PAPERS

Quantization and Dither: A Theoretical Survey*
STANLEY P. LIPSHITZ, AES Fellow,ROBERT A. WANNAMAKER,

AND JOHN VANDERKOOY, AES Fellow**

Audio Research Group, University of Waterloo, Waterloo, Ont. N2L 3Gl, Canada

A theoretical survey of multibit quantization is presented, beginning with the classicalmodel of undithered quantization and proceeding to modern statistical models of undithered, subtractively dithered, and nonsubtractively dithered quantizing systems.
Properties of the error and output signals for each type of system are examined inrelation to the input, using both first- and second-order statistical approaches. The
virtues of various dither signals are assessed for use in practical applications. It is
hoped that this survey can help to clarify the differences between different types ofdithered quantization schemes and to correct some of the common misunderstandings
regarding the effects of dithered quantization.

0 INTRODUCTION
Dither and quantization are among the most frequently
discussed topics in audio signal processing. Dithering
techniques are fast becoming commonplace
in applications where quantization or requantization is requiredin order to reduce the wordlength of audio data. The
literature contains appropriate
recommendations
regarding dither signals which are suitable for audio applications [1], [2]. In spite of the widespread interest
in dither and quantization, a comprehensive theory of
their operation appears not to exist in print. The relevant
theorems are scattered among sundry journals and conference proceedings, and many have not been publishedat all until very recently [3], [4]. This paper attempts
to collect all of the significant theory and to supply
references to its originators as best we can ascertain
them. The treatmentbegins
with fundamental concepts
and proceeds to discuss models of undithered,subtractively dithered,and nonsubtractively
ditheredquantizing systems in a consistent manner which makes
clear both their differences and similarities,

Analog-to-digitalconversion is customarily decomposed into two separate processes: time sampling of
the in'put analog waveform and amplitude quantization* Presentedat the91st Conventionof theAudioEngineering
Society, New York, 1991 October 4-8; revised 1991 October21.
** The authors are members of the Guelph-Waterloo Program for Graduate Work in Physics.
J.AudioEng.Soc.,Vol.40,No.5,1992May

of the signal values in order that the samples may be
represented by binary words of a prescribed length (theorder of these two processes being immaterial). The
sampling operation incurs no loss of information aslong as the input is band-limited [5] in accordance withthe sampling theorem, but the approximating
natureof the quantization operation generally results in signal
degradation.
An operation with a similar problem isrequantization, in which the wordlength of digital data
is reduced after processing in order to meet specifications for its storage or transmission.
An optimal(re)quantizer is one that minimizes the deleterious effects of the aforementioned
signal degradation by converting the audible signal-dependent
artifacts into benigh signal-independent
ones as far as possible.
Quantization
and requantization
possess similartransfer characteristics,
which are generally of either
the rnidtread or the rnidriser variety illustrated inFig. 1. Assuming that these represent infinite quantizers, _ the corresponding transfer functions can be expressed analytically in terms of the input to the quarttizer, w, and the quantizer step size A as

for a midtread quantizer,

or

1
2

] For practical purposes, this simply means that the signal
is never clipped by saturation of the quantizer.
355

LIPSHITZET
AL.

PAPERS

for a midriser quantizer, where the floor operator [ Jreturns the greatest integer less than or equal to its
argument. The step size A is commonly referred to asa least significant bit (LSB) since a' change in input
signal level of one step width corresponds to a change
in the LSB of binary coded output. Throughout thesequel, quantizers of the midtread variety will be assumed, but all derived results have obvious analogs
for midriser quantizers, and all results stated as 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, Q(w),and its input w
q(w) A= Q(w) -

w ,

(3)

where we use the symbol =_ to indicate equality bydefinition. This quantization error is shown as a function.ofw for amidtreadquantizerinFig.
2. It has a maximum
magnitude of 0.5 LSB and is periodic in w with a period
0f 1 LSB.
We' shall refer to systems that restrict the accuracyof sample values using multibit quantization as quantizing systems, of which the_;e exist three archetypes:
undithered, subtractively dithered, and nonsubtractively
dithered: Schematics of these systems are shown in
Fig. 3.
Throughout the sequel we will refer to the system· input as x, the system output as y, and the total errorof the Systemas e, whereA
e = y -xas distinguished(4)from the quantization

2 A stationary random process is one whose statisticalpropertiesaretime invariant,

_V
Q(w)I

proper statistical properties can render the total error
signal audibly equivalent to a steady white noise.
The body of this paper proceeds to examine the threequantizing systems from a theoretical viewpoint. First,
the classical model of undithered quantization is outlinedwith discussion of its limitations in Sec. 2. Sees. 35 survey generally valid statistical models for undithered, subtractively dithered, and nonsubtractively
dithered quantizing systems, respectively. Each of thesethree sections'is divided into subsections which address
the statistics of the total error separately from those of
the system output. In each case, a first-order statisticalanalysis of the error or output precedes a second-orderanalysisdealingwith the relationshipbetweenerroror
output samples separated in time. For the dithered systems, properties of some practical dither signals are
discussed.
2 CLASSICALMODEL OF UNDITHERfiD
QUANTIZATION

error q defined:by Eq. (3). In an undithered quantizing system, thesystem input x is identical to the quantizer input w sothat the total error equals the quantization error [e =q(x)]. In the other two schemes, the quantizer input iscomprised of the system input plus an additive random
signal v, called dither, which is assumed
to bestationary 2 and statistically independent of x. In suchI

systems the quantizer input w = x + v is not a deterministic function of x, and neither is the total error e.
In the subtractively dithered topology the dither signal
is subtracted from the quantizer output, presumably
after this output has been transmitted through somechannel. This subtraction
operation is omitted in a
nonsubtractively
dithered system.
The object of dithering is to control the statisticalproperties of the total error and its relationship to thesystem input. In undithered systems we know that theerror is a deterministic
function of the input. If the
input is simple or comparable in magnitude to thequantization step size, the total error signal is strongly
input-dependentand audible as gross distortion andnoise modulation. We shall see that use of dither with

As mentioned,

Although it is a deterministic function of the input, theclassical model of quantization
treats this error as anadditive random process which is independent of the
input and uniformly distributed, meaning that the total
error values have a probabilityof the form

Q(w)_density function(pdf)

windowfunctionof widthF,
A(b)

Fig. 1. Quantizer transfer characteristics. (a) Midtread. (b)Midriser. Size of 1 LSB is denoted by A.
356

(5)

F
I

system,e = q(x) .

Hr,'isdefinedaswhere
P_(e)the= rectangularl-la(e)Hr(e) A=(a)in an undithered quantizing

,
,

(6)
2 < e _<2otherwise.

(7)

Thepdf of Eq. (6) will be referredto as a uniformor
rectangular pdf.
The mth moment of a randomvariablee with pdf

J.AudioEng.
Soc.,
Vol.40, No.5, 1992
May

PAPERSQUANTIZATION
ANDDITHER

p_(_) is defined as the expectation

E[ Em]_----fa -oc

value of em:deempe(e)where E[ ], the expectation

random process is zero, its variance and second moment
are equal.
(8)

value operator,

is defined

moregenerally
by
E[f]

A=

/

f(e)p_(e)to For
Eq. a(6),randomthe moments
process uniformly
aredistributed according
E[e]

= 0

E[e 2] de .

(9)

(10)

A2

(11)

12

-oc

The zeroth moment of any random process (that is,
E[e°]) 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[(e - E[e]) 2]= E[e 2] - E2[e]. It is clear that if the mean of aE[e m] =

I'
_
[0,

_

,

m evenm odd.

(12)

Eq. (11) is the familiar expression for quantization
error variance (or power) in the classical model.

q(w)
lLXn]

Fig. 2. Quantization error q(w) as a function of quantizer input w for midtread quantizer.
II
II
II
ta X
lt

input
.r

W= X

_

II
I I
II
I I
II
II
II
II

quantizerr_
_d

II
input
x

+

wmx+v

quantizer
Q
II
I I
II
II
II
Il
II

dither,

V

input
x

_
quan[izer
w=x+v

II
II
II
II
II
II
channelFig. 3. Archetypal quantizing systems. (a) Undithered. (b) Subtractively
system input x, quantizer input w, and system output y.
J. Audio Eng. Soc., Vol. 40, No. 5, 1992 May

output_y = Q(w)- v= x + q(x+v)
=X+£

II

,_
+

=x+g

II

+
(C)

Y = Q(x)= x + q(x)

II
II
SI
t I

dither, V(b)_ output· output
Y = Q(w)= x + v + q(x+v)=X+£

dithered. (¢) Nonsubtractively

dithered. Shown are

357

LIPSHITZ ET AL.

PAPERSThis model of quantization error is valid for complex(quasi-random) input signals which are large relative

3.0

signals where, in undithered systems, the quantization
error retains the character of input-dependent distortion-_.0

or noisemodulation.
The nonrandom nature of the error can be demonstrated by using a computer to simulate the undithered
to an LSB. It fails catastrophicallyfor small or simple
quantization of a very simple signal, say, a 1-kHz sine
wave of 4..0 LSB peak-to-peak amplitude. Fig. 4 shows
the system input and output from such a simulation as
well as the resulting total error signal, and the power
spectrum of the system output. Evidence of the input
signal is clearly visible in the total error waveform. In
the power spectrum, many sharp peaks fall at multiplesof the input sine wave frequency, indicating not only
a high degree of nonrandom structure (that is, harmonic

_
_

0.0-.
-_.0

_
_

2.o
-2.0

relation between/................................
_._...._'..

