Perceptual audio compression based on loudness uncertainty

Abstract: A new technique for the determination of the masking effect of an audio signal is employed to provide transparent compression of an audio signal at greatly reduced bit rates. The new technique employs the results of recent research into the psycho-physics of noise masking in the human auditory system. This research suggests that noise masking is a function of the uncertainty in loudness as perceived by the brain. Measures of loudness uncertainty are employed to determine the degree to which audio signals are "tone-like" (or "noise-like"). The degree of tone-likeness, referred to as "tonality," is used to determine masking thresholds for use in the compression of audio signals. Tonality, computed in accordance with the present invention, is used in conventional and new arrangements to achieve compression of audio signals.

Description: This invention relates to methods of and apparatus for encoding an audio signal. Perceptual coding of audio signals involves the concept of "perceptual masking." Perceptual masking refers to a psycho-acoustic effect where a listener cannot hear an otherwise audible sound because that sound is presented to the listener in the presence of another sound (referred to as the "masking signal"). This psycho-acoustic effect has been employed to advantage in several audio compression systems which treat the audio signal - the signal to be compressed - as the masking signal and coding (or quantizer) noise as the signal to be masked. These systems seek to quantize the audio signal with a quantizer stepsize which is as large as possible without introducing audible quantization noise in the audio signal. Naturally, the level of quantization noise which may be introduced without audible effect will be a function of how well a particular audio signal -- the masker -- serves to supply a masking effect. The greater the masking ability of the audio signal, the coarser the quantization may be without introducing audible noise. The coarser the quantization, the lower the bit-rate of the compressed signaL The ability of an audio signal to mask noise in the past has been linked to how tone-like (or, conversely, noise-like) the audio signal is. A given audio signal may fall anywhere along a continuum from "pure tone" to "pure noise." However, audio signals which are more noise-like have been empirically determined to be better at masking quantization noise than audio signals which are more tone-like in comparison. Accordingly, measures of tone-likeness - referred to as "tonality" -- and noise-likeness -- referred to as "chaos" -- have been employed by audio compression systems as a basis of setting of quantizer step size. Examples of such systems include those described in U.S. Patent No. 5,040,217, by K. Brandenburg and J.D. Johnston; U.S. Patent No. 5,341,457, by J.L. Hall and J.D. Johnston; and U.S. Patent Applications Serial Nos. 07/844,804; 07/844,819; 07/844,811 (US-A-5285498; 5592584; 5627938) and U.S. Patent No. 5,227,788. As explained in these references, tonality (or chaos) is used to compute a perceptual threshold which in turn is used to compute the stepsize. Through such measures as tonality (and chaos), these systems have been able to reduce bit-rate without introducing substantial, if any, perceivable degradation of (or quantization noise in) the audio signal. In the past, systems such as those referenced above have computed the tonality measure with use of a tone prediction scheme. Tonality of an audio signal at a given point in time was computed based on how well the audio signal matched a predicted audio signal value at that time, the prediction being a function of past audio signal values. The predicted audio signal value is determined based on an assumption that the audio signal is a pure tone. If the predicted signal value matched the actual value of the signal, the assumption that the actual signal could be well represented by a tone model would be validated and a large value of tonality ( e.g. , one, on a normalized scale) would result If, on the other hand, the predicted signal value did not match the actual signal value very well -- a result which undercuts the original assumption that the signal could be well represented by a pure tone -- a comparatively small value of tonality would result. As such, the signal would be assigned a tonality metric value of less than one, with the exact value being dependent on the degree to which the actual signal value differed from the predicted value. (Chaos, on a normalized scale, is a measure which equals one minus the value of tonality). Although the concept of tonality (and chaos) has been used to advantage in determining quantizer stepsize, the concept is based on an observed effects on masking ability of different types of signals, not an understanding of how such effects are caused in the human auditory system as a result of exposure to such signal. IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, feb 1988, New York, Vol. 6, No. 2, p.314-323, JOHNSTON J.D.: "Transform Coding of Audio Signals Using Perceptual Noise Criteria" discloses a perceptual transform coder for encoding an audio signal. The spectrum of the audio signal is used as an input to quantizers in the coder and also for calculating a perceptual threshold. This threshold estimates the maximum noise that can be inaudibly inserted into the signal. The step sizes of the quantizers for frequency components of the audio signal are determined by a bit-rate calculation process, which maintains the shape of the perceptual threshold. The quantizer step sizes are calculated iteratively to provide a fixed bit-rate coder. Based on such step sizes, quantized frequency components are generated and then compressed for transmission, along with side information. The latter includes information concerning the quantizer step sizes and quantized spectral peak levels. According to one aspect of this invention there is provided a method as claimed in claim 1. According to another aspect of this invention there is provided apparatus as claimed in claim 15. The present invention provides method and apparatus for encoding an audio signal in accordance with a model of noise masking which is based on a measure of loudness uncertainty. Loudness uncertainty is central to a newly developed understanding of how noise masking effects are caused in the human auditory system. In an illustrative embodiment of the present invention, measures of loudness uncertainty are compared with a loudness differential between original and synthesized audio signals in a subband, analysis-by-synthesis framework. For a given coder band, a plurality of different (provisional) encodings (quantization stepsizes) are tried to determine an encoding which achieves a loudness differential which is less than (but not too far below) the loudness uncertainty. Such an encoding (stepsize) is the one deemed appropriate for the band. This stepsize may then be used to quantize the signal in that band. Brief Description of the Drawings Figure 1 presents an illustrative embodiment of the present invention. Figure 2 presents a loudness computation for use in the illustrative embodiment of Figure 1. Figure 3 presents a variable resolution analysis filterbank for use in the illustrative embodiment of Figure 1. Figure 4 presents a quantization processor for use in the illustrative embodiment of Figure 1. Detailed Description Introduction: Loudness Uncertainty in the Human Auditory System When sound is transduced by the cochlea of the human auditory system, it is turned into a neural code. The neural code is close to a non-homogeneous Poisson process. It is well known that for a Poisson point process, the variance of the process's random variable, here the number of spikes, is approximately equal to the mean. Assume that the number of spikes that result from a 50 ms tone burst of 1 kHz is N k . Each time the tone burst is presented to the ear, the number of spikes N k are generated, where k indexes the trial. For example, for the trial, we get N 1 spikes, and for the 10 th , we find N 10 spikes. The mean value of N k for all trials is L and the standard deviation of random variable, N k , is σ N . If the process giving rise to the spike count is Poisson, then σ N = L . Loudness of auditory signals is related to the total number of spikes summed over all of the nerve fibers. Research which has inspired the present invention has shown that the masked threshold is related to the uncertainty (or degree of variability) in loudness, and is directly proportional to σ N . Thus, such uncertainty provides a theory for calculating the masked threshold of signals. This theory first requires the calculation of the partial loudness, which is the loudness along the basilar membrane, and then the computation of the standard deviation of those partial loudnesses based on the Poisson model. The basic philosophy of the present invention is that distortions of the audio signal are only perceptible if they result in a detectable change in relative loudness in a particular frequency region. Relative loudness is the difference between the loudness of the original signal and a signal synthesized based on coded version of that signal (coded signal). This determination is advantageously made in each of the relevant cochlear (or critical) frequency bands associated with the human auditory system, to wit: 1. Passing the audio signal through a bandpass filter. This filter's shape is based on the shape of the human "critical band" filters but may be modified to achieve computational efficiency. 2. Passing the output of the filter through an instantaneous non-linearity. This non-linearity is based on the intensity versus loudness relationship. 