A Theory of Non-Subtractive Dither
Robert A. Wannamaker, Stanley P. Lipshitz∗, and John Vanderkooy∗
Audio Research Group, University of Waterloo, Waterloo, Ontario N2L 3G1, CANADA

andJ. Nelson Wright
Acuson, 1220 Charleston Rd., Mountain View, CA 94039-7393, USA

Abstract
A detailed mathematical investigation of multi-bit quantizing systems using
non-subtractive dither is presented. It is shown that by the use of dither having
a suitably-chosen probability density function, moments of the total error can bemade independent of the system input signal, but that statistical independenceof the error and the input signals is not achievable. Similarly, it is demonstrated
that values of the total error signal cannot generally be rendered statisticallyindependent of one another, but that their joint moments can be controlled andthat, in particular, the error sequence can be rendered spectrally white. The
properties of some practical dither signals are explored and recommendations are
made for dithering in audio, video and measurement applications. Some of theresults presented here are known to a handful of individuals in the engineeringcommunity, but many appear to be unpublished. In view of many widespreadmisunderstandings regarding non-subtractive dither, formal presentation of theseresults is long overdue.
∗

Members of the Guelph-Waterloo Program for Graduate Work in Physics.

1

1

Introduction

Analog-to-digital conversion is customarily decomposed into two separate processes:
sampling of the input analog waveform and quantization of the sample values in orderto represent them with binary words of a prescribed length. The sampling operation
incurs no loss of information as long as the input is appropriately bandlimited, but
the approximating nature of the quantization operation always results in signal degradation. Another common operation with a similar problem is requantization, in which
the wordlength of digital data is reduced after arithmetical processing in order to meet
specifications for its storage or transmission.
Dither, straightforwardly put, is a random “noise” process added to a signal prior
to its (re)quantization in order to control the statistical properties of the quantization
error. This is not a new idea. Subtractively dithered (SD) quantizing systems, inwhich the dither is subsequently subtracted from the output signal after quantization,
have been discussed and used for over thirty years in speech and video processing
applications [1, 2], and a satisfactory theory of their operation exists in print [3, 4].
Non-subtractively dithered (NSD) systems, in which the dither signal is not subtracted
from the output, are a subject of more recent interest. The following provides a brief
history of the theory of NSD quantization.
It must be acknowledged that all theoretical treatments of dithered quantization
owe a substantial debt to the work of Widrow [5, 6, 7], who developed many of the essential mathematical tools while studying undithered quantization. Among Widrow’s
contributions was the “Quantizing Theorem,” a counterpart in disrete-valued systems
to the better known “Sampling Theorem” in discrete-time systems. Important extensions to Widrow’s treatment were made by Sripad and Snyder [8], Sherwood [4] andGray [9].

2Early investigations into non-subtractive dither per se were conducted by one of
the authors, Wright [10], in 1979, resulting in discovery of many of the importantresults which follow. This work remained unpublished until it was brought to theattention of the other authors of this manuscript [11].
The results concerning moments of the error signal were rediscovered independently
by Stockham [12] in 1980, and documented in an unpublished Master’s thesis by
Brinton [13], a student of Stockham’s, in 1984. Stockham, however, has remained
silent on the matter until recently [14].
The properties of non-subtractive dither (and, in particular, triangular-pdf dither)were again discovered independently by two of the present authors, Lipshitz and Vanderkooy, in the mid-1980’s. Vanderkooy and Lipshitz were the first researchers topublish their findings on non-subtractive dither [15, 16, 17, 18, 19], and this prompted
collation and extension of the theoretical aspects by another of the authors, Wannamaker [20, 21, 22, 23, 24, 25]. Lipshitz, Wannamaker and Vanderkooy have publisheda broad theoretical survey of multi-bit quantization treating both undithered anddithered systems [26]. They have also extended the treatment to include an analysis of dithered quantizing systems using noise-shaping error feedback [27, 28, 24] andmultichannel quantizing systems [29].
The results concerning error moments have also been independently discovered byGray [30], using arguments employing Fourier series along with characteristic functions. Gray and Stockham have published a paper on the subject [14].
Although a small number of individuals in the engineering community are awareof the correct results regarding non-subtractive dither, a number of misconceptions
concerning the technique are widespread. Particularly serious is a persistent confusion
regarding the quite different properties of subtractive and non-subtractive dithering(see, for instance, [31, p. 170]). The aim of this paper is to provide a consistent
3

Q(w)

Q(w)

∆

∆
w

w

(a)

(b)

Figure 1: Quantizer transfer characteristics: (a) mid-tread, (b) mid-riser, with ∆denoting the size of one LSB.
and rigorous account of the theory of non-subtractively dithered systems in orderto promote a more universal understanding of this dithering technique. As such, itgreatly extends and elaborates the treatment provided by our earlier presentation [22].

1.1

The Classical Model of Quantization

Quantization and requantization processes possess similar transfer characteristics,
which are generally of either the mid-tread or mid-riser variety illustrated in Fig. 1.
We will assume that the quantizers involved are infinite, which, for practical purposes,
means that the system input signal is never clipped by saturation of the quantizer.
(Some comments regarding the application of dither to 1-bit and sigma-delta converters will be reserved for the Conclusions.) In this case the corresponding transfer
functions relating the quantizer output to its input, w, can be expressed analyticallyin terms of the quantizer step size, ∆:w
1Q(w) = ∆
+
∆ 2


4



for a mid-tread quantizer, orw∆
Q(w) = ∆
+
∆
2




for a mid-riser quantizer, where the “floor” operator, b c, returns the greatest integer
less than or equal to its argument. The step size, ∆, is commonly referred to as a
LSB (“least significant bit”), since a change in input signal level of one step widthcorresponds to a change in the LSB of binary coded output. Throughout the sequel,
quantizers of the mid-tread variety will be assumed, but all derived results have obvious
analogs for mid-riser quantizers and all stated theorems are valid for both types.
Quantization or requantization introduces an error signal, q, into the digital data
stream, which is simply the difference between the output of the quantizer and its
input:
4

q(w) = Q(w) − w,4where we use = to indicate equality by definition. This quantization error is shown
as a function of w for a mid-tread quantizer in Fig. 2. It has a maximum magnitudeof 0.5 LSB and is periodic in w with a period of 1 LSB.
Although q is clearly a deterministic function of the input, the Classical Modelof Quantization (CMQ) [31] holds that the quantization error can be modeled as anadditive random process which is independent of the system input and iid (i.e., that
distinct samples of the error are statistically independent of one another and identically
distributed). The CMQ further postulates that the error is uniformly distributed,
meaning that its values exhibit a probability density function (pdf) of the form
pq (q) = Π∆ (q),

5

(1)

q(w)
[LSB]
0.5

-1.0

1.0

w
[LSB]

-0.5

Figure 2: Quantization error, q(w), as a function of quantizer input, w, for a mid-tread
quantizer.
where the rectangular window function of width Γ, ΠΓ , is defined as 1 , −Γ < q ≤ Γ,4ΓΠΓ (q) =22otherwise.

 0,Such a pdf is referred to as a uniform pdf or RPDF (rectangular pdf ).
The m-th moment of a random process q with pdf pq is defined as the expectation
value of q m , m ∈ Z:
m 4

E[q ] =Z ∞−∞q m pq (q)dq,

where E[ ], the expectation value operator, is defined more generally by
4

E[f ] =

Z ∞−∞f (q)pq (q)dq.

The zeroth moment of any random process (i.e., E[q 0 ]) is identically equal to unity.
The first moment is usually referred to as the mean of the process, whereas the term
variance refers to the quantity E (q − E[q])2 = E[q 2 ] − E 2 [q]. It is clear that if themean of a random process is zero, its variance and second moment are equal.
6

If a quantization error signal is distributed according to Eq. (1), its moments are:E[q] = 0∆2
E[q 2 ] =12m

E[q ] =




(2)
(3)
1
m+1






0,

∆
2

m

, for m even,(4)

for m odd.

Eq. (3) is the familiar expression for the variance of the quantization error in theclassical model.
The CMQ is valid for input signals which exhibit smooth pdf’s and which are large
relative to an LSB [31, 25]. It fails catastrophically for small signals and many particularly simple (e.g., sinusoidal) signals, for which the quantization error retains thecharacter of input-dependent distortion, rather than noise. The mid-tread quantization of a small signal of peak amplitude less than 0.5 LSB provides a simple exampleof this failure: the quantizer output is null, and the quantization error is just the input
sign-inverted. Such an error is not uniformly distributed, iid or independent of the
input. In such cases, application of an appropriate dither can be used to temper thestatistical properties of the error signal.

1.2

Dither: Subtractive vs. Non-Subtractive

Schematics of subtractively dithered and non-subtractively dithered quantizing systems are shown in Fig. 3. In each case we denote the system input by x and thesystem output by y. We thus distinguish the system input from the quantizer input,
which we continue to denote by w and which is given by w = x + ν. ν represents
the dither signal, a strict-sense stationary random process which is assumed to bestatistically independent of x. Similarly, the total error of each quantizing system isdefined as the difference between the system output and system input, and is denoted7dither , ν
+
input
x

Σ

+

w=x+ν

quantizer
Q+(a) subtractively
ditheredΣ

outputy = Q(w) - ν= x + q(x+ν)=x+εdither , ν
+
input
x

+

Σ

w=x+ν

outputy = Q(w)= x + ν + q(x+ν)=x+εquantizer
Q(b) non-subtractively
ditheredchannelFigure 3: Dither quantizing systems: (a) subtractively dithered (SD), (b) nonsubtractively dithered (NSD).

