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7/27/2019 The Intermodulation and Distortion due to Quantization of Sinusoids
1/10
IEEETRANSACTIONS ONACOUSTICS, SPEECH,ANDSIGNALPROCESSING,
VOL.
ASSP-33, NO.
6 ,
DECEMBER 1985
1417
The Intermodulation and Distortion due to
Quantization
of
Sinusoids
NELSON M. BLACHMAN, FELLOW,
IEEE
Abstract-The Fourier series representation of the quantization er-
ror sawtooth yields exact expressions and convenient approximations
for all intermodulation IM) and distortion components produced by
quantization of the sum of two sinusoids whose respective amplitudes
are A and a. The mean-squared values of the IM components are also
calculated in the case where
A
and a fluctuate over several quantization
steps. When A and a are many times the quantization-step size
Q
hese
mean-squared values turn out to be approximately Q4/ (180 **Aa) ex-
cept for high-order IM. The quantization is generally assumed to be
uniform, but nonuniform quantization is also discussed. The case of A
>> Q and a JP Z),
p =
--m
(2)
x t) =
A ( t ) sin
+(t) (3)
we find that,when the input is
with
+(t)= 2nFt + *( t ) ,
the output is
y =
A
sin
+ +
Im
m
= C
A,,
sin p a , 4)
p = l
where
0096-3518/85/1200-1417 01.00 985 I E E E
7/27/2019 The Intermodulation and Distortion due to Quantization of Sinusoids
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1418
IEEE TRANSACTIONS ON ACOUSTICS, SPEECH,AND SIGNALPROCESSING, VOL. ASSP-33. NO. 6, DECEMBER
198
Y
Y
Y
ic
)
id)
Fig.
1. (a) Quantization sta ircase with a r iser a t he origin . (b) Quantiza-
tion staircase with a read centered at he origin. (c) Resolution of the
firs t s ta ircase in to an ideal ramp and the saw tooth quantization error . id)
The same
for
the second.
2 A . Large Inputmplitude
A~
I A +
n = l
laJ~(2nPA) or Odd p
(1
When A >> 1,we can use
[ 2 ]
he asym ptotic expression
and AP = 0 for ev en p is the amplitude of thepth-harm onic
J , ( ~ )
os for >> + 1
output omponent; S 0 for
p
q , and 6,, 1. The
oddymm etry of the quantization staircaseesults in the
(8
generation of onlyddarmonics. for the Bessel function in ( 5 ) o obtain for odd p
The expression
( 5 )
for
A,
is similar to on e of the forms
found by Bennett [4]. Apart from a sign convention, A,(A)
isheth-order Chebyshev transform
[SI
of the quantiza- 7r
= l
tion staircase
y(x)
an d, as he pth Fo urier-series sine coef-
ficient of
y (A
sin
a)?
an be expressed as =
6 P lA
+ ( - 1)(,- I*
h(A) l
~
I n-3 2
m
for A >> p , p odd
A
( A ) =
( - l)(P- I)/*
-
P 3- ST-7/2 y (A cos 0) cos p0 d8 .
(6)
where
Applying the Poisson summation formula to
( 5 )
or putting
A
cos
0
=
x
in (6) and in tegra ting by parts, for
A
> 0 we
get
(7)
where [A] is he argest intege r not exceeding A , and
UP ](cos 0) = (sin p0)/sin
0
is the Chebyshev polynomial
of the second kind,e.g. , U ) = 1, U2 (z ) = 4z2 - 1,
U4(z)= 16z4 12z2 + 1. Equation (7) is similar to an-
othe r of Bennetts expressions
[4]
and is easily used when
A
is not too large. Fig. 2(a) shows
A ,
as a function of
A ,
and Table
I
gives values of f i imes 41
-
A , - A 3 ,
A 5 ,
-A,, A 9 ,
and
- A l J .
is a periodic function of A with unit pe riod , i .e. , a func
tion of
{ A )
=
A
-
[ A ] ,
he fractional part
of
A .
Since this
infinite serie s converges on ly slowly, h ( A ) is not readily
evaluated from (10).However, by applying to (7) the term
of the Eule r summ ation formula [ l ] involving the sum
mand and its derivative as well as its integra l, taking sin-
(1
-
z) E 43
- -
z&/12, andgiving U p -
( k / A
the value
U p - l f
1) =
p
[which cancels the
p
in
t h e
de
nominator of (7)] because AP comes mainly from the erm
of
(7)
with
k
near
f A
when
A >>
1, we obtain the very
good approximation
h(A) = - JqTj -
4 1 6 { A } 2
+
2 0 { A } + 5
3 P
J20+2
(11)
P
7/27/2019 The Intermodulation and Distortion due to Quantization of Sinusoids
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BLACHMAN: MANDDISTORTION FROM QUANTIZATION OF SINUSOIDS
1419
4
h ( A )
0.2
-
-0.2
-0.3
-
0.4
-
(a)
(b)
Fig. 2. (a) The fundamen tal output amplitude
A ,
as a function
of
the am-
plitude A of a single input sinusoid.
(b)
h
( A )
as a function of
the
frac-
tional part
{ A }
of
A .
For odd
p > p ,
h(A) l&
thus fairly accurately describes the
scallops of Al(A) seen in Fig. 2(a), and , in general, for
o d d p
>
1, we have
A, = (
- l ) ( p - ) 2 h ( A ) / & .
B.
Fluctuating
Inpu t
Amplitude
As the frac tional pa rt {A} of the input amplitude A >>
p
fluctuates because
of
modulation or fading, the magni-
tude of
A,,
for odd p
>
1 thus varies between zero (when
{ A } is 0.0668 or 0.6566)andamaxim um of approxi-
mately 0.3751/& (when {A} = 0). If this fluctuation of
A
causes
( A }
to be uniformly distributed between
0
and
1,
the resulting average value of h(A) and, hence, of A, is
zero, while that
of h2(A)
s (2/7r4)r(3), where l(3 ) =
1 +
2 - 3
i
-3 + 4-3
+ -
* * = 1.202 s heRiemannzeta
function, since, on squaring (10) and averaging we get
from the square of each sine and nothing from crossprod-
ucts. Hen ce, the rms value of A , for
A
>>
p
>
1
is 0.157/
&, and the average pow er in the pth-harmonic output is
half the square of this quantity, 0.01234/A.
