BSTJ 60: 7. September 1981: Digital Signal Processor: A Tutorial Introduction to Digital Filtering. (Angelo, E.J. Jr.)

23.2 MB·English

Digital Signal Processor:

A Tutorial Introduction to Digital Filtering

By E. J. ANGELO, JR.
(tare serit rived Juy 11, 1980)

Verplarge-scae inteyrotinn (ext of liga eletroniecreuite has
changed the hardiare aspects of digital filters im a major way 80
‘hot the use of such filters ax componente in commercial systems has
Become both economically feasible and technically desirable. Thus,
large numbers of system engincers and circutt designers ore man
finding « need to learn about the properties of such filters, how they
‘are usd, ah hry rates. This paper tea fra step toward
meeting the need.

1. mtRopUCTION

‘The possibility uf doing fering wad ober signal-procesing opens:
tions by murmerical roeans instead of by traditional analog means has
‘been kon and studied for 2 years r longer. However, unl rently
the bardware for the physical eaizaion of digital Beers nas been,
Ith, powershungry, and expersive, and for thie reason. the diel
‘ker hae no beet suitable for une av a commponent in commercial
systems, Thus, intrest in digital filters has orn lal toa elaively
small number of speciosa doing esearch in thie nd ela Geld,
‘here size, power consumption, and cost are not primary uesidera

Tlowover, vist has changed this condicion drastically. Is rare
the sine, power consumption, ad cost of digital filers to the point
‘where their use as © system: component is both economically fsa
and technically desirable. Asa result, large numbers of ste on
enna nnd circ derigners ate finding « need to learn about digi
Fikers. Theretore, there space for torial material addressed
seins a the nets of these pepe
“Tis puper is an atlempt to meet this nocd and adeveszes syste
cengineets and circuit designers having no previous experience sith
tigtl ers or siplelvista sycems. Homer, they are asnumed to
hhave a good understating of the Laplace transform ad te use with
‘agpela diferent! equations, and eielri rims We hope co provide
ft goud undarstanding of the fundamen of dial filtering and a
Strong fount for farther etady of the subjecr. To reach thes
Sbjectives mast effectively, an allen # mie to avoid all unnecessary
Abstractions: Generali i sacrificed forthe woke of simplicity.

1 ELEMENTS OF DIGITAL FILTERING

"This ert gives an ingoduetion tv dela tering in toms of the
rary evit ehowm in Fig 1. This siple cizest ean be usu lo

llseeate bow ivering ix dane in che distel domain, fo euntrant with
the more usual ease of Filtering inthe analog demain.

“Tho cre deseibed by the fellowing single node equation for
the sing unknown voltage

a

which i fistonder Linear dilfventil equation ip the wnlnown
voltage

Tn thie exttaple, ty and v. azv undersod Ls be information bearing
signals, Por wxampl, ty may be che gulp ollaye of «stv gauge,
the output an aiseleomete, ar the ourpat of « Ielephone wananic
ter The infortation-ssrain, sceseralion. or such i represenced
I i nmplizule ft, Tha voeaga ss cepresents the information ater
ith teen procesred bythe [UC cre filler). The proceed form of
the information, rin ie wore valoable than the orginal form,
‘because for example, hilctrequmneyneine bes been removed frm he

41600 THE BELL SYSTEM IECHWIGA! JOURNAL, SEPTEMBER 1381
eee of aga awe ain he

signal. ‘The amplitudes of the voltages 2, and e» are the physica
analogs ofthe ariginal informalion-—strain, acceleration or epoeeh-—
and the physical sysiem represented by Fig. 1 il lp be an analog
system.

CConsier further the signal rf), fov example. Tn che mathewatical
repmesentation of eg, (2), voltage 1s reiresents a vontinoun ei i
fan represen the values of any and sll weal rambers, and theve arena
real numbers tha eennot be values of» Thi, rzean change esnootly
from any one value to anether without any juny orciseontinutie,
‘Simiorly, time #in og. (2 ropresenc a continu, and it ean ebaage
smoothly trom one value vs tmocher wichout any jumps or dieeatrin-
Licies Morenver when int) presenta physical quai, sit alway
does in the systems under anidy here is efined thas a numerical
vale) for every value of time & ‘These analog, emiderations are
mentioned here because, in nanirnt, muciers are quite ferent In
ltl fers and in dgical systems