8

by
4

ε=y−x
to distinguish it from the quantizer error, q = Q(w) − w.
The total errors introduced by subtractively dithered and non-subtractively ditheredsystems are not identical. In a subtractively dithered system, the dither is subtracted
from the quantizer output to yield the system output. Hence, for such a system:
ε = y−x= Q(x + ν) − (x + ν)
= q(x + ν).
On the other hand, for the non-subtractively dithered system:
ε = y−x= Q(x + ν) − x= q(x + ν) + ν.
In neither case is the total error equal to q(x) as in an undithered system (i.e., one forwhich ν ≡ 0), although in an SD system the total error does equal the quantization
error associated with the total quantizer input w.
It has been shown by Schuchman [3] that the total error induced by an SD quantizing system can be rendered uniformly distributed for arbitrary input distributions
if and only if the dither’s characteristic function or cf (the Fourier transform of its
pdf [32, 33]) obeys a certain condition. Defining the Fourier transform operator, F[ ],by
4

4

F[f ](u) = F (u) =

Z ∞

−∞

9

f (x)e−j2πux dx,

and denoting the dither pdf and cf as pν (ν) and Pν (u), respectively, Schuchman’s
condition is that
Pν



k
∆



=0

∀k ∈ Z0 ,(5)where we take this opportunity to define the set Zn0 as the set of all n-vectors withinteger components with the exception of the zero vector 0 = (0, 0, . . . , 0); i.e., Zn0 =Zn \ 0.
Furthermore, it can be shown [8, 4, 26] that the total error in an SD quantizing
system is statistically independent of the system input if and only if Eq. (5) holds.
Thus, dither obeying Schuchman’s condition renders the error statistically independentof the input and uniformly distributed. In particular, it exhibits a variance of ∆2 /12.
In these regards, then, it resembles the idealized quantization error of the CMQ. The
simplest random process satisfying Schuchman’s condition is one exhibiting a uniformpdfpν (ν) = Π∆ (ν),
whose associated characteristic function is a “sinc” function:
4 sin(π∆u)

Pν (u) = sinc (u) =π∆u.

(Different authors employ slightly different definitions of the sinc function. We willretain the above throughout the sequel.)
It can also be shown [8, 4, 26] that subtractive dither will render distinct samplesof the total error signal statistically independent of one another for arbitrary input
distributions if and only if
Pν1 ,ν2



k1 k2
,
∆ ∆



=0

∀(k1 , k2 ) ∈ Z20 ,(6)

where ν1 and ν2 represent dither values separated in time by τ 6= 0, and wherePν1 ,ν2 (u1 , u2 ) represents their joint characteristic function (the two-dimensional Fourier
10

transform of their joint pdf, pν1 ,ν2 (ν1 , ν2 )). This condition is satisfied by any dither
which is iid, so that Pν1 ,ν2 (u1 , u2 ) = Pν (u1 )Pν (u2 ), and which satisfies Eq. (5). For
instance, this means that a subtractively dithered quantizing system employing iid
dither of uniform distribution produces an iid total error signal whose values are uniformly distributed and statistically independent of the input. The error thus behaves
like a purely additive independent noise process, as postulated by the CMQ. This
beautiful result represents the ideal outcome for a quantization operation.
Unfortunately, subtractive dithering is difficult to use in many practical systems
since the dither signal must be available at each end of the channel. This requires
either the transmission of the dither values or the use of synchronized noise sources(pseudo-random number generators) separated, in general, by both time and distance.
Furthermore, any digital processing of the dithered signal would necessitate processingof the dither prior to subtraction. For reasons such as these, the possibility of using
dither without subsequently subtracting it is frequently of interest.
We will see that non-subtractively dithered systems, as distinct from subtractively
dithered ones, cannot render the total error statistically independent of the input.
Neither can they make temporally separated values of the total error statisticallyindependent of one another. However, we shall prove that they can render any desired
statistical moments of the error signal independent of the input and regulate the joint
moments of errors which are separated in time. The theory underlying these features of
non-subtractive dither is developed in Section 2, and is subsequently used to explore
the properties of some practical dither signals in Section 3. Section 4 explores theimportant special case of quantizing systems in which the available dither is discretevalued, while Section 5 summarizes the most important observations and conclusions.

11

2

Non-Subtractive Dither Theory

We begin by describing the relationship between the total error and the input signal
in probabilistic terms.

2.1

Total Error PDF’s

The dependence of the total error on the system input can be analyzed in terms of
its pdf as a function of a specified input value. This function is referred to as theconditional pdf , or cpdf , of the total error and is denoted pε|x (ε, x) throughout thefollowing discussion.
In order to derive an expression for pε|x (ε, x), we consider a non-subtractively
dithered quantizing system, as in Fig. 3(b), with a specified system input value, x.
The input to the quantizer is w = x + ν, the sum of the system input and thestatistically independent dither process. This sum has a cpdf
pw|x (w, x) = pν (w − x).

Fig. 4 shows that total error depends not only on the system input value, but also
on the value of the dither. In particular, if the input to the quantizer, w, is between−∆/2 and +∆/2, the output will be nil (for a mid-tread characteristic) so that theerror is ε = −x. Similarly, if the input to the quantizer is between +∆/2 and +3∆/2
the output will be +∆, so that ε = −x + ∆. Hence, the pdf of the error for a fixed
input is a series of delta functions separated by intervals of ∆, each weighted by theprobability that w falls upon the corresponding quantizer step:pε|x (ε, x) =∞Xδ(ε + x − k∆)

k=−∞

12

Z ∆ +k∆
2

−∆
+k∆
2

pν (w − x)dw.

yp (w, x)= p ( w-x)
w|x
νw = x +ν
input, x

Figure 4: Cpdf of the quantizer input showing its justification relative to the quantizertransfer characteristic.

13

In the parlance of Widrow [7], the error cpdf is an area sampled version of the quantizer
input cpdf.
Writing the integral in the last equation as a convolution (denoted by ?) of pν witha rectangular window function, ∆Π∆ , it reduces topε|x (ε, x) = [∆Π∆ ? pν ](ε)W∆ (ε + x),
where
4WΓ (ε) =∞X(7)δ(ε − kΓ)k=−∞is a train of Dirac delta functions separated by intervals of width Γ.1 Thus the pdf of
ε is given bypε (ε) =Z ∞−∞pε|x (ε, x)px (x)dx= [∆Π∆ ? pν ](ε)[W∆ ? px ](−ε).

(8)

As discussed in Section 1.2 in association with subtractively dithered systems,
the quantization error, q(w), will be statistically independent of x and uniformly
distributed if the dither statistics obey Schuchman’s condition, Eq. (5). Unfortunately,q(w) is not the total error of a non-subtractively dithered system. Indeed, we will now
show the following:
Theorem 1 In an NSD quantizing system it is not possible to render the total error
either statistically independent of the system input or uniformly distributed for system
inputs of arbitrary distribution.
1

A problem arises in the formalism if the dither is null (pν (ν) = δ(ν)) and the system input occurs

at a quantizer step edge, since the product of the generalized functions W∆ (ν − (2n + 1)∆/2) andΠ∆ (ν) is not conventionally defined. It is shown in [25] that an appropriate definition of this productfor the purposes at hand is 21 [δ(ν − n∆) + δ(ν − (n + 1)∆)].

14

Proof : Eq. (7) makes it clear that pε|x (ε, x) cannot be rendered independent of x byany choice of dither pdf, since the convolution of any dither pdf (which must be nonnegative everywhere) with a rectangular window function yields a function at least as
wide as the rectangular window. Hence, at least one delta function always makes acontribution to the sum, and the position of that delta function is dependent on thesystem input.
Taking the Fourier transform of Eq. (8) we find that the characteristic function of
ε is given byhi

Pε (u) = [sinc (u)Pν (u)] ? W 1 (−u)Px (−u)
∞
X

∆

k
k
k
=
sinc u −Pν u −
Px −
,
∆
∆
∆
k=−∞












(9)

where Px is the arbitrary cf of the input signal and Pν is the cf of the dither. In orderfor ε to be uniformly distributed, this must reduce to sinc (u) for some choice of Pν .
Suppose that this is possible, in which case we obtainsinc (u) =∞Xsinc u −

k=−∞

k
k
k
Pν u −
Px −
.
∆
∆
∆








`
−
,
∆





Now let u = `/∆ where ` ∈ Z0 . Then we have
`
sinc
∆




= 0 = Pxwhich contradicts the assumption that Px is arbitrary. Thus the total error cannot
be made uniformly distributed in a non-subtractively dithered system for inputs ofarbitrary distribution.
2

