500
`
`IEEE TRANSACHONS O N INFORMATION THEORY, VOL. 37, NO. 3, MAY 1991
`
`Dithering and Its Effects on Sigma-Delta
`and Multistage Sigma-Delta Modulation
`
`Wu Chou, Member, IEEE, and Robert M. Gray, Fellow, IEEE
`
`Abstract -The spectrum of the quantization error in a dithered
`sigma-delta modulator and multistage sigma-delta modulator
`is derived under the constraint that the dithering signal does
`not cause overload. The results apply to dc, sinusoidal and more
`general quasi-stationary signals. It is shown in the case of a
`simple sigma-delta modulation that no-overload dithering can
`smooth the error spectrum and can make the quantization error
`asymptotically uncorrelated with the input. It does hot, however,
`make the quantization error white. In the case of multistage
`sigma-delta modulation with the appropriate dithering, the
`quantization error becomes white, even for a system with only
`two-stages. The signal-to-quantization-noise ratio (SQNR) is
`derived for sigma-delta and multistage sigma-delta oversam-
`pled analog-to-digital conversion with additive dithering. Simu-
`lation results are presented to support the theoretical analysis.
`Index Terms -Sigma-delta modulation, dithering, oversam-
`pled A/D conversion.
`
`Q plays a fundamental role in digital communication
`
`I. INTRODUCTION
`UANTIZATION is a nonlinear operation which
`
`and signal processing systems. It has been a subject of
`extensive studies for many decades. Recently oversampled
`sigma-delta modulation and multistage sigma-delta mod-
`ulation have received increasing attention as candidates
`for high resolution analog to digital converters because
`they are robust against circuit imperfections and are well
`suited for VLSI implementation [1]-[ll].
`In all these
`devices, the incoming analog signal is sampled at a rate
`many times higher than the,required Nyquist rate and the
`signal is quantized using a low resolution quantizer (typi-
`cally one bit) at the high sampling rate inside a feedback
`loop. The high rate, but low resolution bit stream is then
`downsampled by a digital decoding filter to produce a
`high resolution digital approximation of the analog input
`signal at the Nyquist rate. The quantization error from
`the l-bit quantizer is accumulated by a discrete time
`integrator in the forward loop of a sigma-delta modulator
`as shown in Fig. 1. The overall quantization noise at the
`
`Manuscript received February 2, 1989; revised November 30, 1989.
`This work was supported by the National Science Foundation under
`Grant ECS83-17981, 8957058 and by a Seed Grant from the Center for
`Integrated Systems, Stanford University, Stanford, CA. This work was
`presented at the ISCAS 1990, New Orleans, LA, May 2, 1990.
`W. Chou is with AT&T Bell Laboratories, 600 Mountain Ave., Mur-
`ray Hill, NJ 07974.
`R. M. Gray is with the Information Systems Laboratory, Department
`of Electrical Engineering, Stanford University, Stanford, CA 94305.
`IEEE Log Number 9041822.
`
`&fintiat
`
`error
`
`ion
`
`I
`
`- I
`
`delay
`
`q n
`quantizer
`output
`
`Fig. 1. Diagram of sigma-delta modulator.
`
`output has a very complicated structure even for dc inputs
`[31, [131, [141.
`Most previous studies of sigma-delta modulation and
`multistage sigma-delta modulation have focused on the
`special cases of dc and sinusoidal inputs. In this paper, we
`study the quantization noise structure of sigma-delta and
`multistage sigma-delta modulation when such inputs are
`dithered by an independent identically distributed (i.i.d.1
`process. It is also common to dither the input by a
`sinusoid that has a frequency far above the band of
`interest. But this type of deterministic dithering can be
`reduced to the case of sinusoidal inputs to sigma-delta
`and multistage sigma-delta modulator, which has been
`studied in [141 and [SI.
`A signal is “dithered” by adding to it a second signal,
`usually from an i.i.d. process (shown in Fig. 2). In some
`systems, the dithering signal is later removed either by
`filtering [32] or by subtraction from the output (subtrac-
`tive dithering) [23]. It has long been used for the purpose
`of smoothing the noise spectrum (as in audio, speech, and
`image applications) and for making the noise spectrum
`independent or less dependent on the input signal level
`(as applied in digital signal processing) 1261, [271, [281.
