PROCEEDINGS OF THE IBEE, VOL.
6 0 , NO. 6, JUNE
1972
681
DiscreteRepresentation of Signals ALAN V. OPPENHEIM,
SENIOR MEMBER, IEEE, AND
DONALD H. JOHNSON
Abstruct-h proceaaing continuou&ime signals by digitalmeans, it is necessary to represent the signal by a digital sequence. There are m a n y ways other than periodicsamplingforobtaining such a sequence. The requirements for such representations and some exof simulatinglinear amplesare discussed withintheframework time-invariant systems. The representation of digital sequences by other digital sequences is also discussed, with particular emphasis on the use of such representations to implement nonlinear a warping of the digital frequency axis. Some applications and hardware implementation of this digital-frequencywarpingaredescribed.
I n Section 11, a discussion of the discrete representationof continuous-time signalsfor thedigitalsimulation of linear time-invariant continuous-time filters is presented, and several examples of such representations arediscussed. In Section 111, discrete representation of discrete-time signals is discussed, corresponding to the represention of one digital sequence by another digital sequence. The development of the representation of discrete-time signals parallels closely the development of the representationof continuous-time signals. For the repreI. INTRODUCTION sentation of continuous-time signals, i t is shown that requiring I T H T H E increasing speed and decreasing cost of the representation to map linear time-invariant continuoustimesystemstolinearshift-invariantdiscretesystemsis digital hardware, digital methods are playing an imequivalent to requiring that the representation map the Laportant role insignal processing. Insomeapplicaplace transform of the continuous-time signals to the z-transtions, the input and output are continuous-time signals but the processing is digital. In suchcases, the input must firstbe form of the discrete-time signal by a substitution of variables. represented by a sequence; after this sequence is processed, Consequently,inadditiontoitsapplicationtosimulation, the output sequence must then be reconverted to a continu- such a representation is useful in instances such as spectral ous-time signal. In other instances(such as spectral analysis), analysis where i t is desirable for the analog frequency axis to the objective of the signal processing is a set of measurements map to the digital-frequency axis. In the representationof discrete-time signals by otherdison a continuous-time signal; the input must be converted to a digital sequence but it is unnecessary to convert back to a crete-time signals, the notion of mapping the z-transform bya continuous-time signal. There is, of course, a variety of ap- substitution of variables assumes the major emphasis. In parplications in which the signals to be processed are inherently ticular, i t is possible to maintain theform of the spectrum for discrete signals. If conceptually useful, they can be viewed as a discrete-time signal but transform the frequency axis in a representing continuous-time signals; however, this often innonlinear manner. The ability to do this has potential applicatroduces additional complications that may not be warranted. tion in a number of contexts, such as unequal resolution and An example of such a signal is arithmetic roundoff noise or vernier spectral analysis, and the correction of the frequency limit cycles generated in a digital filter. distortion inherent in divers' speech due to the effect of presWhenthe signalprocessing involves a continuous-time sure and the contentof their breathing mixture. input or output, the continuous-time signal must be repre11. DISCRETEREPRESENTATION OF sented by a discrete-time signal, i.e., a sequence. The most CONTINUOUS-TIME SIGNALS common procedure when the continuous-time signal is bandIn this section, the discussion is directed toward the simulimited is to choose the sequence values to be samples of the lation of signal processing techniques which can be carried out continuous-time function equally spaced in time. This representation is commonly referred to as periodic sampling. There with a linear time-invariant filter. While there are many examples of signal processing which do notfall into this category, are, however, manywaysotherthan periodic samplingin which a continuous-time function can be represented by a linear filtering does represent a wide class of signal-processing problems. Just as linear time-invariant filtering plays an imsequence. In this papera class of representations of continuous-time portant role in continuous-time signal processing, linear shiftfunctions by sequences is discussed. In addition, a parallel invariant digital filtering plays an important role in digital signal processing. The reason for this lies partly in the fact notion is developed-the representation of a sequenceby other sequences. Many of the results presented appear else- that linear time-invariant systems, either in the continuouscases, are analytically manageable where by the present authors and by others. Thus to some ex- time or in the discrete-time and consequently much insight into processing signals with tent this paper can be considered as a review of these results. such systems has been gained. For this reason i t is desirableto A major objective is to provide this review within a common framework and, toward this end, results of other authors are choose a discrete representation for continuous-time signals which will permit linear time-invariant systems to be implecombined with our own results and point of view. mented digitally with linear shift-invariant discrete systems. This is similar to the condition imposed by Steiglitz in demonManuscript received November 12, 1971; revised March 13, 1972. strating the isomorphismbetween digital and analog signal This work was supported in part by the Joint Services ElectronicsProgram under Contract DAAB07-71-C-0300, in part bythe Advanced processing [I]. Research Projects Agency under Lincoln Laboratory Purchase Order A linear time-invariant filter is conveniently characterized CC-570, and in part by the National Science Foundation under Grant GK-31353. in terms of the convolution integral so that if x ( t ) , y ( t ) , and The authors are with the Department of Electrical Engineering and h(t) are the system input, output, and impulse response, reResearch Laboratory of Electronics, Massachusetts Institute of Techspectively, then nology, Cambridge, Maas. 02139.