The counterintuitive nature of this result is the source of much confusion regarding
NSD systems. For instance, it is tempting to accept the following line of reasoning:15

suppose that a dither satisfying Schuchman’s condition (Eq. (5)) is used so that q isindependent of x. Then, since ν is also independent of x, the total error ε = q +ν is thesum of two random processes both of which are independent of x and thus should be
independent of x as well. This conclusion is flatly false. In an NSD quantizing system,
given the value of q + ν we know that the possible values of x satisfy the equationx = −(q + ν) + k∆, k ∈ Z, so that the distribution of x is highly dependent on q + ν.
To elucidate the source of the problem we may reason as follows: for arbitrary randomvariables q, ν, and x and a fourth ε = q + ν (none of these necessarily representing
quantities in a quantizing system) it is clear thatpε|q,ν,x(ε, q, ν, x) = δ(ε − q − ν).
Thenpε,x (ε, x) ==Z ∞ Z ∞

−∞ −∞
Z ∞
−∞

pε|q,ν,x(ε, q, ν, x)pq,ν,x (q, ν, x)dqdν

pq,ν,x (ε − ν, ν, x)dν,the Fourier transform of which yields the joint cf of ε and x:
Pε,x (uε , ux ) ==Z ∞ Z ∞ Z ∞−∞ −∞ −∞Z ∞ Z ∞ Z ∞−∞ −∞ −∞

pq,ν,x (ε − ν, ν, x)e−j2π(uε ε+ux x) dνdεdx
pq,ν,x (w, ν, x)e−j2π[uε (w+ν)+ux x] dwdxdν,where w = ε − ν,

= Pq,ν,x (uε , uε , ux ).
By definition, ε and x are statistically independent of one another if and only if
Pε,x (uε , ux ) can be written as a product of two functions, one involving uε alone while
the other involves ux alone. From the above we see that this is the case if and only if
Pq,ν,x (uε , uε , ux ) = Pq,ν (uε , uε )Px (ux ).
16

Unfortunately, knowing as we do that Pq,x (uq , ux ) = Pq (uq )Px (ux ) and Pν,x (uν , ux ) =Pν (uν )Px (ux ) is simply not sufficient to ensure satisfaction of the latter condition. Of
course, the result would hold if {q, ν, x} formed a set of independent random variables;that is, if it were the case that
Pq,ν,x (uq , uν , ux ) = Pq (uq )Pν (uν )Px (ux ).
However, this even stronger condition is certainly not met in an NSD quantizing systemwith an arbitrarily distributed input.
Since we have shown that statistical independence of the total error from the system
input is not achievable, we now turn our attention to the possibility of controlling
moments of the error. For many applications, controlling relevant error moments isjust as good as having full statistical independence of the input and error processes.

2.2

A Condition for the Independence of Total Error Moments

The m-th moment of the error signal is the expectation value of εm :E[εm ] =Z ∞−∞εm pε (ε)dε.

It can be shown that these moments may also be expressed in terms of the cf of thegiven random variable as [33]:E[εm ] =(m)

where Pεj m (m)Pε (0),2π(10)denotes the m-th derivative of Pε . (This expression is easily derived bydifferenting with respect to u the definition of Pε (u) as the Fourier transform of pε .)From Eq. (9) we obtain
E[εm ] =



∞
j m X
G(m)
ν
2π
k=−∞



17



k
Px
∆




k
,
∆


(11)

where4Gν (u) = sinc (u)Pν (u).

(12)

Since the cf, Px , of the system input is arbitrary we obtain the following result [26]:Theorem 2 In an NSD quantizing system, E[εm ] is independent of the distributionof the system input, x, if and only if
G(m)
ν



k
∆



=0

∀k ∈ Z0 .

(13)

If the conditions of Theorem 2 are satisfied then, from Eq. (11),m

E[ε ] =

j m (m)Gν (0),
2π
which is precisely the m-th moment of a notional random process with cf Gν and pdf∆Π∆ ? pν , although this is not, of course, the pdf of ε. We can derive the following
expressions for the moments of the total error in terms of the moments of the dither
signal by direct differentiation of Gν (u):E[ε] = E[ν](14)

E[ε2 ] = E[ν 2 ] +m

E[ε ] =

m
bX
2 c

`=0

m
2`

∆2

(15)

12
!

∆
2

2`

E[ν m−2` ].
2` + 1(16)

We emphasize that each of these equations for E[εm ] is only valid when Theorem 2 issatisfied for that particular value of m, and that the validity of one of these equations
does not imply the validity of any others corresponding to different m values.
Eq. (15) merits special comment. It indicates that if the total error variance in an
NSD quantizing system is input independent, then it always exceeds that of an SD
18system (or a system described by the CMQ) by an amount equal to the variance ofthe dither. This characteristic increase in the error power is not problematic in most
multi-bit applications, and the benefits of dithering typically far outweigh the slightnoise penalty.
Two corollaries to Theorem 2 follow.
Corollary 1 In an NSD quantizing system, if the condition of Eq. (13) is satisfied
for any given m, then for any choice of nE[εm xn ] = E[εm ]E[xn ];i.e., εm and xn are uncorrelated.
Proof : We observe that if px (x) = δ(x − x0 ) thenpε (ε) =
=

Z ∞

−∞
Z ∞
−∞

pε,x(ε, x)dx
pε|x(ε, x)δ(x − x0 )dx= pε|x(ε, x0 ).

(17)

By Theorem 2, E[εm ] is independent of the choice of px , and in particular it is independent of the choice of x0 when px (x) = δ(x − x0 ), as above. ThusZ ∞−∞εm pε|x(ε, x)dε = E[εm ]for any x. In this case,m nE[ε x ] ===Z ∞ Z ∞

εm xn pε,x(ε, x)dεdx−∞ −∞Z ∞ Z ∞
−∞
Z ∞

m

−∞



ε pε|x(ε, x)dε xn px (x)dx

E[εm ]xn px (x)dx−∞m= E[ε ]E[xn ].
19

2
In particular, if E[ε] is independent of the distribution of x, then ε and x are uncorrelated in the usual mathematical sense:
E[εx] = E[ε]E[x].
The second corollary is somewhat better known than Theorem 2 itself, but demands
satisfaction of a stronger condition [10, 30].
Corollary 2 In an NSD quantizing system, E[ε` ] is independent of the distributionof the system input, x, for ` = 1, 2, . . . , m if and only if(i)Pν∀k ∈ Z0



k
∆

and



=0

i = 0, 1, 2, . . . , m − 1.

Proof : Proof of the “if” direction follows immediately from repeated differentiationof Eq. (12):G(`)ν (u) =

!
X̀ `
i=0

i

sinc(`−i) (u)Pν(i) (u).

We see that the `-th and all lower derivatives of Gν will all go to zero at u = k/∆,k ∈ Z0 if the first ` − 1 derivatives of Pν do. The “only if” direction is easily provenusing induction, but this requires more space than is justified here. The interested
reader is referred to [25].
2

In most practical applications, we are interested in dither signals which satisfy theconditions of Corollary 2, and it turns out that the conditions of this corollary will be
20

of interest when we examine the statistics of the quantizer output (Section 2.4) andthe special nature of digital dither signals (Section 4).

2.3

Second-Order Statistics of Total Error Values

We now begin an investigation into the joint statistics of temporally separated total
error values, corresponding to input samples separated in time, in order to derive
conclusions about the spectral characteristics of the total error sequence.
Consider two total error values, ε1 and ε2 , which are separated in time by τ 6= 0.
(In the special case where τ = 0, the analysis reduces to that of Section 2.2.) The corresponding system input values will be denoted as x1 and x2 , respectively. Employinga derivation analogous to that of the Section 2.1 we find thatp(ε1 ,ε2 ) (x1 ,x2 )=(ε1 , ε2 , x1 , x2 )
∞
X

∞
X

δ(ε1 + x1 − k1 ∆)δ(ε2 + x2 − k2 ∆)k1 =−∞ k2 =−∞×

Z ∆ +k1 ∆ Z ∆ +k2 ∆
2

+k1 ∆
−∆
2

2

+k2 ∆
−∆
2

pν1 ,ν2 (w1 − x1 , w2 − x2 )dw1 dw2= [∆2 Π∆∆ ? pν1 ,ν2 ](ε1 , ε2 )W∆∆ (ε1 + x1 , ε2 + x2 ),where the convolution is two-dimensional, involving both ε1 and ε2 , and where4ΠΓΓ (ε1 , ε2 ) = ΠΓ (ε1 )ΠΓ (ε2 )and4WΓΓ (ε1 , ε2 ) = WΓ (ε1 )WΓ (ε2 ).
pν1 ,ν2 represents the joint pdf of the dither values, ν1 and ν2 , associated with the
inputs x1 and x2 , respectively.

21

Hencepε1 ,ε2 (ε1 , ε2 ) ==Z ∞ Z ∞

p
(ε1 , ε2 , x1 , x2 )px1 ,x2 (x1 , x2 )dx1 dx2−∞ −∞ (ε1 ,ε2 ) (x1 ,x2 )[∆2 Π∆∆ ? pν1 ,ν2 ](ε1 , ε2 )[W∆∆ ? px1 ,x2 ](−ε1 , −ε2 ).