`The power spectral density of the quantization error in
`a sigma-delta modulator with dc and sinusoidal inputs
`consists of discrete spikes [13], [14]. It is generally be-
`lieved that dithering will remove or reduce the spikes as it
`does in uniform quantization (PCM), but this property
`has not previously been proved. In the case of multistage
`sigma-delta modulation, the quantization error is smooth
`and white for dc and for sinusoidal inputs provided cer-
`tain conditions are satisfied [8]. It is of importance to find
`out whether the conditions required to make the quanti-
`
`0018-9448/91/0500-0500$01.00 01991 IEEE
`
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`Xilinx, Inc. and Xilinx Asia Pacific Pte. Ltd. v. Analog Devices, Inc. IPR2020-01559
`
`

`

`WU AND GRAY: DITHERING AND ITS EFFECTS
`
`W"
`
`Snd'4-H-t
`
`Nonlinear
`System
`
`Filter
`
`output
`
`d h i n g
`% signal
`I
`
`501
`
`&Lm
`
`error
`
`Fig. 2. Diagram of dithering.
`
`I
`
`I
`
`I
`
`zation error white can be eliminated by dithering. Since
`thermal noise will always be present in the sampling
`circuit, the analysis of sigma-delta modulation and multi-
`stage sigma-delta modulation studied under dithering
`will be closer to their true performance.
`In Section I1 we study the quantization error spectrum
`of a sigma-delta modulator under a no-overload dither-
`ing i.i.d. process. In Section I11 we extend the analysis to
`multistage sigma-delta modulation. Decoding filter de-
`sign and simulation results are presented in Section IV.
`
`11. SIGMA-DELTA MODULATION WITH DITHERING
`The basic sigma-delta modulator is a discrete time
`system with an integrator in the forward loop and a one
`bit quantizer in the feedback loop as shown in Fig. 1. The
`nonlinear difference equation governing the system is
`n = 0,
`
`- U , - 1 - q ( u , - , ) + X,-I,
`
`n = 1 , 2 , . . .,
`(1)
`where q(u) is the output from the one bit quantizer as
`defined by
`
`b
`
`i f u 2 O
`
`= ( b , otherwise,
`
`and
`
`- b I E , I b
`E , = 4( U , ) - U , ,
`(2)
`is the binary quantization error. Since U , = q(u,)- E , ,
`the output from the one-bit quantizer q(u,) satisfies the
`difference equation
`q(u,,) = x,-1 + E , -€,-I.
`(3)
`It has been proved in [12] that if the dc input x , falls
`within the range [ - b, b ] , the integrator state U , will be in
`[ x , - b , x , + b ] . Otherwise U , can exceed the bounds and
`overload the one-bit quantizer. This range is called the
`no-overload region of a sigma-delta modulator. If the
`integrator is not overloaded, the feedback sequence of
`b can be used to approximate the dc input via a low
`pass decimation filter.
`We consider inputs of the form
`x , = d , + w,,
`(4)
`is the
`where d , is the original input signal and w,
`dithering signal. It is assumed that w,, is i.i.d. and inde-
`pendent of d , and that d , and w, together are such that
`no overload occurs, i.e., d , + w, E [ - b, bl. Fig. 3 depicts
`
`Fig. 3. Diagram of sigma-delta modulator with dithering.
`
`a sigma-delta modulator with dithering. The no-overload
`requirement can be avoided if there is a saturator or a
`modulo-limiter preceding the sigma-delta modulation.
`Note that this restriction effectively reduces the input
`range from the case where no dithering is present.
`Spectral analysis of sigma-delta and multistage sigma-
`delta modulation can be formulated in the framework of
`quasi-stationary processes as considered in [HI. The class
`of quasi-stationary processes forms a general class of
`deterministic and random processes for which the first
`and second order moments are well defined and to which
`traditional system autocorrelation and spectral analysis
`can be applied. This class includes stationary random
`processes as well as more general block stationary and
`asymptotically mean stationary processes. Also included
`are deterministic periodic and almost periodic signals as
`well as combinations of these signals. Because the dynam-
`ics of sigma-delta and multistage sigma-delta modulation
`are inherently nonlinear, the output can still be very
`complicated and nonstationary even for an input as sim-
`ple as a dc or a stationary process. The idea of quasi-sta-
`tionary processes enables us to handle different compli-
`cated inputs with a unified treatment.
`Following the notation of [ 181, we begin by introducing
`the average operator
`
`i
`
`N
`
`
`
`where E is the expectation, provided the limit exists.
`The signal {e,} is a quasi-stationary process if there is a
`constant C such that
`a) E(e,) = m,(n) where Im,(n)l I C for all n;
`b) E{e,) = Iim,,,l/NC~='=,E(e,) exists;
`c) lR,(n, k ) I s C for all. n, k where R,(n, k ) = E(e,e,);
`d) E(enen+,} = 1imN4, l / N C r = ' = , E ( e , e , + , ) = R,(k)
`exists for each k .
