the I B M T. J. \&’atson Research Center, H. D. Helms is with Bell Telephone Laboratories, Inc., Whip- Telephone Laboratories, Inc., Murray Hill, N . J . 07974. Vorktown Heights, 73. Y. 10598. pany, N . J . 07981. Nyack, N. 1’. 10994. I,. B. Jackson is with Rockland Systems Corporation, West C. M. Rader and C. 1. Weinstein are with M.I.T. Lincoln Lab- oratory, Lexington, Mas:. 02173. (Operated with support from the U. S. Air Force.) K. Steiglitz is with Princeton University, Princeton, pi. J . 08540. 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 generally describes a waveform that is continuous in time (or any other appropriate independent variable) and that belongs to a class that can take on a continuous range of amplitude values. Examples of analog wuveJorms or anulog signals are thosc derived from acoustic sources. Such signals are repre- sented rnathematically as functions of a continuous variable. The functions sin (ut) and the step function au-l(t) are examples of common mathematical functions that could describe “analog signals.” The use of the term “anaIog” in this context appears to stem fronl the field of analog computation, where a current or voltage waveform serves as a physical analog of some variable in a differential equation. 3 ) The term continuous t ime implies that only the independent variable necessarily takes on a continuous range of values. I n theory the amplitude may, but need not, be restricted to a finite or countable infinite set of values (i.e., the amplitude may be quantized). There- fore, analog waveforms are continuous-time waveforms with continuous amplitude. In practice, however, “con- tinuous-time waveforms” and “analog waveforms” are equivalent. Since most signal processing problems have nothing to do with analogs as such, the us11 of the term analog waveform is often ambiguous at the least and may in fact be misleading. T ~ L I S , the term continuous- time waveform is preferable. 4) Discrete time implies that time (the independent variable) is quantized. T h a t is, discrete-time signals are defined only for discrete values of the independent vari- able. Such signals are represented rnathcmatically as sequences of numbers. Those discrete-time signals that take on a continuum of values are referred t o as sum$led- data signals. 5 ) The term digital implies that lmth timc and mlpli-- tude are quantized. Thus a digital system i s one i n w h i c h signals. are represented as sequences o f n u m h - r s which take on only a finite set of values. Thlls onc uses d ig i td when discussing actual phJ-sical realizations (as hard- ware or programs) o f discrete-time signal processing sys- tems, whereas the term discrete t i m e is a h;tt-er modifier when considering mathematical abstractions of such systems in which the effects of amplitud- quantization are ignored, A digital signal or digital ~ ( ~ v e j o s m is a sc- quence produced, for example, b y digital circuitry or by an analog-to-digital converter which is sampling a COIF tinuous-time waveform. In digital signal processing these terms are commonly shortened to signal 01- wave- f o rm. Sometimes the term signal is restricted to being a desirable component of a sequence instead of being used interchangeably with waveform. Noise is either defined as a) an undesirable component of a sequence, or b) a sequence of random variables. 6 ) (Digital) simulatiofz i s the exact or approxinlate representation o f a given system (discrete or contin- IaBIirTm et al.: DIGITAL SIGKAL PROCESSING TERMIXOLOGY 323 uous) called the source system, by a (digital) system y ( n ) = x ( n - m), m > 0 (2) called the object system. 7) Next-state simulation is a method of digital simu- lation whereby the values of the digital system signals m ( 3 ) are represented by nodes in a block diagram representa- tion. Usually, there is a close correlation between blocks in the object system and elements of the source system. The method entails ordering the calculations in the digital system so that all the inputs to each block a t a given sample time are computed before the output is computed. 8) A real-time process is one for which, on the average, the computing associated with each sampling interval can be completed in a time less than or equal to the sampling interval. ,4 program running in 100 times real time requires 100 times as long to process the Sam(: number of samples; i.e., it is 100 times too slow for real time operation. A program ten times as fast as it needs to be could be said to run in 1/10 real time. Obviously, the extra speed can only be used if other computing can be done in the interstices, or if the complete sequences to be processed have been stored beforehand. 9) Throughp.ut rate is the total rate at which digital information is processed by a discrete-time system, measured in bits per second or samples per second. In a multiplexed system, where several signals are processed, we may refer to the throughput rate per signal, measured in bits per second per signal or samples per second per signal. Thus, a multiplexed system which processes 10 signals, each at 1000 bits/s, has a throughput rate of 10 000 bits/s, and a throughput rate/signal of 1000 bits/s/signal. 