(18)

The joint characteristic function of ε1 and ε2 is found by taking the two-dimensional
Fourier transform of Eq. (18) with respect to ε1 and ε2 , resulting in an expression in
the corresponding frequency variables, u1 and u2 :
∞
X

∞
X

k1
k2
sinc u1 −sinc u2 −Pε1 ,ε2 (u1 , u2 ) =∆∆k =−∞ k =−∞1

2

×Pν1 ,ν2







k1
k2
k1 k2
u1 − , u2 −Px1 ,x2 − , −
.
∆
∆
∆
∆









(19)

No choice of dither pdf will allow Eq. (19) to be expressed as a product of two characteristic functions, one involving u1 alone and the other u2 alone, for arbitrary choicesof Px1 ,x2 . Thus ε1 and ε2 cannot be rendered statistically independent for arbitrary
joint input distributions. Let us therefore proceed to investigate the joint moments of
ε1 and ε2 in the hope that we can exercise some control over them by an appropriate
choice of the dither statistics.
The (m1 , m2 )-th joint moment of the two signals of interest is given by:
4

1 m2
E[εm
1 ε2 ] ==Z ∞ Z ∞

2 m2
εm
1 ε2 pε1 ,ε2 (ε1 , ε2 )dε1 dε2−∞ −∞m1 +m2j
2π

1 ,m2 )Pε(m(0, 0)
1 ,ε2

(20)

where
4 ∂

1 ,m2 )Pε(m(u1 , u2 ) =1 ,ε2(m1 +m2 ) P
ε1 ,ε2(u1 , u2 ).
m1
m2∂u1 ∂u2

Substituting Eq. (19) into Eq. (20), one finds that
1 m2
E[εm
1 ε2 ] =∞∞X
j m1 +m2 X
k1 k2
k1 k2
1 ,m2 )
Px1 ,x2 − , −G(m− ,−ν1 ,ν22π∆∆∆∆
k =−∞ k =−∞



1



2

22







(21)

where4Gν1 ,ν2 (u1 , u2 ) = sinc (u1 )sinc (u2 )Pν1 ,ν2 (u1 , u2 ).
At this point we may deduce a theorem which represents a second-order analog ofTheorem 2:
1 m2
Theorem 3 In an NSD quantizing system, the (m1 , m2 )-th joint moment, E[εm1 ε2 ],of two total error values, ε1 and ε2 , separated in time by τ 6= 0, is independent of thesystem input for arbitrary input distributions if and only if
1 ,m2 )G(m
ν1 ,ν2



k1 k2
,
∆ ∆



∀(k1 , k2 ) ∈ Z20 .

=0

(22)

The proof is completely analogous to that of Theorem 2. When Eq. (22) is satisfied,
we have
1 m2
E[εm1 ε2 ] =j m1 +m2 (m1 ,m2 )Gν1 ,ν2 (0, 0),
2π


(23)

so that by explicitly performing the differentiation we can write an expression, analogous to Eq. (16), relating the joint moments of the total error to those of the dither:
1 m2
E[εm
1 ε2 ] =m1
m2bX
2 c bX
2 c`1 =0 `2 =0

m1
2`1

!

m2
2`2

!

∆
2

2(`1 +`2 )E[ν1m1 −2`1 ν2m2 −2`2 ].
(2`1 + 1)(2`2 + 1)
(24)

We attach the caveat that satisfaction of Eq. (24) for some particular m1 and m2 does
not imply its satisfaction for any other values thereof.
If the dither process is iid so that ν1 and ν2 are statistically independent thenPν1 ,ν2 (u1 , u2 ) = Pν (u1 )Pν (u2 ).

23

Then if the conditions of Corollary 2 are satisfied for m = max(m1 , m2 ) we have
1 m2
E[εm
1 ε2 ]

j m1 +m2 (m1 )2)=Gν (0)G(m
(0)
ν
2π
m2
1
= E[εm
1 ]E[ε2 ]




(25)

m2
m
m
m
1
so that εm
1 and ε2 are uncorrelated. In this case, of course, E[ε1 ] = E[ε2 ] = E[ε ].

Hence:
Corollary 3 Any iid non-subtractive dither signal which satisfies the conditions ofCorollary 2 for m = max(m1 , m2 ) will ensure that, for two error values, ε1 and ε2 ,separated in time by τ 6= 0,m1
1 m2]E[εm2 ].
E[εm1 ε2 ] = E[ε(26)In this case E[εm2 ] and E[εm2 ] will be given by Eq. (16). In particular, for an iid
dither with zero mean we note that E[ε1 ε2 ] = 0.
In a digital system, the total error is a discrete-time signal, so that τ = kT where
T represents the sampling period and k ∈ Z. The autocorrelation function of such asignal is defined to be E[ε1 ε2 ](k). The power spectral density (PSD) of a discrete-time
random process is equal by definition to the discrete-time Fourier transform (DTFT)
of its autocorrelation function, where we define the DTFT as
4

FDT [h](f ) = 2T∞X

h(k)e−j2πf kT ,

(27)

k=−∞

where the continuous frequency variable, f , is in hertz if T is in seconds. This definition
is normalized such that the integral of the PSD from zero to the Nyquist frequency,
1
2T , yields the variance of the signal.

24

Using Eqs. (15) and (26) we find that, for an NSD quantizing system using iid
dither satisfying the conditions of Corollary 2 for m = 2, the autocorrelation functionof the error is
2 E[ν 2 ] + ∆ , k = 0,12E[ε1 ε2 ](k) =2E [ν],

otherwise.

Comparing this with the autocorrelation function of the dither sequence,E[ν1 ν2 ](k) =we conclude that
 E[ν 2 ], k = 0,
 E 2 [ν], otherwise,

PSDε (f ) = PSDν (f ) +∆2 T
6so that the total error signal must be spectrally white since the dither is spectrally
white (apart from a dc component if the dither is not zero mean).
It is possible to derive conditions which ensure the satisfaction of Eq. (22) forthe case where m1 = m2 = 1, but which do not require statistical independence ofdistinct dither values [24, 25]. This will allow the use of certain dither signals which
are notspectrally white.
Theorem 4 In an NSD system where all dither values are statistically independentof all system input values,E[ε1 ε2 ] = E[ν1 ν2 ](28)

for arbitrary input distributions if and only if the following three conditions are satisfied:
k1 k2
,
= 0
∆ ∆


k1
Pν(0,1)
,
0
= 0
1 ,ν2
∆


k2
Pν(1,0)
0,
= 0
1 ,ν2
∆

Pν1 ,ν225∀(k1 , k2 ) ∈ Z20 ,(29)∀k1 ∈ Z0 ,(30)

∀k2 ∈ Z0 .

(31)

This can be thought of as a second-order counterpart to Corollary 2 for the simple
case m1 = m2 = 1. When the conditions of the theorem are satisfied, Eq. (28) follows
immediately from direct differentiation of Eq. (23). It is possible to gain further insight
into the meaning of the conditions involved by noting thatE[ε1 ε2 ] = E[(q1 + ν1 )(q2 + ν2 )] = E[ν1 ν2 ] + E[q1 ν2 ] + E[q2 ν1 ] + E[q1 q2 ].
The last term is equal to zero as long as Eq. (29) is satisfied (see Eq. (6) or Theorem 2of [26]) while it can be shown that the second and third terms vanish subject to thesatisfaction of Eqs. (30) and (31), thus yielding Eq. (28). Necessity of the conditions
follows from the arbitrariness of the input distribution.
Suppose that the conditions of both Theorem 4 and Corollary 2 with m = 2 aresatisfied. Then the autocorrelation function of the error is given by
2 E[ν 2 ] + ∆ , k = 0,12E[ε1 ε2 ](k) =E[ν1 ν2 ],(32)

otherwise.

This indicates that the power spectrum of the error will be identical to the power
spectrum of the dither, apart from a contribution due to the k = 0 case, manifested
as an additive constant present at all frequencies (i.e., a white spectral component
introduced by the properly dithered quantization operation). Hence, as before,
PSDε (f ) = PSDν (f ) +

∆2 T
,
6

(33)

except that now the dither PSD is not necessarily white. This will be illustrated by
the discussion of high-pass dither in Section 3.5.