`If e, is a deterministic signal, then E(e,) and E is
`simply a time average. If e, is a stationary random pro-
`cess, then all of the terms in the sum are identical and E
`is just the expectation. If the input sequence is determin-
`istic and periodic or, more geneially, almost periodic 1241,
`then it is quasi-stationary and E is just the limiting time
`average. Most importantly, if (e,} is a quasi-stationary
`
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`Xilinx, Inc. and Xilinx Asia Pacific Pte. Ltd. v. Analog Devices, Inc. IPR2020-01559
`
`

`

`502
`
`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 3, MAY 1991
`
`process, then the autocorrelation
`
`mean
`
`me = E{enj 9
`and the average power
`
`An important property of the normalized binary quantiza-
`tion error sequence from the l-bit quantizer of sigma-
`(6) delta modulator is [12]'
`
`(7)
`
`(8)
`R,(O) = E(e:)
`are all well defined (the limits exist and are finite).
`Furthermore, if the limits exist and are finite, then the
`process is quasi-stationary.
`The power spectrum of quasi-stationary process is de-
`fined as the discrete time Fourier transform of the auto-
`correlation:
`
`m
`
`E , defined
`
`Se( f ) =
`(9)
`Re( n)e-jzTfn,
`n = - m
`where the frequency f has been normalized to lie in [O, 11
`(corresponding to a sampling period being considered as
`a single time unit). Two quasi-stationary processes x,, y ,
`are called asymptotically uncorrefated if for all k integer,
`Y n + k 1 *
`E{ x n Y n + k I = E{ X n I
`( 10)
`The usual linear system input/output relations hold for
`this general definition of spectrum (see Chapter 2 of
`Ljung [lS]).
`we normalize the binary quantization
`in (2) as
`,. E n
`- - < e < - .
`1
`1
`E , = - , 2b
`2 - " - 2
`We wish to prove that the normalized binary quantization
`error sequence is a quasi-stationary process* From (3)~ the
`overall quantization noise from the sigma-delta modula-
`tor is the output of the binary quantization error se-
`quence through a linear time invariant filter with a trans-
`fer function (1 - 2 - l ) . The effect of dithering on the
`overall quantization noise from the sigma-delta modula-
`tor can be derived from the spectrum of the binary
`quantization error sequence from the quantizer provided
`the quantization noise is asymptotically uncorrelated with
`the input.
`We first show that the binary quantization error se-
`quence is a quasi-stationary process when the input signal
`is a constant (dc) dithered by a no-overload i.i.d. process
`wn (that is, the sum d , + w, E [ - b, b]). The argument is
`then extended to more general quasi-stationary inputs.
`From now on, unless otherwise specified, the input is
`x, = d + w, , - b I X, I b ,
`where w, is an i.i.d. dithering process. Since d is a
`constant, the dithered input x, is also an i.i.d. process.
`Define
`
`1
`y = - + -
`X n
`2
`2b'
`
`In order to prove that e, is a quasi-stationary process, we
`need to show that mi = E(;,} and R; =
`exist
`and are finite. By expanding (14),
`
`1
`R; = R ( & ) - m ( , ) + 4.
`( 16)
`Since limn+m a , = A implies that limN+ml/NZ:=lan =
`A , it suffices to prove that
`
`lim E ( ( 6 , ) ) exists
`n - m
`
`(17)
`
`and
`
`( 18)
`lim E ( ( 6 n ) ( 6 n + k ) ) exists.
`n - m
`The key
`idea Of the proof is to show ( ( 6 n ) ? ( 6 n + k ) )
`converges in distribution to a random vector having the
`desired properties. A sequence of random vectors x,
`defined on a probability space with measure
`is said to
`converge in distribution to a random vector x if
`
`= E ( L f ( x ) ) ,
`for all bounded Bore1 sets A and all continuous functions
`f. The limits in (17), (18) can be calculated if the limit
`denote that x, con-
`distribution is known. Let x, I-,
`verges in distribution to x. Denote the characteristic
`function of a one-dimensional random variable
`by
`ax( r ) = E( e J r X ) ,
`and denote the characteristic function for a two-dimen-
`sional random vector U = ( u l , u 2 ) by
`aU( r l , r 2 ) = E( ejriui + l r ~ U z
`1.
