Hearing Research 2 (1980) 115"122
© Elsevier/North-HollandBiomedica~t Press
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GROWTH OF L(f2 - f ~ ) AND L(2f~ - ] ~ ) WITH INPUT LEVEL: INFLUENCE OF
LARRY E. H U M E S
Divisio,~ of, He_aringand Speech Sciences, Vanderbilt University School of Medicine, Nashville,
TN37232, U.S.A.
(Received 25 Seplember 1979; revised 12 November 1979; accepted 13 November 1979)
The present investigation examined the effect of f2/fl (where fl and f2 are the, frequencies of a
two-tone input, f2 > fl) exerted on the slopes of the functions relating simple difference tone level
(LfJ'2 - ]'1)) and cubic difference tone level (L(2ft - f2)) to input level (L I = L2, where L t and L2
represent the levels of the tones at fn and f2, respectively). Two normal-hearir~g young adults served as
subjects. An adaptive 2AFC nonsimultaneous gap-masking paradigm was util:ized to obtain estimates
~ d~ortion product magnitude. Parameters of the two-tone input were: fl = 155(.) Hz;f2/ft = 1.08
aJ'~d 1.41; and LI = L 2 - 3 5 to 85 dB SL. Results revealed that slopes for the functions relating
L(f2 - f l ) and L(2fl - f 2 ) to Lz -L2 were approximately 1.0 dB/dB for f2/fl = 1.08. For f2/fl =
1.41, however, a slope of 2.0 dB/dB was observed for Lff2 - f l ) while a slope of 3.0 dB/dB was
obtained for L(2fl - f2)- It is suggested that some combination of nonlinearities is needed to account
for these data.
Key words: auditory nonlinearity; simple difference tone; cubic difference tone.
INTRODUCTION
When two tones of frequencies f~ and f2 (f2 > f l ) are presented simultaneoudy to the
ear of a normal-hearing listener, a variety of subjective phenomena may be reported by
the listener depending, in part, upon the saturation of the two tones in frequency (f2/fl)
[17]. For 1.1 < f 2 / f l ~ 1.5, for instance, various distortion products can be perceived
readily by the listener. The most prominent distortion products perceptually are the
cubic difference tone (27"1 - f2) and the simple difference tone (f~ - f,~) [4,15].
Several psychophysical investigations of the dependence of distortion product level
(i.e., L(f2 - f l ) and L(2fl - f~)) on various parameters of the two-tone input have been
conducted in recent years [3-9,12,16,18,21-26]. Models of aural nonlinearity have been
developed to describe the data resulting from the~e investigations [1~2,4,22,23]. Of
primary interest in this regard ]has been the dependence of distortion product level o.-.
input level (Ll and L2 where l,i and L2 represent the levels of the tones at f t and f2,
respectively). For the case o f L l = L~, for instaace, the classical power-series non ii~~earity,
described initially by Helmholtz [10], predict~ that L ( f 2 - f ~ ) will grow at a rate of 2.0
116
dB/dB and L(2ft - f2) at a rate of 3.0 dB/dB *. Other classes of nonlinearity, in particular Goidstein's normalized power-series [4] and Smoorenburg's p-law nonlinearity [1,22,
23], predict slopes of 1!.0 dB/dB for the functions relating both L ( 2 f t - / ' 2 ) ancl
L{f2 -- A) to L t = L2.
A ~ecent study of the perception of the simple difference tone provided data which
sug,;,:sted that no single class of nonlinearity could describe the results entirely [I 1,12].
Rather, some combination of nonlinearities appeared to be responsible for the observed
bel'avtor of the simple difference tone. Specifically, for small f2/1"l Or2[fl < 1.16), slopes
of approximately 1.O dB/dB were observed for the function relating L(f2 - / ' t ) to Li =
L2. At larger f2/ft values (f2ffi f> 1.41), on the other hand, slope estimates of approximately 2.0 dB/dB were obtained. Thus, for a two-tone input comprised of tones dose in
frequency (small f2]fl), 4;ither the normalized power-series or the p-law nonlinearity
could a~:count for the data:. For larger [2[ft, howe~er, the behavior of L(J'~ - / ' ! ) was most
consistent with the predictions of the classical power-series nonlinearity.