26

2.4

Statistics of the System Output

It is of interest to express the statistical attributes of the output, y, in terms of thoseof the input, x, since it is frequently required that one be deduced from the other.
We apply the same brand of reasoning as used to determine the cpdf of the total
error in Section 2.1. The values of the quantizer output are restricted to values of k∆,k ∈ Z. Therefore py (y) will consist of delta functions at these locations weighted by theprobability that the quantizer input w = x + ν falls in the range of the corresponding
2k+1
quantizer step, 2k−1
2 ∆ < w <2 ∆. This probability is just the integral of pw (w)over this range where, since x and ν are statistically independent, pw is given by [34]pw (w) = [pν ? px ](w).
Thus we havepy (y) =∞Xδ(y − k∆)k=−∞

Z ∆ +k∆
2

−∆
+k∆
2

[pν ? px ](w)dw= [∆Π∆ ? pν ? px ](y)W∆ (y).
Taking the Fourier transform of this expression yields
Py (u) = [Gν (u)Px (u)] ? W 1 (u)
∆

=

∞
X

k=−∞

Gν

k
k
u−
Px u −
∆
∆









(34)

and soj m (m)Py (0)2π!   m−r
 
∞ X
m
X
m
j r (r) k
j
k
(m−r)
=
Gν
Px
.
r
2π
∆
2π
∆
k=−∞ r=0E[y m ] =





27

(35)

Now, if the first m derivatives of Gν (u) are zero at all non-zero multiples of 1/∆, thenEq. (35) reduces tom

E[y ] =m
X!

m
E[εr ]E[xm−r ],
r

r=0

(36)

where the expectation values of the total error are given in terms of the expectation
values of the dither by Eq. (16). By direct differentiation of Gν (u), the above condition
is easily shown to be equivalent to the condition of Corollary 2. For the special cases
m = 1 and m = 2 we note that
E[y] = E[x] + E[ε] = E[x] + E[ν]E[y 2 ] = E[x2 ] + 2E[x]E[ν] + E[ν 2 ] +∆212where Eqs. (14) and (15) have been substituted for the error moments.2
Proceeding similarly for the joint moments of output values y1 and y2 , separated
in time by τ 6= 0, we find thatE[y1m1 y2m2 ]

=

∞
X

∞
X

m1 X
m2
X

k1 =−∞ k2 =−∞ r1 =0 r2 =0

×

"

m1
r1

!

m2
r2

! "

j r1 +r2 (r1 ,r2 ) k1 k2
Gν1 ,ν2
,
2π
∆ ∆

#

#

(37)



j (m1 −r1 )+(m2 −r2 ) (m1 −r1 ,m2 −r2 ) k1 k2
Px1 ,x2
,
2π
∆ ∆






.

If the indicated partial derivatives of Gν1 ,ν2 are zero for all (k1 , k2 ) ∈ Z20 , ri =1, 2, . . . , mi , i ∈ {1, 2}, then Eq. (37) reduces toE[y1m1 y2m2 ] =

m1 X
m2
X

r1 =0 r2 =0
2

m1
r1

!

!

m2
1 −r1 m2 −r2
E[εr11 εr22 ]E[xm
x2
],
1
r2

(38)

Note that the so-called dither averaged transfer characteristic, E[y|x], is given by
E[y|x] = E[Q(x + ν)|x] =Z ∞Q(x + ν)pν (ν)dν = Q(x) ∗ pν (−x),−∞

which is the convolution of the quantizer staircase with the dither pdf. For the m = 1 case this defines
the line y = x (see illustrations in [16, 17]).

28

where the joint moments of the total error are given in terms of those of the dither by
Eq. (24).
Beginning from Eq. (37) with m1 = m2 = 1 it is straightforward to show that if
the conditions of Theorem 4 are satisfied (i.e., Eqs. (29), (30) and (31)) thenE[y1 y2 ] = E[x1 x2 ] + E[ν1 ν2 ],so that, with the aid of Eqs. (36) and (16), we find that the output has an autocorrelation function
E[y1 y2 ](k) =2 E[x2 ] + 2E[x]E[ν] + E[ν 2 ] + ∆ ,12 E[x x ] + E[ν ν ],
1 2

1 2

k = 0,(39)

otherwise.

Then the spectrum of the output is the sum of the input and dither spectra apartfrom a white noise component, which is contributed by the k = 0 case of Eq. (39).
The latter component is comparable to the white ”quantization noise” posited in the
CMQ. In particular, for a system using a zero-mean dither,
PSDy (f ) = PSDx (f ) + PSDν (f ) +

3

∆2 T
.
6

Error Moments in Some Representative Systems

We proceed to apply the above results to realizable quantizing systems using a variety
of dither signals.

3.1

Null Dither

We begin by considering an undithered system. The pdf of a “null dither” ispν (ν) = δ(ν),

29the Fourier transform of which is equal to unity everywhere. Hence, by Eq. (12),Gν (u) = sinc (u).
No derivatives of this function vanish at non-zero multiples of 1/∆, so no moments ofthe total error (excepting the zeroth) will be independent of the input distribution. Of
course, it is not expected that they would be. We know that in the absence of dither,
the error is a deterministic function of the input. Indeed, the mean value of the error
as a function of the input is identical to the error function, q(x), shown in Fig. 2.

3.2

Rectangular-PDF Dither

Now consider a system using dither with a simple rectangular (i.e., uniform) pdf of
1 LSB peak-to-peak amplitude:pν (ν) = Π∆ (ν),
with a corresponding cf
Pν (u) = sinc (u).
Hence, from Eq. (12):Gν (u) = sinc2 (u).
The first two derivatives of this function are plotted in Fig. 5. The first derivative
clearly satisfies the condition of going to zero at the regularly spaced points stipulated
by Eq. (13), while the second derivative does not (nor do higher derivatives). This
indicates that the first moment of the error signal is independent of the input, but thatits variance remains dependent. These conclusions are borne out by the accompanying
plots in Fig. 5 of the conditional moments
4

E[εm |x] =

Z ∞

−∞

εm pε|x(ε, x)dε,

30

1.0

1.5

0.8
First Moment [LSB]First Derivative [LSB]

2.0

1.0
0.5
0.0
-0.5
-1.0
-1.5

0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8

-2.0

-1.0
-4

-3
-2
-1
0
1
2
3
Frequency Variable, u [1/LSB]

4

-2

-1
0
1
System Input, x [LSB]

2

-2

-1
0
1
System Input, x [LSB]

2

6

0.40

4

0.35
Second Moment [LSB^2]

Second Derivative [LSB^2]

(a)

2
0
-2
-4
-6

0.30
0.25
0.20
0.15
0.10
0.05

-8

0.00
-4

-3
-2
-1
0
1
2
3
Frequency Variable, u [1/LSB]

4(b)
Figure 5: Derivatives of Gν (u) (left) and conditional moments of the error (right) for a(1)

quantizer using RPDF dither of 1 LSB peak-to-peak amplitude: (a) Gν (u) and E[ε|x](2)(both in units of ∆), (b) Gν (u) and E[ε2 |x] (both in units of ∆2 ). The frequency
variable, u, is plotted in units of 1/∆ and the system input, x, in units of ∆.

31

as computed using Eq. (7). The first moment, or mean error, is zero for all inputs,
indicating that the quantizer has been linearized by the use of this dither. The error
variance, on the other hand, is clearly signal dependent, so that the noise power in thesignal varies with the input. This is sometimes referred to as noise modulation and isundesirable in audio or video signals.
If the dither is iid, then (as shown in Section 2.3), temporally separated error
values will be uncorrelated. Thus, short-time error spectra will appear flat but theirlevel will be input dependent.

3.3

Triangular-PDF Dither

The most straightforward means of generating dither signals with more complicatedpdf’s is to simply sum two or more statistically independent RPDF random processes.
For instance, the sum of two such processes, ν1 and ν2 , each of 1 LSB peak-to-peakamplitude, yields a dither with a triangular pdf (TPDF) of 2 LSB peak-to-peak amplitude, since the summation of statistically independent random processes convolves
their pdf’s (see Fig. 6):pν (ν) = [pν1 ? pν2 ](ν)= [Π∆ ? Π∆ ](ν).

(40)

Convolution of pdf’s corresponds to multiplication of the respective cf’s [34], so thatin a system employing this kind of dither Pν is a squared sinc function and Gν is givenbyGν (u) = sinc3 (u).
The first and second derivatives of this function go to zero at the required places, so
this dither renders both the first and second moments of the total error independent32

p1 (x)p2 (x)
1/∆

_ __
∆
2

__
∆
2

*

x

[p

1

1/∆

_ __
∆
2

__
∆
2

x

* p2 ](x)
1/∆

_∆

∆

x

Figure 6: A triangular pdf, formed by the convolution of two rectangular pdf’s.
of the system input.3 The second derivative of Gν is shown in Fig. 7, along withthe second conditional moment of the total error, which is a constant ∆2 /4 for all
inputs, in agreement with Eq. (15). Higher derivatives of Gν do not meet the requiredconditions, so that higher moments of the error remain dependent on the input.
We now proceed to show that a triangular pdf of 2 LSB peak-to-peak amplitudeis the only choice of zero-mean dither pdf which renders the first two moments of thetotal error independent of the input while minimizing the second. We will begin bynoting, from Corollary 2 and the stipulation of zero mean, that
k
= 0,
∆
 
k
Pν(1)
= 0,
∆
Pν





∀k ∈ Z0 ,∀k ∈ Z.

Also, Pν (u) must be equal to unity at u = 0 if it is to be a valid characteristic function,
3

A different proof of this result, using a direct method, was given in [16].

33

0.40

4

0.35
Second Moment [LSB^2]

Second Derivative [LSB^2]

6

2
0
-2
-4
-6
-8

0.30
0.25
0.20
0.15
0.10
0.05

-10

0.00
-4

-3
-2
-1
0
1
2
3
Frequency Variable, u [1/LSB]

4

-2

-1
0
1
System Input, x [LSB]2(2)

Figure 7: Gν (u) (left) and E[ε2 |x] (right) (both in units of ∆2 ) for a quantizer using
triangular-pdf dither of 2 LSB peak-to-peak amplitude. The frequency variable, u, isplotted in units of 1/∆ and the system input, x, in units of ∆.
sincePν (0) =Z ∞−∞e−j2π(0)ν pν (ν)dν = 1.