Equation (2) can be coved easily iy anulyie aun forthe unknown
voltage os then the input voltage 4 snus Eunetion of time, a
exponential fanetion of ime, oF a aeep fonction of time. Ie ean ale he
solved acnytically, bue wieh more dificdiy, when or a eqaaee wave
and als, with sil more difieaty, whew iva more general peo
Tanetion of time

‘When the input signal voltage in Fig. 1 ic more complicated
fanetioe of tine thas i the few examples ete above ts wsuay aot
practical, or even posible, co volve eq. (21 By analytic means, In such
fases, however, ik it posible to obtain an approximate solution by
hnumerical methods. Thee numerical methods provide lw basis for
‘dgial Bering. The numerical ethics uppropaiate t this study are
based on considering only discret values ofthe eigals, voltages
land tin Fig 1, chosen al wniformly spaced instant of time, These
discrete values ofthe signals nr elled samples ofthe signals, Figure
2B shows the waveform of x continuous analog sgnul vellage, sad
Uniformly spaced seiples ofthe signal are nfiatel on this diagram,

IF waveforms forthe analog siganbs», nt ein aq (2) exe, then

DIGITAL FITERING 1501
sequences of samples similar vo (he une in Fig. 2 also exist foro: and
ty Assuming the seme sampling intanie forthe two signal these
Sequences can be represented symbolically, starting et sume instant
designated ¢= 0, a=
nT) 00), 0
(nT) 240), 24), O12, 0 URT 2 ®

1, 827), «0, IAT,

[Nom if the time inenval between ssunple is eufiieny seal, the
Alerivative fae, (2) ean be appeoxinatad at time ¢ = nT by

dx _ dn) ~ hin = VT]

ca T
‘As Tis mach xmaller, the approximation becomes betta. Then, if Tis
tmade sulficienely mall, (2) canbe spproximatet for one instant of
time, = nT, by aubetituliny ef} inca ea. 2) to Ret.

Re Re

any

Equation (6) is linear difference equation thal approximates the
Tinea diferendal equations given By eq. (2) above for ue insane of

c)

ose) + AIT] war

"The difterence ey (6) offers a psibiicy that not offered by the
Alifarenial eg Ii cn be lve expliciy for che response vn)
vw abiain

pie + ale]

1 RevP
Teor Kcr
ain) + bute ~ 197), o
‘This equation gives the present sample of the response volage
tw?), 9 singe sample valve in terme of the present sample of the
inpoc voltage, einT), and the immediately proveding sample of the
response vellage, a(n — UIT]. IC rye) is « krown sample of the
Input, and if fin — 1/7] i known from a previous caleulation of
‘eT, then the presen. valve of tn) ean be calculated by simple

Tthmetic from of (7) assuming, of cours, ha the coeficients @ and.
bare known, This alculation can be repeat fr aueassive values of
tun) and wll ~ 117 0 abton the sequence of output samples for
aint).

Tn tho calevlation deseribed ahve, the present response sample
amin) depends on the prosent input sample vin?) and on the im
imetiataly preceding response sample of" ~ 1)T. Thus, eq. (7) is

in which each calculation of xin) provides
the value of fin — 1/7 forthe net eleulation in the sequence. The
Start-up hehavior for this sys races wn additional question, but one

vin)

41602 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1901
ono verious consequence. Ifthe start-up instant is designate ¢ = 0,
Thich for the aiserete samples corresponds e-n— O, then e0) is
sasumed! tobe known, and in addition, the value of x(~T) ts need
forthe last term in en, (7) wheat n = 0, This lat requires sina
(0 the requirement that the initial conditions be Inown when a
Aiferencial equation ist be sulved, Ifthe value of wT) is known
from phyrical comiderations, there is no problems, If vx") is not
lenown, a reasonable value, such ns zero, can be astusned, and the
sequence of calculations can begin In this case, the Mt few samples
of zsv7) chat are calelated wl depend on the assumed initial value
of afin ~ 117} However, the effet of the assumed initial condition
‘decays with time, and it soon disappears.

