the mathematics of eigenvalue optimization optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge such problems arise often in theory and practice particularly in engineering design and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory this essay presents a personal choice of some central mathematical ideas outlined for the broad optimization community i discuss the convex analysis of spectral functions and invariant matrix norms touching briefly on semidefinite representability and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and lie algebra analogous nonconvex notions lead into eigenvalue perturbation theory the last third of the article concerns stability for polynomials matrices and associated dynamical systems ending with a section on robustness the powerful and elegant language of nonsmooth analysis appears throughout as a unifying narrative thread

k dimensional regularity classifications for s fractals we study subsets e of ri which are hs measurable and have 0 h8 e oo where h is the s dimensional hausdorff measure given an integer k s k n we consider six s k regularity definitions for e in terms of k dimensional subspaces or surfaces of r if s k they all agree with the hk k rectifiability in the sense of federer but in the case s k we show that only two of them are equivalent we also study sets with positive lower density and projection properties in connection with these regularity definitions

local entropy averages and projections of fractal measures we show that for families of measures on euclidean space which satisfy an ergodic theoretic form of self similarity under the operation of re scaling the dimension of linear images of the measure behaves in a semi continuous way we apply this to prove the following conjecture of furstenberg let m n be integers which are not powers of the same integer and let x y be closed subsets of the unit interval which are invariant respectively under times m mod 1 and times n mod 1 then for any non zero t dim x ty min 1 dim x dim y a similar result holds for invariant measures and gives a simple proof of the rudolph johnson theorem our methods also apply to many other classes of conformal fractals and measures as another application we extend and unify results of peres shmerkin and nazarov and of moreira concerning projections of products self similar measures and gibbs measures on regular cantor sets we show that under natural irreducibility assumptions on the maps in the ifs the image measure has the maximal possible dimension under any linear projection other than the coordinate projections we also present applications to bernoulli convolutions and to the images of fractal measures under differentiable maps

effect of level on spectral ripple detection threshold for listeners with normal hearing and hearing loss this study investigated the effect of presentation level on spectral ripple detection for listeners with and without sensorineural hearing loss snhl participants were 25 listeners with normal hearing and 25 listeners with snhl spectral ripple detection thresholds srdts were estimated at three spectral densities 0 5 2 and 4 ripples per octave rpo and three to four sensation levels sls 10 20 40 and when possible 60 db sl each participant was also tested at 90 db sound pressure level spl results indicate that level affected srdts however the effect of level depended on ripple density and hearing status for all listeners and all rpo conditions srdts improved from 10 to 40 db sl in the 2 and 4 rpo conditions srdts became poorer from the 40 db sl to the 90 db spl condition the results suggest that audibility likely controls spectral ripple detection at low sls for all ripple densities whereas spectral resolution likely controls spectral ripple detection at high sls and ripple densities for optimal ripple detection across all listeners clinicians and researchers should use a sl of 40 db sl to avoid absolute level confounds a presentation level of 80 db spl can also be used

marstrand s theorem revisited projecting sets of dimension zero abstract we establish a refinement of marstrand s projection theorem for hausdorff dimension functions finer than the usual power functions including an analogue of marstrand s theorem for logarithmic hausdorff dimension

relations between auditory functions in impaired hearing relations between auditory functions as expressed by coefficients of correlation were studied for a group of 22 sensorineurally hearing impaired subjects with moderate losses in addition to the audiogram we measured frequency resolution temporal resolution and speech reception in quiet and in noise frequency resolution was derived from masking with comb filtered noise and from the psychophysical tuning curve for both paradigms in simultaneous and in nonsimultaneous masking the critical ratio was also determined temporal resolution was determined with intensity modulated noise and from backward and forward masking all tests were performed at 1000 hz correlations among tests were gathered in a matrix and subjected to a principal components analysis it turned out that tests on frequency resolution form a cluster and are approximately independent of audiometric loss furthermore hearing loss for speech in noise is closely allied to frequency resolution whereas hearing loss for speech in quiet is governed by audiometric loss