3. Finally, the loudness of the signal is computed as a moving exponentially weighted average of the output of the non-linearity. The loudness calculation is performed on both the original audio signal and the coded (distorted) audio signal; a normalized difference between the original and coded partial loudness yields a measure of the perceptual distance between the loudness of the two signals over a frequency region. For each frequency region (filter), the difference between the original and coded partial loudness is divided by the square root of the original's loudness ( i.e. , the uncertainty of the original) to yield a detectability measure, d ' / k (i). The "uncertainty" in the loudness of a signal is related to the square root of its loudness (hence the square root in the denominator). With proper choice of parameters (principally, the α's in the partial loudness calculation), d ≥1 represents a distortion that is perceptually detectable whereas d < 1 represents a distortion that is not perceptible. The above method then outlines the basis for the creating a perceptual audio coder whose output is perceptually indistinguishable from the original audio signal. A coder that distorts the audio signal in a manner that maintains all d ' / k (i) < 1 will produce an audio signal that is perceptually indistinguishable from the original. However, if more coding gain (reduction in bit rate) is desired, a coder that minimizes the function max[ d ' / k (i)] for all k and i will produce the most perceptually transparent signal. An Illustrative Embodiment For clarity of explanation, the illustrative embodiment of the present invention is presented as comprising individual functional blocks (including functional blocks labeled as "processors"). The functions these blocks represent may be provided through the use of either shared or dedicated hardware, including, but not limited to, hardware capable of executing software. For example, the functions of processors presented in Figure 1 may be provided by a single shared processor. (Use of the term "processor" should not be construed to refer exclusively to hardware capable of executing software.) Illustrative embodiments may comprise digital signal processor (DSP) hardware, such as the AT&T DSP16 or DSP32C, read-only memory (ROM) for storing software performing the operations discussed below, and random access memory (RAM) for storing DSP results. Very large scale integration (VLSI) hardware embodiments, as well as custom VLSI circuitry in combination with a general purpose DSP circuit, may also be provided. An illustrative embodiment of the present invention operates in accordance with an analysis-by-synthesis subband framework to code an input audio signal, x ( i ). Signal x(i) is broken into frequency components (or "subbands") both for the purpose of perceptual modeling and to obtain a suitable representation for coding. A different subband framework is used for each of these purposes. For the purpose of perceptual modeling, signal x(i) is applied to a cochlear filterbank, while for the purpose of coding efficiency, signal x(i) is applied to a coding filterbank. In the coding filterbank, a segment of M samples of x(i) is broken into M subbands. For a given segment of signal x(i) ,1 ≤i≤M and a given subband k , 1 ≤k≤M , a provisional coding ( i.e. , quantization) is performed. The initial quantization step size is determined based on the loudness of the signal, L k . The provisional coding introduces quantization noise, n k . The provisionally coded audio signal is then used to synthesize an audio signal, x and k = x k + n k . This synthesized audio signal, x and k , having loudness L and k , is then used to determine a coding metric, referred to as the "detectability" metric. The detectability metric represents a comparison of the change in computed loudness, |Δ L k |=| L k - L and k |, as between original and synthesized audio signals, with the uncertainty in loudness for that subband, σ L k = L k . This metric is then compared with a threshold to determine whether coding noise introduced by quantization is perceptually detectable. This comparison can yield one of three possibilities: (i) the coding noise is undetectable without any severe overcoding of the signal; (ii) the coding noise is undetectable with significant overcoding; or (iii) the coding noise is detectable. In the first case, coding for the segment is complete. In the second case, the signal is requantized using coarser quantizer step size, and the detectability metric is recomputed with the same possibilities reexamined. In the third case, a process involving up to two steps will be employed. The first of these steps concerns an attempt at reducing the detectability of the noise without modifying the step size. This is done by modifying the phase relationship between the audio signal, x k , and quantization noise, n k . The audio signal is requantized using the same quantizer step size, and the detectability metric is recomputed. If the detectability metric is now below the threshold, coding is complete for the segment. Otherwise, the quantizer step size is reduced and the audio signal is requantized. The detectability metric is recomputed and the above possibilities reexamined. The Analyzer Figure 1 presents an illustrative embodiment of the present invention which employs analysis and synthesis features. The analysis feature of this embodiment has two components: ( i ) an analysis of the audio signal for the purpose of a loudness computation, and ( ii ) an analysis of the audio signal for purposes of quantization. Regarding the first of these two analysis components, the audio signal to be coded, x ( i ), is provided to a loudness computation processor 10 for a determination of signal loudness in each of a plurality of signal sub-bands. The details of the loudness computation are performed in Figure 2. As shown in the Figure, the loudness computation is presented on a subband basis and can be understood for all K subbands of a cochlear filterbank 11 by focusing on subband k. The number of subbands, K , of the cochlear filterbank depends on how accurately the basilar membrane of the cochlea is to be modeled for a particular audio signal bandwidth. For example, for 20kHz CD-quality audio, the number of subbands in the filterbank may be about 80. Cochlear filterbanks suitable for use in realizing an embodiment of the invention are well known in the art. See, e.g., Allen, 68 J. Acoust. Soc. Am., 1660-70 (1980); and Allen, et al., 95 J. Acoust Soc. Am., 3006 (1994). Signal x ( i ) is provided to each filter of the cochlear filterbank 11. Filter 11- k generates the k th cochlear subband of signal, x k , for subsequent processing. This subband signal is passed through a non-linearity comprising of a squaring function 12- k and a cube-root function 14- k . The resulting signal, x 2/3 k , is then integrated over a 5 ms interval by integrator 16- k . The non-linearity is an approximation for the shape of a loudness growth function and is a good approximation based on current research. However, better approximations of this non-linearity may become available in the future. Also, the 5 ms. integration interval may be varied. The integrated signal is then scaled by a factor α. The scale factor is used to ensure that the detectability metric can be compared to a threshold of approximately 1. Illustratively, assuming that a signal r.m.s. of 1v results in a sound pressure level ≈ 90dB, α = 959.68. The result is a measure of loudness for the subband signal, L k . Thus, the loudness computation 10 outputs a plurality of loudness measures, one for each subband. Regarding the second of the analysis components, the audio signal, x ( i ), is provided to a variable resolution analysis filterbank (VRAF) 70. This filterbank 70 performs a modified discrete cosine transform (MDCT) of the signal for subsequent use in quantization. As is known in the art, the MDCT is a subband decomposition filterbank that provides maximally decimated, uniform frequency resolution subbands. It also allows a perfect reconstruction of the filtered signal and therefore is particularly suited for coding techniques. The variability of the MDCT resolution pertains to the number of subbands it generates ( i.e. , the frequency resolution of the MDCT). VRAF 70 varies the number of subbands according to variability of loudness with time. The details of VRAF 70 are presented in Figure 3. As shown in Figure 3, the VRAF 70 comprises a buffer memory 72 and a conventional variable resolution MDCT processor 74. The resolution of the MDCT processor 74 (i,e., the number of subbands) is dependent upon computed loudness values, L k , over time, and is determined by resolution control processor 60 (discussed below). As such, the audio signal, x ( i ), is buffered in memory 72 in order to allow the loudness computation 10 referenced above to precede the performance of the MDCT on the same segment of the audio signal, x ( i ). Buffer memory 72 stores two consecutive segments of 1024 input samples of the audio signal, and provides sufficient memory and control for a conventional 50% overlap "lapped" transform (or filtering) operation. MDCT processor 74 performs a conventional lapped transform of the buffered segments (2048 samples) with a resolution, M , as determined by a signal from resolution control processor 60. For 48kHz sampling rate of the original audio signal, the VRAF 70 will operate in either of two resolutions. For signal deemed stationary by resolution control 60, a resolution of M = 1024 will be employed. For non-stationary segments (i.e., segments containing sharp transients), a resolution of M = 128 will be employed. As is well known, for a given signal segment, lower frequency resolution implies higher time resolution (i.e., larger number of output subband samples from the MDCT), while higher frequency resolution implies lower