-3.0
0.0

0:2

0.4

0.6

0.8

1.0

o.6

o.8

1.o

0.4
o.6
T_me[.... ]
(c)

o.8

z.0

Time [....
(a)
3.0

_

2.0

this signal and the system input. In-._

harmonic peaks are also present due to aliasing of dis-_0.0...........................

tortion components above the Nyquist frequency
dist°rti°n)(22.05 kHzintheerr°rsignal'butals°astr°ngc°r-_'°_
in this simulation) into the baseband.

_
_

-_.0
-2.0

_ii

3 WIDROW'S MODEL OF UNDITHEREDQUANTIZATION

-3.o
o.o

A generalized statistical model of undithered quantization, valid for inputs with arbitrary statistical properties, was first developed by Widrow [6]-[8] in the
1950s. Widrow realized that quantizing a signal trane-o.2

impulsea fashion function
reminiscent
time
forms its functions
pdf from aincontinuous
to a oftrain
ofsampling, so that recovery of the system input statistics
from those of the system output must require conditions
analogous to those of the sampling theorem [5].
We pause to mention that the theoretical results presented in this section have limited practical significance,since detailed statistical knowledge of the total error
produced by an undithered quantizer is not very helpful
in alleviating the undesirable audible effects of thiserror.Thedevelopmentwhichfollowsis includedboth
for completeness and in order to provide the mathe-___

0.5 __....__
0.0
-0. s

_

-_. 0

_

40

matical machinery necessary for the treatment of dithered systems. In particular, the results of this section
can be applied to subtractively dithered systems almost

_

30

_

20

3.1 Statistics}

0by Pdx(e, x). This bivariate

function

represents

the

3 A more conventional notation is mlAelx), but in manycases this can be confusing since, in fact, P_I_is simply afunction of the two variables e and x.
358[msec ](b)_.0

We begin by deriving Widrow's expression for the
directly
withtheaidofavery
simpletransformation.
pdf of thetotal error,
p_, in termsof theinput pdf Px.
In an undithered system we have seen that the total
error of the system is a deterministic function of the
input. This is reflected in the form of the conditional
probability density function (cpdf) ore givenx, denoted 3

o._
Time

_.s
_

of the Total Error

]

1.5

0.0

0.2

|

_
%
_

t
............

-_0z0 i/t' t]i'_

it iii itili'itiili["_'iif's Frequency_0 [kHz]_s
(d)

it
20

Fig. 4. Results from computer-simulated quantization of 1kHz sine wave of 4.0-LSB peak-to-peak amplitude without
dither. (a) System input signal. (b) System output signal. (c)Resulting total error signal. (d) Power spectrum of system
output signal (as estimated from sixty 50% overlapping Hannwindowed 512-point time records with assumed sampling
frequency of 44.1 kHz; 0 dB represents a power spectral
density of A2T/6, T being the sampling period).
J. Audio Eng. Soc., Vol. 40, No. 5, 1992 May

PAPERSQUANTIZATIONAND
DITHER

pdf of the error for a specified input value. From Eqs.
(1) and (3) the cpdf in question is expressible in termsof the Dirac delta function b asPrix(e, x) = _(e -q(x))= g(e-

A[_

+ _]

+ x)

= AIIa(e)· Wa(e + x) ,

(13)

where

where Px represents the cf of the system input. If theerror is to be uniformly distributed, Eq. (17) must reduce
to a single sine function sin(_rfiu)/(_Au)
centered at
the origin, in which case it will be independent of Px.
We observe that this can occur only under the conditionsof the following theorem:
quantizing system
Theoreml[
The total
is uniformly
error induced
distributedby an ifundithered
and only
if the cf of the system input, Px, satisfies the condition
that
I

Wr(e)

A= _

8(e -

kF)

(14)

Ps(u) I= 0
lu=k/'X

k = -0c

fork
is a train of Dirac delta functions separated by intervalsof width F. The rectangular window function in Eq.
(13) serves to select from the impulse train Wa thatdelta function which is closest to the origin (since[e I _< A/2, of course),
We could easily use Eq. (13) to derive the marginal
pdf of the total error, pt(e), by integrating p4x(e, X)px(X)with respect to x, but it is more instructive to reason
as follows. If -A/2 _<x < A/2, then, by inspection ofFig. 5, we see that the distribution of errors arisingfrom such inputs is equal to Px(-X) between -A/2 andA/2 and zero elsewhere. Similarly, the distribution oferrors arising from inputs x such that --3A/2 _< x <-A/2 is given by that portion ofpx(-X) which resides
between A/2 and 3A/2, but recentered around x -- 0.
We conclude that

pt(e)

=

p_(-e
k=-oo

+ kAL

= AIla(e )

:_
_
px(-e
k=-cc

= AHa(z)· [Wa * pA(-e)

2

+ kA)

,

operator_[_[f](u)=_ F(u)Ae -j2xrut

de

.

=

sin(zrAu),tAu= k=-___ Px (_)-

3._.2 Second-Order Statistics
The statistical relationships between total error values
separated in time are of particular interest since these
determine the power spectral characteristics of the total
error signal. Consider two system input values Xl andx2 occurring at times ti and t2, respectively,
so thatthey are separated in time by x = t2 - ti, where
x % 0. 6 Their statistical relationshipis described bytheir joint pdfpx,,x2(xl, x2), representing the probability
4 This definition is retained throughout the sequel, j =X/_ll.
s Widrow [8] cites a different condition, which is sufficient
but not necessary; namely, Px(u) = 0 for Iai_>1/A. Widrow
calls this requirement"half-satisfaction"of the conditionsof the quantizing theorem (see Theorem 3).
6 In the special case where x = 0, the analysis reduces tothat of Section 3.1.l.

Y

I,._"Pw(x)= Px(X)(16)

Then from Eq. (15) the cf Pt of e is given by

Pt(u)This condition is not actually due to Widrow, s but
to Sripad and Snyder [9]. Note that if the requirementsof Theorem 1 are satisfied, then the error is preciselyof the sort which is postulated by the classical modeland the moments of the total error are given by Eq.
(12).

_

I bY4
f(e)

(18)

(15)

where * denotes the operation of convolution.
The characteristic function (cf) of a random variable
is the Fourier transform of its pdf, and is often easier
to interpret than the pdf itself. We define the Fouriertransform-+3.....

_< e <
2otherwise'= -+1, +2,14;=x* [Wlx(-u)Px(-U)]'rr_ -- - _:/_
sin[wA(uk/A)] '

J. Audio Eng. Soc., Vok 40, No. 5, 1992 May(17)

Fig. 5. Pdf of quantizer input
in undithered quantizing
system,showingitsjustification
relativetothequantizer
characteristic.
359

LIPSHITZET
AL.

PAPERS

that the inputs at times ti and t2 have the indicatedvalues. Since the error is a deterministic function of
the input in an unditheredsystem,system inputs xl and x2 satisfies the conditionthat

We can immediately

write down the cpdf for a pair of error values, callthem si and s2, given the inputs x] and x 2 [see Eq.
(13)1:

Px,,x2(ul, u2) ,,,=k,la = 0,2=a2/afor all integers kt, k2 with (k], k2) _ (0, O) . (26)p(_,,Ol(x,,xg(Sl,= _(el-$2; x], x2)

q(xl))_(e2= A2IIaa(el,-q(x2))s2) · WAA(Eiq-Xl, E2 -Jr X2)(19)

Eq. (25) shows that, subject to the specified conditions, the joint pdf of el and e2 is a product of two
rectangular window functions, one of which is a function(20)of E1 alone and the other of s2 alone. Hence the two
error values are statistically independent and each iswhereuniformly
E[e'_s_]Ilrr(el, e2) _ Hr(el)Hr(e2)Wrr(sl,e2) _Wr(s0Wr(s2)

(El'

e2)

=

--cc

--ac

so that for 'r _ 0 we can write= E[eT]E[s_]

(27)

·

(22)
the joint pdf

p(_.... )l(x,x0(e], e2; xt, x2)

In a digital system the total error is a discrete-time
signal, so that x = kT, where T represents the sampling
period and k is an integer. The autocorrelation functionof such a signal is defined to be the function E[ele2](k)of the lag parameter k. According to the WienerKhinchin theorem [10], the power spectral density(PSD) of a discrete-time signal is equal to the discretetime Fourier transform (DTFT) of its autocorrelation
function, where we define the DTFT as_DT[h](f ) A__2T

h(k)e -j2'fkrk=x px_,x2(X],x2) dx] dx2= A2IIaa(sl,

.

(21)

Now we can straightforwardly
computeof the two error values [see Eq. (15)]:Pel,e2

distributed,f2) · [Was * Px,x2](-ft,

-e2)
(23)

(28)

-cc

and where the frequency variable f is in hertz if T isin seconds. (This definition is normalized such that the
integral of the PS D from zero to the Nyquist frequency,
1/(2T), yields the variance of the signal.)
For an undithered system satisfying the conditions

where the convolution is two-dimensional,
involving
both e t and e2.
Taking the two-dimensional
Fourier transform of Eq.

of Theorems 1 and 2, the autocorrelation
the error is given byfunction(23) with respect to both fl and e2, we find that thefE[12],

k = 0= [E[el]E[f2],otherwiseof(29)
joint cf of et and f2 is given by [see Eq. (17)]U2)= Z

E[ftf2](k)Px,,x, a'a=sin[xrA(ut - kJA)] sin[wA(u2 - k2/A)]xxrA(ul - kt/A )_A(u2 - k2/A)'(24)

two-dimensional

analogof

Theorem 2 In an undithered quantizing system,
the joint pdf p_,,_: of total error values et and e2,
separated in time by x _ 0, is given bye2)=

IIA(Ei)IIA(E2)(25)

if and only if the joint cf Px,,x: of the corresponding
360

.