effective partitioning method for computing generalized inverses and their gradients abstract we extend the algorithm for computing 1 1 3 1 4 inverses and their gradients from 11 to the set of multiple variable rational and polynomial matrices an improvement of this extension appropriate to sparse polynomial matrices with relatively small number of nonzero coefficient matrices as well as in the case when the nonzero coefficient matrices are sparse is introduced for that purpose we exploit two effective structures form 6 which make use of only nonzero addends in polynomial matrices and define their partial derivatives symbolic computational package mathematica is used in the implementation several randomly generated test matrices are tested and the cpu times required by two used effective structures are compared and discussed

computation of the generalized inverse of a polynomial matrix and applications abstract the computation of the generalized inverse of a constant matrix is utilized in finding the generalized inverse and its laurent expansion of a nonregular polynomial matrix the proposed algorithm constitutes a generalization of the algorithm proposed by fragulis et al for regular polynomial matrices and gives rise to numerous applications in multivariable system analysis

a unitary model of pitch perception a model of the mechanism of residue pitch perception is revisited it is evaluated in the context of some new empirical results and it is proposed that the model is able to reconcile a number of differing approaches in the history of theories of pitch perception the model consists of four sequential processing stages peripheral frequency selectivity within channel half wave rectification and low pass filtering within channel periodicity extraction and cross channel aggregation of the output the pitch percept is represented by the aggregated periodicity function using autocorrelation as the periodicity extraction method and the summary autocorrelation function sacf as the method for representing pitch information it is shown that the model can simulate new experimental results that show how the quality of the pitch percept is influenced by the resolvability of the harmonic components of the stimulus complex these include i the pitch of harmonic stimuli whose components alternate in phase ii the increased frequency difference limen of tones consisting of higher harmonics and iii the influence of a mistuned harmonic on the pitch of the complex as a function of its harmonic number to accommodate these paradigms it was necessary to compare stimuli along the length of the sacf rather than relying upon the highest peak alone these new results demonstrate that the model responds differently to complexes consisting of low and high harmonics as a consequence it is not necessary to postulate two separate mechanisms to explain different pitch percepts associated with resolved and unresolved harmonics

a unified approach for the recursive determination of generalized inverses abstract using a unified approach simple derivations for the recursive determination of different types of generalized inverses of a matrix are presented these include results for the generalized inverse the least squares generalized inverse the minimum norm generalized inverse and the moore penrose inverse of a matrix

models of auditory masking a molecular psychophysical approach gilkey et al j acoust soc am 78 1207 1219 1985 measured hit proportions and false alarm proportions for detecting a 500 hz tone at each of four starting phase angles in each of 25 reproducible noise samples in the present paper their results are modeled by fitting the general form of the electrical analog model of jeffress j acoust soc am 48 480 488 1967 to the diotic data the best fitting configurations of this model do not correspond to energy detectors or to envelope detectors a detector composed of a 50 hz wide single tuned filter followed by a half wave rectifier and an integrator with an integration time of 100 to 200 ms fits the data of all four subjects relatively well linear combinations of the outputs of several detectors that differ in center frequency or integration window provide even better fits to the data these linear combinations assign negative weights to some frequencies or to some time intervals suggesting that a subject s decision is based on a comparison of

observer efficiency and weights in a multiple observation task a sequence of seven tones sampled from one of two distributions differing in mean frequency is presented to observers who try to report which distribution was sampled estimates are obtained of the weight or importance given to each tone as a function of its temporal position in five experiments the reliability of the information is varied by changing the variance of the distributions tones with high reliability are sampled from distributions with relatively small variance whereas tones with low reliability are sampled from distributions with relatively large variance results show that the observations are weighted more efficiently when the tones have equal rather than unequal reliability and when the most reliable tones have the greater intensity additional results show that the most intense tones often receive the greatest weight even when those tones have the least reliability