10) A multivate system is a discrete-time system in which there are signals sampled a t different intervals which are usually integer multiples of some basic or fundamental interval. In ( 3 ) , X ( z ) appears multiplied by z-%. This result for m'= 1 accounts for the fact that 2-l is often termed the unit delay operator, since a delay of the sequence by one sample is equivalent to multiplication f the z transform by z- l . ( Similarly, z is often called the unit advance operator.) 2) In many cases, sequences are defined over both positive and negative values of A. In such cases, a some- what more general point of view is called for. In general, the z transform is written as m X ( Z > = x(n)z-a. (4) n=--m I t should be noted that a common usage is to call (1) simply the z transform, and (4) the two-sided z transform. Since (4) is most general, it would seem preferable to refer to (4) as the z transform, and the special case, ( l ) , as the one-sided z transform. 3 ) I t is possible to think of the z transform as simply a formal series whose properties can be tabulated, and which never need be summed. However, it is generally preferable to realize that if certain convergence condi- tions are met, both (1) and (4) are Laurent series in the complex variable z. As such, all the properties of the Laurent series apply. For example, if the series in (1) converges, it must converge in a region I zI > R+. If the series of (4) converges, it must converge in an annular region R+ < I zI < R-, where R+ may be zero and R- may be infinity. The coefficients of a Laurent series are deter- mined by an integral relationship. In the context of the z transform, this relation is 11. Discrete Systems (5) 1) The z transform plays a role in discrete-time system theory analogous to that of the Laplace transform in continuous-time system theory. Two view points re- garding the z transform are common. One is based on what may be termed the one-sided z transform, which is defined as m X ( Z ) = x(n1z-n (1) 9L-0 regardless of the value of x(n) for n < 0.l One applica- tion of the one-sided z transform'is in the solution -of linear difference equations with constant coefficients. Solutions are obtained for the interval Osn < ~0 subject to prescribed initial conditions. These solutions are obtained with the aid of the equations The notation x(%) is used rather than X,, or x(nT) to denote a sequence because of the ease of handling complicated indices, e.g., ~ ( N - 1 / 2 ) . where C is a closed contour inside the region of conver- gence of the power series and enclosing the origin. Equa- tion (5) is referred to as the inverse z transform. 4) In the region of convergence of the series, both (1) and (4) represent analytic functions of the complex variable z. These functions can often be extended by analytic ontinuation everywhere xcept at certain .singular points (poles). Since these singularities of the z transform are characteristic of the particular sequence, i t is common t,o plot their locations in the z plane, (Le., the complex plane determined by the real and imaginary parts of c). I t should be noted that it is often convenient, because of the special functional form which character- izes exponential sequences, to plot singularities in the 2-l plane. Furthermore, some authors define the z trans- form as 8 ( z ) = x(n)zn. (6 ) m n=-w 326 IEEE TRAXSACTIOKS ON AUDIO AND ELECTRO.~COUSTICS, DECEMBER 1972 is called the sampling interval. The highest frequenq- present in x , ( t ) , (defined above) S o , is called the Nypuist frequency. The Nyquist frequency is sometimes called the folding frequency. I t is recommended that the term Nyquist frequency be avoided because of the general confusion with the term Nyquist rate. Furthermore, we recommend that the term folding frequency refer to half of the actual sampling frequency (see Fig. 3). 4) The relationship (13) between the Fourier trans- form of a sequence of samples x,(nT) and the Fourier transform of the continuous time signal xc( t ) is depicted in Fig. 4. Part (a) of this figure depicts a band-limited Fourier transform X , ( j 2 ~ f ) . In Fig. 4(b) and (c) the sampling rate is greater than or equal to the Nyquist rate and we note that the form of X,( j27rf) is preserved to within a constant multiplier 1/T in the frequency range - l / Z T < f < l / 2 T . However, in Fig. 4(d) the signal x,(t) is undersumpled, i.e., sampled at a rate below the Nyquist rate. In this case the Fourier transform of the sequence obtained by sampling is not equal to X,( j27rf)lT due to the fact that some of the other terms in (13) such as Xc(j27rf-j(27r/T)) are nonzero in the frequency range - 1/2T<Ip< 1/2T. One way of viewing this is to say that a set of frequencies in X,( j271-f‘) is in- distinguishable from a different set of frequencies in X,( j21rj-j27r/T). These frequencies are called aliases of one another and the process of confounding frequencies as in Fig. 4(d) is called aliasing. 