1959
`
`PROCEEDINGS OF THE IRE
`
`1219
`
`A Discussion of Sampling Theorems*
`
`D. A. LINDENt, ASSOCIATE MEMBER, IRE
`
`The following transform definitions will be used:
`f +
`F(f) =
`
`w ,
`
`2irf
`
`-00
`
`r+c
`
`f(t) = f
`
`f(t)e-i-tdtI
`
`F(f)eiltdf.
`
`It will be conveniienit to use the niotationl
`
`a(t) * b(t)-3
`
`a(r)b(l - r)dr.
`
`Following the noomenclature of Kohlenberg,2 samiiplinig
`of a time function9 will be designated as first-order if
`the sample points are equispaced. Seconid-order sam-
`pling involves two interleaved sequences of equispatced
`samiipling points.
`
`SAMPLING OF Low-P.Ass FUNCTIONS
`The simplest case is that of a time functioni f(t) whlose
`spectrum F(f) is limited to - W.f < W. The restult of
`sampling the function at regular initervals spaced T
`seconds apart is'0
`/(I) = f(t) E 6b(t - nr) = Z f(fT)6(I - Tr).
`
`(1)
`
`The tranisformli of
`
`Eb(t - IIT) is ,
`
`a(
`
`Multiplication in the timie domaini corresponids to coIn-
`volutioni in the frequenicy domiaini, anid thie first equalitv
`of (1) leads to
`
`1
`
`(
`
`F)
`
`Summary-The convolution theorem of Fourier analysis is a con-
`venient tool for the derivation of a number of sampling theorems. This
`approach has been used by several authors to discuss first-order
`sampling of functions whose spectrum is limited to a region including
`the origin ("low-pass" functions). The present paper extends this
`technique to several other cases: second-order sampling of low-pass
`and band-pass functions, quadrature and Hilbert-transform sam-
`pling, sampling of periodic functions, and simultaneous sampling of a
`function and of one or more of its derivatives.
`
`INTRODUCTION
`S EVERAI, sampling theoremiis have appeared in the
`enigineerinig literature.'-' These miiay be derived in
`aI particularly perspicuous mannier by means of the
`convolution theoremii of Fourier analysis. The samnplinig
`process is regarded as a imiultiplicationi by a periodic se-
`(iuenice of 6-funlctioIns, its couniterpart in the frequenicy
`(lomain being a conivolutioni by a traini of equispaced 6-
`ftunctionis. Interpolation-the recovery of the original
`is viewed in the fre-
`signial fromii
`its sample values
`qIuenicy domluainl as a process of reconistructinig the orig-
`inial spectrum by miieanis of a spectral "winidow." The
`corresponiding time domiiain operation con1sists of the
`conivolutioii of the samiiple imiipulses with the iniverse
`Fourier tranisformii of the winidow fuinction1. This ap-
`lproach has been- used by a iiumber of authors6" to dis-
`cuss the equispaced samuplijig of low-pass functionis. It
`is the purpose of this paper to present a consistent set of
`lheuristic derivationis for a numiiber of additional samil-
`plinig theoremiis.
`
`* Original manutiscript received by the IRE, November 10, 1958;
`revised manuscript received, March 30, 1959. Part of the work re-
`ported here was done under Nat'l. Sci. Found. Fellowship No.
`28,215. Space and facilities were supplied by Office of Naval Res.
`Conitract No. 225(44).
`t Stanford Electronics Labs., Stanford University, Staniford,
`Calif.
`I C. E. Shannon, "Communication in the presence of noise, '
`PROC. IRE, vol. 37, pp. 10-21; January, 1949.
`2 A. Kohlenberg, "Exact interpolation of band-limited funcltions,"
`J. Appl. Phys., vol. 24, pp. 1432-1436; December, 1953.
`3S. Goldman, "Information Theory," Prentice-Hall, Inc., New
`York, N. Y.; 1953.
`9 L. J. Fogel, "A note on the sampling theorem," IRE TRANS. ON
`INFORMATION THEORY, VOl. IT-1, pp. 47-48; March, 1955.
`5 D. L. Jagerman and L. J. Fogel, "Some general aspects of the
`sampling theorem," IRE TRANS. ON INFORMATION THEORY, v7ol.
`IT-2, pp. 139-146; December, 1956.
`6 P. M. Woodward, "Probability and Information Theory, with
`Applications to Radar," McGraw-Hill Book Co., Inc., New York,
`N. Y.; 1955.
