RPX-Farmwald Ex. 1029, p 1
`
`
`
`
`
`1.1 Mathematical Symbols, Notation and Definitions
`1:
`
`an integer
`
`n:
`
`T:
`
`II
`
`II
`
`II
`
`H:
`
`"II
`
`M
`
`an integer
`
`a period of time
`
`a period of time
`
`carrier frequency
`
`modulation frequency
`
`sampling frequency
`
`a finite frequency band
`
`modulation index
`
`J
`
`=n
`
`9(t) =
`
`G(f) =
`
`Bessel function of the first kind
`
`generalised function of time
`
`generalised function of frequency
`
`<=> signifies a Fourier transformation
`*
`
`signifies a convolution integral
`
`a summation over all
`
`'n'
`
`a summation over all
`
`'n' except n=o
`
`combT[
`
`] = E: 6(t—iT) - the sampling function
`i
`(at T intervals)
`
`Zn
`
`s(t)
`
`ReP1[]=
`T
`
`the replication function (at % intervals)
`
`rect(t) =
`
`u(t+k)-u(t-k)
`
`the rectangular function
`
`sinc(t)
`
`= sin(wft)
`nft
`
`+15
`
`Mt) g f 6(t)dt = 1; s(t) = 0, t *0 = the Dirac delta function
`
`RPX—Farmwald Ex. 1029, p 2
`
`RPX-Farmwald Ex. 1029, p 2
`
`
`
`RPX-Farmwald Ex. 1029, p 3
`
`
`
`Pictorially the sampling of a bandlimited signal may
`thus be viewed as an amplitude modulated harmonic series
`as indicated in Fig 1.
`
`9(1)
`
`comb, [gm]
`
`wavalorm
`
`t
`
`l
`
`I
`
`I
`
`l
`
`4-»
`T
`
`0
`0
`bandlimixad
`input
`time samplnd lime output
`
`
`input
`lrzquency replicated
`
`
`IIT RepUT [Gm]
`
`
`
`lit) Inmpllng
`l‘uncliun
`
`spectrum
`
`Gm
`
`f
`
`D
`.__.
`23
`
`°
`
`if”T
`sampling
`frequency
`
`Fig 1
`
`Sampling a band-limited signal
`
`Clearly the limiting condition for separability — or
`recovery of the original information undistortedyis when
`fs > 28 - this is commonly referred to as the Nyquist
`condition [1] and 28 called the Nyquist frequency.
`
`3.2
`
`Sub-Sampling.
`
`If the sampling frequency is not at least twice that
`of the highest component present, or if the signal is not
`strictly band—limited, aliasing distortion [16] occurs as
`the frequency bands overlap.
`aliasing due to sub sampling
`
`aliasing due to unlimited bandwidth
`
`m 1 fi 6
`
`0
`
`0
`f, < 23
`
`Fig 2 Alias distortion in the frequency domain
`
`This phenomenon of non-ideal-separability may also
`be conveniently viewed in the time domain as depicted in
`Fig 1
`
`Aqswwl “mm“
`
`‘
`
`1
`
`I
`
`Fig 3 Alias distortion in the time domain
`
`RPX—Farmwald Ex. 1029, p 4
`
`RPX-Farmwald Ex. 1029, p 4
`
`
`
`RPX-Farmwald Ex. 1029, p 5
`
`
`
`Fig 6
`
`Example spectra
`
`0n the one hand to recover the carrier we might
`to sample at fs>2fc.
`On the other,
`to retrieve the
`expect
`modulation information we have to sample at fs>23 — not
`fs>2(fc+ B)
`1!
`How is this so? Consider the action of
`the sampler:-
`
`..... (7)
`
`.....
`
`signnl
`
`g(t) = [1 + m(t)] cosmct
`
`sampled signal h(t) = combT [coauct] + combT [m(t) coswct]
`
`AI
`
`IV
`
`nu) = % Repl {[Mf) + 14(5)] * % [6(f-fc) + amen}
`
`T
`
`..... (9)
`
`Mi
`
`_I’T
`
`Fig 7
`
`o
`
`'IT='s>25
`
`2,T
`
`Sampler output spectrum for a carrier plus
`modulated carrier
`
`Qualitatively we can argue that a carrier alone
`contains very little information (only amplitude and
`frequency) and occupies zero bandwidth,
`therefore
`requiring a sampling rate << fc. Similarly the
`information bearing energy of the sidebands lies in a
`bandwidth B, and a sampling rate of fs>28 is sufficient.
`In short,
`the process of down conversion can be achieved
`by a sampler of rate >23.
`
`It is clearly necessary to take some care to define
`sampling systems in terms of the information to be
`recovered; for a repetitive wave containing fixed
`amplitude and frequency information we may in principle
`sample as slowly as we wish.
`
`4‘
`
`NON-UNIFORM SAMPLING
`
`4.1
`
`Statement of Constraints.
`
`The sampling theorem tells us that we may recover
`all of the original
`information provided that samples are
`
`RPX—Farmwald Ex. 1029, p 6
`
`RPX-Farmwald Ex. 1029, p 6
`
`
`
`RPX-Farmwald Ex. 1029, p 7
`
`
`
`5(t) = %{1 + 22],. cosZ-nfsit}
`
`.... .