w
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682
PROCEEDINGS OF TEE IEEE, JUNE
(1) where 64 is used to denote continuous-time convolution. Let us denote the discrete representation of f(t), g(t), and h(t) by fn, gn, and h,, respectively. We want fn, g,, and hn to correspond to the input, output, and unit sample response of a discrete linear shift-invariant system,so that gn must be the discrete convolution of fn and h,,: +ro
gn =
fkhn-k
fn
k=-a
*
(2)
1972
convenience we will require that given the set of functions { t$,,(t) 1, the sequencefn representing f(t) is unique. This then requires that the set of functions {r#t(t) } be linearly independent, i.e., that any oneof the functions is notexpressible as a linear combination of the others. In this discussion, we have not assumed that the setof functions { & ( t ) } is complete, but we have assumed thatf(t) is in the space spanned by {4,,(t)}. Equations (1) and (2) can now be used to derive necessary and sufficient conditions on the set of functions (@,,(s) } in (5). Toward this end, let G L ( s ) , F&), and HL(s) denote the Laplace transforms of g ( t ) , f(t), and h(t), respectively. Then (1) can be rewritten as
GL(s) = FL(s)HL(s). (6) where * is used to denote discrete convolution. The validityof (2) for anyfn and h,, implies that the repreFrom (5) and its counterparts for GL(s) and HL(s),i t follows sentation of a continuous-time function by a sequence must that be linear, i.e., if x, and yn are the representations of x ( t ) and +- +o y ( t ) , then %,,+cyn is the representation of x(t)+cy(t) where c gn@n(s) = fkh@v(s)@k(s). (7) is an arbitrary constant. This “linearity property” can be den-rn k-w rived as follows. Let fn denote the discrete representation of f(t) = x ( t ) + c y ( t ) . Then from (1) and (2), If we carry out the substitution of variables n = r + k on the right-hand side of (7), the equation can be rewritten as
2
*
gn = fn
hn n -
since
n -
5
x x++o
gn@n(s) =
fkk-k@n4(s)@k(s).
(8)
-k
But using (2) to express the left-hand side of (8) in terms of and h,,, we obtain
fn
+90
[email protected](s>
that
n -
k-
x
+ o + =
=
n-
Since we require (3) t o hold for any h,,, i t follows that =
fn
Zn
In order that (9) may hold for any sequences f k and require that
+ cy,.
Now let us express an arbitrary sequencef, as a linear combination of weighted delayed unit samples: +o fn
=
fnkn-k@n-k(s)@k(S).
fk6n-k
(9)
k=-m
h,,+
we
@k(s>
(10)
@n(s)@n(s) = @n+n(s)
(11)
@n(s) =
@m-k(s)
or equivalently
-k
from which a general form for the functions {@,,(s) } can be where 6 4 represents a sequence whose values are zero except derived. In particular, for n = R so that 60= 1. Let &(t) denote the continuous-time @n+l(s) - @l(s)@n(s)= 0 functionrepresentedbythesequenceThenbyvirtue of the linearity property previously derived, which is a difference equation in @,,(s) with a general solution of the form = c[@l(s)lrn.
k-
or, for convenience, changing the summation index,
x
Furthermore, from ( l l ) , @o*(s) -@o(s), which is satisfied only for c - 0 and c = 1. Thus the nontrivial solution to (12) is
tcs
f(t>=
fn+n(t>.
(4)
n=--(ID
Alternatively, with FL(s) and @n(S) designating the Laplace transforms of f ( t ) and &(t), then +sr FL(s)
=
fn@n(s>.
(12)
(5)
n -
The relationship between f(t) and fn in (4) can be viewed as a n expansion of f(t) in terms of a set of functions (+,(t) } ; for
@n(s) = [@1(s)ln.