perception of temporal patterns defined by tonal sequences this experiment tested how listeners discriminate between the temporal patterns defined by two sequences of tones two arrhythmic sequences of n tones were played successively n 8 12 or 16 tone duration 35 ms frequency 1000 hz and the listener reported whether the sequences had the same or different temporal patterns in the first sequence the durations of the intertone gaps were chosen at random in the second sequence the gaps were either a the same as the first sequence or b chosen at random discrimination performance increased with the variability of the gap sequences and decreased with the size of the correlation between the sequences a discrimination model based on computation of the sample correlation between the sequences of gaps but limited by an internal variability of approximately 15 ms described observer performance in a variety of conditions

informational processing of complex sound ii cross dimensional analysis a series of experiments investigated listeners ability simultaneously to process information across different acoustic dimensions on each trial the listener heard a pair of brief n tone sequences n 1 to 12 the frequency intensity and duration of each tone in the sequence varied randomly from trial to trial on average the values of these three parameters were greater for one sequence the target than the other the nontarget the listener s task was to identify the target on each trial for an ideal observer in this task d performance grows as the square root n obtained d grew at a rate slightly less than the square root of n close to cube root of n growth was observed when the average difference occurred in only one of the three tone parameter values within a block of trials although performance fell short of ideal optimum weights were consistently given to each tone and each parameter the results are consistent with a model in which performance depends predominantly on the information content of the sounds regardless of how the information is packaged in the stimulus transmitted information is estimated to be 0 9 2 0 bits within a single acoustic dimension 2 1 3 0 bits when distributed across dimensions

an autocorrelation model with place dependence to account for the effect of harmonic number on fundamental frequency discrimination fundamental frequency f0 difference limens dls were measured as a function of f0 for sine and random phase harmonic complexes bandpass filtered with 3 db cutoff frequencies of 2 5 and 3 5 khz low region or 5 and 7 khz high region and presented at an average 15 db sensation level approximately 48 db spl per component in a wideband background noise fundamental frequencies ranged from 50 to 300 hz and 100 to 600 hz in the low and high spectral regions respectively in each spectral region f0 dls improved dramatically with increasing f0 as approximately the tenth harmonic appeared in the passband generally f0 dls for complexes with similar harmonic numbers were similar in the two spectral regions the dependence of f0 discrimination on harmonic number presents a significant challenge to autocorrelation ac models of pitch in which predictions generally depend more on spectral region than harmonic number a modification involving a lag window is proposed and tested restricting the ac repr

factors affecting masking release for speech in modulated noise for normal hearing and hearing impaired listeners the speech reception threshold for sentences in stationary noise and in several amplitude modulated noises was measured for 8 normal hearing listeners 29 sensorineural hearing impaired listeners and 16 normal hearing listeners with simulated hearing loss this approach makes it possible to determine whether the reduced benefit from masker modulations as often observed for hearing impaired listeners is due to a loss of signal audibility or due to suprathreshold deficits such as reduced spectral and temporal resolution which were measured in four separate psychophysical tasks results show that the reduced masking release can only partly be accounted for by reduced audibility and that when considering suprathreshold deficits the normal effects associated with a raised presentation level should be taken into account in this perspective reduced spectral resolution does not appear to qualify as an actual suprathreshold deficit while reduced temporal resolution does temporal resolution and age are shown to be the main factors governing masking release for speech in modulated noise accounting for more than half of the intersubject variance their influence appears to be related to the processing of mainly the higher stimulus frequencies results based on calculations of the speech intelligibility index in modulated noise confirm these conclusions