A2T
PSDdf)-

6

'

(31)

which is obviously constant with respect to frequency
so that the error signal is spectrally white and exhibitsa total power of A2/12 up to the Nyquist frequency.
3.2 Statistics

pe,,%(el,otherwise

Thus its PSD is given by

where Px_,x2denotes the joint cf of xl and x2. We can
now write an obvious
Theorem 1:

(30)
[0,

of the System Output

We now proceed to investigate the relationship between the input and the output of an undithered quantizing system.
d.AudioEng.Soc.,Vol.40,No.5,1992MayPAPERSQUANTIZATION AND DITHER

3.2.1 First-Order Statistics
The output can only assume values which are integer
multiples of the quantization step size A. Referring to
Fig. 5, we see that the probability of an output having
value y = kA, for some specified integer k, is equal tothe probability that the input lies between -A/2 + kA
and A/2 + kA. Hence,

In practice, recovering the pdf of the input is often
unnecessary and it is sufficient to recover the momentsof the input signal from the output. The mth momentof the output signal can be expressed in terms of either
the pdf or the cf of y by 7
E[ym] __

ympy(y) dy

(36)

-_

o0

py(y)

=

_

g(y-

kA)

k=-_

a/2+ka px(X) dx.

=__

J -aY2+ka_

du m (u)

(37)

u=0

(32)
It is obvious that if the quantizing
Borrowing Widrow's terminology,
we say that thequantization operation performs area sampling of the
input distribution. Writing the integral in Eq. (32) asa convolution ofpx with a rectangular window function,
it reduces toPy(Y) =[AIIa* Px](Y) ' Wa(y) ·theoremis satisfied

then the shifted versions of Gx(u) do not overlap, sothat the mth derivative of Py(u) at the origin is determined only by the "baseband" (k = 0) term in Eq. (34).
This is also true, however, subject to the weaker condition that the quantizingtheoremis only half-satisfied(see footnote 5) or the still weaker condition that

(33)
dmGx

Clearly, py(y) cannot be equal to px(y) unless the input
pdf is itself a train of impulses separated by intervalsof A (that is, unless x is already quantized at the LSB

du m (u)level). We will see, however, that the statistical properties of the input can be recovered from the output
subject to certain less restrictive conditions.

If the input statistics

From Eq. (33), the cf of y, Py, is given by
Py(u) : Gx(u) * WI(u)
X
=

_]
_

k w -cc

G_ u(k)'. Px(u)2wwso that by repeated
(34)

(35)

0

fork=

il,

_+2, _+3.....
(38)

E[ym] =

where
G_(u) _ sin(xrAu)_r Au

u=k/A =

obey this condition,

_du

(u)

(39)

u=0

differentiation

thenof Eq. (35) we canx.
In particular,
for theoffirst
moments
can write
expressthe moments
y infew
termsof the we
momentsofthe following useful relationships,
which give rise to
Sheppard'scorrectionsforgrouping:
E[y] = E[x]

(40)
A2

and where P_ is the cf of the input. Hence Py(u) consistsof copies of the function G_(u) separated by intervals
that
0 for lulthatifPxisband-limitedsuch>_ 1/(2A), then the repeated
ofl/A.Px(u)Note,= however,versions of G_(u) do not overlap, allowing recovery

E[y2] = E[x2] + 12m
E[ym] = [m/21()(_)
_]
21
/=0

(41)
E[x m-2t]
2/ 2l + 1

(42)of the input cf (and hence the input pdf) from that of
the output. Indeed, this is [6]-[8]:We emphasize that, in an undithered quantizing system,
each of these equations for E[y m] is only valid when

Theorem 3 (Widrow's Quantizing Theorem)
The
pdfofthe input to an undithered infinite linear quantizing system is recoverable from the pdf of its output
if the cf of the input, P_, is band-limited such that
P_(u) = 0 for lul _> 1/(2A).

Eq. (38) is satisfied for that particular value Of m, andthat the validity of one of these equations does not
imply the validity of any others corresponding to different m values.
Furthermore,these equations show that if Eq. (38)is satisfied for some m, then the ruth moment of y =x + e is the same as that of x plus a statistically independent additive random process with uniform pdf.

Obviously, this theorem closely resembles the sampling
theorem, which allows recovery of an appropriately
band-limited analog signal from discrete-time samples
thereof. The notable difference, of course, is that thequantizing theorem pertains not to time sampling, but
to amplitude quantizing of a signal (that is, to areasampling of the pdf of a signal),
J. Audio Eng. Soc., Vol. 40, No. 5, 1992 May

7 Eq. (37) can be straightforwardly derived by writing= E[e-J2*ruY], expanding the complex exponential
in a Taylor series about y = 0 and using the fact that the
expectationvalue operator is linear.
_[py](U)361

LIPSHITZ ET AL.

PAPERS

It is important to remember, however, that x and c are
not, in fact, statistically independent and that, for anundithered quantizing system, they are deterministicallyEq. (42), relating the joint moments
those of the input:of the output to

We notein passing
relatedas shown
in Fig.that2. by repeated differentiationof Eq. (35) for Gx(u) we can derive from Eq. (38) astronger, but perhaps more practical, condition in termsof the input cf, which ensures that E[y m] obeys Eqs.
(39) and (42) form = 1,2 .....
M:

E[yTy_]

2(1_+22)dipx= 0
dut (u) u=kla

for k = + 1, -+2, +3,

3.2.2 Second-OrderM - l.

(43)* Px,,x2](Yl, Y2) ' WaA(yl,we can write that
A2

E[y_y2](k)rjE[x2] + -= I
12'
[ E[XlX2],

for k=

0(50)

otherwise

A2T* W! _1(rtl,u2)

PSDy(f)= PSDx(f)

+

6

(51)

3.3 Summary of Undithered
Ouantization
It is clear that the results of this section are primarilyof theoretical, rather than practical, interest. All of thejoint cf [see Eq. (34)]Gx,,x2 (u 1, u2)

so that the power spectral density of the output is identical to that of the input apart from an additive white~noise component arising from the quantization
operation;thatis

Y2)

(44)

Pyl,y2( u 1, U2) =(49)

Statistics[A2IIaawith correspondingtrt_]-_2 'J(2/1 + 1)(2/2 + 1)'

Proceeding in a fashion similar to that of Sec. 3.2.1,
we find that the joint pdf of two system output values
Yl and Y2, separated in time by 'r % 0, is given [seeEq.(33)]by
Y2) =r-.r _m-211 vn--21.1

x

In particular,
i = 0, 1, 2 .....

Py,,y2(Yl,= [m/2ltn/2l(
_
_
2Il
1]=0
_2=0
m )( 2/2
n )(_)

,

aa

(45)

where

theorems given impose conditions on the statistics ofthe system input, and such restrictions are usually undesirablein practice.Somenotuncommonsysteminputs
satisfy the conditions of Theorem 1 (for example, aGx,,x2(Ul, u2) A sin(xrAu]) sin('rrAu2)
=
_Aut
*tAu2
' Px,x2(u], u2).

uniformly distributed random input), but the conditionsof the quantizing theorem (Theorem 3) cannot be met(46)

by any system input whose pdf is only nonzero on a
finite interval (although some signals, such as large

We can now write a two-dimensional
analog of thequantizing theorem, namely that the joint pdf of the
input is recoverable from that of the output if Px,,x2(u_,u2) = 0 for [u_ I _> 1/(2A) or la21_> 1/(2A), or both.
Of potentially greater interest, however, is the twodimensional analog of Eq. (39), which allows us torecover the joint moments of the system input fromthose of the output. That is, if

ones with Gaussian distributions, can come very close
to satisfying the conditions [6]-[8]).
There now becomes apparent, however, the possibility ofdithering the system input with a suitably chosen
dither signal v so as to ensure that the quantizer input
w = x + v [Fig. 3(b) and (c)] satisfies some of thex,,___(Ul,U2)
Ou_Ou_

ul=k,/Afor all integers

kl, k2 with (kl, k2) _ (0, 0)

om+nG

= 0

aforementioned conditions. In particular, if the dither
is statistically independent of the system input, thenthe pdf Pw of w is the convolution Pw = Px * P_, andhence its cf is the product Pw = Px ' P_. In this casethe dither statistics can be freely chosen so as to cause
Pw to vanish at the required places, and this accomplishment cannot be undone by any system input x thatu_=k;a(47)

thenis statistically independent of v.
These ideas will now all be quantified

mathematicaltechniquesdevelopedin this section.
4 SUBTRACTIVE(

E[y_y_]

j _m+n

= k_J

_Xi,X

Rm+n('g.