Tht: vast majority ot data obtained on the dependence of L(2.fl - f ~ ) on input level
have b,~en conducted with narrow frequency separations of the input (f2]/'~ < 1.3) and
low.to.moderate input lew,'ls (L~ = L~ < 65 dB SL). Under these conditic~ns, slope values
of approximately !.0 dB/dB have been observed by several investigators [3-9,18,2226l. These data, then, ar~ consistent with predictions made by either the normalized
power-series or p-law nonlinearities.
Smoorenburg [23], however, has reported psyehophysical data from one subject for
wider frequency separatio~a of the two-tone input (f2[fl = 1.41)and input levels up to
85 dB SL. For L~ = L2 < 70 dB SL, L(2fl - f 2 ) varied asystematically in magnitude from
0 to I0 dB SL. For L~ = L2 > 70 dB SL, however, the cubic difference tone grew at a rate
of roughly 3.0 dB/dB. Thus, the:e limite~ data are consistent with the recent observations
on the behavior of the simple d~:fference tone [ 12 l. Specifically, for small f2/fi, data are
tno~t consistent with the prediclions made by the normalized power-series or p-law nonlinearities. For large f2/f~, on the other hand, the classical power-series nonlinearity
appears to be operating. The present study seeks to confirm this hypothesis of dual nonlinearities by measuring L(f2 - f t ) and L(2fl - f 2 ) as a function of LI = L2 within the
same ears and for both small and large f2/fl values.
hiI?TItODS
Two normal-hearing young adults served as subjects for this experiment. At the time
the deta reported here were gathered, both subjects had participated in a study on perception of the simple differenc,~ tone for about 40 h. Thus, both subjects were well trained in
th,~ particular paradigm employed in this study.
The apparatus used in riffs experiment has been described in detail elsewhere [12].
Briefly, all stimuli were to,strutted digitally by a laboratory minicomputer (PDP-I 1/10),
* it is perhaps most appropriate to refer to this as the 'classical truncated power-series' nonliltearity
in that only the trust few terms of the series are eonsideJed. Molnar [15], for instance, has demonstrated that the classical power-series can predict almost ~ny slope value by selecting an appropriate
number of terms and arbitrary coefficients.
117
output through a 14-bit digital-to-analog converter at a sampling rate of 20000 points/s
and low-pass filtered (3000 Hz cut-off frequency, - 9 6 dB/octave rejection rate). Stimuli
were then attenuated, mixed, amplified, and then transduced via an electrodynamic
earphone (TDH-49 earphone mounted in a Grason-Stadler 001 circumaural cushion).
For the signals employed in this experiment, acoustic disto~'tion products at f2 - f l and
2f~ - f 2 were more than 80 dB below the level of the two-tone input as measured in a
NBS-9A 6-cm 3 coupler.
The psychophysical procedure, e~nployed in this study to estimate L ( t 2 - f~)and
L(2f~ - f 2 ) was an adaptive two-alternative forced-choice temporal gap-masking paradigm
[12,22]. The primary reason for utilizing a nonsimultaneous paradigm as opposed to the
more common simultaneous psyehophysical method of cancellation was to avoid the
confounding influences of two-tone suppression on estimates of distortion product level
[ 11,16,17,21,22 ]. The masker wa~; comprised of the two-ton." input (J] and f2 at levels of
L l and L2) while the signal was at a frequency of either/'2 - f l or 2fl -/'2. The signal
was a 25 ms pure-tone with a 12.5 ms cosine-shaped rise--f~dl time centered in a 30 ms
gap between two 225-ms maskers (12.5 ms rise-fall time), each consist!ing of the twotone masker. In this experiment ft = 1550 Hz and f2/fl = 1.08 or 1.41. For the stt~dy of
the simple difference tone, Ll --"L2 was varied from 65 t¢ 85 dB SL in 5-dB steps while
for the cubic difference tone, L1 = L2 ranged from 35 to 85 dB SL in 10-dB steps.