`7R. B. Blackmani and J. XT. Tukey, "The measurement of power
`spectra from the point of view of communication engineering," Bell
`Sys. Tech. J., vol. 37, pp. 185-280, 485-569; January and March,
`1958.
`8 J. R. Ragazzini and G. F. Franklini, "Sampled Data Control
`Systems," McGraw-Hill Book Co., Inc., New York, N. Y.; 1958.
`
`n,
`
`T
`
`)I
`
`1
`E 1F(
`T
`Apart from the weighting factor 1/T, F(f) is seeni to con-
`sist of replicas of F(f) cenitered onl the spectral linles
`b(f-n/r), as illustrated in Fig. 1.21 The possibility of
`recovering the original spectrumn is insured if l1>.2W;
`equality is permissible if F(f) does nlot containi a b-fuLIc-
`
`(2)
`
`9 All time functions are assumed to be real unless specifically des-
`ignated as being complex.
`10 All summations are from -oo to + oo unless otherwise stated.
`11 F(f) is in general a complex function and is iindicated symboli-
`cally in Fig. 1 (a). Weighting factors such as 1 /r will be indicated as
`shown in Fig. I (b).
`
`RPX-Farmwald Ex. 1027, p 1
`
`

`

`1220
`
`PROCEEDINGS OF THE IRE
`
`-fitly
`
`TV. Assuminig that sampling takes place at
`tion at J
`the lowest permiiissible rate, one has 1/r = 2 TI.
`TIhe
`originial spectrumii nmay be recovered by multiplyinig
`F(f) by the spectral window
`funictioni S(f) slhowni in
`Fig. l(c). The equivalenit operationi in the titme domanll
`is the conivolutioni of f(t) by the iniverse FoLurier trmns-
`formii s(t) of S(f), i.e.,
`f(tT)6(t - liT) = I: f(11T)S(/ - HT).
`
`f(t) = s(t) *
`
`Substituting T = 1/2 W anid the funictionial formii of s(t),
`
`(a)
`
`(c)
`
`T
`
`_
`
`F (f I
`
`F
`
`i
`
`i
`
`w
`1~~~
`
`(DRAWN FOR - >2W)
`
`C
`
`,
`
`L
`
`0
`
`t
`
`t
`
`rsKNXN'XN'Xe4,. V--llll
`-<
`--- -
`O
`
`-
`
`-
`
`S (fl
`
`sin 2ir (tV t
`
`Fig. 1--
`
`[i-it-
`
`o(l(1 l-
`
`nllpliiit
`
`I~
`
`of lo\- v'a fun1ctionX.
`
`f(t)= Ef2
`
`27r 1,1'
`
`2W()
`tI
`\2 Tf17
`
`The low-pass funiction f(t) may also be subjected to
`secon-id-order samiiplinig. The twvo interlaced sampllinug
`trainis
`
`and
`
`t
`
`-
`
`f, 5
`
`I -
`
`- a
`
`F (f)
`
`-W
`
`O
`
`W
`
`(Ab
`
`~
`
`\
`
`f
`
`F A>,
`
`-_~~~I
`
`will be designiated by the letters .1 antd B, respectivek.
`The sanmpled funictionis are
`
`f-,(1) =
`
`(
`
`'
`
`(4a)
`
`(d
`
`~~~~~~~~~~~~~~~~~~~f
`
`exp
`
`A
`
`t,R-7r']
`
`2W sin 2$
`
`tof
`
`antd
`
`fB(t) = fE(
`
`+a)6(t-- - a)
`
`(4b)
`
`iSB(f
`
`io //A
`
`\\\ss A
`
`\u
`.v.
`
`~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~f
`
`F'ig. 2-Siecollid-order saiiipli'tg of loxvs-ptls,; fetlictiott.
`
`anid the corresponidinig spectra are giveni by
`ll%6(f - OiF)
`FA(f) = F(f) *
`
`It
`
`(Sa)
`
`Inspection of Fig. 2 yields'3
`
`FB(f) = F(f) *
`
`j, I'l-
`
`b(f -1ff-?
`
`(5b)
`
`exp i,.