the case of the missing delay lines synthetic delays obtained by cross channel phase interaction temporal models of pitch and harmonic segregation call for delays of up to 30ms to cover the full range of existence of musical pitch to date there is little anatomical or physiological evidence for delays that long we propose a mechanism by which delays may be synthesized from cross channel phase interaction phases of adjacent cochlear filter channels are shifted by an amount proportional to frequency and then combined as a weighted sum to approximate a delay synthetic delays may be used by pitch perception models such as autocorrelation segregation models such as harmonic cancellation and binaural processing models to explain sensitivity to large interaural delays the maximum duration of synthetic delays is limited by the duration of the impulse responses of cochlear filters itself inversely proportional to cochlear filter bandwidth maximum delay is thus frequency dependent this may explain the fact puzzling for temporal pitch models such as autocorrelation that pitch is more salient and eas

symbolic computation of weighted moore penrose inverse using partitioning method abstract we propose a method and algorithm for computing the weighted moore penrose inverse of one variable rational matrices continuing this idea we develop an algorithm for computing the weighted moore penrose inverse of one variable polynomial matrix these methods and algorithms are generalizations of the method for computing the weighted moore penrose inverse for constant matrices originated in wang and chen g r wang y l chen a recursive algorithm for computing the weighted moore penrose inverse a mn j comput math 4 1986 74 85 and the partitioning method for computing the moore penrose inverse of rational and polynomial matrices introduced in stanimirovic and tasic p s stanimirovic m b tasic partitioning method for rational and polynomial matrices appl math comput 155 2004 137 163 algorithms are implemented in the symbolic computational package mathematica

a comparison of threshold estimation methods in children 6 11 years of age estimating detection threshold for auditory stimuli in children can be problematic because of lapses in attention and the time limits usually imposed by scheduling restrictions or fatigue data reported here were collected to compare the stability of threshold estimation procedures in testing children ages 6 to 11 in a three alternative forced choice paradigm stimuli consisted of a 1 khz tonal signal and a gaussian noise masker bandpass filtered between 500 2000 hz and presented at 25 db spectrum level the signal was either presented for 400 ms in the presence of a continuous masker simultaneous masking or for 10 ms just prior to a 400 ms masker backward masking for each masking paradigm the 79 correct threshold was assessed via each of three procedures 3 down 1 up adaptive staircase levitt maximum likelihood estimation mle and method of constant stimuli percent correct was measured at the end of the study for a signal 10 db above the previously determined threshold in order to estimate the most appropriate psychometric function asymptote for fitting data collected via the method of constant stimuli both the mle and levitt procedures produced equally stable threshold estimates for both conditions and age groups this was the case despite considerable variability in backward masking thresholds

computation of the inverse of a polynomial matrix and evaluation of its laurent expansion abstract an algorithm is given which constitutes a generalization of the algorithm for the inversion of a pencil se a due to mertzios 1984 in the case of general systems described in differential operator form recursive formulae are obtained for the calculation of both the coefficient matrices of the adjoint of the polynomial matrix as well as for the coefficients of the characteristic polynomial of the polynomial matrix a simple method is also presented that allows the evaluation of the laurent expansion for the inverse of a polynomial matrix of any order specifically an arma model is given for the computation of the coefficient matrices of the laurent expansion in terms of the coefficient matrices of the polynomial matrix without inverting the latter the laurent expansion of a polynomial matrix is used for the analysis and synthesis of the polynomial matrix descriptions that constitute a further generalization of the singular or generalized systems

a molecular description of profile analysis decision weights and internal noise systematic inefficiencies and internal noise in a spectral profile discrimination task were investigated listeners detected a 1000 hz sinusoid added in phase to the central component of a complex consisting of 11 equal intensity sinusoids parameters for a channel model that employs decision weights and internal noise were estimated with molecular psychophysical techniques maximum likelihood predictions of the model were generally within a few decibels of observed thresholds the degree to which an assumption of ideal weights leads to overestimation of internal noise was also assessed