DITHER

2

OuTOu_ (Ul'u2)

/all=0,:=0(48)

In this case we can write an expression
362

using theanalogousto

The first use of subtractive dither must be credited
to Roberts [11] in the early 1960s, who applied it to
picture coding. Roberts added uniformly distributed
random noise of 1-LSB peak-to-peak amplitude (statistically independent of the system input) to a video
J. Audio Eng. Soc., Vol. 40, No. 5, 1992 May

PAPERSQUANTIZATIONAND
DITHERsignal prior to its quantization, and subsequently subtracted the same dither signal from the quantizer's output. He found that the total error of the quantizing
system was uniformly distributed and statistically independent of the input signal.
Prompted by Roberts' discovery, a theoretical in-will be uniformly distributed and statistically independent of the input for arbitrary input distributions
if and only if the cf of the dither, Pr, satisfies thecondition thatvestigation
subtractive
was undertaken
Schuchman of[12],a studentditheringof Widrow's.
Schuchmanby

P,(u) I,=k/a = 0

deriveda conditionon the dither pdf [see Eq. (56)]which guarantees that the total error and the input tothe quantizer are statistically independent of each other.
His work was later recast and generalized to include asecond-order statistical analysis by Sherwood [13].
4.1 Statistics of the Total Error
4.1,1 First-Order Statistics
The quantizer input is w = x + v so that the outputof the system is [see Fig. 3(b)]y = Q(x + v) -v

P£1'E'2(U

(52)

]'

U2)

=

_
kl=--oo

×

PXDX2

_
k2=-oosin[wA(ul,'/TA(u1--(x + v)(53)= AIIa(s)· [Wa * pw](-e)= AHa(e)· [Wa * px * p,](-s)

sin(wAu)
wAu* [WiX (--u) · P_(-u)(54)

Note that the statisticalproperties· P,(-u)].

of the dither can bechosen to control the properties of the error. In particular, it can be shown that [12]:4 (Schuchman's
Condition)
In a subdithered quantizing system the total error

J.AudioEng.Soc.,Vol.40,No.5, 1992May

A

'

(57)

5

In a subtractively

ditheredquantizing

ps,,_2(E1,E2)=

IIa(el)IIa(s2)(58)

for arbitrary input distributions if and only if thejoint cf Pv,,v 2 of the corresponding
dither values vi
and v2 satisfies the condition thatI
Pv,,_,(u], u2) [ u,=k,/a = 0
u2=k2/afor all integers kl, k2 with (kl, k2) _ (0, 0)(55)

Theoremtractively

Pv I' v2

system the joint pdfp_,_2 of two total error values
el and e2, separated in time by _' _ 0, is given by

and

-

,

Theoremwhich is simply the quantization error of the total
quantizer input w. Therefore we can treat the system
as one of the sort described by the analysis of Sec. 3,
but with w rather than x as its input. From Eq. (15)andpw(S) = px(s) * p_(s) (since x and v are statisticallyobtainindependent) we immediatelyPs(u)4.1.2 Second-Order Statistics
Proceeding as for the first-order statistics, we use
Eq. (24) to deduce that for two total error values eland E2, separated in time by x _ 0: s

where P,,._2 represents the joint pdf of dither values
vi and v2, applied to input valuesxi andx2, respectively.
We immediately draw the following conclusion:= q(x + v)

pc(e)This is Clear from Eq. (55), which reduces to a single
sine function centered at the origin under the conditionsof the theorem. In this case, the total error is of theform postulated by the classical model, and its moments
are given by Eq. (12) for a system input of arbitrary
distribution.

-- kl/A)] sin['rrA(u2 -- k2/A)]
kl/A )'rrA(u 2 -- k2/A)x= Q(x + v) -for k = + 1, + 2, + 3 .....
(56)

A

and hence the total error is given bye = y -

I

(59)

We observethat if the conditionsof this theoremare
satisfied, then s] and _2 are both uniformly distributedand statistically independent of each other.
It should be noted that if 1.,] and v 2 are statisticallyindependent of each other, and the cf of each satisfies
Eq. (56), then Eq. (59) will be satisfied. This is thesituation of interest in most practical applications using
subtractive dither.
Subject to satisfaction of Eq. (59), the joint momentss In the special case where x = 0, the analysis reduces tothat of Sec. 4.1.1.
363

LIPSHITZ ET AL.

PAPERSof _] and _2 are given by Eq. (27), reproduced
for reference:
E[e_]

here

= E[_]E[_]

so that E? and e_ are uncorrelated. In particular,
m = n -- 1 and a stationary dither signal,

forHence, the joint moments of the output in terms of the
moments of the input will be given by Eq. (49), andEqs. (50) and (51) will hold. The quantization process
has thus merely added to the input signal a white-noise
process of total power A2/12 (up to the Nyquist frequency).
4.3 Properties

E[_iE2]= E[81]E[e2].

of Practical

Dither Signals

It is naturally of interest to inquire as to which common random signals satisfy the criterion of Theorem
4. Perhaps the simplest imaginable candidate is ditherwith the uniform pdf= E2[e]
=0

(60)
pr(v)

independent of % for x _ 0, where we have assumed
that the dither satisfies the conditions of Theorem 4 so

= '[la(v)(65)

whose corresponding

cf is the sine functionthat the moments of the total error will be those predicted
by Eq. (12) of the classical model. Eq. (60) indicates,of course, that the total error signal will be spectrally
white even if the dither signal is not.
4.2 Statistics of the System Output
4.2.1 First-Order Statistics
For pragmatic reasons, we will not derive an expression for the output pdf, given an arbitrary dither distribution. Instead, we will assume that the dither signal
satisfies the conditions of Theorem 4. Then, since thetotal error is statistically independent of the input, andsince the output is given by y = x + e, we can immediately -write '_'_*mm
Py(Y) = [P_ * Px](Y)-sin(xrAu)
_rAu

(61)

so thatThis cf obviously satisfies the desired conditions. We
conclude that dither of uniform pdf will render the
total error statistically independent of the input anduniformly distributed in a subtractively dithered system.
We assume that values in the dither sequence are statistically independent of one another so that the criterionof Theorem 5 is also satisfied and distinct values in the
total error sequence are statisticallyindependentof
one another (thus ensuring that this sequence meetsthe.........weaker reauirement, of being_ spectrally_ white).
Of course, there are other cf's which meet the re-independent uniformly distributed random processes,
each of I-LSB peak-to-peak
aml31itude, will yield adither whichsatisfiesthe criterion. The summationoperation convolves the pdf's of the random processes

sin(,rAu)
=

'rrAu

'

together, thus multiplying their cf's,
dither will exhibit a cf of the form

Px(u)= G_(u).
(62)
Obviously, the output is precisely the sum of the input
plus a statistically independent,
uniformly distributed
random process. Hence, the moments of the output interms of the moments of the input will be given by Eq.
(42), which, in this case, is valid for all m.

Pr(u)[sin(_rAu)] n= [ _uu J

of Theorem

Py,,y2(Yl, Y2) = [A21-Iaa * Px,,x2](Y], Y2)

5, then(63)

so thatIt2)=

sin(xrAu]) sin(_Au2)
zrAul
*tAu2

= Gx,,x:(u_, u2) ·364. Px .... (u], u2)(64)

so that such a(67)

However, in a subtractively dithered system such dithers
have no inherent advantage over the uniform-pdf ditherof Eq. (65).
4.4 Summary

4.2.2 Second-Order Statistics
If Ph,_ satisfies the conditions

Py,,y2 (ul'(66)

quirement of vanishing at all nonzero multiples of
1/A. For instance, a dither produced by summing n= [AHa * px](y)

Py(u)

Pr(u)

of Subtractive

Dither

The most practicallyimportant theoreticalresults
concerning subtractively
dithered quantizing systems
are that:
1) The total error can be rendered uniformly distributed and statistically independent of the system input
by choosinga dither whichsatisfiesthe conditionsof
Theorem 4.
2) Values of the total error separated in time can berendered statistically independent of one another (sothat the total error signal is spectrally white) by using
a dither whose values, in addition to satisfying Theorem
d. Audio Eng. Soc., Vol. 40, No. 5, 1992 May

PAPERSQUANTIZATIONAND
DITHER4, are statistically independent of one another.
A familiar dither which satisfies all the required conditions is one with a rectangular pdf of 1-LSB peakto-peak amplitude, and whose values are statisticallyindependent of one another. Fig. 6 shows the resultsof a computer-simulated quantization operation performed upon a 1-kHz sine wave of 4.0-LSB peak-topeak amplitude and using this type of subtractive dither.
Shown are the system input and output, the total error,
and the power spectrum of the system output. Notethat the system output resembles a sine wave plus an
independent additive noise, and that no trace of the
input signal is visible in the noiselike total error waveform. Furthermore, the power spectrum of the system
output exhibits no distortion components whatsoever
and indicates that the total error is spectrally white.
(The 0_-dB noise floor in Fig. 6 represents a power
spectral density of A2T/6, which has an integrated noise
power of A'2/12 up to the Nyquist frequency.) These
results should be compared with those in Fig. 4, which
illustrate the signal-dependent distortions produced byan undithered quantizing system with the same system
input signal.
Subtractively dithered quantizing systems are clearly
ideal in the sense that they render the total error an
input-independent additive noise process. The requirement of dither subtraction at the system output,
however, imposes restrictions which make it difficult
to implement in practical audio applications. For example, the dither signal must be available at the output,
and so either the dither must be transmitted along withthe signal or synchronized dither generators must bepresent at either end of the channel. Even more seriously, any signal editing or modification occurring between the original quantization and the subtraction of
the dither necessitates a like operation on the dither
sequence. It is for such reasons that subtractive dither
is generally not a feasible option, and nonsubtractive

3.0
_
-

2.0
_.0

_
_

o.0
-_.o

_

-2.0
-3.o0.0

....

0.2

0.4

0.6

0.8

1.0

Time [msec]
(a)
3.0
_

2.0

_
_

_.o
0.0
-1.0

_

-2.0
-3.0

0.0

0.2

0.4

0.6

0.8

1.0

0.4

[msec ]

0.6

0.8

_. 0

Time[mseo_
(b)

_.s
_

_.o
0.5

-_

0.0
0.5

_

-_.0
-_. 5
0.0

dithering schemes are of interest. Although many of
the same benefits can be achieved (see Sec. 5), the

0.2

Time

(c)

total error variance is inevitably greater, and the beautiful result regarding full statistical independence of_the total erroris unattainable.