Masked thresholds for the signals at/'2 - f~ and 2f~ - / 2 were referenced to growth c,f
masking functions in which the masker in the gap-masking paradigm was a pure-tone ~.t
either f 2 - f~ or 2fs -/'2. In this manner, equivalent levels for the distortion products
(i.e, L(f2 -- fl ) and L(2ft - f2 )) wer ,~-derived [ 12,22 ]. Growth of masking functions were
obtained for masker levels ranging from 10 to 50 dB SL in 10-dB steps.
An adaptive procedure was used to estimate the 70.7% correct point on the psychometric function relating percent correct detections to signal intensity [14]. A single
run consisted of twelve reversals in signal level, the first two of which were discarded. The
ten remaining reversals were averaged to form a single threshold estimate. A minimum of
four such estimates were obtained from each subject for ea:h of the conditions investigated. A 6 dB step size was used during the first two reversals. Step size was reduced to
2 dB for the remaining ten reversals of each run.
RESULTS
Because of the nonlinear nature of the growth of masking functions obtained with
nonsimultaneous procedures for masker levels less than 10 dB SL and greater than 70 dB
SL [19,27], estimates of distortion product level less than 10 dB or greater than 70 dB
are invalid. Estimates of L ( / z - / ' I ) and L ( 2 f l - f 2 ) beyond these extremes, therefore,
have been omitted in the results to follow.
Median L(f2 - f l ) values for each of the two subjects of this study are shown in the
top half of Fig. 1 (panels A and B). For a gi"ren LI = L2 value, the range of individual
L ( f 2 - fs) values never exceeded 5 dB and w~.s usually less than 4 dB. Panel A of this
figure illustrates the growth of L ( f 2 - j ' t ) as a function of L~ = L2 for f2/f~ --- 1.08 while
panel B shows comparable data for f2/fl = 1.41. The parallel dashed lines have been
included in both panels to permit a comparison to slope values 9redicted by v~rious
118
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Fig. 1. L(f 2 .-/1) and L(2fl - I2) in dB SL as a function of Ll = L2 in dB SL. Median data for two
subjects are shown (e, SI; =, $2). In all four panels/'i -, 1550 Hz. In panels A and C,[2//'I = 1.08. In
panels B and D, f2/fl - 1.41. Parallel dashed lines havk,g slopes of 1.0, 2.0 or 3.0 dB/dB have been
included to permit comparison of the data to the predictions of various classes of nonlinearity. In
l:al~els A and B most left peint on abcissa corresponds to L 1 = L2 = 60 dB SL, while most right point
in each panel represents 90 dB SL (5-dB steps marked along abcissa).
c!a~ses of nonlinearity. Note that the data for f2/ft = 1.08 exhibit a slope of approximately 1.0 dB/dB while the data for f2[ft = 1,41 exhibit a slope of roughly 2,0 dB/dB.
The actual slope values for these functions were determined by applying linear-regression
techniques to the i,~dividual data points of the functions. The resulting slope estimates
aad the standard error of these estimates may be found in the top portion of Table I.
119
"FABLEI
SLOPES IN dB/dB FOR THE FUNCTIONS KELATING DISTORTION PRODUCT LEVEL TC
LI =L2
Distortion
product
l'2ff l
L(f2-/t)
1,o8
1.41
L(~f~-j'2)
1.o8
1.41
Subject
Slope
S.E.