moderate cochlear hearing loss leads to a reduced ability to use temporal fine structure information the ability of normally hearing and hearing impaired subjects to use temporal fine structure information in complex tones was measured subjects were required to discriminate a harmonic complex tone from a tone in which all components were shifted upwards by the same amount in hz in a three alternative forced choice task the tones either contained five equal amplitude components non shaped stimuli or contained many components but were passed through a fixed bandpass filter to reduce excitation pattern changes shaped stimuli components were centered at nominal harmonic numbers n 7 11 and 18 for the shaped stimuli hearing impaired subjects performed much more poorly than normally hearing subjects with most of the former scoring no better than chance when n 11 or 18 suggesting that they could not access the temporal fine structure information performance for the hearing impaired subjects was significantly improved for the non shaped stimuli presumably because they could benefit from spectral cues it is proposed that normal hearing subjects can use temporal fine structure information provided the spacing between fine structure peaks is not too small relative to the envelope period but subjects with moderate cochlear hearing loss make little use of temporal fine structure information for unresolved components

detection of simple and complex changes of spectral shape in most of the previous studies see green 1987 concerning the detection of a change in spectral shape or profile analysis the listener s task was to detect an increment to a single component of an otherwise equal amplitude multicomponent background an important theoretical issue is whether listeners sensitivity to more complex spectral changes can be predicted from these results in the present investigation the sensitivity of a single group of listeners to a wide variety of simple and complex spectral changes was determined after collecting the data it was noted that almost all the thresholds could be predicted by a simple calculation scheme that assumed detection of a change in spectral shape occurs when the addition of the signal to the flat multicomponent background produces a sufficient difference in level between only two regions of the spectrum unfortunately this scheme while successful for our limited set of data fails to account for other profile data namely those obtain

level dependence of auditory filters in nonsimultaneous masking as a function of frequency auditory filter bandwidths were measured using nonsimultaneous masking as a function of signal level between 10 and 35db sl for signal frequencies of 1 2 4 and 6khz the brief sinusoidal signal was presented in a temporal gap within a spectrally notched noise two groups of normal hearing subjects were tested one using a fixed masker level and adaptively varying signal level the other using a fixed signal level and adaptively varying masker level in both cases auditory filters were derived by assuming a constant filter shape for a given signal level the filter parameters derived from the two paradigms were not significantly different at 1khz the equivalent rectangular bandwidth erb decreased as the signal level increased from 10to20db sl after which it remained roughly constant in contrast at 6khz the erb increased consistently with signal levels from 10to35db sl the results at 2 and 4khz were intermediate showing no consistent change in erb with signal level overall the results sugge

symbolic and recursive computation of different types of generalized inverses abstract we propose a method and algorithm for recursive computation of different classes of generalized inverses of a given one variable rational matrix and corresponding algorithm for polynomial matrix these methods and algorithms are generalizations of the method for computing the generalized inverses for constant matrices originated in f e udwadia r e kalaba a unified approach for the recursive determination of generalized inverses comp math appl 37 1999 125 130 and the partitioning method for computing the generalized inverses of rational and polynomial matrices introduced in p s stanimirovic m b tasic partitioning method for rational and polynomial matrices appl math comput 155 2004 137 163 m b tasic p s stanimirovic m d petkovic symbolic computation of weighted moore penrose inverse using partitioning method appl math comput 189 2007 615 640 algorithms are implemented in the symbolic computational package mathematica

spectral weights in profile listening the coss analysis suggested by berg b g berg j acoust soc am 86 1743 1746 1989 is applied to a profile listening task the listener s task is to detect an increment in the level of the middle component of an n component spectrum the overall level of the components is randomly selected from a 20 db range on each presentation thus the detection task is essentially one of detecting a change in spectral shape to implement the coss analysis a small perturbation in level is added to each component of the complex coss functions are generated from these perturbations and the spectral weight that the listener assigns to each component is estimated data are reported for n 3 5 and 11 components and for perturbations with standard deviations of 0 5 1 and 2 db the estimated weights are similar to those derived for an optimum detector namely the level at the signal component is compared with the average level of the nonsignal components this result supports the view that profile analysis invol