(13)
Consequently, (13) must be satisfied if the expansion in (5) is to result ina discrete representationfor which continuous convolution is mapped to discrete convolution. The condition on %(s) under which the set of functions {&(t) }, corresponding to } as given in (13), iscompletehas been discussed by Masry, Steiglitz, and Liu [2]. Since a continuous-time function can be represented by i t s Laplace transform anda sequence can be represented by its I-
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OPPEXHEIM AND REPRESENTATION JOHNSON:DISCRETE
OF SIGNALS
transform, the discrete representation as expressed in (4) and (5) can be thought of as a mapping from the Laplace transform of the continuous-time function to the z-transformof the sequence.However,amappingfromtheLaplacetransform to the z-transform cannot always be expressed as a mapping from the s plane to the z plane (z cannot be written as a function of s). If the mapping also preserves convolution, Le., if (13) is satisfied, then the discrete representation of (4) or (5) will correspond to expressing z as a function of s. T o show this result, let the z-transform of the sequence f n be denoted by FD(z) so that
683
The sequencef,, then provides a discrete representation of the continuous-time function f(t). Piovoso and Bolgiano [3] have shown that the Poisson transform has the desired property of mapping continuous convolution to discrete convolution, i.e., the Poisson transform of the convolution of two time functions is the discrete convolution of the Poisson transforms of the components of the convolution. In the case of periodic sampling, the function M ( s ) of (16) +m which specifies the mapping from the s plane to the z plane is F&) = fnz-n. M ( j w ) = ejwT. When w is real, corresponding to the imaginary n=-m axis inthe s plane, the magnitudeof MGw) is unity corresponding to the unit circle in the z plane. Thus the representation Since @,,(s) = [al(s)]", then (5) can be rewritten as resultingfromperiodicsamplingmapsthespectrum of the +m analog signal as viewed on the imaginary axis of the s plane FL(s) = fn{[*I(s)]-l)-n. (15) to the spectrum of the digital signal as observed on the unit n=-m circle. In contrast, the function M ( s ) resulting from the use Comparing (14) and (15), it is clear that the variables z and s of the Poisson transform is M ( s ) = l/(l-s). I t is straightare related by forward to verify that thejw axis in the s plane maps to a circle z = [@1(s)]-1 p M ( s ) . (16) in thez plane with radius onehalf and center at z = 1/2 so t h a t i t passes through the points z = O and I = 1. Consequently, i t A common example of a discrete representationthat preserves is tangent to the unit circle at z = 1 but otherwise does not coincide with the unit circle. Thus when an analog function is convolution for band-limited continuous-time signals is periodicsampling. In thiscase,thecontinuous-timesignal is mapped to the z plane by use of the Poisson transform, its represented by a sequence f n consisting of samples of the signal Fouriertransform does notmapontotheunit circle, but equally spaced in time so that rather onto the circle to which the jw axis in the s plane is mapped. This does not, of course, affect the validity of the fn = T f ( n T ) (17) Poisson transform for simulation since both the system input where T designates the samplingperiod. I t is well known that and the system impulse response are mapped in the same way. As discussed by Steiglitz [l],another example of a discrete periodic sampling can be viewed as an expansion of the conrepresentation with the desired property is based on the use of tinuous-time functionf(t) in the formof (4) with the functions the bilinear transformation in relating the s and z planes. T h e { b ( Q 1 given by bilinear transformation is commonly used for designing fre?r quency selective digital filters [4].This transformation has the &,(t) = sin- ( t - n T ) / r ( t - nT). (18) property that no aliasing results wheni t is used to map a conT tinuous-time filter to a discrete-time filter. On the other hand, The Laplace transform of the functions in (18) converge only the use of the bilinear transformation introduces a distortion on the j w axis. On the j w axis their transforms are inthefrequencyaxis;theimaginaryaxisinthe s planeis mapped nonlinearly onto the unitcircle in the 4; plane. A common design procedure in discrete-time filtering is toderive the filter from a continuous-time design by means of the bilinear transformation while the discrete-time signal is obtained from Thus we see that the transform of thesefunctionshasthe the continuous-time signal by means of periodic sampling. Bedesiredform as givenin(13). The advantage to a discrete cause of the distortion in the frequency axis resulting from representation based on periodic sampling is that the coeffithe bilinear transformation,thisprocedureisgenerallyrecients in the expansion are easily obtained. Furthermore, as stricted to situations inwhich the distortion in the frequency opposed to many other representations, the function f ( t ) is not axis is not important. This is true, for example, in the design required for all t in order to obtain each of