`
`y=exp (i27raWV)
`The results of these conivolutionis are easily visualized:
`sketches of the spectra are shown in Fig. 2.'2 Since all
`time functiolns inivolved in this discussion are real, it
`suffices to conisider their spectra for positive frequenicies
`otnly. The spectral winidow functionis SA(J) anid SB(f)
`mav be determined by the requir-emenit
`
`(,c
`
`WSA(f) + 11SB(f) = I
`
`IS S (f) +
`
`SB(f) = (
`
`whetice
`
`S.1(f) - 5SR*(_f) =
`
`-y
`
`ex
`
`(i
`
`0
`
`1)2
`
`2 W.
`
`sin
`
`1
`
`2
`
`FA(f)SA(f) + FB(f)SB(f) = F(f),
`
`0 < f < IlF.
`
`(6)
`
`(0 < f < 1W).
`
`12 Each spectrLum is showni as the st>um
`of two conipouneuits which
`correspotud to the convoluitions of F(f) with differenit spectral lities
`of the samiiplinig futnction.
`
`1 It is showni in Appenclix I that tieseh e(ljutionis f0lowx mllquiell
`fromii (6).
`
`RPX-Farmwald Ex. 1027, p 2
`
`

`

`1959
`
`Linden: A Discussion of Sampling Theorems
`
`1221
`
`Since
`the corresponding time functions
`real,
`are
`SA,B(-f) =S*A,B(f). The inverse Fourier transforms of
`SA(f) and SB(f) are the interpolating functions
`cos (2irWt - raW) - cos 7raW
`2irWt sin iraW
`
`SA(t) = SB (-) =
`
`(7a)
`
`Finially,
`
`f(t) = SA (t) *fA (t) + S (t) *fB (t)
`
`=
`
`E1(;) SA( -;)
`
`+(-+±i )SA(-tA+ )i .
`
`(7b)
`
`With a = 1/2 W, (7) reduces to (3).
`
`SAMPLING OF BAND-PASS FUNCTIONS
`The spectrum is assumed to occupy the ranige
`Wo.<If <(Wo+W), as sketched in Fig. 3(a). In gen-
`eral, second-order sampling must be used,14 and (4), (5)
`apply. The results of the convolutions are shown in Fig.
`3(b) and 3(c). The spectral window functions SA(f) and
`SB(f) which are required to restore the original spec-
`trum may be computed by a procedure similar to that
`leading to (6).1" The result is indicated in Fig. 3(d). The
`corresponding interpolating functions are'6
`
`F(f)
`
`0
`
`(m-I)W
`
`Wo
`
`m W
`
`(W0.
`
`i
`
`I f
`No
`
`{
`
`FA ff
`
`}
`
`FB f)
`
`SA(-fh)= S* (f )
`
`;
`
`S (f )
`
`SA(-f)
`
`Fig. 3-Second-order sampling of band-pass function.
`
`SA(t) =
`
`cos [27rmaW - 2r(W + WO)I] - cos [2rmaW - 2r{ (2m - 1)W - WO t]
`2rWt sin 27rmaW
`cos [(2m - 1)7raW - 27r{(2m - 1)W - Wo}t] - cos [(2m - 1)7raW - 27rWot]
`27rWt sin [(2m - I)raW]
`
`SB(t) = SA(-t)
`
`(8)
`
`where m is the largest integer for which (mr-1) W < Wo.
`Eq. (7b) applies provided that SA(t) is taken to be the
`function defined by (8). The separation a between the
`two interlaced sampling trains is arbitrary, except for
`the restriction that it may not be an integral multiple
`of 1/2W unless Wo = (mr-1) W. In the latter case, a de-
`velopment based on the first-order sampling of (1) and
`(2) yields the interpolation formula
`
`f
`
`)
`
`n
`
`(=22f(
`s(t
`
`)
`
`(9a)
`
`s(t) =
`
`- [sin 2irmWt - sin 27r(m - 1) WI].
`
`(9b)
`
`14 An exceptional case will be discussed at the end of this section.
`15 The only significant difference lies in the fact that the window
`functions must be computed separately for Wo f< [(2m-1) W
`-2WT0o and [(2m-1)W-2W0]<f<(Wo+W).
`16 This expression differs from (31) of Kohlenberg, op. cit., only
`in notation; using r_2m-1. Kohlenberg's result is obtained.