`For all n, the support of the distribution of ((6n),(6,,+k))
`is a subset of [O,l]X[O,l]. A useful fact regarding the
`multidimensional Fourier-Stieltjes transform is the fol-
`lowing: if (x,} is a sequence of random variables sup-
`ported on [0,llk and if each x, has a distribution with a
`density, then x, -+ x if and only if Qx (27rf) converges to
`QJ21rf) for all f E Z k (see [19], 129). For example, if
`k = 2, the limit of the characteristic function sequence at
`(27r1,,27rf2) for all integer pairs (f1,f2) uniquely deter-
`mines the limit distribution.
`We say a one-dimensional random variable X has a
`lattice distribution if for some real numbers a and b with
`b > 0, the lattice points ( a + kb: k = 0, k 1, . . . } support
`
`'The notation ( x ) here represents the fractional part of x .
`
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`
`

`

`WU AND GRAY: DITHERING AND ITS EFFECTS
`
`503
`
`X. The following proposition is an important property of
`the lattice distribution [22].
`Proposition I: Random variable X has a lattice dis-
`tribution if and only if for some real number r # 0,
`l@x(r)l= 1.
`In practice, dithering signals are chosen from a smooth
`probability distribution in order to improve the overall
`noise spectrum at the output. In this paper, we suppose
`that dithering signals be chosen from a probability distri-
`bution that is smooth in the sense that it has a density. A
`distribution with a density is not a lattice distribution.
`Furthermore, if X is a random variable with distribution
`F having a density, then bDX(r)l < 1 for all r # 0; and
`lim, --rm l@x(r)l + 0 from the Riemann-Lebesgue lemma
`[21]. Hence we can always find an N so large that
`l@x(27rI)l < 1/2 for all integers 12 N . By Proposition 1,
`for all integers 1, we have
`
`We formalize this fact as the following proposition.
`Proposition 2: If random variable X has a distribution
`with a density, then for all integer I, suplzo l@x(27rI)l < 1.
`When the dithering i.i.d. process w, satisfies the condi-
`tion of Proposition 2, the existence of
`
`is the essence of the proofs of the subsequent results.
`Theorem I: Suppose that the input to a sigma-delta
`modulation is a dc level d plus an i.i.d. dithering signal
`w,. Suppose also that w, has a distribution with a density
`and x, = Id + w,l I b so that no overload occurs. Then
`the normalized binary quantization error sequence E",
`converges in distribution to a random variable which is
`uniformly distributed in [ - 1/2,1/2].
`Proof: From (14), we need to show (6,) converges in
`distribution to a random variable uniformly distributed
`I # 0,
`[0,1]. This happens
`in
`if
`for any
`integer
`limn --lm @.(sn>(2rl) = 0 and
`limn ~= @'cs,,(0) = 1. Since
`(Cy:dyi>
`is the fractional part of Cy::yi
`and since yi =
`1/2+ x i /2b = 1/2+(d + wi)/2b will be an i.i.d. process
`having a distribution with a density, we have
`
`I} 11:
`{ [;I:
`j2.rrlyi = n E{eJ2T1Yf)
`
`= E exp
`
`Let QY(27rI) be the common one dimensional characteris-
`tic function. Then there is a positive real number 4 such
`l@y(27rI)l < 4 < 1. Thus we have the following
`that
`
`-
`
`.
`
`
`
`I '
`
`limit:
`
`n - 1
`
`n - I
`
`U[O, 11 and E", -+ U [ - 5, 51.
`Therefore (6,)
`0
`Corollary I: If the no-overload dithering process w, has
`a distribution with a density, then m; = E(;,) = 0, where
`2, is the normalized binary quantization error defined in
`(14).
`Proof: This is a direct consequence of Theorem 1
`and the definition of convergence in distribution.
`U
`In order to evaluate the autocorrelation function of the
`normalized binary quantization error process E",,
`the fol-
`lowing property is useful. A proof is provided in Ap-
`pendix A.
`Lemma I : Suppose that (s,) converges in distribution
`to a random variable that has no point mass at the end
`points (O,l}; and let i(11),i(12)
`be the l,th and 12th
`Fourier coefficients of periodic function ( x >, respectively.
`Then
`E{(sn)(sn+d} = c c 2 ( 4 ) 2 ( 1 2 )
`. E( e 1 2 ~ 1 1 s , + 1 2 ~ ~ ~ s , + k
`1 7 (20)
`provided that either side of (20) has a limit.
`In many cases, the left hand side limit is very hard to
`obtain whereas the right-hand side limit of (20) can be
`evaluated easily. In the subsequent proofs, this approach
`will be used in combination with the weak convergence
`method.