1.038
¢'14~
0.,,,9
2.112
0.076
0.049
0.141
2.419
0.816
0.672
2.760
3.i 14
0.243
0.081
0.063
0.193
0.276
Slope estimates and standard errors were derived from the individual estimates of
L(]'2 - f l ) or L(2fl -f2), not the median values shown in Fig. 1.
The lower half of Fig. 1 (panels C and D) provides comparable data for L(2/s -/'2).
Variability of individual estimates was comparable to that observed for estimates of
L ( f 2 - ]'1)- Again, the dashed lines provide reference slope values to which tile data can
be compared. For ]'2[]'i = 1.08, the slope of the function relating L(2ft - / ' 2 ) to L l = L2
is seen to approximate a value of 1.0 dB/dB. For f2/~q = 1.41, on the other hand, the data
follow a slope of roughly 3.0 dB/dB. The exact slope values for these data are also
provided in Table I.
It is also apparent from Fig. 1 that both L(f2 - f l ) and L(2ft - f 2 ) can be elicited at
much lower input levels for f,.[ft = 1.08 than for fz/ft = 1.41. Generally, input levels in
excess of 70 dB SL (86 and 90 dB SPL re: 20/aPa) are required to produce subjective
tones at f2 - f t or 2/'1 - f 2 that are 10 dB SLor greater forf2[f~ = 1.41.
Finally, comparison of the relative levels of L0r2 - f l ) and L(2fl - f 2 ) for the same
i~tput conditions indi.cates that L(2fl - f 2 ) predominates for fz]fl = 1.08 and low-tomoderate input levels (Lt --L2 < 80 dB SL). For the same f2[fl value, but higher input
levels (LI = L2 >i 80 dB SL), however, both distortion products are approximately equal
in magnitude. For [2/ft = I A I , the relative level of L ( f 2 - f t ) tends to be about 5 to 10
dB below that of L(2fl - f2).
DISCUSSION
The present data confirm the results of previous investigations which demonstrated the
influence of f2/f~ on the slopes of the functions relating distortion product level to input
level [ 12,23]. This influence of f2/ft on the obtained slope values was demonstrated most
conclusively by Humes [12] for L(f2 - f l ) . It was indicated i,. that study, moreover, that
such findings offered a cohesive explanation for the wide range of slope values observed
for L(f2 - f ~ ) b y earlier investigators. In addition, it was demonstrated in that sludy that
f'z/fl alto exerted an influence on the slopes of the functions relating L(f2 - fl) to varia-
120
tions in either L t or Lz. The change in slope, moreover, was cGnsistent with that observed
for Lt = L:. Specifically, for small f2/ft, slopes of appreximately 1.0 dB/dB were
obtained for Lt <~ L2 when LI was varied and Lz fixed (i.e., Lv = Lt and Lf= L2) and
L2 ~<Ll for variations in Lz (Lv = Lz; Lf = L l ) while slopes of 0.0 to -0.4 dB/dB resulted
for Lt > Lz (Lv = Ll) or L2 > Ll (Lv = L2). This behavior is consistent with the predictions of either the normalized power-sei'ies nonlinearity or the compressive p-law nonlinearity [1,2,4,22,23]. At wider frequency separations, of the two-tone input (f2/f~ =
i.41), on the other hand, slopes of roughly 1.0 dB/dB were obtained for Lv ~ L f and
Lv > Lf for Lv = L~ or Lz. Such behavior is consistent with predictions made by the
classical power-series nonlinearity. Thus, the present data on the effect of fz/ft on the
growth of L(J~ -/'~) provide an independent confirmation of these earlier findings.
The results of the present study for L(2ft - f 2 ) are also consistent with the limited
data available that have been obtained under comparable measuring conditions (i.e.,
/'2/fl > 1.3 and L~ = L2 > 65 dB SL). As mentioned previously, Smoorenburg [23] has
reported data which suggest a similar dependence of slope value on f2/f~. That is, for
/,./ft = 1.11 a slope of approximately 0.6 to 0.8 dB/dB was oi~served in one subject for
the function relating L(2ft - 1 2 ) to Ll = Lz. For f2/ft = 1.41 and Li = L2 >t 70 dB SL,
t~n the other hand, a slope of approximately 3.0 dB/dB was observed in this same subject.