To
determine the average powerf the higher harmonics
f o r p
H
A, we observe that all harmonic-distortion com-
ponents com e from the sawtooth qua ntization rror y(x)
-
x, which, for x = A sin (27rFt ith A >> 1, becomes
a slowly frequency-modulatedsawtooth waveform (with
successive teeth having nearly the same duration). Its fun-
damen tal frequency varies from zero (w hen the input si-
nusoid is at one of its peaks and is hus changing only
slowly) up to 27rFA tee th per seco nd (wh en x is passing
throughzeroand schangingmost apidly). Thu s, the
fundamental component of the sawtooth (the n
=
1 term)
yields odd harm onics up to abou t the (27rA)th mu ltiple of
the input frequency, while the weaker higher-frequency
n
>
1) components contribute a small amount of harmonic
output at frequencies higher than 27rAF. Ignoring the lat-
ter, we see that the re are effectively about T A output har-
monics-all of them of odd order.
Because the error
y x) - x
is uniformly distributed be-
tween
--;
nd when A
>> l
withmean-squared value
the sum of the pow ers (me an-squ ared value s) of all of
the harmonic-distortion compon ents must be A Dividing
by nA, we see that the average powerof such a component
is 1/(127rA)
=
O.O2653/A. This exceed s the value 0.01234/
A found ab ove for p > 1 inasmuch as the input sinusoid spends a larger
proportion of the imechangingat atesnear k27rFA
quantization ste ps per second than at rates nearer zero.
According to Woodwards theorem [6],
[
191, he power
1
7/27/2019 The Intermodulation and Distortion due to Quantization of Sinusoids
4/10
1420
IEEE TRANSACTIONSONACOUSTICS, SPEECH,ANDSIGNALPROCESSING, VOL.
ASSP-33.
NO.
6,
DECEMBER 19
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
1 . 0
1 .1
1 . 2
1 . 3
1 . 4
1 . 5
1 . 6
1 . 7
1 . 8
1 . 9
2 . 0
3 . 1
3 . 2
3 . 3
3 . 4
3 . 5
3 . 6
3 . 7
3 . 8
3 . 9
4 . 0
7 . 1
7 . 2
7 . 3
7 . 4
7 . 5
7 . 6
7 . 8
7 . 7
7 . 9
8 . 0
1 5 . 1
1 5 . 2
1 5 . 3
1 5 . 4
1 5 . 5
1 5 . 6
1 5 . 7
1 5 . 8
1 5 . 9
1 6 . 0
0 . 0 5 7 2
0 . 1 4 5 0
0 . 1 6 7 7
0 . 1 5 2 2
0 . 1 0 9 3
- 0 . 0 3 7 9
0 . 0 4 4 8
- 0 . 1 3 6 4
- 0 . 2 4 9 1
- 0 . 3 7 4 5
0 . 0 5 7 2
0 . 1 4 5 0
0 . 1 6 7 7
0 . 1 5 2 2
0 . 1 0 9 3
0 . 0 4 4 8
- 0 . 0 3 7 9
- 0 . 1 3 6 4
- 0 . 2 4 9 1
- 0 . 3 7 4 5
0 . 0 5 7 2
0 . 1 4 5 0
0 . 1 6 7 7
0 . 1 5 2 2
0 . 1 0 9 3
0 . 0 4 4 8
- 0 . 0 3 7 9
- 0 , 1 3 6 4
- 0 . 2 4 9 1
- 0 . 3 7 4 5
0 . 0 5 7 2
0 . 1 4 5 0
0 . 1 6 7 7
0 . 1 5 2 2
0 . 1 0 9 3
- 0 . 0 3 7 9
0 . 0 4 4 8
- 0 . 1 3 6 4
- 0 . 2 4 9 1
- 0 . 3 7 4 5
0 . 0 5 7 2
0 . 1 4 5 0
0 . 1 6 7 7
0 . 1 5 2 2
0 . 1 0 9 3
0 . 0 4 4 8
- 0 . 0 3 7 9
- 0 , 1 3 6 4
- 0 . 2 4 9 1
- 0 . 3 7 4 5
0 . 1 6 9 7
0 . 1 8 4 4
0 . 1 9 5 3
0 . 1 4 9 7
0 . 0 9 6 6
0 . 0 2 8 4
- 0 . 0 5 3 0
- 0 . 1 4 6 1
- 0 . 2 4 9 9
- 0 . 3 6 3 4
0 . 0 7 0 3
0 . 1 5 3 8
0 . 1 7 1 2
0 . 1 5 1 1
0 . 1 0 4 9
0 . 0 3 8 6
- 0 . 0 4 4 0
- 0 . 1 4 0 5
- 0 . 2 4 9 2
- 0 . 3 6 8 7
0 . 0 6 2 3
0 . 1 4 8 6
0 . 1 6 9 3
0 . 1 5 1 9
0 . 1 0 7 6
0 . 0 4 2 2
- 0 . 0 4 0 6
- 0 . 1 3 8 2
- 0 . 2 4 9 0
- 0 . 3 7 1 5
0 . 0 5 9 8
0 . 1 4 6 9
0 . 1 6 8 6
0 . 1 5 2 2
0 . 1 0 8 7
0 . 0 4 3 7
- 0 . 0 3 9 1
- 0 . 1 3 7 2
- 0 . 2 4 8 9
- 0 . 3 7 2 9
0 . 0 5 8 8
0 . 1 4 6 2
0 . 1 6 8 4
0 . 1 5 2 4
0 . 1 0 9 2
0 . 0 4 4 4
- 0 . 0 3 8 4
- 0 . 1 3 6 7
- 0 . 2 4 8 9
- 0 . 3 7 3 6
- 0 . 0 6 7 1
- 0 . 0 9 4 9
- 0 . 1 1 6 2
- 0 . 1 3 4 2
- 0 . 1 5 0 1
- 0 . 1 6 4 4
- 0 . 1 7 7 5
- 0 . 1 8 9 8
- 0 . 2 0 1 3
- 0 . 2 1 2 2
0 . 2 0 5 0
0 . 2 2 4 4
0 . 1 8 0 7
0 . 1 1 4 7
- 0 . 0 3 2 7
0 . 0 4 1 4
- 0 . 1 0 4 8
- 0 . 1 7 3 6
- 0 . 2 3 8 8
- 0 . 3 0 0 1
0 . 1 1 0 1
0 . 1 7 8 9
0 . 1 7 8 0
0 . 1 4 1 2
0 . 0 8 2 9
- 0 . 0 6 9 9
0 . 0 1 0 9
- 0 . 1 5 6 6
- 0 . 3 3 9 0
- 0 . 2 4 6 8
0 . 0 8 0 6
0 . 1 6 0 9
0 . 1 7 3 4
0 . 1 4 8 ;
0 . 0 2 9 0
0 . 0 9 7 7
- 0 . 0 5 3 5
- 0 . 1 4 6 7
- 0 . 2 4 8 5
- 0 . 3 5 7 1
0 . 0 6 8 6
0 . 1 5 2 9
0 . 1 7 0 8
0 . 1 5 0 6
0 . 1 0 4 0
- 0 . 0 4 5 5
0 . 0 3 7 3
- 0 . 1 4 1 5
- 0 . 2 4 8 9
- 0 . 3 6 5 8
TABLE I
& TIMESHE A MPLITU D ES
F
T H E THIRD
O
E L E V E N T H
ARMONICS
ALCULATED FROM
( 7 ) WI TH
T H E
APPROXIMATION
11) TO h ( A ) FOR A V A R I E T YF VALUES F THE SINGLE
NPUT
MP LI TUDEAOR THE
Q U A N T I Z A T I O N
STAIRCASE
F F I G . (a j
COM P ARI S ON OF & TIMESH E D E P A R T U R EF T H E
FUNDAMENTAL
U T P U T AMPLITUDEROM A A N D
approx.