partitioning method for rational and polynomial matrices we propose an extension of the grevile s partitioning method for computing the moore penrose inverse which is applicable to the set of rational matrices also we develop an algorithm for computing the moore penrose inverse of given one variable polynomial matrix which is based on the grevile s method major problems arising in the implementation of this method are repetitive recomputations of the same values and simplification of rational and polynomial expressions which contain unknown variable these algorithms are implemented in the symbolic computational package mathematica

sensitivity to changes in overall level and spectral shape an evaluation of a channel model two experiments involving level and spectral shape discrimination which test an optimal channel model developed by durlach et al j acoust soc am 80 63 72 1986 are described the model specifies how the auditory system compares and or combines intensity information in different frequency channels in the first experiment psychometric functions were obtained for the discrimination of changes in level and discrimination of changes in spectral shape for an eight tone complex sound a variety of different base spectral shapes were tested in some conditions level randomization was introduced to reduce the reliability of across interval changes in level increasing the amount of level variation degraded performance for the level discrimination task but had no effect on the shape discrimination task in all conditions sensitivity to changes in spectral shape was superior to sensitivity to changes in level consequently two models of central noise are evaluated in an attempt to explain these results one in which central noise acts prior to the formation of the likelihood ratio and one in which central noise degrades the likelihood ratio the former model is more successful in accounting for the data in a second experiment the detectability of a level increment to one component of a multitone complex was measured the frequency content of the complex was varied by systematically removing six components from a 23 component complex thresholds were measured for increments at three different signal frequencies a common trend in the data was that when there was a spectral gap directly above the signal frequency thresholds were lowest this result differs from the predictions of a simple channel model and contrasts with results presented by green and berg q j exp psychol 43a 449 458 1991

the role of excitation pattern temporal fine structure and envelope cues in the discrimination of complex tones the discrimination of bandpass filtered harmonic h from inharmonic i tones produced by shifting all components of the h tones upwards by a fixed amount in hz could be based on shifts in the pattern of ripples in the excitation pattern ep or on changes in the temporal fine structure evoked by the tones the predictions of two computational ep models were compared with measured performance one model used auditory filters with bandwidth values specified by glasberg and moore 1990 hear res 47 103 138 and one used filters that were twice as sharp stimulus variables were passband width fundamental frequency harmonic rank n of the lowest component within the passband component phase cosine or random signal to noise ratio snr and random perturbation in level of each component in the tones while the ep models correctly predicted the lack of an effect of phase and some of the trends in the data as a function of fundamental frequency and n neither model predicted the worsening in perfo

effective partitioning method for computing weighted moore penrose inverse we introduce a method and an algorithm for computing the weighted moore penrose inverse of multiple variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices these methods and algorithms are generalizations of algorithms developed in m b tasic p s stanimirovic m d petkovic symbolic computation of weighted moore penrose inverse using partitioning method appl math comput 189 2007 615 640 to multiple variable rational and polynomial matrices and improvements of these algorithms on sparse matrices also these methods are generalizations of the partitioning method for computing the moore penrose inverse of rational and polynomial matrices introduced in p s stanimirovic m b tasic partitioning method for rational and polynomial matrices appl math comput 155 2004 137 163 m d petkovic p s stanimirovic symbolic computation of the moore penrose inverse using partitioning method internat j comput math 82 2005 355 367 to the case of weighted moore penrose inverse algorithms are implemented in the symbolic computational package mathematica

signal frequency uncertainty in spectral shape discrimination psychometric functions psychometric functions were measured for spectral shape discrimination profile analysis with the signal frequency either fixed what we call the signal known condition or randomly varied what we call the signal unknown condition the functions were obtained using an adaptive up down procedure in the signal unknown condition independent tracks for each signal frequency were interleaved within the same block of trials the mean slope of the psychometric function k in log d k log delta l c where delta l 20 log 1 delta p p was 1 13 for the signal known conditions and 1 19 for the signal unknown condition the mean signal to standard ratio at threshold pc 79 4 obtained in the signal unknown condition was 3 to 4 db higher than that obtained in the signal known conditions the psychometric functions of the ideal observer were derived for both the signal known and signal unknown conditions a comparison of the measured and derived psychometric functions suggests that the internal noise component that is statistically independent across frequency channels contributes minimally to the total internal and external noise sources that are involved in the decision process