5 ) Suppose we have a sampled waveform x ( n ) with z transform X ( z ) . We define a new sampled waveform y ( n ) using one of every M samples as the samples of the new waveform, i.e., y ( n ) = x ( M n ) , with 13-6 any positive integer. Clearly, this process is equivalent to sampling a t a lower rate, and i t is to be expected that aliasing may occur. When the aliasing occurs due to “sampling” a discrete-time signal i t is called digital aliasing. I t is readily shown that Y(z) can be written in terms of X ( z ) as 1 M-I M 2=0 y(z ) = - x ( z l / . ~ f e - j ( z x / ~ w ) z 1. (16) IV. Theory and Design of Digital Filters 1) Discrete filters may be divided into two classes on the basis of whether the signal values can take on a continuum of values (sampled-data filters) or a finite set of values (digital filter). Thus we have the following. a) A sampled-dala fi l ter i s a computational process or algorithm by which a sampled-data signal acting as an input is transformed into a second sampled-data signal termed the output. The sampled-data signal is considered only a t a set of points (usually equally spaced in time or space as the independent variable); at these points the signal can take on a continuum of values. ’ . b) A digitul filter is a computational process or algorithm by which a digital signal or sequence of num- - fS - f, -2fo -f, 0 f, 2f0 fs fs f 2 NYQUIST ’ .f FREQUENCY NYQUIST RATE SAMPLING FREQUENCY t FOLDING FREQUENCY Fig. 3. Labeling of terminology concerned with frequencies related to the sampling process. - f S fS 2 (b) - f, 0 f, I 3 fo 2f,=f, (c) f s = Zf, f, < 2f, ALIASING (a) Fig. 4. An example of the effects of various sampling frequencies on the frequency response of the digital signal. hers (acting as input) is transformed into a second se- quence of numbers termed the output digital signal. The numbers are limited to a finite precision. The algorithm may be implemented in software as a computer SUI,- routine for a general-purpose machine or in hardware as a special-purpose computer. The term digital fi ter is then applied t o the specific routine in execution or to the hardware. 2 ) Further complexity of filtering action may he obtained by switching. Thus, a switched filter ,is one .ill RABINER et Ut. : DIGITAL SIGNAL PROCESSING TERYINOL GY 327 which the input and output are simultaneously switched in a definite pattern among a group of input and output ports. The filter being switched may be either of the continuous or discrete types. Examples of switched filters are commutating filters or n-path filters. 3 ) A multiplexed Jilter is a restricted form of a switched filter; commonly a single discrete filter which Fig. 5. Block diagram representation of comb filter. by means of a switching action is made to perform the function of several discrete filters virtually simul- 12) A frequency-sampling Jilter is an F I R filter which taneously. Th:: multiplexing is most commonly done in is designed by varying one or of its D F T ~ coeffi- a time-division manner whereby the input to the dis- cients (called frequency samples) to minimize crete filter is sequentially switched from a number of aspects of the filter?s frequency response. F~~ example, input signals and the filter output sequentially switched the DFT coefficients of a frequency-sampling lowpass in synchronism to a corresponding set of output signal filter are 1.0 in the passband, 0.0 in the stopband, and lines. Thus a single filter may be made to do the work variable in the transition band. one design criterion of many filters by this time division multiplexing. would be to choose the variable coefficients to minimize realized via a recursion relation, i.e., the output samples 13) extraripfile f i l t er (also called ripple of the filter are explicitly determined as a weighted sum filter) is an FIR filter whose frequency response is equi- 4) A recursive filter is a discrete-time filter which is peak stopband ripple. of Past output saInPles as We' as Past and/or Present ripple in both the passband and stopband, and whose input samples. For exaInPle, Y(n) =box(%) + h l x ( f l - l l ) frequency response contains the maximum possible +b2x(n-2) - a a l y ( n - l ) --2y(n-2). number of ripples6 There is no general agreement as to 5) A nonrecursive filter is a discrete-time filter for the appropriateness of this term, and as no recorn- which the output samples of the filter are explicitly mendation as to its usage is made. determined as a weighted sum of past and present input 14) equiripple (optimal) filter is an FIR filter wllich +bzx(n - 2). to some specified frequency spo se characteristic ver whose impulse response h(n) is zero utside some finite lowpass filter the filter may be an extra- samples only. For example, Y(n> =bOx(n)+blx(n-ll) is the unique best approximation in the minimax sense 6 , A finite imfiulse is a any closed subset of the frequency i terval. For the limits, i.e., h(n) = O , for n>N1 and n<N2 With N12 NZ. ripple filter, an equiripple filter with one less than the 7) An in$nite ('IR) is a maximum possible number of ripples, or a filter with the for .which either NI = a or N2 = - or both, in 6). Thus maximum possible number of ripples all except one of the duration of the filter's impulse response is infinite. which are of equal amplitude. restricted to z = 0 or z = a, whereas there are no such realization of an FIR filter of duration samples as a restrictions on the positions of either the poles or zeros cascade of a comb filter and a parallel bank of of I IR filters. plex pole resonators. The filter output is obtained as a 9) The terms recursive and nonrecursive are recom- weighted sum of the outputs of each of the parallel mended as descriptions for how a filter is realized and branches; thk multiplier On the Kth branch being the kth not whether or not the filter impulse response is of finite DFT coefficient of the filter impulse response. duration. (Although I IR filters are generally realized 