`
`It is interesting to note that the repetitive nature of
`the spectra FA(f) and FB(f) of Fig. 3 offers the possibil-
`ity of recovering not the original function but a fre-
`quency-translated version of it. For example, if the
`spectral window of Fig. 4 were used, the corresponding
`time function would represent an upward frequency
`translation of f(t) by W cps."7
`
`QUADRATURE AND HILBERT TRANSFORM SAMPLING18
`The sampling operation may be preceded by prepara-
`tory processing of the time function. The most obvious
`example is the representation of a band-pass function in
`terms of its in-phase and quadrature components, each
`of which may be sampled separately. Let
`f(t) = A (t) cos [wot + AI(t) ]
`
`(10)
`
`17 These remarks apply equally well to the low-pass function of
`Fig. 1. Amplitude modulation could have been achieved by the use
`of a suitable band-pass spectral window.
`18 Goldman, Op. cit.
`
`RPX-Farmwald Ex. 1027, p 3
`
`

`

`1222
`
`PROCEEDINGS OF THE IRE
`
`Ju.ll,}
`
`SA (f)
`
`0
`
`W0+W 2mW-W.
`
`WO+2W
`
`eB P S (R 2)
`2W sin m's
`
`exp [; (2nm+l
`( 2
`
`)]
`
`Fig. 4-Frequency-translation by use of spectral window.
`
`and let its spectrum F(f) be confined to a frequency
`band of width W, centered onfo, as shown in Fig. 5. Pro-
`viding that fo > W, the in-phase and quafrature com-
`ponents
`
`fi(t) = A(t) cos
`
`&(t) and fQ(t) = A(i) sin At(t)
`
`(11)
`
`may be obtained by multiplying f(t) by 2 cos wot and
`-2 sin wot, respectively, and by filtering out the sum-
`frequency components. The corresponding spectra are
`given by
`
`F1(f) = {F(f) * [6(f-fo) + 3(f + fo) }f
`
`FQ(f) = {F(f) * i [6(f- fo) - b(f + fo) I} If
`
`(12)
`
`where the subscript If indicates that the sum-frequency
`components have been discarded. These relations are
`illustrated in Fig. 5. Sincefi(t) andfQ(t) are band-limited
`to - W/2 <f < W/2, each may be sampled at the rate of
`W samples per second. Reconstruction of the original
`function involves separate interpolations of f1(t) and
`fQ(t), multiplication by cos wot and sin wlot, respectively,
`and addition of the results.
`First-order sampling of a band-pass function f(t) and
`of its Hilbert transform
`
`flu(t)
`
`1
`
`7r
`
`r+1f(r)dr
`_0
`f(t)*(--)
`I - Tr
`71
`s
`suffices to determine the function. This result is readily
`obtained by observing that the spectrum FH(f) of
`fH(t) is given by
`
`1
`
`(13)
`
`FH(f) = F(f)Y{5 - } = F(f)[-isgnf]
`
`where F(f) is the spectrum of f(t) and is assumed to be
`limited to the band WO.< If| <(Wo+W). The functions
`f(t) and fH(t) are now sampled at a rate of W times per
`second. Using the results of Fig. 3(b), the periodic
`spectra F(f) and FH(f) may be sketched immediately, as
`
`9.
`
`fo
`
`F (f)
`
`0
`
`* f
`
`-.f
`
`FI (f)
`
`0fX
`
`FQ(f )
`
`2o
`
`2
`
`W
`2
`
`Quadrature sampling. The direction of cross-hatching dis-
`Fig. 5
`tinguishes the positive- and negative-frequenicy parts of F(f) and
`the spectral contributions derived from them.
`
`shown in Fig. 6.1' The required window functions S(f)
`and SH(f) may be determined by inspection [Fig. 6(d) j,
`and the corresponding interpolating functions are
`sin 7rWt
`rWt - cos 2r Wo + 2)1
`
`/
`
`W\
`
`(14a)
`
`s (t) =
`
`sH(t) = -
`
`T
`sin 7rWt
`- sin 2w r Wo +-I .
`
`/
`
`(14b)
`
`These two functions are Hilbert transformiis, as antici-
`pated in the notationi. Finally,
`
`f(t) =
`
`f(--)s(t --) + fH(-) SH (t--) (15)
`
`SAMPLING OF PERIODIC FUNCTIONS20
`While the preceding discussion does not exclude linle
`spectra, its results are not particularly useful for pe-
`riodic functions since the interpolation process is based
`on an infinite number of samples rather than a finiite
`number of points within one period. The necessary
`modifications will be outlined for the low-pass case.
`Letf(t) be a periodic function of period T, which con-
`tains no spectral components above the Nth harmnonic,
`and let the function be sampled at intervals of r seconds.