informational processing of complex sound i intensity discrimination this paper reports on some initial experiments using the sample discrimination paradigm to investigate normal hearing listeners ability to process information in complex nonspeech sounds an important feature of the sample discrimination experiment is that the value of the d i f f e r e n c e to be discriminated randomly varies from trial to trial it is this variation that yields potential information in the present study listeners heard a pair of multitone complexes or sequences on each trial the individual levels of the tones were drawn from two normal distributions differing only in mean the listener s task was to identify the sound having the higher mean tone level for an ideal observer in these experiments performance in d grows as the square root n where n is the number of tones obtained d grew more nearly as the cube root of n regardless of whether the tones were played sequentially or simultaneously or whether they were increased in number from high frequencies to low or from low frequencies to high a preliminary model is proposed in which discrimination performance depends predominantly on the information content of the sounds and is largely independent of the physical dimensions along which the sounds vary information content is defined in terms of the variance of the underlying stimulus distributions and a stimulus equivocation factor that is derived from the data based on this model transmitted information is estimated to be between 1 0 and 2 6 bits

multiple observations and internal noise a multiple observation task was used to evaluate models of internal noise on each trial n tones n 1 2 3 4 6 8 10 or 12 within a block were independently sampled from one of two probability density functions on frequency each was normal in form with a standard deviation of 100 hz the two distributions had means of 1000 and 1100 hz respectively the sample was presented through headphones as n 50 ms tone bursts separated by 50 ms the subjects task was to decide from which of the two distributions the tones were sampled a model partitioning internal variance into peripheral variance added to each observation prior to formulating a decision statistic and central variance added to the decision statistic provided a good description of the data of all four subjects a subsequent study counterbalanced two levels of the difference between distribution means 100 or 150 hz with two levels of the standard deviation 100 or 150 hz increasing did not affect estimates of internal variance however increasing resulted in increased estimates of internal variance for both levels of one interpretation of this result is that external and internal variance are not independent work supported by afosr and nmrdc

spectral and temporal weights in spectral shape discrimination the coss analysis b g berg j acoust soc am 86 1743 1746 1989 was used to estimate spectral and temporal weights of a three component amplitude modulated stimulus in a spectral shape discrimination task in all experiments the task of the observer was to detect an increment in the level of the center component a spectral temporal weight quantifies the relative influence of a spectral component on the decisions of an observer during a specified segment of the total stimulus duration in the first two experiments the signal was added to all three temporal segments of the center component the ideal weights for each component should be the same across temporal segments spectral temporal weights were obtained for four conditions with different stimulus durations in general the estimated weights for each component were not equal at different temporal segments in the third experiment the signal was added to only one of three segments of the center component ideally weight patterns should have changed when the temporal position of the signal segment was altered two stimulus durations 300 and 15 ms were used for the 300 ms condition the signal was added to only the end segment and for all three observers the weight patterns are different from that obtained in experiment 1 with the signal added to all segments for the 15 ms conditions altering the signal position changed the estimated weights for only one observer

perception of the low pitch of frequency shifted complexes when all of the components in a harmonic complex tone are shifted in frequency by f the pitch of the complex shifts roughly in proportion to f for tones with a small number of components the shift is usually somewhat larger than predicted from pitch theories which has been attributed to the influence of combination tones smoorenburg j acoust soc am 48 924 941 1970 experiment 1 assessed whether combination tones influence the pitch of complex tones with more than five harmonics by using noise to mask the combination tones the matching stimulus was a harmonic complex test complexes were bandpass filtered with passbands centered on harmonic numbers 5 resolved 11 intermediate or 16 unresolved and fundamental frequencies f0s were 100 200 or 400 hz for the intermediate and unresolved conditions the matching stimuli were filtered with the same passband to minimize differences in the excitation patterns of the test and matching stimuli for the resolved condition the matching st