A h ( A ) f i A , - A ) - A3 V Z A
5
- 6 A 7 A9 - f i A l l
0 . 0 4 0 30 . 0 2 8 8. 0 2 2 40 . 0 1 8 3
0 . 0 5 6 90 . 0 4 0 7. 0 3 1 60 . 0 2 5 9
0 . 0 6 9 70 . 0 4 9 8. 0 3 8 70 . 0 3 1 7
0 . 0 8 0 5
0 . 0 9 0 0
0 . 0 9 8 6
0
. l o65
0 . 1 1 3 9
0 . 1 2 7 3
0 . 1 2 0 8
0 . 3 5 7 3
0 . 1 9 8 5
0 . 0 5 2 7
- 0 . 0 5 1 3
- 0 . 1 1 6 7
- 0 . 1 5 2 3
- 0 . 1 6 6 0
- 0 . 1 6 4 3
- 0 . 1 5 1 7
- 0 . 1 3 1 8
0 . 2 0 1 9
0 . 2 2 2 2
0 . 1 7 3 8
0 . 1 0 1 4
0 . 0 2 3 4
- 0 . 0 5 1 5
- 0 . 1 1 9 0
- 0 . 1 7 7 3
- 0 . 2 2 5 5
- 0 . 2 6 3 9
0 . 1 2 2 2
0 . 1 8 5 9
0 . 1 7 8 6
0 . 1 3 5 8
0 . 0 7 3 1
- 0 . 0 0 0 8
- a
. 0 8 0 4
- 0 . 1 6 2 1
- 0 . 2 4 3 5
- 0 . 3 2 2 8
0 . 0 8 8 1
0 . 1 6 5 8
0 . 1 7 4 7
0 . 1 4 6 0
0 . 0 9 2 9
- 0 . 0 5 9 3
0 . 0 2 2 9
- 0 , 1 5 0 3
- 0 . 2 4 7 1
- 0 . 3 4 9 6
- 0 . 0 5 7 5
- 0 . 0 6 4 3
- 0 . 0 7 0 4
- 0 . 0 8 1 3
- 0 . 0 7 6 1
- 0 . 0 8 6 3
- 0 . 0 9 0 9
- 0 . 0 7 0 0
- 0 . 2 6 2 6
- 0 . 3 0 9 1
- 0 . 2 7 0 3
- 0 . 1 9 7 3
- 0 . 1 1 8 2
- 0 . 0 4 6 2
0 . 0 1 3 6
0 . 0 6 0 1
0 . 0 9 4 2
0 . 2 2 8 7
0 . 3 0 3 7
0 . 1 1 2 0
- 0 . 0 7 5 3
0 . 0 0 5 7
- 0 . 1 2 8 9
- 0 . 1 5 7 7
- 0 . 1 6 5 8
- 0 . 1 5 7 7
- 0 . 1 3 7 4
0 . 1 8 2 1
0 . 2 1 4 4
0 . 1 7 5 2
0 . 1 0 7 8
0 . 0 3 1 2
- 0 . 0 4 4 7
- 0 . 1 1 4 6
- 0 . 1 7 5 5
- 0 . 2 2 6 0
- 0 . 2 6 5 5
0 . 1 8 3 3
0 . 1 1 7 4
0 . 1 3 6 7
0 . 1 7 8 0
0 . 0 7 4 9
- 0 . 0 7 8 9
0 . 0 0 1 1
- 0 . 1 6 1 3
- 0 . 2 4 3 5
- 0 . 3 2 3 5
0 . 0 4 4 7
0 . 0 5 0 0
0 . 0 5 4 8
0 . 0 5 9 2
0 . 0 6 3 3
0 . 0 6 7 1
- 0 . 0 2 4 3
0 . 0 7 0 7
- 0 . 0 5 3 9
0 . 0 7 3 4
0 . 1 9 0 7
0 . 2 5 2 9
0 . 2 6 4 6
0 . 2 4 2 0
0 . 2 0 0 3
0 . 1 5 0 6
0 . 3 1 5 8
0 . 1 0 0 0
- 0 . 0 4 7 3
0 . 1 1 6 6
- 0 . 1 4 1 1
- 0 . 1 7 4 2
- 0 . 1 6 4 6
- 0 . 1 2 9 0
- 0 . 0 8 0 5
- 0 . 0 2 8 6
0 . 0 2 0 6
0 . 2 5 1 2
0 . 2 3 2 0
0 . 1 4 9 5
- 0 . 0 2 3 3
0 . 0 5 5 7
- 0 . 0 9 7 2
- 0 . 1 7 4 7
- 0 . 1 4 5 6
- 0 . 1 8 6 2
- 0 . 1 8 2 6
0 . 1 5 5 5
0 . 2 0 3 0
0 . 1 7 7 8
0 . 1 2 0 4
0 . 0 4 8 6
- 0 . 0 2 7 6
- 0 . 1 7 1 5
- 0 . 1 0 2 2
- 0 . 2 3 3 2
- 0 . 2 8 6 0
- 0 . 0 3 6 6
- 0 . 0 4 0 9
- 0 . 0 4 4 8
- 0 . 0 4 8 4
- 0 . 0 5 1 8
- 0 . 0 5 4 9
- 0 . 0 5 7 9
- 0 . 1 8 2 1
- 0 . 0 4 3 3
0 . 0 3 8 6
0 . 0 6 2 5
- 0 . 1 3 3 9
- 0 . 0 4 6 5
- 0 . 1 9 7 0
- 0 . 2 3 0 0
- 0 . 2 3 6 6
- 0 . 2 2 3 6
0 . 1 1 9 1
- 0 . 1 3 5 1
- 0 . 2 4 5 6
- 0 . 2 3 3 7
- 0 . 0 7 1 0
- 0 . 1 6 1 2
0 . 0 1 1 5
0 . 0 7 4 2
0 . 1 1 3 5
0 . 1 3 0 7
0 . 3 0 8 6
0 . 2 1 6 4
0 . 0 8 6 7
- 0 . 1 0 1 7
- 0 . 0 2 4 5
- 0 . 1 4 4 0
- 0 . 1 5 6 2
- 0 . 1 4 4 9
- 0 . 1 1 6 7
- 0 , 0 7 8 0
0 . 2 0 0 3
0 . 2 2 0 7
0 . 1 6 9 8
0 . 0 9 3 9
0 . 0 1 3 2
- 0 . 0 6 1 9
- 0 . 1 2 6 2
- 0 . 1 7 7 0
- 0 . 2 1 3 6
- 0 . 2 3 6 3
spectral density of a sinusoid of slowly varying frequency,
e.g. , any component ( l lnr ) sin [ 2 n r A sin (27rFt +
q ]
f
the sawtooth, is approximately proportional to the proba-
bility density function of the instantaneous frequency f,
which is he re f
=
2nrAFIcos (2rFt + \E)
When multi-
plied by the power
1 / (2n2 r2 )
f this componen t, the prob-
ability density function o ff yields
n-27r-3(4n2r2A2F2 -
f 2)-12
for
f < 2 n r A F .