time resolution (i.e., smaller number of subband samples from the MDCT). Switching between resolutions is accomplished by conventional window switching techniques. See, e.g., Edler, "Coding of Audio Signals With Overlapping Block Transform and Adaptive Window Functions," 43 Frequenz, 252-56 (1989). Thus, MDCT processor 74 generates a vector of output M subband samples (MDCT coefficients), s m , 1 ≤m≤M , for each input segment of input samples from buffer 72. As is conventional, lower frequency resolution operation of processor 74 will yield more output vectors for a given input segment, with each vector being of a smaller length (reflected in the size of M). Output vectors are provided to quantization processor 80, discussed below. Resolution control 60 determines which of two frequency resolutions for the VRAF is appropriate for a given segment of the audio signal. Those of skill in the art will recognize that a greater number of frequency resolutions could be chosen for use with the embodiment for enhanced coding efficiency. Also, non-uniform frequency subbands could be employed. Resolution control 60 buffers the loudness measures (provided by computation 10) for each cochlear subband, L k , 1≤k≤K, corresponding to a given segment. For each of the subbands, control 60 forms the absolute difference between the minimum and maximum values for L k (in that band) for the segment and compares it against a threshold. Illustratively, this threshold is 100. Control 60 counts the number of subbands where this difference exceeds the threshold. If this count is larger than a predetermined number of bands, it indicates that the segment should be classified as non-stationary. Consequently, signal M is set to indicate a need for the lower resolution MDCT ( M = 128). Otherwise, the higher resolution MDCT is used M = 1024). Illustratively, the predetermined number is 20, indicating that at least 20 of the K = 80 cochlear filter bands are experiencing sharp loudness transitions. For the purpose of quantization processor 80, the MDCT subbands are grouped into a smaller set of coder bands, indexed by j . The grouping of MDCT subbands into coder bands approximates the critical band divisions of the human auditory system, as is conventional in the art. J. D. Johnston and K. Brandenberg, "Wideband Coding-Perceptual Considerations for Speech and Music," Advances in Speech Signal Processing, edited by S. Furui and M. M. Sondhi (1992). There are two such groupings corresponding to the two resolutions in frequency employed by the VRAF 70. See Appendix 1. Grouping refers to the fact that each MDCT subband of a group will be quantized at the same stepsize and will have the same phase control treatment. (Illustratively, where N is the number of coderbands, N=49 for M=1024 and N=15 for M=128.) Quantization processor 80 receives output vectors from VRAF 70 and quantizes these vectors according to a set of initial loudness signals output from loudness computation 10 and output from the differential loudness detectability processor 20. Quantization is performed for each vector received from VRAF 70. Processor 80 comprises quantizer step size control 85 which is coupled to phase control 83 and quantizer 86. Phase control 83 is further coupled to a memory 87, referred to as the waveform dictionary, which provides signals for modifying coder band data to effect phase variations between corresponding quantizer noise signals and original audio signals. Quantizer step size control 85 receives an initial quantizer threshold from processor 82 for each vector based upon subband loudness signals received from loudness computation 10. Quantizer step size control 85 also receives information from DLDP 20 pertaining to those coder subbands which may require either phase control or modified quantizer step size. Coder subband information is provided by mapping function 84 which selects appropriate coder subbands associated with a particular cochlear subband with use of a stored relationship (mapping) 81 between cochlear subbands and coder subbands. The coder band to cochlear band mapping memory 81 is a two-way associativity map between the K cochlear subbands and N coder bands, i.e. , it is a K×N matrix whose (i,j) th element is 1 if the i th cochlear band and j th coder band associate with each other. This is determined by looking at the magnitude response of cochlear filter, i. If the magnitude response of i th filter is more than -25dB (illustratively) of its peak response within the passband of j th coder band, the associativity is true. There are two associativity maps corresponding to two VRAF frequency resolutions (M). The mapping 81 is used by both mapping function 84, which generates a list of coder bands associated with a particular cochlear band, and by the initial stepsize processor 82, which needs a list of cochlear subbands associated with a particular coder band. Before quantization of coder band signals associated with each incoming vector from the VRAF 70 can occur, an initial quantizer step size, Δ j , for each coder band needs to be calculated. This calculation is performed by initial stepsize processor 82. Processor 82 begins by computing the average energy in a coder band, I j , for each j . This average energy is equal to the sum of the squared values of the MDCT subband signals belonging to the coder band, j , divided by the number of MDCT subbands in the coder band. Next, a minimum loudness measure is determined. This is done by first determining the minimum loudness value over time for each group of M coefficients (corresponding to the incoming vector from VRAF 70) of each cochlear subband, where M is the above- referenced frequency resolution. Next, based on the mapping from memory 81, the minimum of the previously determined minimum loudness values associated with the coder band is determined for each coder band, L min / j . The noise power for each coder band, N j , is illustratively computed as follows: N j = ( L min j + L min j ) 3 - I j . The initial quantizer stepsize, Δ j , for each coder band is determined as follows: Δ j = 12 N j . For each coder band, processor 80 quantizes all MDCT coefficients corresponding to the coder band with conventional uniform quantizer 86 employing stepsize Δ j . Phase control 83 performs no function at this time. Quantized values are further compressed using a conventional lossless encoding technique, such as Huffman encoding 100, to generate a coded bitstream. This bitstream is then made available to the synthesis portion of the coder. The Synthesizer The synthesis portion of the analysis-by-synthesis scheme of the illustrative embodiment begins at the noiseless decoder 110. Decoder 110 performs the reverse of the procedure applied by the noiseless coder 100 (i.e., decoder 110 performs conventional Huffman decoding). The output signal of decoder 110 corresponds to the output signal of quantization processor 80. This signal is then dequantized by dequantizer 120. Dequantizer 120 performs the reverse of the quantizer process performed by processor 80 given side information of quantizer step size and phase control. The output of dequantizer 120 is a set of subband signals which corresponds to the output of VRAF 70. This set of subband signals is then applied to a inverse MDCT processor (VRAF -1 ) 130 which performs the inverse of the process performed by VRAF 70 given side information concerning the frequency resolution, M . The result is a signal x and ( i ), which is equivalent to the original audio signal plus plus additive quantization noise ( i.e. , x and ( i )= x ( i )+ n ( i )). Signal x and ( i ) is applied to loudness computation processor 30 which is identical to the loudness computation 10 discussed above. The result is a set of time-varying subband loudness signals, L and k , which represents the loudness associated with x and ( i ) ( i.e. , the loudness associated with the sum of quantization noise added to the original audio signal.) Significantly, the result of this loudness computation depends on the phase relationship between the original audio signal and the noise signal. Said another way, the degree to which these signals constructively interfere as a result of relative phase will affect overall loudness, L and k . Thus, it is not merely relative signal and noise power (or energy) which is important. The illustrative embodiment actually exploits relative phase relationships in the coding process, as discussed below. Differential Loudness Detectability Loudness signals L and k provided by loudness computation 30 and loudness signals L k provided by loudness computation 10 are both supplied to differential loudness detectability processor (DLDP) 20 for the computation of a time-varying, subband metric, d ' / k ( i ). This metric is given by the following expression: d ' k (i) = L k ( i )- L k ( i ) L k ( i ) , where L k ( i ) is an estimate of the uncertainty associated with auditory neural encoding of the loudness (see Appendix 2.) Such uncertainty may be represented by the standard deviation of a Poisson process, , having a mean L k . Thus the standard deviation is given by L k . The ratio facilitates a comparison of differential loudness to the measure of uncertainty associated with loudness (such a comparison could be made in a number of alternative ways). When differential loudness is larger than L , then the coding noise will have a high probability of being audible (detectable); otherwise, coding noise will have a low probability of being audible. Values of d ' / k ( i ) for all i of a given segment of audio signal are compared to a detectability threshold, illustratively equal to one, and the differences stored. This comparison is made for all subbands independently. If all of the values of d ' / k ( i ) are below one but not below a second threshold, illustratively 0.9, the segment