.;

40
3o
20

5 NONSUBTRACTIVE

DITHER

We now examine the possibility of using dither without subsequently subtracting it. Early investigations
into nonsubtractive dither were conducted by Wright[14]in 1979,resultingin the discoveryof manyof theimportant results that follow. This work has remained
unpublished, and has only very recently become known
to the authors [15], [16].
The results concerning conditional moments of theerror signal were rediscovered
independentlybyStockham [17] in 1980, as documented in an unpublished Master's thesis by Brinton [18], a student ofStockham's,in 1984. Stockham remained silent on thematter until recently [4] for commercial reasons (triangular-pdf dither was used in the Soundstream digital
J.AudioEng.Soc.,Vol.40,No.5, 1992May

0
_

-_0
....
s · Frequency
_0

_s

[kHz]

20

(d)
Fig. 6. Results from computer-simulated quantization of 1kHz sine wave of 4.0-LSB peak-to-peak amplitude using
rectangular-pdf
subtractive dither of 1-LSB peak-to-peakamplitude. (a) System input signal. (b) System output signal.
(c) Resulting total error signal. (d) Power spectrum of system
output signal (as estimated from sixty 50% overlapping Hannwindowed 512-point time records with assumed sampling
frequency of 44.1 kHz; 0 dB represents a power spectral
density of A2T/6, T being the sampling period).
365

LIPSHITZET
AL.

PAPERS

recording/editing
system in the early 1980s).
The properties of nonsubtractive
dither were again
investigated independently by two of the authors, Lipshitz and Vanderkooy, in the mid-1980s. Vanderkooy
and Lipshitz appear to have been the first researchers
to publish their findings on nonsubtractive
dither [1],[2], [19]-[21](and in particularon triangular-pdf
dither), and this prompted collation and extension of
the theoretical aspects by another of the authors, Wannamaker [16], [22], [23]. The ultimate outcome has
been the publication of a detailed analysis of nonsub-

consider a nonsubtractively dithered quantizing systemwith a specified input value x. The input to the quantizer
is w = x + v, which has the cpdf
pw[x(W,x) = P(_+,)lx(w, X) .
Since x and u are statisticallywrite (see Fig. 7)(71)
independent,

we canpw¢(W, x) = p_l_(W, x) * pvl_(W, x)

tractively dithered systems coauthored with Wright [3].
The treatment of the subject presented hereafter rep-

= _(w -x) · p,(w)resents an abridgement of [3], omitting many of themathematicaldetails and some of the more esoteric= p_(w -

x) .

(72)

results.
Recently the results concerning conditional moments
have again been independently discovered by Gray [24],using a somewhat different formal approach from theone adopted herein. Gray and Stockham are currently
preparing a paper on the subject [4].
Although a handful of individuals in the engineeringcommunity are aware of the correct results regarding
nonsubtractive
dither, a number of misconceptions
concerning the technique are widespread. Particularly
serious is a persistent confusion of subtractive
andnonsubtractive dithering, which have quite different
properties (see, for instance, [25, p. 170]). We will
see that nonsubtractively dithered systems cannot render_h_, ..................
tt_tol¢'rr_r statistically . indevendentof the input.
u,,__Neither can they make temporally separated values of
the total error statistically independent of one another.
They can, however, render certain statistical momentsof the total error independent of the system input, andregulate the joint moments of total error values which
are separated in time. For many applications this is as
good as full statistical independence.

We observe that the total error depends on the valuesof both the input and the dither. In particular, if the
input w = x + v to the quantizer is between - A/2 and+A/2, the output will be nil (for a midtread characteristic) so that the error is e = -x. Similarly, if the
input to the quantizer is between +A/2 and +3A/2, theoutput will be + A so that c = -x + A. Hence, the pdf
of the error for a fixed input is a series of delta functions
separated by intervals of A, each weighted by the probability that w falls on the corresponding quantizer step:

5.1 Statistics of the Total Error
5.1.1 First-Order Statistics
The quantizer output in a nonsubtractively

Eq. (74) makes it clear that p_l_(_, x) cannot be rendered independent of x by any choice of dither pdf,

quantizing__(_ + x - kA)k=-_
fA/2+kAx

pMw - x) dw .

_,_,_-aJ2+ka
Writing the integral in Eq. (73) as a convolution of p_with a rectangular window function, it reduces to oP_lx(e, x) = [AHa * p,](c)· Wa(_ + x) .

(74)

ditheredsystem is given by [see Fig. 3(c)]y = Q(x + v)

P_lx(e, x) =9 Note that Eq. (74) reduces to Eq. (13) if dither is not
used [that is, ifp_(v) = b(v)].
(68)

sothatthe totalerroris givenby

y

e=y-x
= Q(x + v) - x(69).----T_= q(x + v) + v .

(70)

_1

Obviously
totalalsoerror
is not simply
the quantization
error alone,thebut
involves
the dither.
This factsomewhat complicates the statistical treatment of nonsubtractively dithered systems. We begin by investigating the conditional pdf of the total error given thesystem input.
In order to derive an expression
366

for P_lx(e, x), we

,

[

I

.............Pwlx(W,x) = Pr(W-x)'
w=x+v/ x_.]\\/

input, x

Fig. 7. Conditionalpdfof quantizerinputin nonsubtractively
dithered quantizing system, showing its justification relative
to the quantizer transfer characteristic.
d. Audio Eng. Soc., Vol. 40, No. 5, 1992 May

PAPERSQUANTIZATIONAND
DITHER

since the convolution of any dither pdf (which must
be nonegative everywhere) with a rectangular windowfunction yields a function at least as wide as the rectangular window. Hence, at least one delta function
always makes a contribution to the sum, and the positionof the delta function is dependent on the input. 10 Furthermore it can be shown [3] that the marginal pdf ofthe errorp_(e) is not uniform for arbitrary inputs.
Since we have seen that statistical independence of
our attention to the possibility of controlling moments
the total error from the input is not achievable, we turnof the error.
The mth conditional moment of the error signal given
x is defined in the obvious fashion:E[_ mix[_fa, x) d_-_ _mp_lx(£= (j)radmp_lxx)u=o_
du-_- (u_,Taking the two-dimensional

·

]

E[e]

= E[v[

E[e2]= E]v2]

E[ _m]

=

(81)
A2
+ 12

(82)

E

(83)

E[vm-2l]

tm/21(2l)(A)2lt=0

21 + 1

which, again, are related to Sheppard's corrections forgrouping. We emphasize that each of these equations
for E[£ra] is valid only when Theorem 6 is satisfied for

(75)

that
particularvalue does
of m,
and
that thethe validity
validity ofof theseequationsnot
imply
of oneanyothers corresponding
to different m values.

(76)

dithersatisfying
conditions
ofproper
Theorem
6, theEq. variance(82)
clearlyshows
that
nonsubtractiveerror
is thegreaterthanwiththatof classicalor total
sub-

Fourier transform 11(withrespect to both e and x) of Eq. (74) and substituting
into Eq. (76) yields the Fourier transform with respect
to x of E[emlx]:[ E[_mlx]

so that we can derive useful expressions for the momentsof the total error in terms of the moments of the dither
signal by direct differentiation of Eq. (79):tractively dithered quantization
dither variance.

(namely, A2/12) by the

The following weaker theorem is really a corollary
to Theorem 6, but is somewhat better known [14],[24]. It follows immediately from repeated differentiation of Eq. (79), resulting in binomial expansions
involving
P_(u) and theindependentsine function.
Theorem the7derivatives
E[_llx] isoffunctionallyof
xforl=
1,2,...
, M if and only if

(Ux)

(j)m
(77)

diPv
du[ (u) Iu=k/a =0

fork=

IfE[eml x] is to be independent of x, we require that
its Fourier transform reduce to a constant times a singledelta functionat.the origin. Imposingthis restriction
on Eq. (77) yields the following

theorem:i = 0, 1, 2.....

....

;

M -

1.
(84)

5,1.2

Theorem 6 In a nonsubtractively
dithered quantizing system, E[e mIx[ is functionally independentof x if and only if-+1, +2,-+3Second-Order

Statistics

Consider two total error values el and e2 which areseparated in time by x _ 0,12 and the two corresponding
input signal values x] and x2. Using a derivation analogous to that of Sec. 5.1.1, we find that

I

dmG_
dum

](_ 1, E2)(u)[i u:k/a= 0

for k = + 1, +2,-+ 3, . .
(78)

where

=

' Waa(_l[A2FIAA * P .....
+ xl, _2 + x2) ,(85)

toAsdiscussedin Sec.4 inassociationwithsubtractively
dithered systems, the quantization error q(w) will be statistically independent of x and uniformly distributed if the dither

Gu(u) A sin(-rrAu) . P_(u) .
=axAu
If the conditions

P(el,_2)l(xi,x2)(El,£2; Xl, X2)of Theorem(79)

6 are satisfied,then from

Eq. (77),

E[smlxl = E[_ m] =

m I

dmG_
dura (u) u=0

J. Audio Eng. Soc., Vol. 40, No. 5, 1992 May(80)

statistics obey Schuchman's condition, Eq. (56). Unfortunately q(w) is not the total error of a nonsubtractively ditheredsystem.
]l This approach is due to Brinton [18], who also used it
in deriving an expression for the conditional moments. Unfortunately his method entailed two invalid assumptions:
1)thatthe inputwasuniformlydistributed,and2) thatthe
rectangular window function in Eq. (74) represented the pdf
of a quantization error which was statistically independentof 12In
the dither.
the special
that of Sec.
5.1.1. case where 'r = 0, the analysis reduces to367

LIPSHITZ ET AL.