16) A Ka~man filter (discrete time) is a linear, but 8 ) I t should be noted that the poles of F I R filters are 15) A fyequency-sampling realization is a means of recursively, and are generally non- possibly time-varying discrete-time filter with the prop- recursively, I I R filters can be realized nonrecursively erty that it provides a least mean-square estimate and FIR filters can be r alized recursively.) of a (possibly vector-valued) discrete-time signal based lo) A t ra~sversa l filter is a (either continuous or on noisy observations. The statistical description of the discrete) in Which the output signal is generated by problem is such that the Kalman filter has a recursive weighted by a set Of weights termed the tap gains- If the servations and old The filter design may be signal are by a tapped d lay line based on a more general c iterion, using a non- summing a series of delayed versions of the input signal implementation, using a linear combination of new ob- then the filter is termed a tapped delay line filter. quadratic loss function. Its essential features are that its design is based on a statistical criterion in the time Or difference Of input and Output Of a Of domain, and that it is, in general, time varying. If the units and unit gain yielding a transfer characteristic filter is further restricted to be time invariant i t becomes H(z) = 1 k z--*' (see Fig. 5 ) ; this filter has hi" zeros of the Wienerfilter. transmission equally spaced on the unit circle in the 11) A JiZter is a comprised Of the sum z plane thus giving rise to a frequency characteristic having M equal peaks and Af real frequency zeros. See Section VI-1 for a definition of DFT. See Section IV-29 for a definition of ripple. 328 IEEE TRANSACTIONS O S AUDIO AND EI.ECTROSCOUS?’ICS. DECEMBER 1972 ...e t bN Fig. 6. Block diagram representation of direct form 1 for an Nth-order system. 17 ) ‘The forms for realizing digital filters include the following: a ) Direct f o r m 1 (shown in Fig. 6) where biz-’ ’Y & - I i=O For convenience in showing the realization, the order of the numerator and denominator are set to be the same. Direct form 1 uses separate delays for both the numera- tor polynomial and the denominator polynomial. In certain cases, e.g., floating-point additions, the results may depend on the exact ordering i n n;hich the additions are performed. b) Direct form 2 is sho\vn i n Fig. 7. Direct form 2 has been called the canonic jorm because it has the minimunl number of multiplier, adder, and delay elements, but since other c-onfigurations also have this property, this terminology is not recommended. c) Cascade canonic fo rm (or series j o rm) , which is shown in Fig. 8, where and Hi(z) is either a second-order section, i.e., 1 + b1iz-l + b2iz-’ 1 + a1iz-1 + u2iz-2 Hi(2) = > or a first-order section, i.e., and bo is implicitly defined in (17), where K is the integer part of (N+1)/2. d) Pnmllel canonic form, which is shown in Fig. 9, where K H ( z ) = c + H&) (21) i=l where H,( z ) is either a second-order section, i.e., Fig. 7. Block diagram representation of direct form 2 for an Nth-order system. *-+-+pJ bo ... * Fig. 8. Block diagram representation of the cascade form. C m Fig. 9. Block ’ - diagram representation of the parallel form. and K=integer part of (Nfl),” and C is proportional to b s as defined in (17). 18) The individual second- and first-order sections of the cascade and parallel forms are generally realized in one of the direct forms. 19) Transpose configurations for all of the above forms can be obtained by reversing the directions of all signal flow (i.e., by reversing the directions of all arrows) and by interchanging all branch nodes and summing junc- tions. The resulting circuits have the same transfer func- tions but different roundoff noise and overflow proper- ties. 20) When the transfer function of a high-order filter is decomposed into a cascade connection of lower order filter sections by distributing the pole and zero factors among the lower order sections, then pairing refers to the associating of a specific zero factor with a specific pole factor to form an elemental or individual section. Ordering refers to the sequence or order in which the indi- vidual sections are connected in cascade to form the composite higher order filter. Varying the pairing and ordering can dramatically change the noise properties and dynamic range of both discrete and continuous filters. As an example, if Z or a first-order section, i.e., D j ( z ) RABISER et al. : DIGITAL SIGNAL PROCESSING TERMINOLOGY 33 I transformation or standard z ) is a technique in which the impulse response of the derived digital filter is identi- cal to the sampled impulse response of a continuous-time filter. If the continuous-time filter has a transfer func- tied k=O then the requirement that h(n) = h,(t) 1 t=ny, o 5 n I ca (36) implies that H ( z ) is obtained from the partial fraction expansion of H,(s) by the substitution (37) It can be shown that Thus, impulse invariance is only satisfactory when H,(sj is band limited. If as in most instances, H,(s) is not sufficiently band limited, H ( z ) is an aliased version of H,(s) . Therefore, this technique is primarily used for narrowband filter designs or else the transformation is applied to the cascade combination of a guard jilter and .Another important point is clear from (38). Due to the 