`
`with the introduction of a sinusoidal sampling
`deviation this takes on the deterministic form:-
`
`T
`
`sd(t) — -1-{l + 223i cos(wsit + Bsinmmt)}
`= :1; :1 + 2221 Jnm cos(ims + wank}
`
`.....
`
`Where Jn(B) are Bessel functions of the first kind
`B is the modulation index
`
`mm is the modulation frequency
`
`Sampling a bandlimited signal with this wave gives:-
`
`sd(t).g(t) <=>% {5(f) +2Jn(8)[ 5(f-ifS-nfm)+6(f+ifs+nfm)]} *G(f)
`
`i,n
`
`. . . . .
`
`
`
`Fig 9
`
`Sampling spectra with deterministic
`perturbation
`
`the
`Hence provided f; is made sufficiently large,
`deviation Bfn,is constrained, and we have a knowledge of
`is (ie when the samples were taken) we can recover g(t)
`intact. when fs>>fG - the highest frequency in the
`bandlimited signal - we need only consider the i — 0
`component
`in isolation:-
`
`J (a)
`
`sd(f)*G(f)[i=o= 0;” +2“ “T
`
`[G(f)-nfm) + cu + nemn
`..... (14)
`
`RPX—Farmwald Ex. 1029, p 8
`
`RPX-Farmwald Ex. 1029, p 8
`
`
`
`RPX-Farmwald Ex. 1029, p 9
`
`
`
`Againnseparability is possible provided; fs >> f6
`and f5 >> (£3 + o), otherwise a record of all sample
`amplitude and times has to be made as per the
`deterministic case.
`
`5
`
`PRACTICAL LIMITATIONS
`
`5.1 Aperture Distortion.
`
`The Dirac 5 function is physically unrealizable and
`in practice we have to content ourselves with more modest
`functions [29,30]. These are not only limited in their
`amplitude and width, but also in their shape. Although a
`rectangular pulse is unrealiseable - as is a rectangular
`filter in the frequency domain, we use this ideal as a
`convenient approximation to demonstrate the practical
`limitations imposed by finite amplitude and width sampling
`pulses.
`
`lamb,- [gm] 0 nml‘lrl
`
`
`
`Efihun-ma(Vfl
`
`Fig 12 Aperture distortion model
`
`With this imperfect sampling pulse the output
`both amplitude scaled and time domain smeared by the
`convolution process, which leads to a reduction of
`effective bandwidth in the frequency domain:-
`
`1: ainc (f1)
`rect (%) <=> )1? Rep1 [G(f)] .
`combT [g(t)] *
`"f
`
`w M w
`
`perfect
`sampling
`
`smearing
`function
`
`perfect
`spectrum
`
`bandwidth
`reduction
`
`thus suffers a frequency domain
`The sampler output
`droop dependent upon the particular shape of the sampling
`pulse.
`For all practical pulses of interest the resulting
`distortion is bounded - best to worst - by sine and sincz,
`ie the pulse shape lies between the rectangular and
`triangular.
`The resulting amplitude distortion introduced
`by these functions, as well as that for a Gaussian pulse
`are given in Fig 13.
`
`RPX—Farmwald EX. 1029, p 10
`
`RPX-Farmwald Ex. 1029, p 10
`
`
`
`RPX-Farmwald Ex. 1029, p 11
`
`
`
`@W'fl‘mloss of sharp
`
`loss of nan rims
`and transients
`
`amplitude scaling only
`
`transitions
`
`Fig 15 The importance of sampling width
`
`5.2
`
`Synchronised Sampling.
`
`When dealing with periodic signals the sampling
`process works provided the sampling frequency and signal
`frequency are not directly (or closely) related.
`To
`illustrate this feature let us consider the case when:-
`
`_
`2m:
`g(t)—cos [nT]
`
`..... (19)
`
`where n is an integer and f5 = %
`
`now comb
`
`cos m:- <= >‘—1 Re
`n'l‘
`21'
`
`T
`
`P;
`T
`
`5(f - —1)+ 5(f + '—1)
`11']?
`nT
`
`- - - - -
`
`do
`term
`
`t
`
`0
`
`samples repeatedly taken
`at the same point in
`\he repetitive waveform
`
`mnoellation of
`sideband;
`
`Fig 16 Synchronised sampling of a sinusoid
`
`This clearly results in a repetitive sampling of the
`same value in the wave which manifests itself as sideband
`cancellation in the frequency domain. Recovery of the
`original wave is thus impossible and hence the sampling
`theorem requirement f5>23 .
`For
`the case of closely
`related sampling and period time - is 'n'
`is 'not quite'
`an integer — a slow roll or progression becomes evident.
`
`2m:
`__
`1
`1
`1
`combT[cos m]< - > E Repl[6(f -—(n+A)T+6(f+-(m)]
`T
`
`. . . . .
`
`The resulting sampling process thus suffers from a
`"beat frequency", which in terms of any measurement or
`observation is undesirable and should be avoided.