PAPERS

where the convolution is two-dimensional,
involving
both el and 82. Here P,,,,2 is the joint pdf of two dither
values vi and v2 corresponding to the inputs x] and x2,
respectively.
Hence,
p_,,_2(el, 82) =p(e,,%)l(Xl,X:)(81,--oowhich no longer depends on the joint pdf of the input.
In this case we can write an expression analogous toEq. (83), relating the joint moments of the total error

E2; Xl, x2)Px,,x2(Xl,x2)

dXl dx2--o0= [A2IIaa* ph,,2](e],82) · [Was * Px,,x2](-8],

o_

kl

-82)

(86)

so that(sin['rrA(ul-- kl/A)]

kl=-o* k2=-oc

x sin['rrA(u2;a_u2 -- --k2-_)k2/A)]
P.,,_ ( gl - X
kl ' u2 Clearl);, no choice of joint dither cf Ph,_ will allow
Eq. (87) to be expressed as a product of two characteristic functions, one involving u l alone and the otheru2 alone, for arbitrary choices of thejoint
input cf
Px, ,x2. We therefore conclude that E1 and 82 cannot be
rendered statisticallyindependent for arbitrary joint

.

_)

(87)

to those of the dither:E[e_8_]

=

_

_

m

inputdistributions.Let us thereforeproceedto investigate the joint momentsof _l and_2in the hopethat
we can exercise some control over them by correct
choice of the dither statistics.
From Eq. (87) we proceed to calculate
ments of E1 and 82, finding thatthe joint mo-

E[8_'8_1 = _1
(j_m+no_o_

k]
A'

×

_'_

_]

k_=-® k2=-o_om+nGvl,v 2

OU_OU_ (Ul,P_l,_: (_)

Note that if vi and v2 are statistically independent,
and each satisfies Eq. (78), then 8'_ and 83 are uncorrelated (that is, E[8?e_] = E[e'f]E[e_]) and Eq. (90)represents
is automaticallya stationary
satisfied.random
Furthermore,
process, ifE[v'_v_l
the dither=
E[vm]E[vn].
In particular, for m = n = 1, using Eq.
(92) we see that under these conditionsfA2

E[v 2] + -= /12 '[E2[v],

E[ele2](k)

where

(92)

(2/] + 1)(2/2 + 1)

(88)

k = 0
(93)
otherwise

sothatGh,,2(ub u2) A sin(*rAu0=_rAul

sin(xrAu2),tAu2'Ph,_(u],

u2).

(89)
Now iforn+n__Vi,V

E[v_n_21_v__212]xU2 ) ' ul=-kl/A

I u2=-ka/a

n

2ll
2/2
[mi2]
[n/2J ()()(_)2(1_+12)
ll=O /2=0[2= PSD_(f)

A2T
+ _-

(94)

Thus the power spectrum of the total error is whitesince the dither spectrumis white. This will be thecase in most practical situations where proper nonsubtractive dither is used. The possibility of using nonwhiteI

Ou_Ou_ (ul, u2) u,=k,/a = 0,_=k_/_
for all integers

PSD_(f)kl, k2 with (kl, k2) % (0, 0)(90)

dithers is explored in detail in [3], which shows that,
under certain conditions, some such dithers can satisfy
the requirements of Theorem 6 while yielding a total
error spectrum which is the sum of the dither spectrum
and a white-noise component.

then
5.2 Statistics[ j _m+nof the System Output

5.2._ First-Order Statisticsc)m+nGU2)

E[eT8_]= !k,_/!

au'_Ou_ (ul,

u,=0
u:=0
(91)

368

We now turn our attention to the output of the system.
Applying the same reasoning used to determine thecpdf of the total error in Sec. 5.1.1, we find that theJ. Audio Eng. Soc., Vol. 40, No. 5, 1992 May

PAPERSQUANTIZATION AND DITHER

cpdf of the output is given by
Pylx(Y, x) -- [AHa * p,](y- x) · Wa(y) ·where the joint moments of the total error are given interms of those of the dither by Eq. (92). In particular,
note that if these conditions are satisfied for m = n = 1,
then

(95)

Hence,
E[yly2]
Py(Y) == E[xlx2]+ E[_l_2](102)

Pylx(Y, X)px(X) dx-o¢so that, substituting= [AHa * pv * p_](y) · Wa(y) .

(96)

E[y]y2](k)Moving to the Fourier transform domain,
Py(u) = [G_(u)Px(u)] * Wi(u)=E[x 2] + 2E[x]E[v]Xo_[E[XlX2]

Eqs. (83), (92), and (99), we have++E[v 2] + 12'

E[viv2],k = 0otherwise.

k

(103)
Hence, under these conditions,zeromean,andhence

if the dither signal has

A2TE[yml =_®

(7)[(j)r __]k =-oor=O

Il j.X[_2'l'r]

m-r

_drGv
du (u) ]dm_rp
dum_rxeSDy(f)= PSDx(f) + PSDv(f) + - 6(104)
so that the spectrum of the output is the sum of the(u) ][ ,=k/a(98)ponent anddither
input
contributedspectra,by theapartk = from
0 termawhite-noisecomin Eq. (103).
5.3 Properties of Practical Dither Signals

Now if the first m derivatives of G_(u) vanish at allnonzero multiples of l/A, then Eq. (98) reduces toE[y m] =E[_r]E[x

_(7)
r=o

m-gl

Theorem 8 A nonsubtractive dither signal generated
by the summation of n statistically independent zero-(99)

LSB peak-to-peak amplitude, renders the first n momean rectangular-pdfrandomprocesses, eachof 1ments of the total error independent of the system
input, and results, for n _> 2, in a total error powerof (n + 1)A2/12.

where the expectation values of the total error are given
in terms of the expectation values of the dither by Eq.
(83). By direct differentiationof Gv(u), this condition
is easily shown to be equivalent to the condition ofTheorem 7 with M = m.
5.2.2 Second-Order

This must be the case since the addition of n suchrandom processes convolves their pdf's, hence multiplying their cf's and yielding

Statistics[sin(xrAu)] n+l
G_(u) = L _uu
J'Proceedingusual
fashion,that thejoint
moments inof theoutputvaluesYl andweY2,findseparatedintime by, _ 0, are given byE[y_'y_I =____kl=--°°

k .....

[(
X

v
0 s=0

J _ m+n-r-sv,,,, 2
OqU_OU

Om+n-r-sPx''x2_,_/

OqU_t-roqu_'-s

]
(UI'

U2)

(105)

[

(Ui,

//2)

u,=k,/A 'u2=k2/A(100)

If the indicated partial derivatives of G_,,, 2 are zerofor all integers kl, k2 with (kl, k2) ya (0, 0) and for
r = 1, 2 .....
m and s = 1, 2, . . . , n, then Eq.
(100) reduces
E[y_ty_] =tor:0 __=0 (7)()

the first n derivativesn E[ele2]E[x]r ss

m-r x2n-s ]

,
(101)

O.Audio Eng. Soc., Vol. 40, No. 5, 1992 May

of which will consist entirely ofSince this function goes to zero at the required places,
terms
containingnonzeropowersofsin(wAu)/(_Au).
the first
n moments of the error will always be independent of the input. Higher derivatives will not share
369

LIPSHITZET
AL.

PAPERS

thisproperty,
Furthermore, it is important

forwhich
to note that using rec-tangular-pdf noises of peak-to-peak amplitude not equal
to 1 LSB (or, rather, not equal to an integral numberof LSB) will not render error moments independent of
the inputsince the zeros of the associated sinc functions
will not fall at integral multiples of I/A (see illustrationsin [1]).
We proceed to examine two important examples of
nonsubtractive dither pdf's. First, consider a systemusing dither with a simple rectangular (that is, uniform)
pdf of 1-LSB peak-to-peak amplitudepr(v)= Ha(v)

G,(u)

!

(107)

The first three derivatives of this function, and thecorresponding
moments as a function of the input, areplotted in Fig. 8. The first derivativeclearly satisfies
the condition of going to zero at the regularly spaced
intervals stipulated by Eq. (78), while the second derivative and higher derivatives do not. This indicatesthat the first moment of the error signal is independentof the input, but that its variance and higher moments
remain dependent. These conclusions are borne out by

(106)

2.0

[sin(_ Au)] 2= L 'rrAu J

,:

1.0

1.5

0.8
0.6

1.0

0.a
.,q

0.2

0.5

>

o.o

_
°

o.o

{2

-0.5

*J

-0.2

_4

-0.4

,_

-0.6
-1.5
'
-2.0
-4

-0.8
i
i
-3
-2
Frequency

L
I-10
Variable,

I
1
u

m
2
[1/LSB]

I
3

-1.0
-2

-1
System0
Input,

x

[LSB]

1

-1
SystemInput,

x

[LSB]

'
-1
SystemInput,

x

[LSB]

(a)
6

,:

0.40

4

[

0.35

2

._

o
-2
-4

tn

_
,J

o.3o

_

0.25

_
o

0.20

_

0.15

u
o
o_

0.10

-6
-8

-4

i
J
-3
-2
Frequency

0.05
,
i
-1
0
Variable,

i
1
u

i
2
[1/LSB]

i
3

0.00
-2

1

(b)
!

0.15

_'

0.10

lO
._

_

o.o5

_

o.oo

_
7:

-o.o5

O--'''''"'"-o

-10

-0.10

-20

-4

i
I
-3
-2
Frequency.
I-10
Variable,

I
1
u

I
2
[1/LSB]

.

-0.15
-2

4

1

(c)
Fig. 8. Derivatives of G_(u) (left) and conditional moments of error (right) for quantizer using rectangular-pdf dither of 1LSB peak-to-peakamplitude.
(a) dG./duand E[elx](in(c) d_G./du 3 and E[e3lx](in units of A3). Frequency
variable370

units of A). (b) d2G_/du 2 and E[e2lx](in unitsofu is plottedin units of 1/A and input x in units of A.