We conclude that the dither cf and its first derivative are completely specified at all
integer multiples of 1/∆. According to the Generalized Sampling Theorem [34], this issufficient to uniquely specify Pν (u) for all u if pν (ν) is ∆-bandlimited (i.e., if pν (ν) = 0
for |ν| ≥ ∆). Since the pdf of Eq. (40) is ∆-bandlimited, and its corresponding cf
satisfies all the given conditions, it must be the unique pdf in question.
It remains to be shown that any dither pdf which is not thus bandlimited willproduce a greater error variance. Since this variance is assumed to be constant withrespect to the input, it is sufficient to show that this holds for a single input value.
We will do so for x = ∆/2.
pε|x (ε, x) for x = ∆/2 is obtained from Eq. (8) using px (x) = δ(x − ∆2 ) (seeEq. (17)). As is shown in Fig. 8(a) it consists of two equally weighted delta functionsat ε = ±∆/2 when triangular-pdf dither of 2 LSB peak-to-peak amplitude is employed.
34

p (ε , x)
ε |x
1
2

p (ε , x)ε|xx =∆ /2

1
2

e-2

e-3
- ∆
2

∆
2

- 5∆
2

ε

- 3∆
2

e-1

e1

- ∆
2

∆
2

x = ∆ /2
e2
3∆
2

e3
5∆
2

ε

(b)

(a)

Figure 8: pε|x (ε, x) evaluated at x = ∆/2 for systems using (a) a triangular-pdf ditherof 2 LSB peak-to-peak amplitude and (b) a wider dither pdf (the delta functions
possess the indicated weightings).
Use of a wider dither pdf will result in the appearance of more delta functions in theerror’s cpdf, as shown in Fig. 8(b), where we denote the weighting of the delta functionat ε = ±(2i − 1)∆/2, i ≥ 1, by e±i , so thatpε|x ε,

∆
2



=

∞ 
X



ei δ ε − (2i − 1)

i=1

∆
2





+ e−i δ ε + (2i − 1)

∆
2



.

(41)

We proceed by expressing the fundamental condition that the integral of this pdf must
equal unity:(e1 + e−1 ) +∞X(ei + e−i ) = 1.

(42)

i=2

Now, by direct integration of Eq. (41), we have
∞ 
X

∆ 2
E[ε |x = ∆/2] =(2i − 1)(ei + e−i )
2
i=1
2

=



∞
X
∆2
(e1 + e−1 ) +(2i − 1)2 (ei + e−i ) .
4
i=2

"

#

(43)

Substituting Eq. (42) yields∞
X∆2
E[ε2 |x = ∆/2] =1+4i(i − 1)(ei + e−i ) ,
4
i=2

"

#

which is always greater than ∆2 /4 since the ei ’s must be positive. We have thus shown
the following:
35

Theorem 5 The choice of zero-mean dither pdf which renders the first and second
moments of the total error independent of the input, such that the first moment is zeroand the second is minimized, is unique and is a triangular pdf of 2 LSB peak-to-peakamplitude.

3.4

The Sum of n Independent Rectangular-PDF Random Processes

Theorem 6 A non-subtractive dither signal generated by the summation of n statistically independent RPDF random processes, renders E[ε` ] independent of the system
input distribution for ` = 0, 1, . . . , n, and results in a total error variance, for n ≥ 2,of (n + 1)∆2 /12.
This must be the case since the use of n such dithers givesGν (u) = sincn+1 (u),the first n derivatives of which will consist entirely of terms containing non-zero powersof sinc (u). Since this function goes to zero at the required places, the first n momentsof the error will always be independent of the input. Higher derivatives will not share
this property [25]. Dithers of this form are sometimes referred to as nRPDF so that,for instance, TPDF dither may also be referred to as 2RPDF.
It is important to note that using uniformly distributed processes of peak-to-peakamplitude not equal to one LSB (or, rather, not equal to an integral number of LSB’s)
will not render error moments independent of the input since the zeros of the associatedsinc functions will not fall at integral multiples of 1/∆ (see illustrations in [16]).
Finally, it is easily shown from the Generalized Sampling Theorem that the (n∆/2)bandlimited dither pdf which renders the first n moments of the total error independentof the input is unique, and must therefore be the pdf of Theorem 6.

36

p (x)
η

p (x)
ν

x

x
PRN

+

η

ν

Σ
_

White
Rectangular-pdf
Input

z -1

High-Pass
Triangular-pdf
OutputFigure 9: High-pass dither generator.

3.5

High-Pass Dither

A very simple discrete-time noise generator capable of producing dither with a highpass spectrum is shown in Fig. 9. The system contains a pseudo-random number
generator, marked PRN, producing iid, uniformly distributed random numbers, anda one-sample delay element marked z −1 . The output is the difference between thepseudo-random number most recently generated by the PRN, ηn , and the previous
one, ηn−1 ; that is, νn = ηn −ηn−1 . The (first-order) pdf of the resulting dither sequenceis triangular (TPDF), since it results from the summation of two statistically independent RPDF sequences, albeit one of these is simply a delayed version of the other.
This means that all the beneficial effects of the TPDF dither discussed in Section 3.3
will also be associated with dither thus generated. Such high-pass TPDF dither may
be preferable in some audio applications since it is less audible than spectrally white
TPDF dither due to the ear’s reduced sensitivity at high frequencies (although these
dithers have equal variances of ∆2 /6). Similar comments apply regarding reduced error visibility in imaging applications. Furthermore, the use of high-pass TPDF dither
is more computationally efficient since it requires the calculation of only one new37

RPDF random number per sample as compared to two when iid TPDF dither is used.
In order to investigate the spectral characteristics of the total error associatedwith this sort of dither, we must derive an expression for pν1 ,ν2 (ν1 , ν2 ) as defined in
Section 2.3. Suppose that the sampling period of the system is T . For time lags|τ | > T , the dither values are statistically independent so thatpν1 ,ν2 (ν1 , ν2 ) = pν1 (ν1 )pν2 (ν2 )= [Π∆ ? Π∆ ](ν1 )[Π∆ ? Π∆ ](ν2 )andPν1 ,ν2 (u1 , u2 ) = sinc2 (u1 )sinc2 (u2 ).
The non-trivial cases are those for τ = ±T . Consider two successive dither values(i.e., τ = T ):
ν1 = η1 − η0
ν2 = η2 − η1 .
Thenpν1 ,ν2 ,η0 ,η1 ,η2 (ν1 , ν2 , η0 , η1 , η2 )= pν1 |(ν2 ,η0 ,η1 ,η2 ) (ν1 , ν2 , η0 , η1 , η2 )pν2 |η0 ,η1 ,η2 (ν2 , η0 , η1 , η2 )pη0 ,η1 ,η2 (η0 , η1 , η2 )= δ(ν1 − η1 + η0 )δ(ν2 − η2 + η1 )pη0 (η0 )pη1 (η1 )pη2 (η2 ).
Taking the Fourier transform of this expression with respect to all variables present
yieldsPν1 ,ν2 ,η0 ,η1 ,η2 (u1 , u2 , w0 , w1 , w2 )=Z ∞ Z ∞ Z ∞−∞ −∞ −∞

e−j2πu1 (η1 −η0 ) e−j2πu2 (η2 −η1 ) pη0 (η0 )pη1 (η1 )pη2 (η2 )×e−j2π(w0 η0 +w1 η1 +w2 η2 ) dη0 dη1 dη2= Pη0 (w0 − u1 )Pη1 (w1 + u1 − u2 )Pη2 (w2 + u2 ).
38

Desired marginal cf’s can be obtained from a given joint cf by simply setting theunwanted variables to zero, sincePx,y (u, 0) =Z ∞ Z ∞−∞ −∞e−j2π(xu+y(0)) px,y (x, y)dxdy = Px (u).

Thus we havePν1 ,ν2 (u1 , u2 ) = Pη0 (−u1 )Pη1 (u1 − u2 )Pη2 (u2 ).
Proceeding similarly for the case of τ = −T , we find thatPν1 ,ν2 (u1 , u2 ) = Pη0 (−u2 )Pη1 (u2 − u1 )Pη2 (u1 ).
For our purposes,Pη0 (u) = Pη1 (u) = Pη2 (u) = sinc (u)so that for both cases (τ = ±T ):Pν1 ,ν2 (u1 , u2 ) = sinc (u2 − u1 )sinc (u1 )sinc (u2 ).
Finally, using τ = kT , k ∈ Z, we can write thatPν1 ,ν2 (u1 , u2 ; k) = sinc (u2 − u1 )sinc (u1 )sinc (u2 ),

sinc2 (u1 )sinc2 (u2 ),k = ±1,(44)|k| > 1.