Putting
f
=
pF
in this approximation to the power spec-
tral density of the nth-harmonic compon ent of the saw-
tooth and multiplying by the spacing 2F between the odd
harmonics of the input frequency F that are generated by
quantization, we get ET l / ( n 3 r 4 A ) for the average of the
power A; of the pth-harmonic output for
p 2 n r A ,
as indicated by
the foregoing ap proximation for the power sp ectra l den -
sity. Th is sharp drop occurs primarily n ear
p
=
2 r A ,
as
. the n = 1 ter m is the principal contributor to
A p .
In th
neighborhood of
p = 2n7rA >>
1, JJ2n rA ) is better ap
proximated by Airy functions [7], but
X )
suffices for th
lower harmonics, whichwill beour principal intere s
since, when
p >> 1,
Ap sin
[2prFt +
p\k(t)] will gener
ally be very wideband.
C.
Fractional-Integral
Relationship
There is an nteresting relationship between the Fourie
series
(10)
for h(A) and the Fourier series in
(1)
for th
error sawtooth which can be expressed in erm s of the We
integral of order
[X] . To
understand it we need only t
observe that
( LIZ)
cos (cz + 0) = cm cos (cz
+ I +
i m r ) (12
when m is any nonnegative integer and can be interprete
as a (-m)fold integ ral when m is a negative integer. T h
usual rules regarding successive differentiations and n-
tegrations remain valid when
(12)
is extended to all value
of m and, in particular, to
m
=
-$.
We thus see that
h ( A
7/27/2019 The Intermodulation and Distortion due to Quantization of Sinusoids
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BLACHMAN: IM AND DISTORTION FROM QUANTIZATION
OF
SINUSOIDS
142
1
TABLE I
(Continued)
3 1 . 1
3 1 . 2
3 1 . 3
3 1 . 4
3 1 . 5
3 1 . 7
3 1 . 6
3 1 . 8
3 1 . 9
3 2 . 0
6 3 . 1
6 3 . 2
6 3 . 3
6 3 . 4
6 3 .5
6 3 . 6
6 3 . 7
6 3 . 9
6 3 . 8
6 4 . 0
1 2 7 . 1
1 2 7 . 2
1 2 7 . 3
1 2 7 . 4
1 2 7 . 5
1 2 7 . 6
1 2 7 . 7
1 2 7 . 8
1 2 7 . 9
1 2 8 . 0
2 5 5 . 1
2 5 5 . 2
2 5 5 . 3
2 5 5 . 4
2 5 5 . 5
2 5 5 . 6
2 5 5 . 7
2 5 5 . 8
2 5 5 . 9
2 5 6 . 0
5 1 1 . 1
5 1 1 . 2
5 1 1 . 3
5 1 1 . 4
5 1 1 . 5
5 1 1 . 6
5 1 1 . 7
5 1 1 . 8
5 1 1 . 9
5 1 2
. O
0 . 1 6 7 7
0 . 1 5 2 2
0 . 1 0 9 3
0 . 0 4 4 8
- 0 . 0 3 7 9
- 0 . 1 3 6 4
- 0 . 2 4 9 1
- 0 . 3 7 4 5
0 . 0 5 7 2
0 . 1 6 7 7
0 . 1 4 5 0
0 . 1 5 2 2
0 . 1 0 9 3
0 . 0 4 4 8
- 0 . 0 3 7 9
- 0 . 1 3 6 4
- 0 . 2 4 9 1
- 0 . 3 7 4 5
0 . 0 5 7 2
0 . 1 4 5 0
0 . 1 6 7 7
0 . 1 5 2 2
0 . 1 0 9 3
0 . 0 4 4 8
- 0 . 0 3 7 9
- 0 . 1 3 6 4
- 0 . 2 4 9 1
- 0 , 3 7 4 5
0 . 0 5 7 2
0 . 1 4 5 0
0 . 1 6 7 7
0 . 1 5 2 2
0 . 1 0 9 3
0 . 0 4 4 8
- 0 . 0 3 7 9
- 0 . 1 3 6 4
- 0 . 2 4 9 1
- 0 . 3 7 4 5
0 . 0 5 7 2
0 . 1 4 5 0
0 . 1 6 7 7
0 . 1 0 9 3
0 . 1 5 2 2
0 . 0 4 4 8
- 0 . 0 3 7 9
- 0 . 1 3 6 4
- 0 . 2 4 9 1
- 0 . 3 7 4 5
approx.
A
h ( A )
f i A l - A )
- S A
\ m A - m
3
- f i A
7
0 . 0 581
0 0611
0 . 0 7 2 6. 0 8 6 8 0 . 1 058 0 . 1 2 9 2
0 . 1 5 5 6. 1 6 4 9. 1 7 6 6. 1 8 9 8
. 1 4 5 9
~. ...