differentiation of generalized inverses for rational and polynomial matrices our basic motivation is a direct method for computing the gradient of the pseudo inverse of well conditioned system with respect to a scalar proposed in 13 by layton in the present paper we combine the layton s method together with the representation of the moore penrose inverse of one variable polynomial matrix from 24 and developed an algorithm for computing the gradient of the moore penrose inverse for one variable polynomial matrix moreover using the representation of various types of pseudo inverses from 26 based on the grevile s partitioning method we derive more general algorithms for computing 1 1 3 and 1 4 inverses of one variable rational and polynomial matrices introduced algorithms are implemented in the programming language mathematica illustrative examples on analytical matrices are presented

decision rules of listeners in spectral shape discrimination with or without signal frequency uncertainty a correlation technique was used to assess the decision rules of three listeners in two cases of spectral shape discrimination tasks in one case the signal frequency was fixed and in the other it was randomly varied within each block of trials in order to estimate the decision rule of the listeners the experimenter superimposed random level perturbations on each frequency component upon each stimulus presentation over many trials correlation coefficients were computed between the random perturbations and the binary responses of the listeners and were expressed as a function of signal level for both the fixed and random signal cases the measured correlation functions were in reasonably good agreement with those predicted based on the likelihood ratio decision rules thus the listeners appeared to use information nearly optimally in discriminating spectral shapes this investigation demonstrated that the correlation technique can be used to reveal the decision rules for cases where the decision statistics are nonlinear functions of the observations

informational processing of complex sound iii interference in this study theoretical results from information theory and detection theory are applied to provide a formal analysis of the interaction of target and context uncertainty on the discrimination of brief multitone sequences the experiments employ a sample discrimination task in which the performance of an ideal observer is held constant while the relative variability of the target t and context c is varied listener performance in these experiments was less than ideal but increased monotonically with t c whether the task was to discriminate the frequency or the intensity of the target performance was also constant for a constant t c regardless of the target s position in the sequence the results are consistent with a class of models in which the decision variable is a weighted sum of the tone values comprising the sequence weighting functions computed from the trial by trial data suggest that limits in performance result from trial by trial variability in the weights the decision variable also

pitch discrimination of diotic and dichotic tone complexes harmonic resolvability or harmonic number three experiments investigated the relationship between harmonic number harmonic resolvability and the perception of harmonic complexes complexes with successive equal amplitude sine or random phase harmonic components of a 100 or 200 hz fundamental frequency f0 were presented dichotically with even and odd components to opposite ears or diotically with all harmonics presented to both ears experiment 1 measured performance in discriminating a 3 5 5 frequency difference between a component of a harmonic complex and a pure tone in isolation listeners achieved at least 75 correct for approximately the first 10 and 20 individual harmonics in the diotic and dichotic conditions respectively verifying that only processes before the binaural combination of information limit frequency selectivity experiment 2 measured fundamental frequency difference limens f0 dls as a function of the average lowest harmonic number similar results at both f0 s provide further evidence that harmonic number not absolute frequency underlies the order of magnitude increase observed in f0 dls when only harmonics above about the 10th are presented similar results under diotic and dichotic conditions indicate that the auditory system in performing f0 discrimination is unable to utilize the additional peripherally resolved harmonics in the dichotic case in experiment 3 dichotic complexes containing harmonics below the 12th or only above the 15th elicited pitches of the f0 and twice the f0 respectively together experiments 2 and 3 suggest that harmonic number regardless of peripheral resolvability governs the transition between two different pitch percepts one based on the frequencies of individual resolved harmonics and the other based on the periodicity of the temporal envelope