Example 1

‘The simple Ger of Fig. 1 is represented approximately by the
Aiterance . (T,

ae) = ansin) + Deda —T%
‘Suppose cha the input to the filer
BT)= 1, =o,

=o, ro,

and suppose that
ber) = 0
‘then, the reponse 127) can be calulated as follows
nl =a+0= a, forn=0
eT) 04 ba = ab, form
e:Q7) 0% blab) ~ at form
(GP) = 0+ deb") — ab fen

tale) 0+ bode NTI ub, for,

1 follows from eqs. (6) awl (7) hal Une coeticiente @ and B ate
nonnegative and len than wily. Ths, the response sin decays
‘with time and tends to zero aan incronses without lat. Thi problem
{is treated again in Seeion 42 from 2 dire pint of ew

In the approximate representation of the civcuit in Fig. 1 by the
liference ea, (7), he time continuum doer not ea; ie has been

Dismal FILTERING 1500
replaced hy a aequence of discrete instants of time. Tho sequence of
Eales ov), for example, i 2 dinerete-time sighs, in contrast ro
the corresponding wf), which it @ continuous-time signal The dis

cecetime signal is defined (hati tas a specific numerical valuel
bac the digeroteinstancs of cme ¢ = m7, whoraas the continuous:

time signal i defined for every iestant of time, The samples of the
tisretetime signal ex(n7), lor exempl, have finite amplitudes snd
zero time duration,

‘e shown above, the vali ofthe present response spl ain) in
the circuit shown in Fig, 1 can be ealeulated fom cq. (7) by using
Simple arinetie. Only mltiplication and adiition (of signed num:
brs) ate require, and i ix pticlany usefal inthis study (o think of
this arithmetic as being vero on a pocket-sized cleetronic cae
lator. This true because the ealculsior ta gil machine that has
‘much in cannon with the digital ler. Al ofthe cams on the right
hhand side of ea. (7) ave entered into the calculator in digital form
(Goimal ligt) through the keyboard, and the result of the calculation,
tan is presented in digital form (lceimal digit agin) on the output
‘display ofthe ealelatar.

“Any givon electron ealelator hs a fted number of digits in its
coucpur display, and fe follows from this fel. that only a finite number
‘tabscrete values can be diplayed onthe output. Thus, ifthoealeutator
isused to evaluate la) foun ey 7), then fea longer Feprewent
‘contour of values; ica represent only a finite numberof discrete
‘alues gives, by the digits dleplayed on the output ofthe ealelator.
"The information ia thi ense i not represenced by the amplicude of
hysieal variable, but rather, i is represented by che digits displayed
by the ealeulator. Therefore, by detniion, the information is notin
tnalog form, end because of fe form, 3 i called digital information.
‘The at of digits reprenetng the information is called a digital signal.
In the case of Binary aystems, the information i represented by Une
binary-digit (it) pattems easoeiated with hinsry numbers.

‘Since the dgial signal producod by Ue calculator and by the digital
filter can have only 8 finite numberof "allowed" values, the signal is
‘quantize, One remale of quantization tthe introduction of random
‘rrors called quantization noise, Anuther reault ia the existence of
‘nonlinear feedback loops in muy digital Mters with the Lkeibood of
SolEsustaining oellations called limit cycles. However, these are prob
Temsaswocinted with the design and performance ofthe hardware used
te realize digital filters and ie isnot appropriate to discus them here.
Detaled treatments of these problems are given in Refs {ubrough 3.
"The reminder ofthis yaper saoumes that che number of digits avall-
able for repretencing signal s unlimited,

Tn effec, sampling the analog sgal shown in Fig. 2 changes the

41608 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1681
continwoustime reprsuntation of te signal to diserete-tine repre
sentation. Converting the analog sample amplitudes in Fig, 2 ta equi
lent digital values changes the continuous-aznplitude representation
toa discreteamplitude representation,

To summarize developmencs up ta this poin, the creult of Fig. Ls
chosen ax a simple filer to be studied for the purpose of geting an
Intraduction ta the ideas of digital sltering, The raat is analyzed by
Standard techniques to obtain an analog representation in terms ofthe
diferencia equation (2). Thon, we imagine that tle wgnal voltages
tnd v, are sampled co obtain the difference 2, (7) as an approximation
to the differential equation. Next, we envision an eleeconieealetator
for evaluating 69. (7) by numerical methods, Data is entered into the
calculator in digital fore, and the result of one ealeulatinn is the value
fone saraple ofthe flier sesponse vin) in digital fore. This eyele
tf ealelation ix repented snccessively with successive samples of the
input volage nin7) to oblain successive values nf the output voltage
wn