`Theorem 2: If the input is a dc with level d and the
`i.i.d. no-overload dithering process w, has a distribution
`with a density, then the normalized binary quantization
`error process 2, is a quasi-stationary process with
`1
`
`m
`
`m
`
`
`
`I 1 = - m l 2 = - m
`
`j = ( k ) = [ ? ' - p@Jkl(27rI),
`
`if k = 0,
`if k # O ,
`
`(21)
`
`47r2 t # O
`where aY(27rI) is the common one-dimensional character-
`istic function of y , = 1/2 + ( d + wn)/2b.
`Proof: Recall that E", = 1/2 - (6,) and, by Theorem
`1, (6,) converges to a random variable that is uniformly
`distributed in [0,1>. Notice that Lemma 1 can be applied
`here to evaluate E{(6n)(8n+k)}, i.e.,
`m
`
`m
`
`E{(an)(an+k)}=
`
`1 I - - - m 1 2 = - m
`
`i ( ' 1 ) i ( I 2 )
`
`. E ( e j 2 ~ l l S n + j 2 ~ l
`2 S n+k},
`
`(22)
`
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`
`

`

`504
`
`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 3, MAY 1991
`
`provided either side has a limit. The following argument
`proves that the right-hand side limit of (22) indeed exists.
`From the definition of the characteristic function,
`E{ ej2.rrl~% +jz.rrW, +k 1 = @ ~ ~ S ~ ) , ( S " + ~ ) ) ( 2 7 r I l , 2 7 r Z 2 ) , (23)
`we have
`
`4 )
`@ ~ ( S l , " ) . ( S l , " + ~ ~ ) ( ~ l ~
`
`Proofi By Theorem 2 and the basic relation between
`the covariance function and the power spectrum,
`m
`S;( f) = RE( k)e-j2irkf
`
`- m
`
`n - 1
`
`i = O
`n + k - 1
`
`*
`
`i = n
`n - 1
`
`i = O
`
`n + k - 1
`
`"+
`
`= n ~ ( e x p [ j2T(ll + 12)yi])
`n E(ex~[j2.rrlz~il}
`= n @,(2dlI + 12)) ,n @y(27r12). (24)
`i = O
`i = n
`Since @,,(27rZ) of 1 # 0 is of a magnitude strictly less than
`q and q < 1, hence ((a,),
`G and
`E( e j2.rrllS, +j2.rrl S n+k) = QG( I,, I,)
`
`- -
`
`I, + I, f 0
`
`(25)
`This implies that the right-hand side limit of (22) exists
`and equals to
`
`c 2(11)2(~2)@&19~2) = c k(11)k(l2)@G(47~2)
`I , = -1,
`1
`= - +
`1
`--@Jk'(2571),
`4
`[ # O 4n2z2
`
`4 , 1 2
`
`which is finite. Therefore we have
`
`I N
`
`1
`
`1
`- F@327r1),
`
`if k = 0,
`
`if k # O .
`
`(27)
`
`0
`Corollary 2: If the input satisfies the conditions of
`Theorem 2, the normalized binary quantization error P, is
`a quasi-stationary process with power spectrum
`1
`1
`
`
`By (11) the relation between the normalized binary
`quantization error 2, and the unnormalized binary quan-
`tization error E , of the system is P, = E , /2b. The binary
`quantization error spectrum of the sigma-delta modula-
`tion therefore has the form
`
`Several conclusions can be drawn immediately from
`these results.
`If the dithering process satisfies the condition of
`Theorem 2, then the normalized binary quantization
`error power spectrum is continuous. If no dithering
`signal is present, then the normalized binary quanti-
`zation error power spectrum will be discrete spikes
`as proved in [13]. Thus dithering indeed smoothes
`the binary quantization error spectrum.
`The power spectrum of the normalized binary quan-
`tization error consists of two parts: one part corre-
`sponds to a white noise process and the other part is
`a continuous function of the frequency
`
`which is superimposed on the spectrum of the white
`noise process. Since for all integers 1,
`SUP&01@,(2~~)1 = 4 < 1,
`the second part of the spectrum is bounded by a
`convergent geometric series ( 1 / 1 2 ) ~ ~ + ,qlkl =
`q/6(1- 4), which provides a means of estimating
`the inband overall quantization noise power of the
`sigma-delta modulation.
`For an input as simple as a dc, the binary quantiza-
`tion error spectrum still depends on the input signal
`level, even with dithering. This can be seen from the
`second part of the power spectrum of E,, which is a
`function of the dc level d . Different dc levels d give
`a different shift on the distribution of y = 1/2+
`( d + w,)/2b
`in CO, 11.