. 0 5 7 2
0 . 1 4 5 0
0 . 1 6 8 2
0 . 1 5 2 4
0 . 1 0 9 4
0 . 0 4 4 7
- 0 . 0 3 8 0
- 0 . 1 3 6 5
- 0 . 2 4 8 9
- 0 . 3 7 4 0
0 . 0 5 8 2
0 . 1 4 5 6
0 . 1 6 8 2
0 . 1 5 2 5
0 . 1 0 9 5
0 . 0 4 4 8
- 0 . 0 3 7 8
- 0 . 1 3 6 4
- 0 . 2 4 8 9
- 0 . 3 7 4 1
0 . 0 5 8 3
0 . 1 4 5 5
0 . 1 6 8 3
0 . 1 5 2 6
0 . 1 0 9 6
0 . 0 4 5 0
- 0 . 0 3 8 2
- 0 . 1 3 6 5
- 0 . 2 4 8 7
- 0 . 3 7 4 4
0 . 0 5 7 5
0 . 1 4 6 5
0 . 1 6 9 0
0 . 1 5 4 1
0 . 1 0 9 3
0 . 0 4 6 4
- 0 . 0 3 6 4
- 0 . 1 3 6 7
- 0 . 2 4 9 2
- 0 . 3 7 1 8
0 . 0 5 9 3
0 . 1 3 8 0
0 . 1 6 6 3
0 . 1 0 3 5
0 . 1 5 4 6
0 . 0 4 5 6
- 0 . 0 3 8 0
- 0 . 1 3 4 6
- 0 . 2 4 9 9
- 0 . 3 6 9 4
0 . 1 6 9 4
0 . 1 5 1 6
0 . 1 0 6 9
0 . 0 4 1 2
- 0 . 0 4 1 6
- 0 . 1 3 8 9
- 0 . 2 4 8 9
- 0 . 3 7 0 1
0 . 0 6 0 4
0 . 1 6 8 8
0 . 1 4 7 3
0 . 1 5 2 1
0 . 1 0 8 2
0 . 0 4 3 1
- 0 . 0 3 9 6
- 0 . 1 3 7 6
- 0 . 2 4 8 9
- 0 . 3 7 2 2
0 . 0 5 9 2
0 . 1 4 6 5
0 . 1 6 8 4
0 . 1 5 2 3
0 . 1 0 9 0
- 0 . 0 3 8 7
0 . 0 4 4 1
- 0 . 1 3 6 9
- 0 . 2 4 8 9
- 0 . 3 7 3 3
0 . 0 5 8 2
0 . 1 4 5 9
0 . 1 6 8 3
0 . 1 5 2 5
0 . 1 0 9 3
0 . 0 4 4 3
- 0 . 0 3 8 4
- 0 . 1 3 6 8
- 0 . 2 4 8 7
- 0 . 3 7 4 1
0 . 0 5 8 1
0 . 1 4 4 9
0 . 1 6 8 2
0 . 1 5 1 4
0 . 1 0 9 8
0 . 0 4 5 5
- 0 . 0 3 8 0
- 0 . 1 3 6 8
- 0 , 2 4 9 0
- 0 . 3 7 3 5
0 . 1 7 1 7
0 . 1 4 9 7
0 . 1 0 1 6
0 . 0 3 4 2
- 0 . 0 4 8 6
- 0 . 1 4 3 6
- 0 . 2 4 8 7
- 0 . 3 6 2 2
0 . 0 6 5 1
0 . 1 6 9 9
0 . 1 5 0 6
0 . 1 5 1 2
0 . 1 0 5 7
0 . 0 3 9 7
- 0 . 0 4 3 1
- 0 . 2 4 8 9
- 0 . 1 4 0 0
- 0 . 3 6 8 3
0 . 0 6 1 5
0 . 1 4 8 0
0 . 1 6 9 0
0 . 1 5 1 9
0 . 1 0 7 7
- 0 . 0 4 0 4
0 . 0 4 2 4
- 0 . 1 3 8 1
- 0 . 2 4 8 9
- 0 . 3 7 1 4
0 . 0 5 9 6
0 . 1 4 6 8
0 . 1 6 8 6
0 . 1 5 2 2
0 . 1 0 8 6
0 . 0 4 3 7
- 0 . 0 3 9 2
- 0 . 1 3 7 2
- 0 . 2 4 8 9
- 0 . 3 7 2 9
0 . 0 5 8 3
0 . 1 4 6 1
0 . 1 6 8 2
0 . 1 0 9 2
0 . 1 5 2 2
0 . 0 4 4 6
- 0 . 0 3 8 6
- 0 . 1 3 6 6
- 0 . 2 4 9 1
- 0 . 3 7 3 5
0 . 1 7 4 4
0 . 1 4 6 2
0 . 0 9 3 4
0 . 0 2 3 5
- 0 . 0 5 8 8
- 0 . 1 5 0 0
- 0 . 2 4 7 7
- 0 . 3 4 9 9
0 . 0 7 2 1
0 . 1 7 1 6
0 . 1 5 5 3
0 . 1 4 9 7
0 1018
0 . 0 3 4 4
- 0 . 0 4 8 3
- 0 . 2 4 8 7
- 0 . 1 4 3 4
- 0 . 3 6 2 4
0 . 0 6 4 9
0 . 1 5 0 5
0 . 1 6 9 9
0 . 1 5 1 2
0 . 1 0 5 8
- 0 . 0 4 3 0
0 . 0 3 9 8
- 0 . 1 3 9 9
- 0 . 2 4 8 9
- 0 . 3 6 8 5
0 . 0 6 1 4
0 . 1 4 8 0
0 . 1 6 9 0
0 . 1 5 2 0
0 . 1 0 7 7
0 . 0 4 2 4
- 0 . 0 4 0 4
- 0 . 1 3 8 1
- 0 . 2 4 8 9
- 0 . 3 7 1 5
0 . 0 5 9 8
0 . 1 4 7 1
0 . 1 6 8 3
0 . 1 0 8 6
0 . 1 5 2 1
0 . 0 4 3 7
- 0 . 0 3 9 2
- 0 . 1 3 7 4
- 0 . 2 4 8 9
- 0 . 3 7 3 1
0 . 1 7 7 0
0 . 1 4 0 5
0.0818
0 . 0 0 9 2
- 0 . 0 7 1 9
- 0 . 1 5 7 6
- 0 . 2 4 5 2
- 0 . 3 3 2 7
0 . 0 8 1 5
0 . 1 6 1 5
0 . 1 7 3 4
0 . 1 4 7 5
0 . 0 9 6 4
0 . 0 2 7 3
- 0 . 0 5 5 2
- 0 . 2 4 8 1
- 0 . 1 4 7 8
- 0 . 3 5 4 3
0 . 0 6 9 5
0 . 1 5 3 6
0 . 1 7 1 0
0 . 1 5 0 2
0 . 1 0 3 2
- 0 . 0 4 6 6
0 . 0 3 6 2
- 0 . 1 4 2 3
- 0 . 2 4 8 8
- 0 . 3 6 4 5
0 . 0 6 3 5
0 . 1 4 9 4
0 . 1 6 9 4
0 . 1 5 1 3
0 . 1 0 6 3
0 . 0 4 0 5
- 0 . 0 4 2 2
- 0 . 1 3 9 4
- 0 . 2 4 9 1
- 0 . 3 6 9 6
0 . 0 6 0 4
0 . 1 4 7 1
0 . 1 6 8 4
0 . 1 0 7 6
0 . 1 5 1 5
0 . 0 4 2 4
- 0 . 0 4 0 5
- 0 . 1 3 8 4
- 0 . 2 4 9 5
- 0 . 3 7 2 4
0 . 1 7 8 4
0 . 1 3 1 7
0 . 0 6 6 3
- 0 . 0 8 7 2
- 0 . 0 0 8 7
- 0 . 1 6 5 3
- 0 . 2 4 0 3
- 0 . 3 1 0 2
0 . 0 9 3 2
0 . 1 7 5 3
0 . 1 6 8 9
0 . 1 4 4 2
0 . 0 8 9 3
0 . 0 1 8 5
- 0 . 0 6 3 5
- 0 . 2 4 7 0
- 0 . 1 5 2 9
- 0 . 3 4 3 9
0 . 0 7 5 1
0 . 1 5 7 3
0 . 1 7 2 0
0 . 1 4 8 9
0 . 0 9 9 6
- 0 . 0 5 0 8
0 . 0 3 1 8
- 0 . 1 4 5 1
- 0 . 2 4 8 8
- 0 . 3 5 9 8
0 . 0 6 6 3
0 . 1 5 1 3
0 . 1 7 0 2
0 . 1 5 0 6
0 . 1 0 4 8
0 . 0 3 7 7
- 0 . 0 4 4 8
- 0 . 1 4 1 3
- 0 . 2 4 9 0
- 0 . 3 6 7 7
0 . 0 6 0 5
0 . 1 4 6 9
0 . 1 6 8 1
0 . 1 0 6 5
0 . 1 5 0 5
0 . 0 4 0 0
- 0 . 0 4 2 7
- 0 . 1 3 9 6
- 0 . 2 4 9 9
- 0 . 3 7 2 5
in (10) is JS/a times the halfth-order integral of the error
sawtooth with A replacing x .
From (12) it follows thathalfth-order ntegration s
equivalent to low-pass filtering with a frequency response
lly52.f; such a filter has impulse response 1/& for t >
0
and zero for