`In
`
`RPX—Farmwald EX. 1029, p 12
`
`RPX-Farmwald Ex. 1029, p 12
`
`
`
`RPX-Farmwald Ex. 1029, p 13
`
`
`
`5.4
`
`Sampling Jitter and Noise.
`
`A11 practical sampling schemes suffer from
`fundamental circuit limitations and signal uncertainties
`that give rise to both amplitude noise and phase jitter.
`Given that every care is exercised to minimise these at
`the design stage, further reduction is possible by signal
`averaging along both the amplitude and time axes, as well
`as more sophisticated image processing [31,35]. Broadly
`speaking averaging techniques achieve performance
`improvements by the voltage addition of the wanted signal
`and power addition of the noise/jitter as depicted in
`Fig 19.
`
`sampled signal
`
`sampled noise
`
`resuham0
`o
`° 0
`
`samplel
`
`:
`
`: -4HJeee:i6199-
`samPIeN
`I
`V}
`i
`l
`-*¥J
`i L**
`oquul
`
`NV
`
`O
`o
`O
`
`O
`
`41:;3—11111rerta —4er——91TLL_
`+ o
`o o o o ta
`0|
`O o
`o
`O
`Jatr-ILo-1r-1y
`-IHJ
`Na
`o
`000 tr
`
`+
`
`o O
`
`O
`
`+
`
`Fig 19 Signal averaging example
`
`2
`Original S/N = (g)
`2
`
`("V3
`no
`
`New SIN =
`
`..
`
`The improvement attained is thus a 10 log n dB.
`
`For small deviations sampling jitter may be
`considered to be a noise-like process and, by virtue of
`phase to amplitude conversion,
`the above averaging
`description is applicable. However, amplitude and time
`axis averaging may be applied independently by operating
`on the full sample array [31,36—38].
`
`6
`
`A FINAL NOTE ON THE PRINCIPLES
`
`Although we have concentrated on time domain
`sampling alone, it should be recognised that many of the
`developments described have a dual role in the frequency
`
`RPX—Farmwald EX. 1029, p 14
`
`RPX-Farmwald Ex. 1029, p 14
`
`
`
`RPX-Farmwald Ex. 1029, p 15
`
`
`
`18.
`
`19.
`
`20.
`
`21.
`
`22.
`
`23.
`
`24.
`
`25.
`
`26.
`
`27.
`
`28.
`
`29.
`
`30.
`
`31.
`
`32.
`
`33.
`
`34.
`
`35.
`
`36.
`
`37.
`
`38.
`
`Papoulis: 1965, Probability Random Variables and
`Stochastic Processes. HcGraw—Hill, New York.
`
`Middleton, D.: 1960, Introduction to Statistical
`Communication Theory. McGraw—flill, New York.
`
`Nahman, N.S.: 1983,
`
`IEE Trans IM—321 , 117-124.
`
`Nahman, N.s.: 1978, Proc IEEE, 661 , 441-454.
`
`& Aitchison, C.S.: 1980,
`Sarhadi, M.
`1611 , 350-352.
`
`IEE Elec Lett,
`
`& Aitchison, C.S.: 1979,
`Sarhadi, H.
`Brighton, England, 345-349.
`
`IEE Proc EMC,
`
`& Guillaume, H.E.= 1981, Deconvolution
`Nahman, N.S.
`of Time Domain Waveforms in the Presence of Noise.
`US NBS, Technical Note 1047.
`
`Andrews, J.R.: 1973, Trans IEEE,
`
`IM-221 , 375—381.
`
`& Nahman, N.S.: 1964,
`Frye, 6.3.
`V01 IM-131 , 8-13.
`
`IEE Trans,
`
`Sugarman, R.: 1957, Rev Sci Inst, 2811 , 933-938.
`
`& Piersol, A.G.: 1971, Random Data:
`Bendat, J.s.
`Analysis and Measurement Procedures. Wiley, New York.
`
`Grove, W.M.: 1966,
`
`IEEE Trans, MTT—l4112, 629—635.
`
`Tielert, R.: 1976,
`
`IEE Elec Lett, 121 , 84-85.
`
`Jones, N.B.: 1982, Digital Signal Processing.
`Control Engineering Series 22. Peter Perigrinus.
`
`Isaacson, R. et a1: 1974, Electron, 19-33.
`
`Bucciarelli, T. and Picardi, G.: 1975, Alta
`Freguenza, XLIV1 , 454—461.
`
`& Wilde, R.: 1978, Nachrichtentechnik
`Kinneman, G.
`Elecktronik, 281 , 337-339.
`
`Reed, R.C.: 1977,
`
`IEE Euromeas Conf Pub 152, 101-103.
`
`Haralick, R.M., Shanmucam, K.
`IEEE Trans, SHE-31 , 610-621.
`
`& Dinstein, 1.: 1973,
`
`Stuller, J.A.
`1148-1195.
`
`& Kurz, R.: 1976,
`
`IEEE Trans Comm,
`
`Nahman, N.S.: Proc IEEE, 55/6, 855-864.
`
`RPX—Farmwald EX. 1029, p 16
`
`RPX-Farmwald Ex. 1029, p 16
`
`
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