`The condition for making the binary quantization
`error white is
`1
`= 0,
`~ 5 @ ; ~ ~ ( 2 7 r 1 )
`
`for all k # 0.
`
`l # O
`This can happen if and only if @Jk1(2aZ) = 0 for all I ,
`k # 0, which means that y, = 1/2+(d + w,)/2b
`is
`
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`
`

`

`WU AND GRAY: DITHERING AND ITS EFFECTS
`
`uniformly distributed in [0,1]. Thus d + w, must be
`uniformly distributed in [ - b, b ] . In that case, the
`range of permitted dc input is reduced to zero.
`Therefore, dithering can not make the binary quan-
`tization error white if the input has a nonzero range.
`We next show that, for a fixed dc input, dithering
`makes the normalized binary quantization error El,
`asymptotically uncorrelated with the input.
`Corollary 3: Given the assumptions of Theorem 1, the
`normalized binary quantization error El,
`is asymptotically
`uncorrelated with the input process x, = d + w,, and
`E{ ~ , 8 , + ~ } = 0,
`for all k.
`(31)
`Furthermore, x, and El,, are asymptotically independent
`in the sense that the joint random vector ( x , , El,+k) con-
`verges in distribution to a random variable G with a
`distribution F, X U [ - 1 / 2 , 1 / 2 ] where F, is the common
`one dimensional distribution of x,.
`The proof of Corollary 3 is provided in Appendix B.
`Theorem 1 can be generalized to the case of determin-
`istic quasi-stationary input {d,}, where y, = 1 / 2 + (d, +
`w,)/2b is an independent process, provided the i.i.d.
`dithering process wn has a distribution with a density. The
`proofs are almost identical because of the following fact.
`Propsition 3: If an i.i.d. dithering process w, has a
`distribution with a density and is independent of the input
`process d,, then for all integers 1,
`
`I supl@w,(2al)l< q < 1 .
`l # O
`This implies the following general form of Theorem 1
`for deterministic quasi-stationary inputs.
`Theorem 3: If a deterministic quasi-stationary input
`{d,,) is dithered by a no-overload i - i d Process w,, which
`has a distribution with a density, then the normalized
`binary quantization error El, converges in distribution to a
`random variable uniformly distributed in [ - $, $1.
`To generalize Theorem 2, we need an extra condition
`upon the input deterministic quasi-stationary process {d,}
`
`in order to have the right-hand side limit of Lemma 1
`exist. We require the following condition.
`Condition I: if for any integer k 2 0, the normalized
`input sequence
`
`505
`
`= 1 , 2 , . . . }
`
`+%)In
`
`dn dn+l ...
`-+-+
`2b
`2b
`
`2b
`
`has a distribution in [0,1); or in other words, for each
`integer k 2 0, the limit
`1
`N
`
`
`exists for all integers 1.
`(33)
`This is true, for example, when {d,) is periodic. In that
`case,
`
`dn+l ... + h)}
`2b
`2b
`
`dn
`-+-+
`2b
`
`is also periodic and has a discrete asymptotic distribution
`in [0, 1).
`Theorem 4: If the deterministic quasi-stationary input
`{d,) satisfies Condition 1 and is no-overload dithered by
`an independent i.i.d. dithering process w,, which has a
`distribution with a density, then the normalized binary
`quantization error process 8, is a quasi-stationary process
`
`1
`
`if k = 0,
`1 1 - 7 1 @ k k ' ( 2 ~ l ) @ d , k ( 2 ~ l ) ,
`if k # 0,
`4 T 2 I + O
`
`
`
`(34)
`where @ , , , ( 2 ~ 1 ) is the one-dimensional characteristic
`function of the i.i.d. process (w, + b ) / 2 b and @d,k(2Ti) is
`the characteristic function of the sequence {(dn / 2 b
`* * * + dn+k-l / 2 b ) l .
`Proofi From Theorem? and 8, = 1 / 2 - (a,), Lemma
`1 can be used to evaluate E{(8,)(8,+k)). We have
`
`+
`
`Xilinx, Inc. and Xilinx Asia Pacific Pte. Ltd. Exhibit 1026 Page 6
`Xilinx, Inc. and Xilinx Asia Pacific Pte. Ltd. v. Analog Devices, Inc. IPR2020-01559
`
`

`

`506
`
`w"
`
`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 3, MAY 1991
`
`sigma-delta modulator with dithering. The difference
`equations describing the m-stage sigma-delta modulation
`(MSM) are as follows [8]:
`
`( U * ,
`
`n = 0,
`
`W i , n = {
`L output
`
`x,,
`E i P l , , ,
`
`e i , , = q ( u i , , ) - ui,,,
`
`n = 0,l;.