`Fig. 1 applies with W = N/ T. The inequality 1/r> 2W
`= 2N/T cannot be satisfied with the equal sign since this
`choice would destroy the identity of the spectral line at
`f= N/T. The lowest acceptable rate of equispaced
`sampling is therefore given by r = T/(2N-+-1). The re-
`
`19 The similarity between Figs. 5 and 6 is evidenit. These sketches
`illustrate the close connection between quadratture samiipling and the
`present procedure.
`20 Goldman, op. cit.
`
`RPX-Farmwald Ex. 1027, p 4
`
`

`

`(m-l)W
`
`WO
`
`mW
`
`W0i+W
`
`(m+l)W
`
`I
`
`I
`
`TZZIZZZZZZZZZA
`
`(a)
`
`(b)
`
`(c)
`
`__~~
`
`S
`
`f)
`
`1(d)
`(d'L
`
`s H(f)
`L
`
`(2m-I) W-Wo
`
`I
`[~~~111112'I
`
`I
`11111111
`
`>2wX
`
`0- f
`
`o f
`
`0-
`
`I.f
`
`w r
`
`IH (f)
`
`hlTn-rrnTlT7.]d
`
`N
`
`T
`
`N + 12
`T
`Fig. 7-Sampling of periodic low-pass function.
`
`O
`
`1959
`
`Linden: A Discussion of Sampling Theorems
`
`1223
`
`(SHOWN FOR N-4)
`
`(a)1
`
`____
`
`_N+I-_N
`T( T
`
`F(f )
`
`44-
`
`I",
`
`ii 11~
`
`WEIGHTING FACTOR =
`1 /11
`
`t
`
`T
`
`N N+
`T I T
`
`0
`
`SO()
`
`(b)
`
`I111 I1 tiT 1
`
`SAMPLING OF A FUNCTION AND ITS DERIVATIVE
`Simultaneous sampling of a funiction and of its deriva-
`tive yields two periodic spectra from which the original
`spectrum may be recovered by appropriate spectral
`windows. It is assumed that the spectrum of (d/dt)f(t)
`is given by i2irfF(f). The procedure will be illustrated
`for band-limited, low-pass functions. Writing F+(f) and
`F(f) for the positive and negative frequency parts of
`F(f), the spectra of f(t) and f'(t) are sketched21 in Fig.
`8(a). The spectra of the sampled functions
`
`WO
`Fig. 6.-Hilbert transform sampling.
`
`W0+ W
`
`fA(t) = f(t) E 6 (t
`
`W
`
`sultinig spectrum is sketched in Fig. 7(a). Since there are
`gaps between successive replicas of F(f), the spectral
`window is not uniquely determined. The window func-
`tion S(f) shown in Fig. 7(b) has the advantage of provid-
`ing independent sampling since the corresponding inter-
`polating function s(t) has zeros at all sampling points
`but one. Eq. (3) may now be applied with obvious
`changes of notationi:
`
`f(t) = E f(nT)s(t - nT);
`
`+00
`
`n1 =-30
`
`T
`= T
`2N + 1
`
`s(t) =
`
`sin 2r(
`
`)I
`
`2r< t
`
`(16a)
`
`(16b)
`
`Sincef(t) is periodic with period T, (16a) may be written
`as
`
`f(t) =
`
`where
`
`2N
`
`n,1=)
`
`f(nr)p(i - nr)
`
`sin (2N + 1) -t
`~~00 ~~~T
`p(t) = , s(t-kT) =
`
`(17)
`
`(21V + 1) sin-t
`T
`
`aind
`
`11
`
`t
`
`it
`
`fB(t) = f(t) E 6
`w
`are shown in Fig. 8(b). Using the condition of (6), one
`obtains in the range 0 <f < W,
`WF+(f)SA(f) + i27rfWF+(f)SB(f) = F+(f)
`WF_(f - W)SA(f)
`+ i2r(f - W)WF_(f - W)SB(f) = 0
`
`(18)
`
`(19)
`
`whence
`
`SAW(f)=
`
`-
`
`0 <f < w
`
`1
`SB(f) = i2rW
`
`0<f<W.
`
`(20)
`
`The initerpolating functions are
`/sin 7rWl\2
`rWt )
`
`SA(t)=
`
`SB (t) = ISA (t)
`
`so that
`f(t) = fA(t) * SA(t) + fB(t) * SB(t)
`
`(- Wf
`
`SA (t--)
`
`(21)
`
`(22)
`
`The last equality is proved in Appendix 11.