1/T multiplier, digital filters derived by impulse invariance have a gain approximately 1/T that of the continuous-time filter. This is generally compensated by multiplying each factor in the partial fraction expansion by T , so that the digital filter will have approximately the same gain as the continuous-time filter from which it was derived. h j Bilinear transformation (also called the bilinear z transform, the bilinear z transformation or z form) is a technique used to circumvent the aliasing problem of the impulse invariant technique. This approach uses the algebraic transformation Hds) . (39) to derive the system function of the digital filter as H ( z ) = U,(sj I ~-(~/*)(l-~-l)/(l+z-'). - (40) This transformation has the ffect of mapping the entire s plane into the z plane in such a way that the left-half modifications can be made to deal with multiple order poles. This formulation assumes that all poles are distinct. Appropriate s plane maps into the inside of the unit circle and the right-half s plane maps to the outside of the unit circle. This results in a nonlinear warping of the frequency scale according to the relation % T a a T 2 2 __ - tan - (41) where wc is the continuous-time frequency variable and wd is the discrete-time frequency variable. Because of this warping of the frequency scale, this design tech- nique is most useful in obtaining digital designs of filters whose frequency response can be divided into a number of pass and stop bands in which the response is essentially constant. Generally i t is necessary to take appropriate account of the warping of the frequency scale. c) Matched z transform (also called the matched z transformation, mapping poles and zeros, or matched z ) is a technique based on mapping the poles and zeros of the continuous-time filter by the substitution s - si -+ 1 - e*i*z-l. (42) This means that the poles of H(z ) will be identical to those obtained by impulse invariant method, however the zeros will not correspond. 32) In the context of designing a discrete-time system and especially a digital filter, an optimization technique is a procedure for minimizing a prescribed performance function based on design requirements. An example is the design of a discrete-time filter to have the minimum mean-square deviation from a desired frequency-domain characteristic. An iterative optimization technique is a procedure for generating successive approximations con- verging (hopefully) to an optimum. This is opposed to an analytical design technique, which yields a closed form solution, such as the Chebyshev design for a lowpass filter. V. Finite Word length Effects-A/D, D/A Conversion 1) A digital-to-analog ( D / A ) converter is a device t ~ h i c h operates on a digital input signal s(nT) t o produce a continuous-time output signal s ( t ) ideally defined by s(t) = s(nT)h(t - nT) (43) n where h(t) characterizes the particular converter. For example, h(t) is a square pulse of duration T for a zero order hold D / A converter. The D / A converter is usu- ally followed by a linear time-invariant low-pass con- tinuous-time filter called a postfilter. The combination of D/A converter and postfilter is called a reconstru.ction device or reconstruction jilter. 2) An analog-to-digital ( A / D ) converter is a device which operates on a continuous-time waveform to pro- 332 IEEE TRANSACTIONS ON AUDIO AND ELECTROACOUSTICS, DECEMBER 1972 duce a digital output consisting of a sequence of numbers each of which approximates a corresponding sample of the input waveform. Expressing the numerical equiva- lent of each sample by a finite number of bits (instead of the infinite number required to completely specify each sample) is the quantizing inherent in the conversion process. The error produced by quantizing is called quantizing noise or A / D conversion noise. Representation of Numbers 3) Various systems are used to represent the numbers in a digital filter. In $xed-point number representation, the position of the binary (or decimal) point is assumed fixed. The bits to the right of the (fixed) binary (or decimal) point represent the fraction part of the number and the bits to the left represent the integer part. For example, the binary number 011.001 has the value OX2~+1X2~f lX2~+0X2-1+0X2-~ f lX2-3 . 4) A seating-point number is formed by two fixed point numbers, the mantissa7 and the exponent. The floating-point number is equal to the product of the mantissa with the quantity resulting when a given base is raised to the power denoted by the exponent. The base is the same for every floating-point number in the digital filter. Consequently, the numerical value of an entry in a specified position in the mantissa is deter- mined by the exponent. The mantissa is generally nor- malized to be as large as possible but less than some number (e.g., 1.0). For example, 0.1 X l o2 is legitimate, whereas 0.01 X l o3 and 10.0 X loo are usually considered to be illegitimate floating-point decimal representations of the number 10. The most commonly used base is two (binary representation). The base 16 (hexadecimal repre- sentation) is used in some general purpose computers. The base 8 is called octal representation. 