`
`0 ,
`
`
`
`i = 1,2;
`
`- , m ,
`(40)
`
`i = l ,
`i = 2 , 3 ; . * , m ,
`
`n = 0 , 1 , . - . ,
`
`(41)
`
`Fig. 4. Diagram of m-stage sigma-delta modulator with dithering.
`
`since sup,,, 1@,,,(2rrl)l< 1. From Lemma 1, we have
`
`(36)
`0
`Corollary 4: The normalized binary quantization error
`E^, of the sigma-delta modulation for the input of Theo-
`rem 4 is a quasi-stationary process with power spectrum
`S ; ( f ) = - + -
`
`Corollary 3 can'also be generalized to the case of a
`quasi-stationary deterministic process.
`Corollary 5: If the i.i.d. no-overload dithering process
`w, has a distribution with a density and the input d , is a
`bounded deterministic quasi-stationary process, then the
`normalized binary quantization error process g,
`is asymp-
`totically uncorrelated with the input process x , = d , + w,,
`such that
`
`E{ x,E^,+~} = 0,
`for all k .
`(38)
`A proof of Corollary 5 is provided in Appendix C.
`The principal application of this generalization is to the
`case where the input is a sinusoidal wave. The methods
`and results in this section for quasi-stationary determinis-
`tic inputs can be applied to the case when the input (d,)
`is a stationary random process. The corresponding con-
`clusions can be obtained via a similar argument, which we
`will not illustrate here.
`111. MULTISTAGE SIGMA-DELTA MODULATION
`WITH DITHERING
`Multistage sigma-delta modulation (also called cascade
`or MASH sigma-delta modulation) consists of a cascade
`realization of several sigma-delta modulators combined
`with a linear network. Fig. 4 is the diagram of an m-stage
`
`and
`
`= (!
`if u 2 0 ,
`b, otherwise,
`where ui,, is the ith integrator output, UT is the initial
`state of the ith integrator, wi,, is the input to the ith
`stage, ei,, is the binary quantization error sequence of the
`ith stage, and x , E[- b,b] is the input sequence to the
`system. From (1) and (401,
`q ( u i , n ) = W i . n - l + e i , n - ei,n-1
`i = 1,
`x,-1+ E l , , - E l , , - l ,
`+ ei,, - E ~ , , - ~ , i = 2,3; .,m.
`E ~
`-
`~
`,
`~
`-
`~
`
`(42)
`
`=i
`
`Define the backward difference operator to be
`
`i = l
`
`a ,
`
`
`
`and V25, = 5, -25,-, +
`For example, V'5, = 5, -
`t n - 2 .
`Note that V' has z transfer function (1 - 2-l)". The
`linear combinatorial network is defined as
`Z, = C ( - l)i-lVi-lq(ui,n-m+i), m = 2 , 3 , . .
`m
`(43)
`where z , is the output of the MSM. The relation of the
`output and the binary quantization error sequence is
`2, = x,-" + ( - 1)" -lvmem,+.
`(44)
`Since V" is a time invariant linear operator with transfer
`function (1 - z - ' ) ~ , the effect of dithering on the overall
`quantization noise of MSM can be derived from the
`spectrum of the mth stage binary quantization error se-
`quence em,,. As in Section 11, we denote the normalized
`binary quantization error sequence from the mth stage by
`Zm,, = em,, /2b, - 1 I gm,, I 1. Let
`xi
`1
`n - ]
`y . = - + -
`' 2
`2 b '
`i,=O
`i,=O
`For m-stage sigma-delta modulation with an arbitrary
`input x, E [ - b, b], the normalized binary quantizer error
`
`a,,,= c
`
`i , - 1
`
` C Yi,.
`
`*
`
`-
`
`-
`
`(45)
`
`Xilinx, Inc. and Xilinx Asia Pacific Pte. Ltd. Exhibit 1026 Page 7
`Xilinx, Inc. and Xilinx Asia Pacific Pte. Ltd. v. Analog Devices, Inc. IPR2020-01559
`
`

`

`WU AND GRAY: DITHERING AND ITS EFFECTS
`
`507
`
`-l)m-l(am,,).
`
`(46)
`
`of the final (mth) stage has the form [8]
`1
`gmm,,=( - l ) m - l - - (
`2
`We study the case of MSM wth a deterministic quasi-sta-
`tionary input d, no-overload dithered by an i.i.d. process
`w,. In this case, the input has the form x , = d, + w, with
`x , E [ - b, bl.