It is straightforward to check that this joint cf satisfies all three conditions of Theorem 4.
Using Eq. (44) and the knowledge that TPDF dither has a variance of ∆2 /6, wefind using Eq. (20) that the autocorrelation function of the dither under consideration
is




1,




k = 0,∆2
E[ν1 ν2 ](k) =× − 12 , k = ±1,

6



 0, otherwise.
39

This corresponds to a simple high-pass power spectral density
PSDν (f ) =∆2 T[1 − cos(2πf T )],3

where the frequency variable, f , is in units of hertz if T is in seconds. Then, according
to Eq. (32),



1,




k = 0,∆2
E[ε1 ε2 ](k) =× − 13 , k = ±1,

4



 0, otherwise,

which corresponds to a power spectral density of
PSDε (f ) =∆2 T[3 − 2 cos(2πf T )].
6

This is simply the high-pass spectrum of the dither, plus a white “quantization noise”
component of ∆2 T /6 (which has a total power of ∆2 /12 up to the Nyquist frequency,
12T ) in agreement with Eq (33).

Of course, it is possible to imagine many other spectrally shaped dither signals.
The properties of such signals have now been investigated in detail. In particular,
there is the following theorem, proven and extensively illustrated in [25, 27]:
Theorem 7 In an NSD quantizing system using dither of the form
νn =

∞
X

ci ηn−ii=−∞where η is an iid nRPDF random process, the total error will be wide-sense stationary
and independent of the system input with a PSD given by
PSDε (f ) = PSDν (f ) +under the following conditions:

40

∆2 T
6

1.

for each ` ∈ Z0 there exists an i such that of ci and ci+` one is zero and theother is a non-zero integer,and2.

either η is nRPDF with n ≥ 1 and there exist at least two distinct values of isuch that ci is a non-zero integer, or η is nRPDF with n ≥ 2 and there exists atleast one value of i such that ci is a non-zero integer.

In particular, simple high-pass TPDF dither satisfies the above conditions.

4

Digital Dither

Some comment is required concerning the special nature of requantization operations,in which the binary wordlength of data is reduced prior to its storage or transmission.
This operation takes place entirely within the digital domain, so that both the input
and dither signals are discrete valued due to the finite wordlengths available in practical
digital systems. The continuous pdf’s discussed thus far are unattainable in a purely
digital scheme so that the properties of true digital dither signals require further
investigation.
The following discussion represents a theoretical complement to empirical resultspresented in [16]. It is not intended to be exhaustive, but merely to demonstrate that
there is no great difficulty in extending the results obtained for analog systems todigital ones, and to illustrate how this may be done. In particular, the discussion will
be restricted to a treatment of first-order statistics with the extension to second-order
being straightforward.
Consider a quantizing system which applies digital dither to digital data before
removing its L least significant bits. We will use δ to denote the magnitude of an LSB
41

of the higher-precision signal which is to be requantized, and∆ = 2L δ
for an LSB of the requantized output.
Let us consider the following digital dither pdf
pν (ν) = δp̃ν (ν)Wδ (ν),(45)

where p̃ν (ν) represents an absolutely integrable function which serves as a “weighting”
for the impulse train. p̃ν is assumed to be normalized such thatZ ∞−∞pν (ν)dν = δ∞Xp̃ν (`δ) = 1.

`=−∞

For instance, p̃ν might be the pdf of a dither of order n, such as an nRPDF dither, inwhich case it is straightforward to show using Poisson’s summation formula [35] thatp̃ν has the above normalization. In general, however, p̃ν need not correspond to a pdf
since it need not subtend unit area.
Taking the Fourier transform of Eq. (45) we find thatPν (u) ==hiP̃ν ? W 1 (u)
δ

∞
X

P̃ν

`=−∞

`
u−
δ





(46)

where P̃ν (u) is the Fourier transform of p̃ν (ν). Note that even if P̃ν satisfies theconditions of Corollary 2 (for some m), Pν will not, due to the modulation of P̃ν (u)by the impulse train W 1 (u). Fortunately, we do not require that these conditions beδsatisfied in a digital system, since the requirement that E[εm |x] be constant for all
values of the system input is not of interest. Instead, we require only that the moments
be constant for a subset of all conceivable x values, namely {x|x = nδ, n ∈ Z}, which42

includes all values that are representable in the digital system. Thus we assume thatthe pdf of the system input can be expressed in the formpx (x) = δp̃x (x)Wδ (x)(47)

where p̃x is a continuous function normalized such that the integral of Eq. (47) is unity.
Then
Px (u) = [P̃x ? W 1 ](u)
δ

∞
X

=

P̃x

`=−∞

`
u−
.
δ





(48)

Now, from Eq. (12), we have4Gν (u) ==sin(π∆u)Pν (u)π∆u∞sin(π∆u) XkP̃ν u −.
π∆u k=−∞
δ(49)

Then, from Eq. (9),Pε (u) =

∞
X

∞
X

Gν

k=−∞`=−∞kk + 2L `
u−
P̃x −
∆
∆





!

so thatj m (m)Pε (0)
2π!
 m X


∞
∞
X
j
k
k + 2L `
(m)
=
P̃x −
Gν
−
.
2π
∆
∆
k=−∞`=−∞

E[εm ] =





(50)

The only way that this quantity can be independent of P̃x is if we require that
(m)
Gν



k
∆



=0

for all k ∈ Z except possibly for those values of k such that
43

k
∈ Z.
2L

(51)

That is, the indicated derivative must vanish for all integral values of k except thosewhich are integral multiples of 2L , the value of this derivative being immaterial in thelatter cases. In order to see that this is so, note that if a dither is chosen such thatEq. (51) holds then many terms vanish from Eq. (50), leavingm

E[ε ] =



∞
j m X
G(m)
ν
2π
k=−∞



  X
∞
k

δ

P̃x

`=−∞

`
.
δ

 

Now, from Eq. (48) we know thatPx (0) =

∞
X

P̃x

`=−∞

`
δ

 

= 1.

This leaves
E[εm ] =



∞
j m X
G(m)
ν
2π
k=−∞



k
,
δ

 

(52)

which does not depend on the input distribution. The necessity of Eq. (51) follows
from the arbitrariness of P̃x (apart from its normalization). Furthermore, by inspection
Eq. (52) is precisely the m-th moment of a notional random variable with pdf∆Π∆ ? pν (ε)Wδ (ε),
2Lalthough this is not, of course, the pdf of ε. Some algebraic manipulation of thisexpression, exploiting the discrete-valued character of ν, reveals that it is equivalent
to the following:∆Π∆ · Wδ (ε) ? pν (ε).
2L


(53)

This may be regarded as the pdf of a notional random variable which is the sum of
the dither and an independent discrete-valued “quantization noise”.
Note also that in the limit as δ → 0 (i.e., as L → ∞) Eq. (51) becomes Eq. (13),the condition of Theorem 2 for analog systems.
44

Returning to Eq. (49) and differentiating, we have∞ XmXm dr sin(π∆u)
dm GνkP̃ν(m−r) u −
(u)
=
.
m
r
r du
du
π∆u
δ
k=−∞ r=0

!









(54)

If P̃ν meets the conditions of Corollary 2, then all terms in Eq. (54) involving thederivatives of P̃ν go to zero at the places required by Eq. (51) except for the single(r = 0) term involving the m-th derivative. Fortunately, this term involves the zeroth
derivative of the leading sinc function, which goes to zero at all the required places.
This yields the following theorem:
Theorem 8 For a digital NSD system in which requantization is used to remove the
L least significant bits of binary data, E[ε` ] is independent of the input distributionfor ` = 1, 2, . . . , m, if a non-subtractive digital dither (with the same precision as the
input data) is applied for which(i)P̃ν∀k ∈ Z0

and



k
∆



=0

i = 0, 1, 2, . . . , m − 1.

This theorem is the digital counterpart of Corollary 2. It is interesting to note thatno such counterpart exists for Theorem 2 in terms of P̃ν .
We observe that using a dither of higher precision than the input signal is of nobenefit. For instance, a dither cf which satisfies the conditions of Eq. (51) with m = 1
for L = 8 will also satisfy them for L = 4, but for a quantizing system in which theprecision is reduced by only four bits there is no advantage associated with this cf over
one which only satisfies the conditions for L = 4.
Frequently, dithers in digital systems will be given a 2’s-complement [31] representation and thus will exhibit a mean which differs slightly from zero. This will be45

reflected in the appearance of a small non-zero mean error which, of course, will be
input independent if an appropriate dither pdf has been chosen.
To express the moments of the system output we impose the conditions of Theorem 8 upon Eq. (35), obtainingm

E[y ] =
=

m
X

r=0
m
X
r=0

m
r

!
!