A2).

d.Audio
Eng.Soc.,
Vol.40,No.5,1992MayPAPERSQUANTIZATION

the plots of the moments themselves. The first moment,
or mean error, is zero for all inputs, indicating that thequantizer has been linearized by the use of this dither,
The error variance, on the other hand, is clearly signal

DITHER

In a system using this kind of dither, P_lx(e, x) involves
the convolution of three rectangular window functions,so that G,(u) is given bydependent, so that the noise power in the signal varies
withthe modulation
system input.
is sometimes
referred toas noiseandThisis audibly
undesirable.
Now consider a triangular-pdf
dither of 2-LSB peakto-peak amplitude resulting from the sum of two independent rectangular-pdf
noises v] and v2, each of
1-LSB peak-to-peak amplitude:G,(u)[sin(_Au)]3= L _rAu J

'

(109)

The first three derivativesof this function, and thecorresponding moments as a function of the input, areplotted in Fig. 9. The first and second derivatives ofthis function go to zero at the required places, so thisdither renders both the first and second moments of thetotal error independent of x. The second moment of
the total error is a constant A2/4 for all inputs, in agree-

p_,(v) = [Ph * P_2](v)= [HA * Yla](v) .

AND

(108)

2.0

1.0

0.6
1.o
0.4
0.5
._
0.0

-

-0.5

_
o

0.2
0.0

_

-0.2
-0.4

-1.0

0.6
-1.5

-0.8

2.0

i
i
-3
-2
Frequency

-4

i
,
-1
0
Variable,1u2[1/LSB]

3

-1.O
-2

4

-1
System1
Input,

x

[LSB]

x

[LSB]

(a)

--

4

0.35

_

2

%

0.30

0

E

o.25

o
_
_

0.20
o.i8

-2 ...................... i-4....................................................... -i

C_

m

6

o.io

-8

0.05
I
-3

-10
4

i
-2

Frequency

i
-1

0

1

2

Variable,u[1/LSB]

0.00
-2

3

I
0

,
-1
SystemInput,,-1
Systemi0Input,

,
1

2

(b)
0.15
40
_'

0.10
20

c_
_

o

0.05

o.oo

cl
-20

_

-0.05
-0.10

-40
,
-4

-3
-2
Frequencyi-1
Variable,

i
'
3
u

-0.15
-2

[1/LSB]

,
1
x

[LSB]

Cc)
Fig.
LSB

9.
3

Derivatives

peak-to-peak
3

of

G.Cu)

Cleft)

amplitude.
3

Ca)

40,

May

and

conditionaldGz/duJ

andmoments
E[slx]

of
(in

error

units

(right)
of

A).

for

quantizer(b)d2G_/du2

using

triangular-pdf

and

E[s21

x]

(in

dither
units

of

of

2-

A2).

(c) d G,/du and E[¢ Ix] (in units of A ). Frequency variable u is plotted in units of 1/A and input x in units of A.
J. Audio

Eng.

Soc.,

Vol.

No.

5, 1992

371

LIPSHITZET
AL.

PAPERS

ment with Eq. (82). Higher derivativesof Gv(u) do
not meet the required conditions, so that higher moments
It can be shown [3] that triangular-pdf dither of 2LSB peak-to-peak amplitude is unique and optimal in
the sense that it renders the first and second momentsof the total error input independent, while minimizing
the second moment. That is, when used in a nonsubof
the errorditheredremain quantizing
dependent system,on the input.
tractively
this dither incurs
the least possible increase in the rms noise level of any
dither which eliminates input-dependent distortion andnoise modulation.
5.4 Summaryof Nonsubtractive

3.0

_

1.0

._

o.0

_
__

-1.0
2.0 /__......_.........._.
..................
-2.0
-3.0
0.

Dither

The results of greatest practical importance concerning nonsubtractively dithered quantizing systems
areasfollows:

o.4
o. 6
Time [msec]
(a)

o. 8

_. 0

0.2

o.4
Time
(b)

0.8

1.0

3.0

_

2.0

dithering, unlike subtractive dith-_1.0

ering, cannot render the total error independent of thesystem input. It can render any desired conditional

7_

0.0

moments of the total error independent provided thatcertain conditions on the cf of the dither are met (seeTheorem 6).
2) Nonsubtractive dithering, unlike subtractive dithering, cannot render total error values separated in time
statisticallyindependent of one another. It can, however,regulate the joint moments of such errors. In particular,

_
_

-2.0
-2.0

white
it
can
render[see discussion
the power
leading
spectrumof
tocanEq.
the(94)].
total anyerrordesired
signal3) Nonsubtractive
dithering
rendermoments of the system input recoverable from thoseof the system output, provided that the statistical attributes of the dither are properly chosen (see Sec.
5.2). This includes joint moments of system inputs
separated in time, so that the spectrum of the input canbe recovered from the spectrum of the output.
4) Propernonsubtractiveditheringalways results ina total error variance greater than A2/12 [see Eq. (82)].

_
_
._

1'5f_
0._
, ]IL

_
i
_

0.s

1) Nonsubtractive

0.2

-3.0
0.0

[

'

0.6
[msec]

t ii_t_l Alill IIIIllltl]
k I_h
Iil,I

,d[

o.0

1.0
-2.s
.0

0.2

5) Triangular-pdf dither of 2-LSB peak-to-peak amplitude incurs the least increase in the rms total error

0.4
Time

0.6

[msec]

0.8

level of any nonsubtractive dither which eliminates
input-dependent distortion and noise modulation.
Fig. 10 shows the results of a computer-simulated
quantization operation performed upon a 1-kHz sine
wave of 4.0-LSB peak-to-peakamplitude and using
the aforementioned triangular-pdf dither. Shown are

_

40

._

30

?_

20

the systeminput and output, the total error, and thepower spectrum of the system output. Note that vestigesof the input signal are clearly visible in the total error
waveform, indicating that the two signals are not statistically independent. Also, the system output does
not resemble a sine wave plus an independent additivenoise. Surprising as it may seem, listening experiments(see Sec. 6) show that the total error signal of Fig.
10(c) sounds like a constant white noise, independent__,o ..........................................................

of the nature of the input signal (with which it is indeed
uncorrelated), and that Fig. 10(b) sounds identical toa noisy sine wave. Furthermore,
the power spectrumof the system output, in Fig. 10(d), exhibits no distortion
components and indicates that the total error is spectrally
372

1.0

(c)

-10'

, .
20
15
2o
Frequency [kHz](d)Fig. 10. Results from computer-simulated quantization of 1kHz sine wave of 4.0-LSB peak-to-peak amplitude using
triangular-pdf nonsubtractive dither of 2-LSB peak-to-peakamplitude. (a) System input signal. (b) System output signal.
(c) Resulting total error signal. (d) Power spectrum of system
output signal (as estimated from sixty 50% overlapping Hannwindowed 512-point time records with assumed sampling
frequency of 44.1 kHz; 0 dB represents a power spectral
density of A2T/6, T being the sampling period).
05

J. Audio Eng. Soc., Vol. 40, No. 5, 1992 May

PAPERSQUANTIZATIONAND
DITHER

white. These results should be compared with those in
Figs. 4 and 6, which illustrate the results of quantizing
a sine wave using undithered and subtractively ditheredsystems, respectively. In particular, it should be notedthat the noise floor in Fig. 10(d) is up by 4.8 dB relative
to that of Fig. 6(d) due to the tripling of the noise
spectral density, in accordance with Eq. (104).
Some comment is required concerning the special
nature of requantization.
In a purely digital system,
random processes exhibiting the continuous pdf's described in this section are not, strictly speaking, available since not all real numbers are representable using
a finite number of binary digits. In fact, digital dither
pdf's of necessity resemble discretized or "sampled"versions of the continuous pdf's (rectangular, triangular, etc.) described. It is not immediately obviousthat such dithers will retain the desirable properties of
their analog counterparts with respect to rendering total
error moments independent of the system input. It isrigorously proven in [3] that such dithers do indeed
retain these properties, and empirical evidence (and aless general proof) corroborating
this conclusion ispresentedin [1].

a need for formal psycoacoustic
tests of this sort involving many participants under carefully controlled
conditions.
We recommend the use of nonsubtractive, triangularpdf dither of 2-LSB peak-to-peak amplitude for mostaudio applications requiring multibit quantization or
requantization operations, since this type of dither renders the first and second moments of the error signal
constant with respect to the input, while incurring theminimum increase in error variance. This kind of dither
is easy to generate for digital requantization purposesby simply summing two independent rectangular-pdf
pseudo-random processes, which are easily generated
by linear congruential algorithms [23], [26]. The resuiting digital dither can be used to feed a digital-toanalog converter for analog dithering applications. 13
This paper has attempted to underscore the closemathematical relationship between time sampling andamplitude quantization. In closing, we emphasize thatappropriate dithering prior to (re)quantization
is asfundamental as appropriate antialias filtering prior to
sampling--bothserve to eliminate classes of signaldependenterrors.