t > 1 ,
and it follows from (13) and from Woodward's heorem
that, for
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IEEE TRANSACTIONSNCOUSTICS, SPEECH, ANDIGNALROCESSING,
VOL.
ASSP-33.
NO. 0, DECEMBER 1985
Apq = 6p16qoA
+ GpoSqla+
,,(2n7rA)Jq(2nna).
L
n = I
n n
(17)
Th us, afte r quantization (15) includes intermodulation and
distortion products whose frequencies have the form p F +
qf, w h e r e p a n d
q
are integers with an odd, positive sum.
Th e second input sinusoid turns each spectral line of ( 4 )
into a comb of lines with spacing IF
-
f if the two input
frequencies
F
and
f
are relatively close together. In the
same way that w e found there we re of the order of
7rA
significant lines in the spec trum of
( 4 ) ,
we see that each
comb will contain roughly 47ra lines since Jq (2na) n (17)
becomes negligible when / q / s much larger than
2na
(note
that q can be even, odd, positive, and negative), and (16)
thus effectively includessomething ike 4n2Aa spectral
lines altogether. Dividing the total quantization-noise
power by thisnumber, we ind that heaverage power
of each of these lines is oughly 1 / (48n2Aa) , ut again this
value overestimates th e streng th of the lines for which
lpI
and
191
are small and underestimates those with
jpl
near
27rA and 191 near
2 n a .
The sum of the output components
in each comb ends tohave a pulse-train envelope with
21F
-
f pulses per unit time occurring whenever ( t )
-
4 ( t )
is a multiple of n .
A . Large Input Amplitudes
When A and
a
re both large, we can use
(8)
for both
of the Bessel functions in
(17) .
[Note that, when
A
and a
exceed 1 the arguments of all of the Bessel functions
exceed 27r, which is already large en ough to make (8) use-
ful if p and q are not large]. Expressing the resulting prod-
uct of two cosines as half the cosine of the sum of the
arguments plus half the cosine of the difference, we find
that, except for the ca ses of input-frequency output com-
ponents ( i .e. , p = 1
q
= 0 and p = 0,
q
= 1 ) which
require an additional term A or a ,
d A a
where
and { z > s the fractional part of z . The lower l imit on the
integral can equally well be 0 or 1, with the sam e result,
but g ( z ) cannot b e expressed in term s of a finite number
of elementary functions (however, it is expressible
[ 9 ]
n
term s of dilogarithms
[7 ,
p.
10041
of complex arguments).
Note that f z ) and g ( z ) are periodic with a unit period,
shown in Fig.
3,
and are a H ilbert-transform pair.
Fig. 3.
. f z )
and g(z) as functions
of
the fractional part {z} of z. When A
>>
1
+ p , a >>
1
+ q , and p + 4 is odd, the amplitude
A,,,
of any 1M
product is
+ 4 / ( ~
a)imes
the
sum or difference
of f (A + a)
and
g
( A
- a ; , , = 0 w h e n p + q is even.
Th e right-hand expressions in (19) and ( 2 0 )are obtained
by double integration of the second derivatives of the Fou-
rier eries.The se second derivatives are he real and
imaginary parts of the geometric series
- C y
exp
(
2 n n z )
whose sum is
-
: ctn TZ plus terms that oscillate at a
frequency hatgrows nfinite as N -,
co
andhence are
removed by the integration. For { z > 0, the impulses at
integ er values of z can be neg lected. Constants of integ ra-
tion are deter min ed by the zero average value (integral
over one period).
Equation
( 2 0 )
s well approximated by
and by
( 2 2
These two expressions together with the identity
g(l -
= - g ( z )
provide accurate values of
g ( z )
for all values of
z . For example, for z = $, 21) gives 0.0232365 and ( 2 2
gives 0.0232012, while from ( 2 0 ) he tru e value is G / ( 4 n 2
=
0.0232017,
where
G
is CatalBn's constant [lo]. Thus,
( 2 2 )
gives better results than
(21),
and it can even be used
up to z =
$,
where it gives -0.00007 instead of the cor-
rect value, zero.
B. Fluctuating Large Input Amplitudes
Since
f ( 4 ) =
1
2-4
+
3-4
+ *
= n 4 / 9 0
andsince
crossproduct term s average out to zero, the mean-squared
values off(z) and g ( z ) are both Shf 2 z ) dz = g 2 ( z )dz =
112880.