‘Consider che eate in which the input to the filter ia Fig is» voice
signal To represent this signal wilh quod accuracy by a sequence af
Samples, tho signal must be ramped bout 10000 times per second
(Tho sampling is examined in detail in Sootion 3.21 This fact implies
{at for each one-second interval of speech, ebout 10,000 samples of
the response e{n7) must be ealculated. although the calelation af
‘ach eurput sample i imple enough, ealeulaing 10,000 vf Uhem wich
‘manually operated calcula takes guite 2 while

‘The etage Is now sct for the introduction of the digital signal
processor a a means forimplementing digital filters, The dial signal
processor i a digital devire (binary digits) that has been especialy
designed to perform the arithmetic equired in the repetitive evalu
tion of che diference equation desreliol ubuve It dows the arithmetic
automatically st very high speed uraler program control. When pro
grammed to solve eq (7) it can accopt a now input signal amp in
ligtal form, calulate the corresponding ceeponce ample, and deliver
‘tho response to the output all in lows than 8 ga. ‘Thu, the signal
proceasor can receive anew input sample and calculate the correspon

response ina tin fraction of che interval between aeoesive inp
samples, an interval of 100 us at 10000 asmplea per second, I follows
from these fact that the processor, with ita lazing speed, can operate
in veal time, solving eq. (2) nlm instantly forthe response (0 euch,
Jnput aample and Usen wiiting for the next input sample to come
long. Thus, by solving eg. (7) in fea time, the processor produces in
sampled digital form the same response to the input signal vs ws the
fcr in Fig. 1 within the accuraey permitted by the approximations
involved in deriving en. (7) and in sampling

DIGITAL FILTERING 1505.
“The fiter shown in Fk. is an analog computer thar colves the
ferential aq.) in neal time. Siaaey, tbe digital sigmal processor
ise digital computer thal solves rhe itfrence eg 7) in real ime. To
the extent that e9,(7)is a good approximation te, (2, the processoe
{se quod approximation tothe fiver in Fig.

"The ideas developed ubove make the concepts of digital Mtering and
live use of the distal signal proresaor for its relation seem to be
‘ite simple, While the basic ideas are inlond perfectly sraghtfor.
‘tard, the implementation of high-performcce Mtaa by these means
ina reliae system environment presents u challenge.

rst, the question of how well difference eq. (3) approximates
liferential eq. (2) hs been taised above, Insofar aa the filtering
‘operation is concerned, this question requires a deailed answer and a
Inore complete mathetnafiea! formulation of he problem than we have
{0 far given, ‘The remainder of the paper concorns this and related
problema

‘Second, the simplicity of the digcal Alter ax prewnled above is
genuine, bot in u way it is deceptive. The starting point for the
proventation above i chosen co bypnes al of the challenging preimi-
fry work thal ts now to put the rel wryinnering problem in:o a
form that can be inplemmnced hy the digital sinal procesor.

“The linear difference equstiot ithe central element in the concept
ofthe digital filter. Tn the exaroplereprecented in Fig. 1 the diference
‘equation is oblainel hy approximating che differential. (2), which,
fn bara, is obtained deel from the event attmed in Fig 1. How
ter bw design of ral flere rary thi kind of staring poinc. For
‘reson whi stan party fom lechnologiealheriago and partly from
Inathematicl taetaility, Fler desig usually start with w scifi
tio af the frequency eharucersties chat are desired ofthe filer Pasi=
band charaeterties and frequencies, top-band charactoristes and
frequencies, delay charaeterstis, andthe Ike "Then inorder (o obtain,
s realization in digal form of a lier having these specified character-

ies it x aeceaary to determine, in nome way, a licar diference
fquaton describing 9 fleer having the specified charwctersties, The
‘igtal signal processor ix en programmed to evaluate this diference
‘equation by arithmetic wperations.