will likely not include audible coding noise in a frequency range close to the passband of cochlear subband k and will not be severely overcoded. Coding for the segment is complete if this test is satisfied for all k. The value of the second threshold affects how efficiently available bits will be utilized in coding, as well as coder complexity. Should the value of any d ' / k ( i ) either exceed one (indicating the detectability of coding noise) or be below the second threshold (indicating significant overcoding), signals are passed to quantization processor 80: (i) k, indicating the cochlear subband needing further processing; and (ii) the condition of overcoding or undercoding. Quantizer Stepsize Control in the Quantization Processor Quantization processor 80 provides additional processing of MDCT coefficient data grouped according to coder bands according to the conditions discussed above. There are two modes of additional processing corresponding to the two conditions of noise detectability and overcoding. The first mode, corresponding to unmasked noise, involves phase control processing by processor 83 in combination with waveform dictionary 87 and, if needed, subsequent requantization with reduced stepsize. The second mode, corresponding to overcoding, involves requantization at increased stepsize. Prior to any remedial processing for these conditions, each cochlear subband index, k, provided by DLDP 20, is mapped into a list of one or more coder band indices j by mapping function 84 employing conventional table look-up in map 81 (described above). Processing for Unmasked Noise For all coder bands j provided by mapping function 84 which require remedial action for unmasked noise, the phase control process 83 referenced above is sequentially applied for each band. This process adjusts the phase of the audio signal relative to a quantization noise signal by subtracting a scaled vector of random values (provided by a waveform dictionary 87, discussed below) from the vector of MDCT coefficients corresponding to the coder band j . The waveform dictionary 87 is scaled (multiplied) by the value of the quantizer stepsize for this band. The result of the subtraction is a vector which is then requantized by quantizer 86. Quantizer 86 uses the same stepsize as previously used for this coder band. The output of the quantizer 86 is applied to compression 100, and further to the synthesis process (110, 120, 130, and 30) to regenerate synthesized loudness, L and k . DLDP 20 updates detectability measures d ' / k , which are then compared to previously computed d ' / k to determine whether any of the d ' / k which were previously above one have now been reduced without making any of the previously masked cochlear bands unmasked. If any of the previously masked cochlear bands have become unmasked, the phase control vector subtraction is disregarded. Otherwise, the number of cochlear subbands which have had reduced d ' / k is stored and the next dictionary entry is tried. Once all dictionary entries have been tried, the entry which has the largest number of reduced d ' / k values is identified and associated with the coder band j . The modified MDCT coefficients (modified due to vector subtraction) associated with the best entry are stored. If all entries result in unmasking one or more previously masked cochlear bands, then the MDCT coefficients corresponding to this coder band are left unchanged. An additional synthesis process is performed using quantized MDCT coefficients corresponding to the best entry to update the d ' / k values for that entry (which may have been lost during the above-referenced search). All the remaining coder bands j which require processing for unmasked noise are then processed as discussed above. This results in the best set of d ' / k based on phase control processing (and no changes to quantization stepsize). If all d ' / k are below one, remedial processing for undercoded coder bands is complete for this MDCT vector. However, if any of the d ' / k of the best set is above one (threshold value), requantization with smaller stepsize will be required. Prior to requantization, the list of coder bands that require processing to remedy unmasked noise is updated based on the best set of d ' / k values in the fashion described above (see discussion of mapping function 84). For the coder bands that are thus identified to require further processing for unmasked noise, any phase adjustment activated at a previous step is deactivated now. Quantizer stepsize is reduced for all of the coder bands in the updated list. The stepsize is reduced by multiplication with a constant less than one, illustratively 0.9. Requantization of all coder in the list is then performed. The synthesis process is also performed, culminating in the recalculation of the d ' / k values. The list of coder bands which require remedial action for unmasked noise is updated as previously discussed. If this list is empty, remedial action and coding for undercoded coder bands is complete. If the list contains coderband entries, a new round of phase control is performed, followed by quantizer stepsize reduction, if necessary, and so on. The end result is a set of d ' / k values which are all less than one. Waveform dictionary 87 contains subdictionaries, each associated with a coder band size. Each subdirectory contains normalized vectors of a fixed length whose entries are generated by a Gaussian random number generator. As is clear from Appendix 1, the coder bands are of size 2 k , k=2,3,4,5,6. There is a subdirectory of dimension 2 k -2 for each k. Processing for Overcoded Coder Bands For all coder bands j provided by mapping function 84 which require remedial action for overcoding, quantizer stepsize is increased to relieve the condition. Processing of such coder bands will occur one band at a time. For each such coder band, quantizer stepsize will be increased by a factor greater than one, illustratively, 1.1. The coder band MDCT coefficients are then requantized and the synthesis process is performed. Updated values of d ' / k are computed and checked to determine whether any such values is greater than one. If not, the stepsize is again increased and the synthesis repeated. If any value of d ' / k exceeds one (as a result of any application of increased stepsize), the stepsize immediately preceding this condition is chose as the final stepsize for this coder band. All other overcoded coder bands are processed in this fashion in sequence. Quantization Processor Output The output of the quantization processor 80 for each vector of MDCT coefficients will include quantized MDCT coefficients and side information for each coder band. Side information will include (per coder band) quantizer stepsize, an index to a dictionary entry selected by the phase control processing (an index of zero indicates no phase control processing for this coder band), and (per vector) information for resolution control of the inverse MDCT processor 130. In addition, processor 80 outputs a flag indicating that current processor 80 output is for use in transmission over a channel. This flag, T x , is provided to transmitter 140 which provides coder output as a coded bit stream over the communication channel. The channel may comprise an actual transmission channel (such as a wired or wireless transmission channel, a telecommunications network, the Internet or other computer networks, LANs, WANs, MANs, etc.) and/or a storage medium (such as a compact disk (CD), CD-ROM, semiconductor memory, magnetic tape or disk, optical storage, etc). A Decoder for Use with the Illustrative Encoder Because the encoder is an analysis-by-synthesis encoder, the synthesizer portion of the encoder contains a decoder suitable for use with the encoder. This decoder comprises the combination of the noiseless decoder 110, dequantizer 120, and inverse MDCT processor 130, as discussed above with reference to the synthesizer portion of the encoder presented in Figure 1. Other Embodiments Although a specific embodiment of this invention has been shown and described herein, it is to be understood that this embodiment is merely illustrative of the many possible specific arrangements which can be devised in application of the principles of the invention. Numerous and varied other arrangements can be devised in accordance with these principles by those of ordinary skill in the art . For example, an embodiment of the present invention may estimate a measure of loudness uncertainty as a measure of the variation of signal amplitude over time. This measure may be of the signal itself or a related quantity, such as signal energy. Such measures may be employed in traditional tonality approach to noise masking. See, e.g., U.S. Patent US-A-5 699 479 published after the priority date, entitled "Tonality for Perceptual audio Compression Based on Loudness Uncertainty," assigned to the assigned hereof and filed on even date herewith . In addition, it will be apparent to those of ordinary skill in the art that the illustrative embodiment presented above is a variable bit-rate compression technique. Embodiments of the present invention which operate at fixed bit-rates may be provided through the use of conventional "rate-loop" mechanisms, such as those discussed in the Brandenburg and Johnston publication referenced above. Similarly, the illustrative embodiment may be extended to provide for the coding of stereo audio using well-known sum/difference encoding techniques. See, J. D. Johnston and A. J. Ferreira, "Sum-Difference Stereo Transform Coding," Proceedings of IEEE, ICASSP , San Francisco, II-569-572 (1992). APPENDIX I Coder Band Upper Limits:- (I) M = 1024: Band limit in subband index limit in frequency 1 4 93.750000 2 8 187.500000 3 12 281.250000 4 16 375.000000 5 20 468.750000 6 24 562.500000 7 28 656.250000 8 