Because of the opposite symmetries of these two
functions, their productf(A
+ a ) g ( A a )
becomes zero
when averaged over fluctuations of either input am plitude
regardless of the value of the other. Hen ce, when A >>
1
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BLACHMAN:
1M
ANDDISTORTIONFROMQUANTIZATION OF SINUSOIDS
1423
and a >> 1 fluctuate ndependen tly over several quanti-
zation ste ps, the mean-sq uared value of each of the lower-
order distortion an d interm odulation prod ucts the average
value of ;A;,) is
1/(180r2Aa) .
This value may be compared
with our previous estimate, 1/(487r2Aa),which is biased
upward by the effect of the higher-order products.
As A and
a
vary, A,,, attains its largest valuewhen both
f ( A +
a )
and
g ( A
-
a )
have their maxim um values, viz.
0.0417 and 0.0257, respectively. Hence, A,, is at most
0.0858/&.
This maximum may be compared with the
rms value
1 / ( 9 0 ~ * A a ) ~0.0336/&.
Notice that both
values are well below one quantization step when
A
and
a
are even moderately large; e .g.
,
he rms values is
49.5
dB
below it if
A = a =
10.
Intermodulation is often measured, on the other hand,
with equal input amplitudes. When A = a , (18 ) becomes
simply A,, =
* 4 f ( 2 A ) / r A .
If heseamplitudes emain
equal but fluctuate so that the fractional pan of 2A is uni-
formly distributed between 0 and
1 ,
the average value of
pl,, isonly 1 / ( 3 6 0 ~ ~ A ~ ) ,ince here s no contribution
from
g ( A
-
a .
C. One Strong and One Weak Input
Exact, closed forms
[11]-[14]
n terms of the complete
elliptic integrals
K ( k )
and
E(k)
have been found for the
output-component amplitudes A,, when A 4-
a
< 1 and
the quantizer accordingly acts like a hard limiter. Being
the Fourier coefficient [11 of order p , q of y ( A sin
+
a
sin
d), A,,
can presumably be expressed in increasingly
complicated closed forms involving
K ( k )
and
E ( k )
for in-
creasing values of A and
a .
For
A
>> 1, these forms would
be cumbersome both to derive and to use, and
so
we now
seek instead to approximate (17) for A >> 1 >> a as in
For
a
1
are then negligible in comparison with those for I 141
=
1 .
The m ost important of the latter is A2, 1 , since the fre-
quency
2F -
falls in the same band as the input fre-
quencies
F
and f. For it we have the approximation
1 2
[ I ]
P I
m
A2, I E -2a J2(2nrA)
for
a 1 >>
u
as a function of the fractio nal part { A } of th e ampl itude
A
of
the trongernput ignal.This urve lsoepresentsimes the
departure from unity of the voltage gain experienced by the w eak input
signal.
Notice that this differentiation yields the halfth derivative
of the quantization staircase when
A >> 1 .
Again using
(8) ,
we find that
(2 3 )
becomes
for A
>>
1
>>
a .
(2 6 )
Alrhough
this
series converges
only
very
slowly (and di-
verges when A is an integer), this expression [ 2 ] shows
A 2 , -1 o be proportional to l / & times a zero-mean peri-
odic function when
A >> l >> a .
Study of the first and
last terms of
(2 4 )
shows the source
of
this behavior to be
the variation due to the presence of the weak signa l in the
length
of
time the input spends n its extreme quantization
steps, the strong input being parabolic n he neighbor-
hood
of
its peaks.
Eulers summ ation formula with term s through the first
derivative yields a good approxim ation [11 to
( 2 4 ) ,
and
hence to
( 2 6 ) ,
viz.,
A 2 , - 1 5 -
[ ~-
4 8 ( A } 2
+
8 4 { A } + 35
P
2 4 ( { A }
+ )32
for A >> 1 >> a ,2 7 )
which is shown in Fig.
4.
Integration
of (2 7 )
with respect
to ( A } from
0
to 1 gives (2 9 & - 4 1 ) / (1 2 r & )
=
0.000323/&, showing that ( 2 7 ) is a good but not perfect
zero-mean approximation to
( 2 6 ) .
The other important output component with 141 =
1
is
that of frequency
f
nd am plitude
Aol .
Putting p = 0 and
q = 1 in
(17), using the same approxim ation
J I ( 2 n r a )
E
n m or a > 1 ,
we again get the infinite series ( 2 6 ) , and hence we have
Aol
E
a + A Z , - ~
or
A >>
1
>> a .
Thus, Fig.
4
also
describes the variation with
A
of the difference between
the inp ut and output am plitudes of the s econd signal when
A
>> 1
>>
a .
If
a
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IEEETRANSACTIONSONACOUSTICS,SPEECH,ANDSIGNALPROCESSING. VOL. ASSP-33, NO. 6, DECEMBER
19
Aol G c a / [ a ( A 2
-
k2)Il2] for
a
1
the integral (29) aga in gives
us
the halfth-
order integral
of
the derivative of the quantization stair-
case, i.e., its halfth derivative.
Likewise, (25), when applied to (6) with p = 1, yields
- A
U
A2,-I G __ 1 ( 2 x 2 - A 2 ) y ' ( x ) & / (A 2 - x2)'12
a A 2
- A
for
a > 1 and show that it is the quantization-step size near
the ex trem es of the excusions of the input that principally
determine the distortion and intermodu lation, these rela-
tionships are only special cases of the more exa ct, mo re
general transf orm relationships (29) and (30 ), which show
Aol and
A 2 ,
- to be related to the zeroth- and second-order
Chebyshev transform s [5] of
y
(x),espectively, for a
1.
(32)
2 R
A
n = ~
Because Jl (2 n a a ) has not been approximated here by the
first term,
rzaa,
of its power-series expa nsion, (32) con-
verges well for all a.
Since J , ( z ) reaches a peak
of
0.5819 when
z
=
1.84 and
passes through zer o when z = 3.83, oscillating th ereafte r
as described by (8), it can be dequately approxima ted for
I
z
I
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BLACHMAN: IM AND DISTORTION FROM QUANTIZATION OF SINUSOIDS
1425
If
(T
=
1,
the exponential factor here is
-
71 dB for n =
1 and dim inishes so fast as n increa ses that it suffices to
retain just the first term of the summ ation. That term is
seen to be extremely sma ll, suggesting that in the pres-
ence of even a small amount of wideband noise (which is
readily eliminated by output filtering), the intermodula-
tiondue oquantization hould be undetectable; even
without the noise the foregoing results show it to be very
weak unless both
A
+
1 and B > 1.