`
`21 These sketches are equivalent to (8) of Fogel, op. cit.
`
`RPX-Farmwald Ex. 1027, p 5
`
`

`

`1224
`
`PROCEEDINGS OF THE IRE
`
`July
`
`F ( f )
`F (f)% F
`
`/0/~
`
`F+(f)
`
`W
`
`0f
`
`2i
`
`r f F+(f )
`
`-W
`
`0
`II
`i2vrf F(f)
`I
`
`(a)
`
`(b)
`
`(f > ()
`
`Z Sr(r)(f) [i2ir(f - k/T)] = TbO,k,
`r=O
`) [kllitl(i) + R d
`k = k0ni2t(n),
`( R -
`it = °,1 21, 4, *
`* *, (R -1)
`(R odd)
`- 0, 2, 4, .
`. .,R
`(R even)
`where 60,k is one or zero according as k is zero or nioi-
`zero, and where k ,.,(n) is the smlallest integer- such that
`n-R
`
`kinill(n) >
`
`The interpolating functions s(r)(t) are then obtained as
`the inverse Fourier transforimis of the S(r)(f), and
`drf(mT)
`R
`Z
`r=O n
`
`S (r)(t -MT)
`
`dtr
`
`1()
`
`= drf(m ) S(r)
`.TdLr=;
`dir
`
`mT)
`
`APPENDIX I
`If the positive and negative-frequency parts of F(f)
`are designated as F+(f) and F_(f), [where F+*(-f)
`= F_(f) ], one has in the interval 0 <f < W
`FA(f) = WF+(f) + WF(f - W),
`W
`
`FB(f) = WF+(f) +
`
`F_(f - W).
`
`Substituting into (6),
`F+(f) [WSA4(f) + WSB(f) - 1]
`
`-
`W
`+ F_(f - W) WSA(f) + - SB(f) = 0.
`
`There is, in general, no functional relationship between
`F+(f) and F_(f- W) = F+*(WW-f); equating to zero the
`coefficients of F+(f) and F_(f- W), one obtainis the two
`equations following (6).
`
`APPENDIX II
`
`cc
`
`0
`
`p(t) =
`
`s(t - kT) = s(t) *
`k=-ook=o
`
`e( - kT)].
`
`The corresponding spectrum is
`
`00
`
`P(f) = S(f) ko T II1)
`
`k
`
`2N +1 k=-N (
`
`T)
`
`i2xrf F_(f )
`
`t
`
`,WF tf )
`
`X
`
`ALi
`
`-,I
`
`/
`
`WF (f-W)
`
`I FA (f )
`
`g
`
`i 2 v f W F+ (ft
`
`FB (f )
`
`i27 (f-W)WF- (f-W)
`
`Fig. 8-Samplinig of a function and its derivative. The direction of
`cross-hatching distinguishes the positive- and negative-frequency
`parts of F(f) and the spectral contributions derived from them.
`
`The more general case of first-order sampling of a real
`low-pass function and its first R derivatives may be
`treated by similar methods. The derivation is straight-
`forward, but somewhat lengthy; it is given in Appendix
`III and leads to the following results. The function- and
`its R derivatives are sampled at intervals of r
`(R
`+ 1) /2 W seconds. The spectral window function S(r)(f)
`for the rth derivative (r=0, 1,
`, R) is obtained in
`2(R+1) segments, each of width W/(R+1), startinlg at
`f = -W:
`
`R
`
`S ((f)
`
`E Sn (r)
`n=-(R+l)
`Each segment represents a separate problem; however
`the following relations reduce the number of functions
`which must be determined:
`f) = [Sn (r)(f)]*
`S-(n+1) (r) (
`S2m+l (r) (f) = S2m(r) (f)
`(R odd)
`S2m+l (r) (f) = S2m+2 (r) (f)
`(R even).
`The Sn(r)(f) for the remaining (R+1)/2 (R odd) or
`(R+2)/2 (R even) values of n are found by solving the
`following sets22 of equations:
`
`22 Each set consists of (R+±) equations, corresponding to the
`Sn(R).
`(R+1) unknown functions S()
`
`The last equality may be verified by inspection of the
`window function S(f) shown in Fig. 7(b). Finally,
`
`RPX-Farmwald Ex. 1027, p 6
`
`

`

`1959
`
`Linden: A Discussion of Sampling Theorems
`
`1225
`
`F (f )
`
`F FI
`I
`
`F~ 'F
`
`1
`
`F
`
`F
`
`I.