32 750.000000 9 36 843.750000 10 40 937.500000 11 48 1125.000000 12 56 1312.500000 13 64 1500.000000 14 72 1687.500000 15 80 1875.000000 16 88 2062.500000 17 96 2250.000000 18 104 2437.500000 19 112 2625.000000 20 120 2812.500000 21 128 3000.000000 22 144 3375.000000 23 160 3750.000000 24 176 4125.000000 25 192 4500.000000 26 208 4875.000000 27 224 5250.000000 28 240 5625.000000 29 272 6375.000000 30 304 7125.000000 31 336 7875.000000 32 368 8625.000000 33 400 9375.000000 34 432 10125.000000 35 464 10875.000000 36 496 11625.000000 37 528 12375.000000 38 560 13125.000000 39 592 13875.000000 40 624 14625.000000 41 656 15375.000000 42 688 16125.000000 43 720 16875.000000 44 752 17625.000000 45 784 18375.000000 46 816 19125.000000 47 848 19875.000000 48 880 20625.000000 49 1024 24000.000000 (I) M = 128: 1 4 750.000000 2 8 1500.000000 3 12 2250.000000 4 16 3000.000000 5 20 3750.000000 6 24 4500.000000 7 32 6000.000000 8 40 7500.000000 8 48 9000.000000 10 56 10500.000000 11 64 12000.000000 12 80 15000.000000 13 96 18000.000000 14 112 21000.000000 15 128 24000.000000 APPENDIX II 1 Introduction 1.1 Need for a masking model The problem of determining the masked threshold for an arbitrary masker is a basic problem in psychophysics which has many important engineering applications. First, the masked threshold is needed in speech and music coders to determine the bit allocations [35, 22, 39] . Second, the masked threshold is important in calculating the loudness of a signal [13, 37, 45] . Third, masking is important in defining the critical bandwidth. Fourth, the masking is required when calculating the articulation index when determining speech intelligibility [11] . 1.2 Central issue: Tone versus noise maskers Presently there are no well developed methods for accurately predicting the masked threshold for arbitrary maskers. For example, there are no models that can account for the large difference in the masking ability of tones and narrow bands of noise of equal intensity. Measurements of the masked threshold for a tone probe whose frequency The relationship between masking, JND in intensity, and loudness is centered on the masker frequency show up to a 23 dB increase in threshold if the masker is a narrow band of noise versus an equal intensity tone [7, 8] . This difference in masked threshold can be seen in Fig. 2 of Experiment I where we measure masked audiograms using a 70 dB SPL tone masker and a 70 dB SPL, 120Hz wide, random noise masker centered at 1kHz. A similar difference in masking efficacy between tones and equal intensity narrow bands of noise is observed if the probe is a wide band noise [44] or a click [20] . Thus an understanding of the effect of masker bandwidth on masked threshold is a central issue in the development of a general model of masking. The difference in the masking ability of tones versus equal intensity narrow band noises is particularly confounding for a variety of published methods that compute masked threshold based on the energy spectrum of the masking stimulus. For example, most techniques for calculating the loudness of an arbitrary sound rely on the assumption that loudness is directly related to the pattern of excitation in the auditory nerve [12, 13, 37, 45] . All of these methods infer the profile of neural activity produced by a sound from the psychophysical masked audiogram when that sound is used as the masker. The area under the masked audiogram is transformed into a representation of the neural activity [13] or "specific loudness" [45] which, when integrated across the spectrum, yields an estimate of the total loudness of the sound. Fletcher and Munson, who were the first to publish a method for loudness calculation, noted an inconsistency when using masked audiograms to infer neural excitation. Although tones and sub-critical bandwidth (SCBW) noises of equal intensity have nearly the same loudness [9] , the area under their masked audiograms are quite different (see Fig. 2). The difference in the masking properties of signals with discrete spectra (e.g. multi-tone complexes) [12] versus signals with continuous spectra [13] (e.g. noise or speech) prevented Fletcher from developing a unified model of the relationship between masking and loudness. Zwicker's method for calculating loudness, although based on the same concepts, ignores the difference in the masking properties of tones and noises by treating all signals as noise-like [45] . Speech and music coders which exploit the masking properties of the input sound to hide quantization noise are also hampered by the difference in masking efficacy of tones versus noises when computing the masked threshold [35, 22] . Developers of these coders struggle with the problem of defining the two classes of signals, tone-like versus noise-like, as well as identifying the two classes in sub-bands of the input signal. If a unifying model that relates the masking properties of noise signals to tone signals existed, the issues currently faced by these coders when categorizing signals might be eliminated. 1.3 Statement of the problem Our ultimate goal is to develop a model of masking that predicts the masked threshold for arbitrary signals. In this paper we examine cases in which (1) the masker ranges in bandwidth from zero (i.e. pure tone masker) up to the critical bandwidth (i.e. SCBW noise maskers), and (2) the probe is a pure tone or a noise of the same bandwidth as the masker. Masking stimulus ≡ m T = m ( t ) t ∈ [0, T ]. Probe stimulus ≡ p T = p ( t ) t ∈ [0, T ]. All signals are centered at 1kHz. Our results suggest that the difference in masked threshold between tone and noise maskers is related to the intensity change that results when the probe is added to the masker. If I [ m T ] is the intensity of the masker and I [ m T + p T ] is the intensity of the masker plus probe, then the intensity change ΔI is given by The right-most term in Eq. 5 represents the correlation between the masker and probe signals. Thus the intensity increment resulting from adding the probe to the masker depends not only on the intensity of probe, I [ p t ], but on the correlation between m T and p T , as well. The relative intensity change Δ I / I , also known as the Weber fraction in JND I tasks, is given by If the masker and probe are frozen (i.e. deterministic) signals, the intensity increment will be greatest for positively correlated masker and probe signals and least for negatively correlated signals. If the masker or probe are random signals, as is the case for tones masked by random noise, the correlation between masker and probe will vary amongst trials. Therefore, the intensity increment is random when a tone is added to random noise as opposed to a constant intensity increment when a tone is added to another tone or frozen noise. Frequently, and incorrectly, it has been assumed that the masker and probe signal are uncorrelated when calculating the intensity increment. Before describing the experiments we will review the issues relevant to making the connection between masking and intensity increments. 1.4 Case of masker and probe of same bandwidth We begin with the case where the probe signal is a scaled copy of the masker signal. namely p ( t ) = α m ( t ), where α is a scale factor. In this case, where the signal is masking itself, the observer's task becomes one of detecting a just noticeable difference in intensity (JND I ) when the probe is added to the masker. The Δ I is related to α as Δ I = (2α + α 2 ) I [ m T ] ≈ 2α I [ m T ] (for small α), 1.4.1 JND I examined using detection theory The detectability of intensity increments has been examined using detection theory [14, 6] . The basic idea behind signal detection theory is that the observer bases their judgments on a decision variable which is a random function of the stimulus. For example, the decision variable must be a function of the stimulus intensity in a JND I task. The uncertainty associated with the subjective ranking of a stimulus is reflected in the distribution of the decision variable. Detection theory shows that the variance of the decision variable limits the subject's ability to distinguish between stimuli that differ only in their intensity [14, 15, 6, 26, 32, 23, 19] . 1.4.2 Δ I as a measure of decision variable variance Masking and JND I are typically measured in a two-interval forced choice (2-IFC) paradigm, where one interval contains the masker m T and the other interval contains the masker plus probe m T + p T . The order of presentation of the two intervals within a trial is randomized and the subject is asked to identify which interval contains the probe. The probe level is varied until the level corresponding to a given detection criterion (e.g. 76% correct) is located. Several auditory detection models assume that the decision variable in a JND I tasks has a Gaussian distribution [14, 15, 6, 32] with a mean value that is linearly related to the intensity (I) of the signal. For example, in a 2-IFC JND I task, the interval corresponding to the standard signal will have one distribution mean whereas the interval corresponding to the higher intensity signal will have a slightly greater mean. If the distributions in the two intervals have identical standard deviations, the subject will correctly identify the more intense signal 76% of the time when the distance between the means is equal to the standard deviation of the distribution (i.e. d' = 1). Since the mean value of the decision variable is monotonically related to the intensity of the signal, the difference in intensity between the two intervals (i.e. Δ I as defined in Eqs. 3- 5 and Eq. 9) at 76% correct performance is a direct measure of the standard deviation of the decision variable's distribution. 