5) The representation of block floating-point numbers is determined by examining all numbers in a block (i.e., array). The largest number is represented as an ordinary floating-point number with a normalized mantissa. The remaining numbers in the block use the exponent associ- ated with this largest number. This use of a single ex- ponent for the whole block saves memory. This type of arithmetic is popular in realizations of the fast Fourier transform. Representation of Negative Numbers 6) The discussion so far has dealt with the repre- sentation of nonnegative numbers. There are three com- mon systems used for representing signed numbers. The representation of positive numbers is the same in thes-e three systems. The first, and most familiar, is sign and magnitude, i.e., the magnitude (which is, of course, positive) is represented as a binary number and the sign is represented by an additional binary digit in the lead- as the term mantissa commonly used i n logarithm tables. The defini- ’ The term n1;mtissa as defined here is unfortunately not the same tion presented here is dictated by its extensive occurrence in the literature: ing position which, if 0 corresponds to a + and if 1 corresponds to a - (or vice versa). Thus , for example, in sign and magnitude 0.0011 represents 3/16 and 1.001 1 represents -3/16. Two related representations of signed numbers are ones complement and twos com- plement. In each of these systems a positive number is represented as in sign and magnitude. For Iwos-comple- ment representation the negative of a particular positive number is obtained by complementing all the bits and adding one unit in the position of the least significant bit. For example, -(0.0110) would be represented in twos complement as (1.1001)+(0.0001) = 1.1014). A carry out of the sign bit is neglected in the addition, so that - (0.0000) = (1.1111) +(O.OOOl) =O.OOOO. For ones- complement representation the negative of a given posi- tive number is obtained simply by complementing all the bits. 7) The choice of representation for negative numbers in a particular system is based almost entirely on hard- ware considerations. With ones-complement and twos- complement numbers, subtraction can be performed conveniently with an adder. For example, in twos com- plement, the difference A-B is formed by simply adding to A the twos complement of B. Finite Word Length Effects 8) Even though the input to a digital filter is repre- sented with finite word length (e.g. through A/D con- version), the result of processing will naturally lead to values requiring additional bits fo, their representation. For example, a b-bit data sample multiplied by a b-bit coefficient results in a product which is 2b bits long. If in a recursive realization of a filter we do not quantize the result of arithmetic operations, the number of bits required will increase indefinitely, since after the first iteration 2b bits are required, after the second iteration 3b bits are required, etc. Two common methods are used to eliminate the lower order bits resulting from arithmetic operations in a digital filter. a) Truncation is accomplished by discarding all bits (or digits) less significant than the least significant bit (or digit) which is retained. b) Rounding of a number to b bits, when the num- ber is initially specified to more than b bits, is accom- plished by choosing the rounded result as the b-bit num- ber closest to the original unrounded quantity. When the unrounded quantity lies equidistant between two adjacent b-bit numbers, a random choice ought to be made as to which of these numbers to round to. For example, 0.0101 1 rounded to three bits would he 0.01 1 ; but 0.01010 rounded to three bits can be chosen as either 0.011 or 0.010, and the choice should be random. In many situations, however, one can choose to always round up in this midway situation with negligible effect on the accuracy of the computation. 9) Roundof error (or roundof noise) or truncation error (or truncation noise) is caused by rounding off or truncating the products formed within the digital filter. RABINER et al.: DIGITAL SIGNAL PROCESSING TERMINOLOGY 333 The roundoff or truncation error is sometimes well puting, or otherwise forming, the discrete Fourier trans- modeled as a random process. On the other hand, if the form of the sequence. data sequence to a recursive realization of a digital filter 2) From the definition, the sequence { f ( O ) , f ( l ) , . . . , consists of constants (e.g., zero) or some other periodi- f ( N - 1 ) 1 has the DFT* f F(O) , F(1) , . . , F ( N - 1) 1 cally repeating samples, the roundoff or truncation error is periodic and causes a deadband efect or limit cycle in ~ ( h ) = j (n)e- i (2~/~7)nk. the filter output. A common type of limit cycle s a n-0 zero-input limit cycle where the output of a digital filter set to zero. Dither is a sequence of numbers that is added its DFT by the ‘peration to ’the input to a recursive digital filter to meliorate th 1 N-1 deadband effect. Even though dit er inc eases th mean- j(n> = - F ( ~ ) ~ ~ ( z T / N I ” ~ (46) square error in the output, it can disrupt the pattern of roundoff errors causing the deadband effect, thereby giving a sequence of N samples ( f ( o > , f(l) , . . . , permitting the output to return to zero. f ( N - 1) 1 as the inverse discrete Fourier t ansform of number that is too large to be represented in the arith- inverse discrete ~~~~i~~ transformation or IDFT, and is metic used in that filter. If no compensation is made for remarkably similar in form to the discrete ~~~~i~~ trans- the overflow then large errors in the filter output can formation. result either in the form of transients or of overjiow oscil- 4) some authors have defined the DFT in related but lations. A technique used to compensate in part for over- different ways, involving e i ( ~ T / ~ ) n k , or a multiplicative flow is saturation arithmetic where a sum tha t is too factor of 1 / ~ or 1/dr. B,, considering the expressions sentable number in thefilter. 