More recently, Weber and Mellert [24] performed a detailed s~udy of the dependence
of L(2fl - f 2 ) on Lt = L2 for various values of f2/fl. Input levels, however, did not
exceed a value of roughly 60 to 65 dB SL (75 dB SPL at 2000 Hz) nor was f2[fl varied
beyond 1.33. Nonetheless, their data for f2//'l = 1.33 and LI = L2 t> 65 dB SPL are net
inconsistent with a slope of approximately 3.0 dB/dB (in particular, their Fig. 7).
The foregoing discussion of the effect of f2[ft on the growth of L ( 2 f t - f 2 ) for
increase in input level (Ll = L2)observed by previous investigators has been somewhat
oversimplified. Although a linear function relating L(2.fl - fz) to LI = L2 usually obtains
for/~ tf~ ~ 1.15, comparable data oblained for f2/fl > 1.15 are characterized typically
by a r~onmonotonic dependence of L(2ft -/'2) on LI = Lz. This nonmonotonic behavior
is us~mlly such that cubic difference to~e level grows monotonically with Lt = L2 up to
moderate intensities, then decreases with further incr~,ments in Lt = L2, only to increase
once again with continued increase in input level. Tht:s, the function relating L(2/'t -/'~)
to L t = L2 has a 'notched' appearance at some f2/f~ Jalues with the minimum occurring
at moderate ir~tensities. The exact Ll = L2 value at w~ich the minimum occurs varies
somewhat with [2/.fi [24]. As indicated :~bove, the ;lope of the function on the upper
side of the minimtun, however, is typically in the order of 3.0 dB/dB for l'2[ft ~ 1.33.
Although not entirely understood, the nonmonotonic behavior of L(2ft - f 2 ) has been
explained by two differing hypotheses. One hypothesis suggests that the minimum
represents the interference of a 'backward-traveling' or 'reflected' intracochlear distortion product ~:20,241, while the second hypothesis suggests that L(2ft -/'2) represents
the vectorial ~;um of two separate distortion processes [24]. The latter hypothesis is
certainly consistent with the two differing slope estimates ob.,~rved in this study for small
and large .fJfl.
In summary, the dependence of L(fz - ft) and L(2ft - I"2) on L~ = L2 ob,~erved in this
investigation, and the effect of fz/I't on t~s dependence, are consistent with the findings
121
of previous investigators. For wide frequ~,~ncy separations of the two-tone input (f2/f~
1.41), the classical power-series nonlinearity appears to best describe the dependence of
L(fz-]'1) and L(2ft-fz) on LI = L2. For more narrow separations of the two-tone
input (fz/fl <~ 1.08), the behavior of both distortion products is more in line with the
predictions made by the normalized power-series or compressive p-law nonlinearity. Thus,
some combimttion of nonlinearities must be invoked to account for all the features of the
psychophysical data on two-tone aural nonlinearity [11 ]. It should be noted, however,
that evidence for the operation of the ch~ssieal power-series nonlinearity at large f2/fl has
not been obtained for f t < 750 Hz, despite the investigation af f2/fl values as large as
1.61 [13]. Hence, generalization of the present findings to low (<750 Hz) or high frequencies ( > 2 0 0 0 Hz) cannot be made at this time.
ACKNOWLEDGEMENTS
The author would like to express his gratitude to Dr. Fred Wighttnan for graciously
making his laboratory facilities availat le for the conduct of this experiment. This resear:h
was conducted while the author was a predoctoral University Fellow in the Audiology
PIogram, Northwester, University, Evanston, I11.. Thanks are expressed to Barb Coulson
fcr typing the manuscript.
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