1.4.3 Internal versus external sources of uncertainty The variance of the decision variable in JND tasks may be decomposed into an internal and external source of uncertainty [6, 26, 32, 4] . The internal uncertainty is an inherent characteristic of the auditory system (e.g. uncertainty contributed by the stochastic neural representation) and represents a fundamental limit on the best performance possible (e.g. in a tone JND I task). External uncertainty is contributed by the signal. For example, the intensity of a finite duration sample of Gaussian noise varies randomly about some mean value thus providing an external source of signal level uncertainty. Buus demonstrated that the JND I for a sub-critical bandwidth (SCBW) Gaussian noise is reduced if the noise is frozen rather than random from interval to interval at sound levels greater than 60 dB SPL [4] . Buus interprets the Δ I / I for the frozen noise as a measure of the internal uncertainty (subjective uncertainty) and the Δ I / I for the random noise as a measure of the combined internal and external uncertainty (stimulus uncertainty). The Δ I / I of a tone was found to be the same as that of an equal intensity frozen SCBW noise, which is consistent with this interpretation [4] . 1.4.4 Previous models of decision variable variance Green derived an approximate solution for the intensity distribution of a band limited, time truncated Gaussian noise [14] . Under the assumption that the value of the decision variable is linearly related to the intensity of the stimulus (the "energy model") and the implicit assumption that the internal uncertainty is dominated by the external signal intensity uncertainty, Green shows that such a detection model is qualitatively consistent with psychophysically measured performance. However, the quantitative predictions are consistently lower than observed human performance. The difference between the predicted and observed values is on the order of 5 dB [14] . De Boer attributed this failure to a lack of an internal source of uncertainty and reformulated Green's model by including an internal source whose variance was proportional to the stimulus intensity [6] . Although de Boer's results indicated an improvement in the qualitative fit of the revised model to psychophysical data, the range of values of the detectability parameter and of the constant of proportionality relating internal variance to stimulus intensity (which, in some cases, assumed negative values) indicated an overall failure of de Boer's model to quantitatively account for the data [6] . In this paper, we present experimental results that show that these "energy models" fail to predict human performance because the auditory system is not an energy detector. Our results suggest that the non-linearities in the auditory system give rise to yet another source of uncertainty ( waveform uncertainty ) that is not present in an energy model. This additional source of decision variable uncertainty accounts for the failures of the energy model. 1.5 The relation between JND I and masked threshold Earlier it was pointed out that when the masker and probe signals are identical, the masking task is equivalent to finding the JND I of the stimulus. Stiller [27] was one of the first to point out clearly this "operational similarity" between masking and JND I experiments. Subsequent publications by other authors indicated the acceptance of his hypothesis that masking and JND in intensity are fundamentally related [16, 5, 2, 17] . Despite the plausibility of such a connection between masking and intensity increments, however, there appears to be an absence of a general theory that quantitatively links these two phenomenon, thereby permitting the quantitative prediction of masking results from JND I data. A relationship between masking and JND I can be established if it can be shown that the decision variable is a function of the same stimulus attribute(s) in these two tasks. It is clear that the decision variable in a JND I task must be a random function of only the stimulus intensity because this is the only attribute of the signal that is changed within a trial. However, the decision variable in a masking task could be a function of the shape of the stimulus waveform as well as the stimulus intensity since both of these attributes change when the probe is added to the masker. The question is, how much of an improvement in detection performance can we expect if the information contained in the details of the signal waveform is utilized in addition to the information in the signal intensity? 1.5.1 Waveform shape changes associated with adding tones to random SCBW noise provide no additional information We hypothesize that there are at least two perceptual cues available to the observer in a masking task in which the masker is a SCBW noise and the probe is a tone. One cue is the change in signal intensity (e.g. loudness cue) and the second cue is the change in signal waveform (e.g. tonality cue). Both cues are clearly perceptible for tone levels above masked threshold. However, we are interested in identifying the stimulus attribute(s) responsible for determining the absolute lowest threshold of perception. To resolve the perceptual cue issue we conducted two sets of experiments. In Experiment II we conducted 2-IFC masking experiments in which the intensity cue was controlled and compared the measured masked thresholds to the JND I of the masking stimuli. According to Eq. 5, the intensity change Δ I resulting from adding the probe to the masker depends on the correlation between the probe and masker. Normally the correlation between a tone probe and a random noise masker is random, hence the resulting Δ I is random. However, we digitally generated our tone probe and random noise maskers so that the tone always added in phase to the noise masker to produce a constant rather than random Δ I . Details about the stimuli are given in the Methods section. The results of Experiment II show that the tone probe level at masked threshold produces an intensity increment (Δ I ) that is equivalent to the JND I of the masking SCBW noise stimulus. This result establishes a fundamental relationship between masking and just noticeable differences in intensity. Experiment III was designed to unambiguously identify intensity and not waveform as the stimulus attribute responsible for determining the absolute lowest masked threshold. In Experiment III we measured masked thresholds and JNDI of the masking stimuli using a 3-IFC paradigm and compared the results to masked thresholds for the same stimuli in a 2-IFC paradigm. In the 3-IFC paradigm, subjects were instructed to "pick the interval that sounds different". The observed thresholds for tones masked by SCBW noise were identical in the 2-IFC and 3-IFC paradigms. The similarity of results in the two paradigms demonstrates that subjects utilized the absolute lowest threshold perceptual cue in the 2-IFC paradigm because the subjects could use either an intensity or waveform cue to detect the "different" interval containing the tone probe in the 3-IFC paradigm. Combining the results of Experiments II and III, we conclude that stimulus intensity alone can account for the absolute lowest threshold for detecting tones in noise. Thus, we have identified one source of the increased threshold for signals masked by SCBW noise versus tones - the difference in JND I for the two maskers. Since the JND I for a random SCBW noise is greater than that for a tone, we would expect that the probe would have to be more intense to produce a detectable intensity change when masked by random SCBW noise versus a tone (or frozen SCBW noise). A second and equally important source of the increased threshold is that adding a tone to random noise results in a random intensity increment as opposed to the constant intensity increment in JND I tasks. 1.5.2 Another source of external uncertainty: Masker-Probe correlation Besides measuring thresholds for detecting tone probes added in phase to random noise maskers we also measure thresholds without controlling the correlation between probe and masker (i.e. a "normal" masked threshold experiment) in Experiments II and III. The results show that masked thresholds are greater when the correlation between probe and masker are uncontrolled. Thus, the random Δ I caused by adding a tone probe to the random noise masker with no phase control provides yet another source for decision variable uncertainty. This Δ I uncertainty contributes to elevate the threshold for probes masked by noise. 