`As in the previous section, we are trying to show that
`((am,,),
`converges in distribution to a certain
`random vector. In the case of MSM, however, this ran-
`dom vector is uniformly distributed in [O,l]X[O,l] and
`independent of the input signal level as long as w, has a
`distribution with a density. We start with some facts about
`nested sums.
`Proposition 4:
`C * * * C Xil= gm,o(n)Xo + gm,,(n)Xl
`
`n - 1
`
`i , - 1
`
`i , = o
`
`i , = o
`
`r
`
`+ ... + gm,n-m(n)Xn-m,
`where gm,n-m(n) = 1 and for any fixed j, with
`if n = j + m ,
`if n < j + m,
`gm,j(n>= 0,
`h m , j ( n ) , if n > j + m ,
`
`where hm,j(n) is an integer coefficient polynomial of
`degree m - 1 in n, provided that j is smaller than n.
`Proof: By induction on m, when m = 2,
`, - I
`i , - 1
`
`x i l = ( n - l ) X o + ( n - 2 ) X , + ... + X n - , .
`Suppose the results hold for k = m. If k = m + 1, then
`C Xi,
`C
`
`i , = O i l = O
`
`n - 1
`
`i , - 1
`
`i , + , = O
`
`i , = O
`
`n - 1
`
`i , + , = O
`
`n - 1
`
`n - 1
`
`= C [ gm,o(im+l>Xo + gm,l(im+1)X1
`+ ... + gm, i , +, -,Xim + - m ]
`= X o C gm,O(im+l)+Xl C
`gm,I(im+l)
`i,+ = m + 1
`i , + l = m
`+ ...
`+ Xm,, - m - I *
`If g(im+,) is an integer coefficient polynomial in i m + ,
`with degree m -1,
`
`then C~m-+~=lg(im+l) is an integer
`coefficient polynomial of degree m (see 0.121 of [251).
`
`Hence Zym-+:=kg(im+l) is a degree m, integer coefficient
`polynomial of n, provided that k is smaller than n. This
`ends the induction.
`0
`
`I
`
`=i 1,m1umnm-' + P,,-,( n),
`
`Proposition 5: Let +(n) = umnm + U , - ,nm-' + . . . +
`U , be a polynomial of degree m where n is an integer.
`Then, for all integers 1, and 1, with (l1, 1,) # (O,O),
`the
`following equality is valid:
`M n ) + I,+(n + 4
`( 1 , + l , ) a m n m + Pm-,(n), 1,+1,#0,
`1, + 1, = 0,
`where Pm-,(n) and P,_,(n) are both integer coefficient
`polynomials in n with a degree less than or equal to
`m - 2 and m - 1, respectively.
`Proof: This is a direct consequence of the application
`of the binomial formula.
`0
`Theorem 5: For MSM with m 2 2 stages, if a determin-
`istic quasi-stationary signal d, is no-overload dithered by
`an i.i.d. process w,, which has a distribution with a
`density, then the normalized mth stage binary quantiza-
`tion error sequence gm,, converges in distribution to a
`random varible uniformly distributed in [ - 1/2,1/2].
`Proof: We only need to show that (am,,) converges
`in distribution to a random variable uniformly distributed
`[O, 11. From (45), am,, = gm,,(n)y0 + gm,,(n)y1 +
`in
`* * * + gm,n-m(n)yn-m. For any fixed i , gm,i(n) is an inte-
`ger coefficient polynomial in n with degree m - 1. Given
`an i.i.d. process w,, having a distribution with a density,
`when d, is a deterministic signal, y , = + ( d , + w,)/26
`is
`a function of independent random variables w,. Hence
`is an independent process having a distribution with
`( y,)
`a density. This implies that the characteristic function of
`(am,,) at 27rl with 1 integer is
`@(s m n , ) ( 2 ~ 1 ) = @y,(2rrgm,o(n)1)@,1(2~gm,l(n)1)
`. . . @ y n - , ( 2 ~ g m , n - m O 2 ) ' ) *
`The limit will converge to 0 if I # 0 and will equal to 1 if
`1 = 0 because Proposition 3 implies that each factor is
`strictly less than q, q E (0,1), and converges to 0 as n goes
`to infinity.
`0
`Theorem 6: For MSM with m 2 2 stages, if a determin-
`istic quasi-stationary input {d,} is no-overload dithered by
`an i.i.d. process w,, which has a distributio