the sequence values of f,,. The major disadvantage to this representation is that of digital filters with piecewise constant frequency characteristics. i t requires that the functionf(t)be band-limited. Whenwe are If we wish to use the bilinear transformation for simulation considering simulating a continuous-time system digitally, the when the filter characteristics are not piecewise constant, the frequencyresponse of thecontinuous-timelinearsystemis same nonlinear distortion must be applied to the frequency usually not truly band-limited. In addition, periodic sampling characteristics of the input signal and the output signal; both corresponds to a linearrelationshipbetweenanalogfrethe signals and the systems must be mapped to discrete signals quencies and digital frequencies. As we will see in a later disand systems using the bilinear transformation. Since the bicussion, i t is sometimesadvantageoustohave a nonlinear linear transformation corresponds to a mapping from the s relationship between analog and digital frequencies. plane to the e plane with Anotherexample of a discreterepresentation that preserves convolution isthe Poisson transform. For a continuoustime function f ( t ) that is zero for t =
..J: 2*
[
w
a
f(t) using only a finite number of nonzero terms by N-
As an approximation, let us assume that the effective width of the spectral window W(Q)is sufficiently small so that over
with the exact representationof f(t) given as
the width of the window, the bilinear frequency transformation of (35) can be approximated by a linear characteristic. Then (36) can be approximated by
OD
Cfn+n(t>* n-0
If the coefficients fnd are chosen to equal the coefficients for n = 0, 1 , N - 1, and since
fn
then f A ( t ) will correspond to an approximation to f ( t ) that minimizes the weighted integral square error given by
S,
+=
E
= error =
t [ f ( t > - fd(t>]zdt.
(32)
Hence, generally speaking, truncation weights errorsfor large to ap-
t more than for small t so that we would expect fA(t) proximate f(t) more closely as t increases.
More generally, rather than obtaining the finite duration sequencefnd fromf, by simple truncation, i t can be obtained by “windowing” : fnA
=
wnfn
(33)
-
where wn= 0 for t~ =
7
2 arctan - - 2 arctan -
FA(Q) = ---J-=W(Q
- cu)F(cy)da
(34)
i.e., multiplication of the squence fn by the window w,, corresponds to “smearing” the transform of the sequencef, with thetransform of w,. Alternatively, we can view thisas a “smearing” of the spectrum of the original time function in such a way that spectral resolution decreases as the frequency increases. T o see that this is so, let fd(t) denote the continuous-timefunctionrepresentedbythe sequence fnw, and let P ( w ) and G ( w ) denote the Fourier transforms offA(t) and f ( t ) , respectively. From (21) with s = j w and z=eJ*, the rela-
The argument ( w - 8 ) [ 2 a / ( a P + w 2 ) ]can be interpreted as a linear scaling of the spectral window, while maintaining the shape so that the window becomes wider as w increases. T h u s the distortion due to truncating the representation by the application of a window can be viewed in terms of ‘smearing” the analog spectrum in such a way that spectral resolution decreases as the analog frequency increases. One of the advantages of a discrete representation based on the bilinear transformation, as developed above in contrast to periodic sampling, is that the analog signal need not be band-limited. On theotherhand, periodic sampling is easier to implement and theperiodic samples can be obtained without buffering, in the sense that periodic sampling is a memoryless transformation. Bilinear sampling requires the entire waveform to be available before any of the values inthediscretesequencecan be obtained,corresponding to infinite buffering of the input. For the simulationof an analog filter i t is possible to avoid the need for infinite buffering of the input by sectioning. T o illustrate the procedure, let us consider an analog filter with impulse response h ( t ) , system function H ( s ) , and input ~ ( t ) . The discrete filter used for the simulation will haveaunit sample response It, where, assuming the system is causal,
Now the entire input x ( t ) can be expanded in terms of the (+,(t) 1, yielding a sequence of coefficients X,, this sequence filtered, and the resulting outputyn used to construct thecontinuous-time output y ( t ) as
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PROCEEDINGS OF THE IEEE, JUNE
686
1972
Equation (42) can be viewed as a n expansion of the sequence in terms of a set of sequences { # k * , n } , with the sequence y representingthe coefficients intheexpansion.ThustheseHowever, as has already been discussed, this requires infinite quences { # k , n ) play the same role in (42) that the functions IMO 1 play in (404. buffering since no output values can be obtained until all of Since the mapping from f(t) t o j,, and also from f(t) to y the inputis available. As an alternative, let us divide the input preservesconvolution, i t follows t h a t a n expansion of (42) ~ ( t into ) sections of equalduration T so that,assuming must also preserve convolution, i.e.. if f,,=fi,,, * ft,, where * .(t) = 0, t