1.6 Purpose of the study We hypothesize the the threshold for any SCBW probe masked by any SCBW masker of equal or greater bandwidth, be it random or deterministic, can be predicted if the distribution of internal uncertainty (i.e. the tone JND I ) is known. Our goal in this paper is to develop a model that can accurately predict the masked threshold for a narrow band probe in the presence of an arbitrary SCBW masker. Such a model would, amongst other things, explain the approximately 20dB difference in masking efficacy of tones versus noises. The endeavor is based on the hypothesis that the decision variable, when one is detecting the presence of a tone in noise, is solely a function of the stimulus intensity as in the JND I task. Therefore, the central issue we investigate is the relationship between the intensity increment at threshold for a tone masked by a SCBW masker and the JND I of the masker. The detectability of intensity increments is governed by the statistical distribution of the decision variable. There are internal and several external sources that determine the distribution of the decision variable. Table 1 decomposes these sources of decision variable uncertainty. Previous "energy models" fail to quantitatively predict the increase in decision variable variance when the masker is a random noise versus a frozen stimulus. Although these energy models included external sources of uncertainty contributed by the chi-squared intensity distribution of band-limited, time-truncated Gaussian noise, we believe that the failures are due to the omission of an additional source of external uncertainty, the waveform uncertainty. Therefore, we designed our experiments to dissect out the external uncertainty contributed by the intensity variability of Gaussian noise, thus leaving only the waveform uncertainty and a small amount of intensity uncertainty due to the stimulus on/off ramps. To this end, our random noise stimuli are generated by summing sine waves of constant intensities but random phases. Such a noise stimulus has constant intensity (constant spectral level) but a random waveform from interval to interval. Thus, unlike a random Gaussian noise stimulus, whose intensity randomly varies from interval to interval, any uncertainty in the decision variable due to our constant spectrum level random noise can not be ascribed to intensity variability. Further, using such a noise stimulus permits control over the correlation between the probe and masker signals thus enabling us to study the contribution of masker-probe correlation uncertainty to the variance of the decision variable. Table 2 summarizes the experiments conducted. In Experiment I we verify that our constant spectrum level random noise reproduces the tone versus noise masking results previously observed using Gaussian random noise [7, 8] . Experiment I involves measuring the masked audiogram for (1) a fixed phase tone masker, (2) a 120 Hz wide flat spectrum random noise masker, and (3) a random phase tone masker. All maskers have a center frequency of 1kHz and are presented at 70 dB SPL. Audiograms are measured using a 2-IFC paradigm. In Experiment II we measure thresholds for tones masked by SCBW maskers and the JND I of the SCBW maskers using a 2-IFC paradigm. The masker bandwidths range from 0 Hz (i.e. pure tone) up to 120 Hz (i.e. flat spectrum random noise masker). Again, all maskers have a center frequency of 1kHz and are presented at 70 dB SPL. The masked threshold measurements are divided into two groups: In the first group, the correlation between the probe and the masker is not controlled; therefore, adding the probe to the random noise results in an intensity increment that is random from trial to trial. In the second group of masking experiments, the correlation between the tone and masker is controlled to give a constant intensity increment from trial to trial while the masking noise waveform remains random. Experiment III is identical to Experiment Ilexcept that a 3-IFC rather than 2-IFC paradigm is employed to measure the thresholds. Finally, we propose a model and test its validity by predicting all the masked threshold data from the JNDI data. 2 Methods 2.1 Stimuli All stimuli were digitally generated at a 40kHz sampling rate. The stimulus duration was 100 milliseconds. A five millisecond cosine ramp was used to smoothly gate the stimuli on and off. 2.1.1 Maskers Figure 1 (upper row) depicts the amplitude spectra of the three different stimuli used as maskers. Masking stimuli had bandwidths (BW) of zero (i.e. pure tone), 40Hz, and 120Hz. The narrow band noise (NBN) stimuli were constructed by summing sine waves having equal amplitude (see Fig. 1) and randomly chosen phase assignments drawn from a uniform distribution. New phase assignments were drawn for each interval and trial. The masking stimuli were always presented at 70 dB-SPL. thus the amplitude of the sine wave components of the NBN are inversely related to the bandwidth of the stimulus (see Fig. 1). The frequency increment between adjacent sine waves composing the masker is determined by the fundamental frequency of the stimulus. Since the stimulus duration was 100 msec, the fundamental frequency was 10Hz. A 40Hz bandwidth signal centered at 1kHz contained five components at frequencies 980, 990, 1000, 1010, and 1020 Hz as shown in Fig. 1. 2.1.2 Probes Two types of threshold measurements were obtained: (1) JNDI of signals with bandwidths equal to 0, 40, and 120 Hz, and (2) thresholds for tones in the presence of background maskers with bandwidths 0, 40, and 120 Hz. The middle row of Fig. 1 depicts the amplitude spectra of the probes used in these two detection tasks (probes are represented by gray vectors whereas maskers are represented by black vectors). For the JND I experiments, the probe is simply a scaled copy of the masker. Adding the probe to the masker produces an intensity increment in the signal (Eq. 8- 9). For the masking tasks of Experiments II and III the probe was always a 1kHz pure tone (i.e. the probe has the same frequency as the center component of the NBN masker). For the masked audiograms of Experiment I the probe was a variable frequency tone. The phase of the 1kHz tone probe relative to the center component of the NBN masker was of particular interest in the masking tasks of Experiments II and III. Let the vector P represents the magnitude and phase of the probe p T and the vector M represents the magnitude and phase of the center (1kHz) component of the masker m T . The lower row of Fig. 1 is a vector diagram of the three different relative phase (re ) conditions between P and M for which we collected data. The probe was added to the masker component in phase (re = 0), out of phase (re = π radians), or with a random phase relationship ( re ∈ [0,2π) ). The length of the vector symbolizes the magnitude of the signal while the relative angle between the M and P vector symbolizes the relative phase relationship. In Fig. 1, the magnitudes of the masker and the resultant signal (S=M+P) have been held constant across all three cases while the probe magnitude depends on the relative phase of M and P. This depiction emphasizes the differences in probe magnitude necessary to achieve the same signal (S) magnitude as the relative phase between M and P is varied. 2.2 Hardware Stimuli were digitally generated in real time on a 486 PC. An Ariel DSP-16 converted the signals to analog and was followed by a 15kHz low pass filter (Wavetek Brickwall Filter). The transducer was constructed by cannibalizing a Yamaha YH-2 headset and placing one of the drivers in an enclosure similar to a Sokolich driver. A piece of airline headset tubing 80 centimeters in length was used to deliver the acoustic stimulus from the driver to the ear. This tubing was terminated with an Etymotic ER-14 foam ear tip. The foam plug was inserted into the auditory meatus of the subject to complete the closed field sound delivery system. Calibration was carried out using a B&K sound level meter and a B&K 4157 coupler. We used SYSid (Ariel Corporation) to measure the distortion in the acoustic system and verified that the level of the distortion products would not influence the results. 2.3 Data collection protocol Thresholds were measured using either a two interval forced choice (2-IFC) paradigm and/or a 3-IFC paradigm. The 100 msec presentation intervals were separated by 500 msec; there was a 500 msec waiting period before the start of the next presentation after the subject entered their response to the previous trial. A 3-up/1-down adaptive procedure was used in the task to force the algorithm to concentrate measurements near the d' = I point of the psychometric curve. Subjects were instructed to choose the interval containing the probe. Visual feedback informed the subject whether they selected the correct interval. 2.4 Data analysis To locate the probe level corresponding to 76% correct, a psychometric function was fit to the data. We assumed that the psychometric function could be approximated by a cumulative normal form when the probe level was expressed in dB-SPL. Rather than fit a cumulative normal to the data, the following logistic function approximation was used [18] : P correct ( L dB ) = 1 m (1 + m - 1 1+ exp (-( L dB - L M )/L S ) ) where P correct is the probability of a correct response, LdB is the probe level in dB, m is the number of alternatives in the m-IFC task (e.g. m =2 or 3 in our experiments), L M is the "midpoint" of the psychometric function corresponding to P correct ( L M ) = (1 + m )/2 m , and L S is the "spread". In the context of our model, approximating the psychometric functions by a cumulative normal that is a function of probe SPL (dB) is crude at best. Equation 10 is simply a sigmoidal function which provided a visually acceptable fit to the measured psychometric data. Since almost all of the data was clustered around the midpoint of the psychometric function, attempting to find a more accurate (and more complicated) form for the psychometric function would not have resulted in a significant improvement of the fit. A maximum likelihood procedure was used to find the parameters of Eq. 10 that provided the optimal fit of the function to the data. An estimate of the probe level corresponding to P correct = 76% was obtained from this function. To obtain a sufficient number of measurements above and below the 76% criterion to accurately determine Ls while concentrating most of the points in the immediate vicinity of 76% correct to accurately determine L m , the "down" step size was 2 dB until the tenth reversal occurred, then 1 dB until the twentieth reversal occurred and, finally, 0.5 dB until the end of the block of trials. A block of trials ran until 30 reversals occurred. Each of the experiments generally consisted of I block yielding an average of ∼ 125 measures per experiment. In some cases, the experiment was repeated and the results of the next block of trials were combined with the previous block of trials. Thus, each threshold estimate was based on the subject's response to approximately 125 to 375 trials. No data was discarded or selected. 