Although the noise makes the quantization staircaseook
very much like a straight ramp for the input signals, there
is still a mean-squared d eparture from the ide al ramp equa l
to about
A,
which manifes ts itself alm ost entirely in the
form of signal
X
noise output [161. Returning to
(17)
with
anadditional actor J r ( 2 n m ) fornarrow-bandadditive
Gaussian input noise and an additional subscript r on Ap4
and to (16) with a correspond ing additional term in the
argument of the sine, we see that the variationf
Jo(2nncy)
about tsmeanvalue xp ( - 2 n (T ) produces up-
pressed-carrier modulation of the signals distortion and
intermodulation by the amplitude
y
of the no ise. Th e term s
with r
>
0 involve phase modulation as well as amplitude
modulation by the noise because of the additional term in
the argument of the sine in (16).
When the spectru m of the noise is substantially wider
than that of the signa ls, all of these signal
X
noise com-
ponents are noiselike and cannot be mistaken for signals.
Their wider spectra permit filteringo attenuate them, but,
even without it, they a re likely to be mask ed by the un-
avoidable undistorted output noise when the nput noise
has a wide spectrum.
2 2 2
V.
CONCLUSIONS
By m eans of a transform-dom ain analysis we have ex-
tended the rangeof input-signal amplitude s for which sim-
ple ex pressions give good approxim ations for the ampli-
tudes of intermodulationproducts,output ignals,and
harmonics. Our results are for the quantization staircase
of Fig. l( a), but those for that of Fig. l( b) can be obtained
by merely inserting the factor ( - 1) into the summations.
Some of the foregoing results have been verified by com-
puter simulation [
171.
All inputs and outputs have been expressed in units of
quantization-step size; if these steps have width and height
Q
volts, the replacem ent of and
y
by
x / Q
and
y / Q
will
allow the input
x
and the output to be expressed n volts.
In he case of nonu niform quantization, he analyses n
Sections 11-D and 111-C show that, when one input signal
is much stronger than the other, the low-order intermo-
dulation and distortion are largely d etermined by the s tep
width at the extreme values of the stronger signal, since
there the quantization error traverses relatively few saw-
teeth per second.
We have seen that the presence of even a small amount
of additive input noise should reduce the intermodulation
and d istortion to undetectable levels. Jus t as a seco nd in put
signal breaks each harmo nic produced by the first input
signal into a comb of spectral lines, the addition of noise
breaks each line of each comb intomany components only
one of which is pure intermodulation while all the rest are
noiselike. Significant intermod ulation and distortion may
rem ain, however, if the cen ters of the quantization steps
do not lie on a straight line. Curvature f the overall stair-
case w ould produce the well-known effects of nonlinear
distortion [3], [5], [6], [14]-[16].
When the input to the quantization process is sampled
and the output is suitably low-pass filtered, intermodu la-
tion, and distortion frequencies exceeding half the sam -
pling rate are aliased to frequencies below it, yielding the
quantization noise that has been the subject of other in-
vestigations [3], [4], [181, [191
;
in [191 W oodw ards theo-
rem[6] is appliedas nSection 11-B to he term s of a
Fourier series for the quantization erro r lik e that in (1) in
order to simplify the analysis.
ACKNOWLEDGMENT
I am grateful to Prof. R.
N.
Bracewell of Stanford Uni-
versity,Stanford,CA, orpapers nd eferences on-
nected with the fractional calculus and to
D.
R. Morgan
of AT&T Bell Laboratories, Wh ippany, NJ, for his very
careful reading of earlier versions of this paper and for
many valuable uggestions egarding ts mprovement,
particularly the idea of using the sinusoidal a pproxim ation
for
J , ( 2 n ~ a )
n
( 3 2 )
when a
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1426 IEEERANSACTIONS ON ACOUSTICS,PEECH,ND SIGNAL PROCESSING, VOL. ASSP-33,O.
6 ,
DECEMBER 19
[ I61
N.
M. Blachman, The signal
X
signal, noise
X
noise, and signal
X
noise output of a nonlineari ty ,
ZEEE Trans.
Inform.
Theory,
vol. IT-
14, pp. 21-27, Jan. 1968.
[17]D.
R .
Morganand A.Aridgides,Discrete-t imedistortion analysis
of quantized sinusoids,
ZEEE Trans. Acoust ., Spe ech , Signal
Pro-
cessing, vol. ASSP-33, pp. 323-326, Feb. 1985.
[IS] W. R. Bennett , Spectra of quantizedsignals, BellSyst. Tech.
J . ,
[I91 T. A.
C.
M. Claasen and A . Jongepier , Model for the power spectral
density of quantization noise,
ZEEE Trans. Acoust., Speech, Signal
v01. 27, pp. 446- 472, July 1948.
Processing, VOI.
ASSP-29.
pp. 914-917. Aug. 1981.
ment of electroacoustic transducers and with the analysis of sonar syst
designs. From 1945 to 1946 he worked at the Cruft Laboratory of Harva
on signal and noise problems in radio communication, particularly FM.
a
member of the T heory Groupof the A ccelerator Project at the rookhav
NationalLaboratory,Upton, NY, from1947 o 1951, hewas concern
with the theory and design of the Cosmotron, B rookhavens
3-GeV
prot
synchrotron. From 1951 to 1954 he was a mem ber of the Staff
of
the Ma
ematical Sciences Division of the Office of Naval Research, Washin gto
DC. administering
the
program of supported research in the fields
of
co
puters and mathematics . In 1954 he joined what is now the Western Di
sion
of
the GTE Government Systems Corporation, Mountain View, C
where he is now a Senior Scientis t and Consultant on sta t is t ical commu
cation theory. From 1958 to 1960 and from 1976
to
1978 he took leaves
absence
to
do scientific liaison work in the field of electronics
as a
memb
of the Staff of the London Branch of the United States Office of Naval R
search . In 1964-1965 he aught communication heory, in Spanish. at h
EscuelaTCcnica Superiorde ngen ie rosdeTe lecommicac idn .Madr id ,
and at the Facultad de Ciencias of the University of Madrid under the Fu
bright program while on sabbatical leave from GT E Sylvania. He has
a
taught at Stanford University and in the off-campu s program s
of
the Un
versity of Maryland and the University of California.
Dr. Blachman is a Fellow
of
the Am erican Association for the Advanc
men t of Science and the Institution of Electrical Engineers and
a
memb
of the Acou stical Society of Am erica, the Am erican Statistical A ssociati
the Optical Society of Am erica, the In stituteof Mathematical Statis t ics ,
Mathematical Association of America, he Society of Industrial and Ap-
plied Mathematics , the U.S. National Commissions C
and
E of
URSI,
a
Sigma
Xi.
He holds an extra-class am ateur radio operators l icensc.