The most sual may of designing « digital ter isco stare with
classical analog filler henry. Given areaizble sel of frequency char
‘cteristice, classical theory ean be used vo derive the analog transfer
function for an analog fter having the specified eharacceritis. Cor
responding to this transfer function there i always llferenial
‘equation euch as the ane in eg, for example. In principle ic would
be possible to proce! ag in the example in Fig. 1 and use chis
liferential equation ta derive an approximately equivalent difference
‘equation. However an alvemative proveshre proves tobe more fruitful.

41808. THE SELL SYSTEM TFCIIMICAL JOURNAL, SEPTEMBER 1981
[Byers differential equation relating an output signal and its derive
tives to am input senal ond ite devintives gives rte, throwgh the
Laplace transform, to am xeloy transfer oneion, Sis, we
stn in Section TV, every diference equation relatiag Ue erent
tnd ast values of ah outpuc sgaal tothe present and past values of
fn inp signal gives ie, chrough the 2 transform, toa digital anster
Function. Tis shown in the vemainder of this paper that he digital
tranafer function ia related to the digital filter and ita frequency
‘haraterstice in much the same way se the analog teansferfueton
is related Go the analog fier, The = imnsform isa special form of Che
Laplree transform that it developed in some detail in Section 3.1

Furthermore, we show in Section VT that an analog transfer funeson
at le Lanford ino digital Crna fet imal wa ta
he frequency charactevitics of che two functions are related in a
presiely known manner. Thus in many eases clawieal techniques can
lie ure! to derive @ protolyne analog traneferFunelion tat can be

nnaformed into a digital transfer fanetion having Uke Aesied fre

‘quency characeristex The diferanee equation corresponding (this
‘dg tranefer function can be derived early, se shown in Section 44

dnd it ean then be implemented with hardware to obtain a physical
realization ofthe digital er.

"Vhe thd cutlind ahve ia the most common, fut he ony
‘athed used for designing digital filters. The remainder of chia paper
ives the details ofthe metho. Hovtever, the treatment is necesatly
Inuredustory, and makes no attempl to provide any expertise in the
Feld (Seo Refs. 1 through for ela tentmen. ofthe steel)

lL THE £ TRANSFORM FOR SAMPLED SIGNALS

Section T introduced digital iering in terms of a simple example,
‘The example reveals samme of the approximations involved in digttal
filtering, ane pointe out the need for a more comprehensive maine
savieal formulation of the problom ‘The 2 transform is che mathe
‘matical tol that i extensively ured for this purpose. "The objective of
‘his action i tm presen the 2 transiorm and to develop its properties
tw the extent required by this paper.

{8:1 Dofinton an elementary properties ofthe 2 transform

Ths general, digial filtering and digital signal processing concern
processing sghale thet aze characterized by a sequence of values,
‘mally igtantaneous samples of continuous-time signal, ebotnze
‘uniflomaly spaced in time, Sueh signal re diecursed in Section IL. Por
Ue purpoce of this sty, we consider only signals thet are zero for
‘me f less than some instant designated ¢ = 0.1 is also assumed that
Ue signe to be sampled is vontinuous al every sampling insta,

DIGITAL FILTERING 1507
eroepc posh at t= 0 in which cage the value of the signal at ¢ =
32 taken by convention.

Figure 2 shows & contiouous-cane signal and sequence of disciete
samples of chat signal. The frst step in che method used here for
Aleveloping che 2 transform is to note that signal samples of zero
duration cannet exist in any phyeies aystem. Any sampling operadion
that is implemented in hardware is necescarily atsociated with a
holding operation that produces signal samples of nonzero duration,
The moet widely used semople-and-bold ercut has the fra shove in
Fig 2a, and when its inpoc voltae ithe concinuous time signal voltage
show in Vig. 2, the cieuit produces a stainarep outpot-voltage
waveform shown as % in Hg. db. It is also important to nove thal
Aigcl filters are ohen followed by a digital-to-anslog converter to
provide an output sigoal in analog form. In mary eaces the converser
Produces a stalstep augpbe wavefoem like the one shoven in Fig
‘Ths slaiatep waveforms play 2 ene role i the analysis of digital
Millers woof sampled-data systems in ener

‘A reat deal of valuable information about the sampling process and
Ihe susratep ouput wavelorm ran be obtained from the Laplace
lrasfoenr ofthe stirs nigh. This iransfort is

van = [sane va e

oa

Fi 2 Same nl ini ft Ess 0 Wee

1508 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMOER 1901