`
`-W
`
`o
`
`~ F1IF
`F,
`H
`
`IF
`
`IF
`
`3
`21
`
`=
`
`R+I
`
`W
`
`f
`
`(f-Ir)
`
`>
`
`-3
`
`2
`
`r
`
`r
`
`T
`
`r
`
`2
`
`r
`
`3
`
`r
`
`(ILLUSTRATION DRAWN FOR R=4)
`Fig. 9- Sampling of a function and its first R derivatives.
`
`the summation may be restricted to those integral val-
`ues of k which satisfy the inequality
`-(R + 1) < (n - 2k) <R.
`For each n, there are therefore (R+1) values of k, start-
`inlg with kr11in(8n); the latter is the smallest integer which
`satisfies
`
`kniin(n) >
`
`n-R
`2
`In order to recover F(f) from the (R+ 1) spectra
`F()r(f), each F(r)(f) is multiplied by a spectral winidow
`functioj S(r)(f). One then demands that
`
`(26)
`
`S(r)(f)F(r)(f) = F(f).
`
`(27)
`
`RE
`
`7-0
`
`Sincef(t) was assumed to be real, the S(r)(f) are spectra
`of real functions, and it suffices to consider positive fre-
`quenicies onily. Each of the (R+1) positive-frequency inl-
`tervals nmust be considered separately so that (27) repre-
`sents (R +1) separate equations;
`
`S.(r))(f)F5(r)(f) = F, (f);
`
`n = 0, 1,
`
`R (28)
`
`RE
`
`r=O
`
`where S0(r)(f) represents the nth segmlenit of S(r)(f), with
`(29)
`S_(nl) (r)(-f) = [Sr (?)(f) *.
`
`Substituting (25) into (28),
`kICmin (n)+R
`R
`
`S. (r)(f)
`
`T
`
`E
`k=kA nin(n)
`
`Dk[(2wfi)rFn-2k(f)] = F,.(f)
`
`=0, 1, **,R. (30)
`Interchanginig orders of summation,
`kmi n (70 +R
`R
`E Dk[Fn_2k() ]
`E Sn(r)(f)Dk[(27rfi)r]
`k=kmin (n)
`r=o
`
`= TFn(f).
`(31)
`Since the Fn(f) are independent, the coefficient of each
`Dk [Fn_2k1 must be identically zero. For each value of n,
`(31) thus provides (R+ 1) equations
`
`N1^T
`
`2+1 k=-N
`
`irt
`sin (2X + 1)-
`
`7rt
`(2N + 1) sin-
`T
`
`APPENDIX III
`It will be assumed that the spectrum of the rth deriva-
`tive is (i2rf)rF(f). The function and its first R deriva-
`tives are sampled at intervals of r _ (R + 1) /2 W seconids.
`Their spectra are therefore conivolved with the impulse
`functioni train
`
`' Z3f k)
`Each of the (R + 1) spectra extends fromi -W to W, and
`will be divided into 2(R+1) intervals of width W/R+1
`= 1/2r, starting atf= - W. Let Fn(f) be equal to F(J) in
`the nth interval and zero outside it, i.e.,
`
`(23)
`
`R
`
`FR(f)= E Fn(f).
`
`(24)
`
`n=- (R+1)
`Since f(t) is assumed to be real, F-(l+j) (-f) = Fn*(f). Fig.
`9 shows the spectrum F(f), the numbering of its (R+1)
`intervals, and the convolving train of impulse functions.
`The convolution process is visualized in terms of
`erecting replicas centered on the impulse functions. It is
`easily seen that a replica of F(f), centered on the im-
`pulse function atf = k/r, will contribute to the (2k+j)th
`interval the function
`
`Fj(f
`
`k-
`T
`
`Using the notation
`
`D[g(f)]
`
`k)
`
`a replica of F(f) centered on b(f-k/T) will contribute to
`the function I/TDk [Fn-2k(f) ]. Let
`the nth interval
`F(r)(f) be the spectrum obtained from the convolutioll
`of (i27f)rF(f) with the train on impulse functions of (23),
`and let FJ(r)(f) be its nth segment, i.e.,
`
`R
`F(r)(f) = E F(r
`n=-(R+1)
`
`Then it follows from the preceding discussioni that23
`1k'i,(,n)+R
`E Dk [(i2rf)Fl.-2k (f)
`nin(n)
`Since Fn-2k(f) vanishes outside the interval (- W, W),
`
`Fn ((f) =
`
`k=
`
`(25)
`
`"I Eq. (25) is equivalent to (14) of Fogel, op. cit.