1/N or l /dx and the possible us of ei(2T/N)nk in other and smallest signals which can be represented in the the other definitions of the IDFT. filter with a given fidelity criterion. Unfortunately the 5) suppose for an ~ - ~ ~ i ~ ~ sequence we are interested fidelity criterion is often vague or unspecified. in computing its DFT, and suppose N is a composite within a digital filter produces a resultant noise a t the output of the digital filter. A signal-to-noise ratio can be A 7 = T I X r2 X . . . X r, (47) defined in this context, for example, as the ratio of the ideal mean-squared output signal (filter output in the where the r; are a set of factors of N , not necessarily absence of any rounding) to the mean-squared output prime factors. Of the various algorithms for computing noise due to rounding or truncation. Expressing this such a DFT, some require a number of operations pro- ratio in bits, as (1/2) log3 of the ratio, gives an approxi- portional to NE‘, rl (since the word proportional al- mate indication of the number of accurate bits in the lows considerable latitude, it is not necessary to be too filter output. Different definitions of signal-to-noise specific about the meaning of “operation”); such algo- ratio may be appropriate in different contexts. rithms are called fast Fourier transforms (FFT).g An 13) Another effect of finite word length is coeficient important special case is when quantization error (or parameter quantization rror), y l = y z = . . . which occurs when the coefficients of a digital filter, = r , = 2 initially specified with unlimited accuracy, are quantized SO that by rounding or truncation. Coefficient quantization error appears as error in the digital filter’s response ( .g., impulse response, transfer function, frequency response, etc.). For fast Fourier transf rms in this case, the propor- N- 1 (45) remains periodic and nonzero, after the input has been 3, I t Possible to recover the original sequence from N k=o 10) Overjiow occurs when a digital filter computes a { F ( O ) , F O ) , * . ‘ , F(N- 1) 1. The operation is called large to be represented is set equal to the largest repre- for the DFT and IDFT i t is evident that the constants 11) Dynamic range is the ratio between the largest definitions of the DFT can easily be compensated for in 12) The roundoff or t uncation noise introduced integer P rk 2 log2 N . k-1 tionality is to N logz N. VI. Discrete Fourier Transforms and the FFT 1) For a sequence of N numbers, possibly complex, the discrete Fourier transform ( D F T ) is another sequence of exactly N numbers which are the values of the z trans- form of the original finite sequence for N values of z , specifically = , 9 ( 2 a / N ) k ~ , K = 0, 1, . * . , N - 1. (44) Discrete Fourier transformation is the operation of com- 6 ) A subclass of FFT algorithms is known which use high speed convolution techniques to compute the DFT of a sequence through a formula in which i t is expressed as a convolution. Examples of such algorithms are the chirp z transform and the prime algorithm. 7) In order to classify different FFT algorithms and efficients rather than F(ei@.’N)k). cably in the literature. For convenience, the notation of F(k) is used to denote DFT co- -a The word transform instead of algorithm is embedded ineradi- 336 IEEE TRAKSACTIONS ON AUDIO AND ELECTROACOUSTICS, DECEMBEK 1972 the lengths of the two sequences, cyclic convolution can be made to yield the same result as ordinary convolu- tion. 5) This use of the fast Fourier transform to compute discrete convolutions is sometimes calledfast convolution or FFT convolution. This technique can be asily adapted for computing acyclic (i.e., nonperiodic) corre- lation functions. In this form, it is called fast correlation or FFT correlation. 6) If one of the two sequences is much shorter than the other, the longer sequence can be sectioned into pieces whose discrete convolutions can be computed separately. These discrete convolutions can be com- bined to produce the discrete convolution of the whole sequence. (Sectioning is used because i t reduces the re- quired amounts of computation and memory.) One of these sectioning techniques (overlap-save or select-save) involves computing the inverse DFT of the product of the DFT’s of a) N samples of the input sequence, and b) the shorter sequence augmented with a sufficient number of zeros so that its sequence contains N samples. (Usually N is at least twice as large as the length of the shorter sequence.) Some of the members of the sequence resulting from the inverse DFT are members of the sequence formed by the desired acyclic (i.e., nonperi- odic) convolution. (This number of members equals one more than the number of zeros originally augmenting the shorter sequence.) The longer original sequence is advanced by this number of members. Iterating this process gives the whole convolution. The overlap-add technique for sectioning uses a similar technique but additionally requires adding shifted sequences of partial convolutions. 