∞ 
X
j r

2π

k=−∞

G(r)
ν

  "kδ #

j m−r (m−r) k
Px
2π
δ


m
E[εr ]E[xm−r ],rwhere we have observed from Eq. (48) that Px (u) is periodic with period 1/δ so thatfor any k ∈ Zj m−r (m−r) k
Px
2π
δ


 

=



j m−r (m−r)Px(0) = E[xm−r ].
2πE[εr ] is given by Eq. (52).
The treatment presented above is most appropriate to dithers generated entirely inthe digital domain using, for instance, pseudo-random number generation algorithms.
In particular, we have shown that whenever the weighting function p̃ν corresponds tothe pdf of an analog nRPDF dither the associated digital dither with pdf given by
Eq. (45) shares the beneficial properties of its analog counterpart.
In the case where a digital dither signal is generated by fine quantization of ananalog dither signal, the details of the derivation change only slightly. The forms ofthe theorems, however, remain the same, with P̃ν representing the cf of the analog
signal. This can be seen directly using Eq. (34) (with null dither), for the pdf of thedigital dither generated by quantization will bepν (ν) = [δΠδ ? p̃ν ](ν)Wδ (ν)
with cfPν (u) =sin(πδu)P̃ν (u) ? W 1 (u).
δ
πδu


46

This expression should be compared with Eq. (46). Note that if P̃ν satisfies theconditions of the theorems, then so will the quantitysin(πδu)P̃ν (u).
πδuThus far, the behavior of the quantizer at step edges (i.e., when w = 2k−1
2 ∆, k ∈ Z)
has not been explicitly considered. This is not a problem if the signals in question arecontinuous-valued. In this case the addition of dither will ensure that the quantizer
input resides at a quantizer-step edge with zero probability. On the other hand, if
digital signals are in use, the probability that the quantizer input resides at a stepedge is always greater than zero. In this instance it makes a considerable difference
to the quantizer output (and total error) whether the quantizer rounds up, down,
or stochastically (up or down with equal probability) at these edges. Technically, it
can be shown [25] that the above formalism yields correct predictions if a stochastic
quantizer is used.
The extension of the results to deterministic (i.e., non-stochastic) quantizers employs a simple trick. Consider, for instance, the consequences of choosing a quantizer
which always rounds up at step edges (a similar argument applies to quantizers whichround down). We note that if a (dc) virtual offset τ such that 0 < τ < δ is introduced
into the dither signal, the quantizer output is unaffected except that quantizer inputs residing at step edges are consistently rounded up. We can thus analyze digitally
dithered systems with deterministic requantizers using such a notional dc offset, which
is a purely mathematical device without physical counterpart. It can be shown [25]that Theorem 8 holds precisely as before. Eq. (52) holds if the virtually offset ditherpdfpν (ν) = δp̃ν (ν − τ )Wδ (ν − τ )47

is used in the calculations. In this case, the expression (53) becomes5∆Π∆ (ε − τ )Wδ (ε) ? pν (ε + τ ).
2L
Conclusions

The following conclusions bear repeating:
1. Non-subtractive dithering, unlike subtractive dithering, cannot render the total
error statistically independent of the system input, but it can render any desiredconditional moments of the total error independent of the input distribution
provided that certain conditions on the cf of the dither are met (see Theorems 1
and 2). In particular, a nRPDF dither will render the first n moments of thetotal error input independent.
2. Non-subtractive dithering, unlike subtractive dithering, cannot render total error values separated in time statistically independent of one another. It can,however, regulate the joint moments of such errors and, in particular, it canrender the power spectrum of the total error signal equal to the power spectrumof the dither signal plus a white “quantization noise” component (see Theorem 4
and Eq. (33)).
3. Non-subtractive dithering can render any desired moments of the system input
recoverable from those of the system output, provided that the statistical attributes of the dither are properly chosen (see Section 2.4). This includes joint
moments of system inputs separated in time, so that the spectrum of the input
can be recovered from the spectrum of the output.
4. Proper non-subtractive dithering always results in a total error variance greaterthan ∆2 /12 (see Eq. (15) and Theorem 6).
48

It is also worth noting that, since the dither is simply an additive signal which is independent of the system input, we are free to add it at any time prior to (re)quantization.
In particular, once a signal is properly dithered, other signals (which are statisticallyindependent of the dither) may be added to it and the resulting total signal will still
be properly dithered for (re)quantization purposes.
For audio signal processing purposes, there seems to be little point in rendering any
error moments other than the first and second independent of the input. Variations inhigher moments are believed to be inaudible and this has been corroborated by a large
number of psycho-acoustic tests conducted by the authors and others [13, 21]. These
tests involved listening to a large variety of signals (sinusoids, sinusoidal chirps, slow
ramps, various periodically switched inputs, piano and orchestral music, etc.) which
had been very coarsely requantized (from 16 bits to 8 bits) in order to render therequantization error essentially independent of low-level non-linearities in the digitalto-analog conversion system used for listening purposes. In addition, the corresponding
error signals (output minus input) were used in listening tests in order to check forany vestiges of audible dependence on the input. Using undithered quantizers resulted
in clearly audible distortion and noise modulation in the output and error signals.
Rectangular-pdf dither of 1 LSB peak-to-peak amplitude eliminated all distortion, but
the residual noise level was found to vary audibly in an input-dependent fashion. Whentriangular-pdf dither of 2 LSB peak-to-peak amplitude (either white or high-pass)
was employed, no instance was found in which the error was audibly distinguishable
from a steady random noise entirely unrelated to the input. Admittedly, these tests
were informal, and there remains a need for formal psycho-acoustic tests of this sort
involving many participants under carefully controlled conditions.
We recommend the use of spectrally-white triangular-pdf (TPDF) dither of 2 LSB
peak-to-peak amplitude for most audio applications requiring non-subtractively dithered49

multi-bit quantization or requantization operations, since this type of dither renders
the first and second moments of the total error signal constant with respect to the system input while incurring the minimum increase in error variance. This kind of dither
is easy to generate for digital requantization by simply summing two independent
rectangular-pdf (RPDF) pseudo-random processes, each of 1 LSB peak-to-peak amplitude, which may easily be generated using a linear congruential algorithm [21, 36].
The resulting digital dither can be used to feed a digital-to-analog converter for analog dithering applications. It should be noted, however, that many analog signals
and digital conversion systems exhibit a Gaussian noise component which is of large
enough amplitude to act as a satisfactory dither without the requirement of an explicit
dithering operation [7, 17].
High-pass TPDF dither is of interest for audio processing since it yields a total
error which is audibly quieter than that associated with iid TPDF dither. It also
can be generated with greater computational efficiency, since only one new pseudorandom number needs to be calculated per sampling period instead of two. Other
spectrally-shaped dithers can also be used [24, 25]. The use of spectrally-shaped
dither will usually be superseded, however, by the powerful technique of noise shaping
in applications where the total audibility of the error signal needs to be reduced [37, 38].
This technique employs error feedback in order to spectrally shape the total error ofa quantizing system, including the white component arising from a properly dithered
quantization. The necessity of and criteria for proper dithering of such systems havenow been explored in some detail [27, 24].
With regard to video signals, at least the first two moments of the total error
should be rendered independent of the input by the use of an appropriate dither.
Some evidence exists that input-dependent variations in the third moment of the total
error can be perceptually significant in some video signals [13], but the effects of such
50

variations are probably not noticeable in most cases.
For signal measurement and statistical signal analysis applications in which signal
moments are being measured, appropriate dither (possibly of a quite high order) must
be used to render the input signal statistics correctly determinable from the statisticsof the quantized output, in accordance with Eqs. (36) and (38).
Some of the results obtained above for multi-bit systems may be applied to 1bit quantizers with certain caveats. For instance, a 1-bit quantizing system using afull-scale RPDF dither signal will exhibit a total error signal which is zero-mean andspectrally white whenever its peak input value always remains less than the peak dither
amplitude. In this instance, the 1-bit quantizer may be regarded as a RPDF-ditheredmulti-bit mid-riser quantizer whose peak input amplitude is restricted to less than∆/2, so that the analysis presented above applies without modification. Granted, theamount of dither required is large, but this may be acceptable when no distortion istolerable and the oversampling ratio is high. A sigma-delta converter topology may
be employed to move the noise power out-of-band so long as the dither is iid (in order
to avoid statistical dependences between the dither and input which would otherwise
be introduced by the error feedback [24, 27]). The potential benefits of using lower
dither amplitudes in such systems have been explored in [39].
We maintain that the use of appropriate dither prior to (re)quantization is asfundamental as the use of an appropriate anti-aliasing filter prior to sampling—both
serve to eliminate classes of signal-dependent errors. In each case this is accomplishedby protecting the system from inputs which, if left unmodified, may introduce such
errors.

51

6

Acknowledgments

Stanley P. Lipshitz and John Vanderkooy have been supported by operating grants
from the Natural Sciences and Engineering Research Council of Canada.

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[2] Jayant, N.S. and L.R. Rabiner, “The Application of Dither to the Quantizationof Speech Signals,” Bell Syst. Tech. J., vol. 51, pp. 1293–1304 (1972 July–Aug.).
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52

[8] Sripad, A.B., and D.L. Snyder, “A Necessary and Sufficient Condition for Quantization Errors to Be Uniform and White,” IEEE Trans. Acoust., Speech, Signal
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[12] Stockham, T.G., private communication (1988).
[13] Brinton, L.K., “Nonsubtractive Dither,” M.Sc. Thesis, Dept. of Elec. Eng., Univ.
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in Digital Systems with Dither,” J. Audio Eng. Soc., vol. 32, pp. 106–113(1984 Mar.); correction ibid., p. 889 (1984 Nov.).
[16] Lipshitz, S.P., and J. Vanderkooy, “Digital Dither,” presented at the 81st Convention of the Audio Engineering Society, J. Audio Eng. Soc. (Abstracts), vol. 34,
p. 1030 (1986 Dec.), preprint 2412.
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53

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