`
`RPX-Farmwald Ex. 1027, p 7
`
`

`

`1226
`
`PROCEEDINGS OF THE IRE
`
`Jitly
`
`Thus
`
`(R odd)
`S2,ln1(r) = S2M (r)
`S2?±+l(r) - S2m+ (r)
`(R even).
`It is therefore sufficienit to solve (32) for eveni valuLes
`of n so that there are (R+1)/2 or (R+2)/2 sets of
`equationis, accordinig to whether R is oddi or even.
`
`ACKNOWLEDGME NT
`I wish to thanik Dr. N. 1'1. Abramsoni for niumiierous
`helpful discussions anid suggestionis, anid for his careful
`reading of the maniuscript.
`
`Sn (r) (f)Dk[(2rfi))r] = 'rbo,k
`, [kmin(n) + RI
`
`RE
`
`r=O
`k = kmini(n),
`n = 0, 2
`, R
`(32)
`where 30,k is one or zero according as k is zero or 11011-
`zero.
`Inspection of (26) shows that for odd R,
`kmin(O) = klijw(1)7
`kmin(2) = kmin(3), etc.,
`while for even R,
`knill(l) = kmin(2),
`
`k,1i.(3) = k,.i,(4), etc.
`
`An Application of Piecewise Approximations to
`Reliability and Statistical Design*
`
`HARRY J. GRAY, JR.t, MEMBER, IRE
`
`Summary-If a random variable can be expressed as a weighted
`sum of other random variables having known distributions which can
`be approximated piecewise by, for example, polynomials, the distri-
`bution of the random variable can be obtained, relatively easily, by
`the use of the algorithm described in this paper.
`
`INTRODUCTION
`I N many systems, such as missile, computer, or coIn-
`trol systems, there may arise a need for the determi-
`nation of the probability of failure due to the grad-
`ual deterioration of the system components. Associated
`with this need is the determination of the probability
`that a specified characteristic of the system or a part of
`the system will be outside of acceptable limits on ac-
`counlt of a chance unfavorable combination of com-
`ponenit
`specific
`values. Examples of
`characteristics
`might be: the delay of a pulse circuit, the phase margin
`in a feedback control system, the gain of a linear ampli-
`quantities all of which are functions of the values
`fier
`of the components involved such as resistances, capaci-
`tanices, vacuum tube transconiductances, and plate re-
`sistances, etc. Denote the characteristic by T and the
`x,.
`values of the components involved by x1, x2,
`,
`Then
`
`T = T(x1, X2, .
`
`Xn).
`
`(1)
`
`* Original imianuscript received by the IRE, July 2, 1958; revised
`maniuscript received, March 6, 1959.
`t Moore School of Electrical Engineering, Philadelphia 4, Pa.
`
`It is often possible to express sufficiently accurately the
`deviation 5T of the characteristic T from some niomiiinal
`value in terms of the deviations of the component val-
`ues, Axi, from their meani values as follows:
`6T = ai5x1 + a2bX2 + *
`(2)
`* + a,,6x,1.
`*
`, a,, canl be determiined either
`The numbers, a,, a2,
`by experiment or by calculation. Eq. (2) mzay be re-
`written:
`5T/To = blxll/xlo + b2bX2, X2O + *
`bi= aixio/To
`i = 1,~2, *
`*
`w1
`(3)
`, X0o are the "meani" values of
`where To, x1o, x20,
`, xno) 1. Eq. (3) canl
`T(xio, X20,
`T, x],
`, xX,. [To
`be considered as expressing the percentage change in the
`characteristic resulting from certain percentage chanlges
`in the componenits involved, as the equality is not
`affected by multiplying both sides by 100. The problem
`theni becomies one of determining how t is distributed
`kniowing how the (i are distributed where
`
`+ b,.6xn/xnO;
`
`(4)
`41 +42 + '
`' *+07
`and t=bT/To, (ibj=xilxio; i=1, 2,
`n, the meani
`of ti is zero for i= 1, 2,
`, n, and the mean of t is zero.
`The (i are assumed to be independent ranidomn vari-
`ables.1
`1 The assumption that the means of t and (i are zero is not neces-
`sary, bLht simplifies the (liscuissioni that follows.
`
`RPX-Farmwald Ex. 1027, p 8
`
`