7) A window is a finite sequence, each element of which multiplies a corresponding element of the main sequence. (This is called windowing.) The sequence of products formed by this element-by-element multiplica- tion is often more useful than the main sequence. The Fourier transform of a typical window (sometimes called the spectral window) consists of a mainlobe, which usually contains a large percentage of the energy in the window, and sidelobes which contain the remaining energy in the window. 8) Windows can be used in estimating power spectra. In the direct method, the power spectrum is estimated by computing the square of the absolute value of the DFT of the windowed sequence. The DFT of the windowed sequence is the convolution of the DFT’s of the window and the original sequence. This convolution smoothes the input power spectrum, consequently values of the power spectrum at frequencies separated by less than the width of the rnainlobe of the spectral window can- not be resolved. In addition to this limit on resolution, the estimate of the power spectrum may contain significant leakage, Le., erroneous contributions from components of the power spectrum at frequencies possibly distant from the frequency of interest because of the nonzero energy in the spectral xy-indow sidelohes, 9) Windows are useful in determining the coefficients of a finite impulse response digital filter. In this case, the original sequence consists of samples of the impulse response corresponding to a transfer function which is approximated by the Fourier transform of the sequence of pairwise products; the product sequence is used as the coefficients of the finite impulse response digital filter. 10) Windows are used also in the indirect method of computing a power spectrum. In this method, the se- quence consisting of samples of the autocorrelation func- tion is multiplied by the window. The DFT of the resulting sequence is an estimate of the power spectrum. 11) Windows can be used also in estimating cross spectra where the estimates are obtained by multiplying the products of the DFT’s of two or more distinct sequences. 12) The determination of a finite impulse response described by an ordinary convolution is called decon- volution or FIR identification. Acknowledgment The authors would like to acknowledge the comments and criticisms of this paper provided by G. D. Bergland, C. H. Coker, D. L. Favin, B. Gold, 0. Hermann, S. Lerman, A. V. Oppenheim, H. 0. Pollak, and H. F. Silverman. Alphabetic Index of Terms A A/D Conversion Noise: V-2 Aliasing: 111-4 Amplitude Response: 11-9 Analog: 1-2 Analog Signals: 1-2 Analog Waveforms: 1-2 Analog-to-Digital Converter: V - Z Analytical Design Technique: IV-32 B Base: V-4 Base R: VI-13 Bilinear Transformation: IV-31 Binary Representation: V-4 Bit Reversal: VI-12 Block-Diagram: 11-11 Butterfly: VI-15 Block Floating-point h:umbers: V-5 Rutterworth Filter: TV-27 C Canonic Form: IV-17 Cascade Canonic Form: IV-17 Causality: IV-21 Causal: IV-22 Chebyshev Filters: 1V-27 Coefficient Quantization Error: V-13 Comb Filter: IV-11 Continuous Time: 1-3 Cooley-Tukey : VI-9 Cyclic or Circular Discrete Conyolutigrl: V I 1 4 D DC Point: 111-2 Deadband Effect: V-9 Dgrimationvin-Frequency: VI-8 RABINER et d.: DIGITAL SIGNAL PROCESSING TERMINOLOGY 337 Decimation-in-Time: VI-9 Deconvolution: VII-12 Digital: 1-5 Digital Aliasing: 111-5 Digital Filter: IV-1 Digital Filter Realizations: IV-17 Digital ImDulse: 11-5 o------ - 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 Discrete Time: 1-4, 1-5 Iliscrete Fourier Transformation: Vi-1 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 Fast Fourier Transform (FFT) : V I 4 FFT Correlation: VII-5 FFT Convolution: VI14 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 Samples: IV-12 Frequency Response: 11-9 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 In-Band Ripple: IV-30 Impulse Response: 11-5 Indirect Method: VII-10 Infinite Impulse Response (IIR): IV-7 In-Place: VI-12 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 Overflow: V-10 Out-of-Band Ripple: IV-30 Overflow Oscillations: V-10 Overlap-Add: VII-6 Overlap-Save: VI 1-6 P Parallel Canonic Form: IV-17 Pairing: IV-20 Parameter Quantization Error: V-13 Passband Ripple: IV-29 Periodic Discrete Convolution: VII-3 Phase Factors: VI-7 Positive Frequency: VII-1 Phase Response: 11-9 Postfilter: V-1 Prime Factor Algorithm: VI-11 Principle Root of Unity: VI-14 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 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 : 1 11-3 Sampling Rate: 1 11-3 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 Sidelobes: VII-7 Series Form: IV-17 Sign and Magnitude: V-6 Signal: 1-5 Signal-to-Noise Ratio: V- Spectral Window: VII-7 Source System: 1-6 Stability: IV-21 Stopband Ripple: IV-30 Switched Filter: IV-2 System Function: 11-9 12 T Throughput Rate: 1-9 Tapped Delay Line: IV-10 Transition Band: IV-28 Throughput Rate per Signal: 1-9 Transpose Configurations: IV-19 Transition Ratio: IV-28 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 : 1 11-4 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 W Waveform: 1-5 Wiener Filter: IV-16 Window: VII-7 Windowing: VII-7 1 2; Plane: 11-4 z-1 Plane: 11-4 z Transform: 11-1 Zero-Input Limit Cycle: V-9