6 CONCLUSIONS

7 ACKNOWLEDGMENTS

For audio signal processing purposes there seems to
be little point in rendering any moments of the total
error other than the first and second independent of the
input. Variations in higher moments are believed to be

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

inaudible, and this has been corroborated by a large
number of psychoacoustic
tests conducted by the authorsand others [18], [23]. These tests involved listening
to a large variety of signals (such as sinusoids, sinusoidal
chirps, slow ramps, various periodically switched inputs, piano and orchestral music) which had been requantized very coarsely (to 8 bits from 16) in order torender the requantization
error essentially independentof low-level nonlinearities
in the digital-to-analog
conversion system through which the listening took
place. In addition, the corresponding total error signals(output minus input) were used in listening tests inorder to check for any audible dependences on the input,
Using undithered quantizers resultedin clearly audible
distortion and noise modulation in the output and error
signals. A subtractively
dithered quantizing systemusing rectangular-pdf
dither of 1-LSB peak-to-peakamplitude Was found to eliminate all audible input dependences in the error signal, which was discovered
to be audibly equivalent to a steady white noise. A
nonsubtractively dithered quantizing system using thesame dither eliminated all distortion, but the residual
noise level was found to vary audibly in an inPut-dependent fashion. When triangular-pdf dither of 2-LSB
peak-to-peakamplitude
was used, noinstance was foundin which the error was audibly distinguishable
from asteady white noise entirely unrelated to the input, although the level of this noise was somewhat higher
than that observed in the subtractively dithered system,
Admittedly, these tests were informal, and there remains

8 REFERENCES

J. AudioEng.Soc.,Vol.40,No.5,1992May[1] S. P. Lipshitz and J. Vanderkooy,"Digital
Dither," presented at the 81 st Convention of the AudioEngineering Society, J. Audio Eng. Soc. (Abstracts),
vol. 34, p. 1030 (1986 Dec.), preprint 2412.
[2] J. Vanderkooy and S. P. Lipshitz, "Dither in
Digital Audio," J. Audio Eng. Soc., vol. 35, pp. 966975 (1987 Dec.).
[3] R. A. Wannamaker, S. P. Lipshitz, J. Vanderkooy, and J. N. Wright, "A Theory of Non-Subtractive
Dither," IEEE Trans. Signal Process., submitted for
publication.
[4] R. M. Gray and T. G. Stockham, "Dithered
Quantizers,"
IEEE Trans. Inform. Theory, submitted
for publication.
[5] E. T. Whittaker, "On the Functions which Are
Represented by the Expansions of the InterpolationTheory," Proc. R. Soc. Edinburgh, vol. 35, pp. 181194 (1915).
[6] B. Widrow, "A Study of Rough Amplitude
Quantization by Means of Nyquist Sampling Theory,"
Ph.D. thesis, Dept. of Elect. Eng., M.I.T., Cambridge,
MA (1956 June).
[7] B. Widrow, "A Study of Rough Amplitude_3It should be noted, however, that many analog signals
and digital-to-analog conversion systems exhibit a Gaussiannoise
component inherent
which isditherof large
enoughtheamplitude
to actas a satisfactory
without
requirementofan explicit dithering operation [8], [2], [23].
373LIPSHITZ ET AL.

PAPERS

Quantization by Means of Nyquist Sampling Theory,"
IRE Trans. Circuit Theory, vol. PGCT-3, pp. 266276 (1956 Dec.).
[8] B. Widrow, "Statistical Analysis of AmplitudeQuantized Sampled-Data
Systems," Trans. AIEE, pt.
II, vol. 79, pp. 555-568 (1961 Jan.).
[9] A. B. Sripad and D. L. Snyder, "A Necessary
and Sufficient Condition for Quantization Errors to Be
Uniform and White," IEEE Trans. Acoust., Speech,
Signal Process., vol. ASSP-25, pp. 442-448(1977Oct.).
[10] J. S. Lim and A. V. Oppenheim,
Advanced
Topics in Signal Processing (Prentice-Hall, Englewood
Cliffs, NJ, 1988).
[ 11] L. G. Roberts, "Picture Coding Using PseudoRandom Noise," IRE Trans. Inform. Theory, vol. IT8, pp. 145-154 (1962 Feb.).
[12] L. Schuchman, "Dither Signals and Their Effecton Quantization Noise," IEEE Trans. Commun. Technol., vol. COM-12, pp. 162-165 (1964 Dec.).
[13] D. T. Sherwood, "Some Theorems on Quantization and an Example Using Dither," in Conf. Rec.,
19th Asilomar Conf. on Circuits, Systems, and Cornputers (Pacific Grove, CA, 1985 Nov.).
[14] J. N. Wright, unpublished manuscripts (1979June-Aug.).
[15] J. N. Wright, private communication (1991
Apr.).
[16] S. P. Lipshitz, R. A. Wannamaker, J. Vanderkooy, and J. N. Wright, "Non-Subtractive
Dither,"presented at the 1991 IEEE Workshop on Applicationsof Signal Processing to Audio and Acoustics, New Paltz,
NY (1991 Oct. 20-23), paper 6.2.

[17] T. G. Stockham, private communication (1988).
[18] L. K. Brinton, "Nonsubtractive
Dither," M.Sc.
thesis, Dept. of Elec. Eng., Univ. of Utah, Salt Lake
City, UT (1984 Aug.).
[19] J. Vanderkooy and S. P. Lipshitz, "Resolution
Below the Least Significant Bit in Digital Systems withDither," J. Audio Eng. Soc., vol. 32, pp. 106-113(1984 Mar.); correction, ibid. (Letters to the Editor),p. 889 (1984 Nov.).
[20] J. Vanderkooy
and S. P. Lipshitz,"Digital
Dither: Signal Processingwith ResolutionFar Below
the Least Significant Bit," in Proc. AES 7th International Conf.: Audio in Digital Times (Toronto, Ont.,
Canada, 1989 May), pp. 87-96.
[21] S. P. Lipshitz and J. Vanderkooy, "High-Pass
Dither," presented at the 4th Regional Convention of
the Audio Engineering Society, Tokyo (1989 June); in
Collected Preprints (AES Japan Section, Tokyo, 1989),
pp. 72-75.
[22] R. A. Wannamaker, S. P. Lipshitz, andJ. Vanderkooy, "Dithering to Eliminate Quantization
Distortion," in Proc. Annual Meeting Can. Acoustical Assoc. (Halifax, N.S., Canada, 1989 Oct.), pp. 78-86.
[23] R. A. Wannamaker, "Dither and Noise Shaping
in Audio Applications," M.Sc. thesis, Dept. of Physics,
Univ.ofWaterloo,Water!oo,Ont., Canada(1990Dec.).
[24] R. M. Gray, private communication(1991
Apr.).
[25] N. S. Jayant and P. Noll, Digital Coding of
Waveforms (Prentice Ha!l, Englewood
Cliffs, NJ,
1984).
[26] D. Knuth, The Art of Computer Programming,
vol. 2, 2nd ed. (Addison-Wesley, Reading, MA, 1981).

THE AUTHORSr

S. P.Lipshitz

R. Wannamaker

Stanley P. Lipshitz is a professor in both the departments of Applied Mathematics and Physics at the
University of Waterloo in Waterloo, Ontario, Canada.
There, in addition to the normal teaching associatedwith a university position, he conducts his own andsupervises graduate student research in audio and electroacoustics as a member of the university's Audio
Research Group. Prior to joining the faculty of the
University of Waterloo in 1970, he studied in South
Africa, the country of his birth, where he received his
374

J. Vanderkooy

Bachelors (1964), Masters (1966) and Ph.D. (1970)
degrees in applied mathematics and physics. He is now
a Canadian citizen.
Dr. Lipshitz is a fellow of the AES. He served as agovernor of the Society for the periods 1984-1986 and1987-1992, and as its president for the year 19881989. Other society memberships include the IEEE,
the Acoustical Society of America, and the CanadianAcoustical Association.
He has presented numerous technical papers at AES. J. Audio Eng. Soc., Vol. 40, No. 5, 1992 May

PAPERS
conventions,both in North America and overseas, ona wide range of topics including amplifier design,
psychoacoustic,
loudspeakercrossover design, electroacoustic transducer measurement, acoustics, anddigital signal processing for audio. He has participatedin many educational workshops and seminars on audio
topics including loudspeaker measurement,
stereo microphone techniques, and the fundamentals of digitalaudio,
Dr. Lipshitz's publications have appeared frequently
in the AES Journal and elsewhere. His current researchinterests include transducer design and measurement,
digital signal processing for audio, and the characterization and design of surround-soundsystems. He has
consulted for a number of companies on audio-related
questions.
e
Robert Wannamaker was born in 1967 in Ontario,
Canada. In 1988 he received the degree of B.Sc.E. inengineering
physics from Queen's
UniversityinKingston, Ontario. He was granted the degree of M.Sc.
in physics by the University of Waterloo in 1990 for
research on dither and noise-shaping conducted under
the aegis of the Audio Research Group. Since graduating, he has continued to work with the ARG as aJ. AudioEng.Soc.,Vol.40, No.5, 1992May

QUANTIZATIONAND
DITHER
research assistant. He plays a pretty mean piano andsome guitar as well.
®
John Vanderkooy was born in The Netherlands in
1941, but received all of his education in Canada, with
a B.Eng. degree in engineering physics in 1963 and a
Ph.D. in physics in 1967, both from McMaster University in Hamilton, Ontario. For some years he followed his doctoral interests in low-temperature physics
of metals at the University of Waterloo, where he iscurrently a professor of physics. However, since thelate 1970s, his research interests have been mainly inaudio and electroacoustics.
A fellow of the AES and a member of the IEEE, Dr.
Vanderkooy has contributed a variety of papers at conventions and to the Journal. Together with his collegue
Stanley Lipshitz and a number of graduate and undergraduate students, they form the Audio Research Groupat the University of Waterloo.
Dr. Vanderkooy's current interests are digital audio
signal processing,measurement of transfer functions
with maximum-length
sequences,transducers,
diffraction of loudspeaker cabinet edges, and most recently
sub-surface
analysis techniques
using maxiumumlength sequences.

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