2.5 Subjects We used four subjects in our experiments. All of the subjects had significant previous experience in psychoacoustic tasks. Subject #1 was a male in his 30s; subject #2 was a male in his 50s; subject #3 was a female in her 20s; subject #4 was a male in his 50s. All subjects had normal audiograms except subject #4 who displayed about a 20dB presbyacusis hearing loss. The first two subjects were the first and second authors, respectively. 3 Results 3.1 Experiment I: Masked audiograms for tone versus noise maskers 3.1.1 Frozen phase tone masker To verify that our constant spectrum level random noise maskers possess similar masking properties to bandpassed Gaussian noise, we repeated the experiments of Egan and Hake [7] , and Ehmer [8] . Masked audiograms were measured for a tone masker and for a 120Hz wide constant spectrum level random noise masker using a tone probe. Both maskers were centered at 1kHz and presented at an overall SPL of 70 dB. The tone probe and tone masker had the same phase (i.e. frozen) from interval to interval. The random NBN masker had a different waveform from interval to interval. Figure 2 displays the results. Figure 2 shows the same qualitative result as that demonstrated previously by Egan and Hake who used bandpassed Gaussian noise rather than a constant spectrum level noise masker. That is, the masked threshold at the center frequency of the masker is 20 dB greater for a constant spectrum level random noise masker than for a tone masker. The lower curve shows threshold in quiet. Several other features in Fig. 2 are also consistent with Egan and Hake's results [7] . In contrast to the large difference in masked thresholds near the maskers' center frequency. masked threshold away from the center frequency is relatively similar for the tone and SCBW noise masker. Egan and Hake ascribed the dips and peaks in the tone masked audiogram to beat listening. 3.1.2 Random phase tone masker The experiment summarized in Fig. 2 was repeated. however, this time the phase of the tone masker was varied randomly from interval to interval. The tone masker's phase was drawn from a uniform distribution. By randomizing the phase of the tone masker we have made it more like a random noise masker which has this property. Figure 3 displays the resulting masked audiograms. The masked audiogram for the 120Hz wide noise is identical to that shown in the previous figure. Comparing the masked audiogram for the tone masker in Fig. 2 with that in Fig. 3, we note the following differences: For probe frequencies distant from the tone masker frequency, the contour of masked thresholds is smoother with a random phase tone masker as compared to a fixed phase tone masker. (The dips and peaks associated with the fixed phase masker are apparently due to constructive and destructive summation of probe and masker distortion products on the basilar membrane, not beats as Egan and Hake had suggested.) The contour of the random phase tone masked audiogram closely follows the profile of the NBN masked audiogram. Masked thresholds for the random phase tone are slightly greater than for the NBN for probe frequencies distant from the masker frequency. Relative to the 120Hz NBN masker, the masked threshold for a 1kHz probe is approximately 10 dB greater if the masking tone has random phase as opposed to 20 dB less when the masking tone has a fixed phase. All of these observations are consistent with the hypothesis that part of the difference in masked thresholds between tone and noise maskers may be due to the different correlations between the probe and masker, and different correlations with the distortion products. 3.2 Experiments II and III: Measurements of masked threshold and JND I at 1kHz for various bandwidth maskers 3.2.1 Experiment II: 2-IFC task Just noticeable differences for intensity increments from 70 dB-SPL were measured in the four subjects. A plot of the observed thresholds versus the bandwidth of the stimulus appears in Fig. 4 as a dashed line. Note that the threshold for a JND in intensity tends to be 3dB to 10dB greater for a constant spectral level random noise masker ( BW > 0) as compared to a fixed phase tone masker (BW=0). An energy model would not predict a difference in JND I between tones and our constant energy noise. The difference between tone and noise JND I is subject dependent. Detection thresholds for tones in the presence of 70 dB-SPL maskers also appear in Fig. 4 connected by solid lines; the maskers are identical to the stimuli used in the JND I experiments. There are three curves for the tone masked thresholds representing the three different relative phase relationships between the probe and corresponding masker component: in phase, out of phase, and random relative phase (see bottom of Fig. 1). The three relative phase cases correspond to conditions under which the correlation between the masker and probe is maximally positive (in phase addition). maximally negative (out of phase addition), and random correlation (random relative phase addition). The results of Fig. 4 show that the masked threshold is very sensitive to the nature of the correlation between probe and masker. Indeed, the masked threshold can vary by as much as 10 dB for masker bandwidths approaching a critical band, and by as much as 30 dB for masker bandwidths approaching zero, depending on the correlation. To investigate whether the change in intensity caused by adding the tone to the masker was sufficient to account for the performance at masked threshold, the intensity increment at threshold in the masking task was compared to the intensity increment in the JND I task. In Fig. 5, the data corresponding to the two lower-most curves in Fig. 4 has been replotted in terms of the relative intensity change Δ I / I when the probe is added to the masker. Ordinate values ( Δ I / I ) in Fig. 5 are computed according to Eq. 7. Figure 5 shows that the Δ I s in the JND I and masking tasks (in phase addition) are identical. Thus the threshold for detecting the tone probe in the presence of the masker is achieved when the intensity increment caused by adding the tone in phase to the masker is equal to the JND I of the masking stimulus. A probe tone of larger amplitude is required to achieve the same intensity increment if the tone is not in phase with the corresponding frequency component in the masker. This intuitive result, which can be deduced from Eq. 5, is illustrated in the phaser diagram of Fig. 1 where the length of the probe vector had to be increased as the relative phase between the probe and masker increased to achieve the same signal amplitude. Figure 6 plots the relative intensity increment Δ I / I as a function of the probe intensity for two extreme cases of correlation between the masker and probe. The upper graph depicts the case where the masker is a pure tone (i.e. BW = 0 Hz) and the lower graph depicts the case where the masker is a narrow band of noise ( BW = 40 Hz). The solid line represents the case where the probe is added in phase to the masker (the condition illustrated by the lower left drawing in Fig. 1) whereas the dashed line represents a probe added π radians out of phase to the masker (the lower middle drawing in Fig. 1). A horizontal dash-dot line has been drawn at the level corresponding to a just noticeable intensity change (about ±0.57 dB for BW=0, and ±1.2 dB for BW=40). Vertical dash-dot lines have been drawn where the in phase and out of phase curves pass through the +0.57 dB and +1.2 dB lines, respectively. The probe intensities corresponding to these intersections are the same as those measured for the in phase and out of phase cases for the 0 and 40 Hz wide maskers in Fig. 4 for subject #1. 3.2.2 Experiment III: 3-IFC task Figure 6 also illustrates how psychophysically measured masked threshold is biased by the measurement procedure. Say the subject is instructed to choose the interval containing the probe in a 2-IFC task. A reasonable criterion would be to pick the interval that sounds more tonal. However, for probe intensities near masked threshold. the similarity between Δ I / I s in the masking and JND I tasks shown in Fig. 5 suggests that the subject relies on an intensity cue. In this case the subject will guess that the more intense interval is most likely to contain the probe. Figure 6 shows that the intensity of the stimulus may actually decrease when the probe is added to the noise. Thus the subject may pick the incorrect interval even though they correctly identified the more intense interval. On the other hand, if the subject is instructed to identify the interval containing the probe in a 3-IFC task, the subject will select the interval that sounds most different from the other two intervals. Hence the subject will tend to correctly identify the interval containing the probe whether the addition of the probe decreases or increases the intensity by a just noticeable amount. Most importantly, a 3-IFC experiment would demonstrate whether the waveform change associated with adding a tone to a noise masker (e.g. tonality cue) may provide a lower threshold cue for detecting the presence of the probe. To resolve the issue as to whether a waveform cue may provide a lower threshold cue, the masking and JND I experiments were repeated using a 3-IFC paradigm in which the subjects were instructed to select the interval that sounds different. The results of the 3-IFC experiments appear in Fig. 7 where they are plotted against the results from the 2-IFC experiment (previously shown in Fig. 4). Thresholds in Fig. 7 are for a detection criterion of d' = 1, corresponding to 76% correct in the 2-IFC paradigm and 63% correct in the 3-IFC paradigm. For all subjects there are only two data points for which the results differed between the 3-IFC and 2-IFC paradigms; these points correspond to the case where the probe is added out of phase or with random phase to a tone masker. In the remaining cases, thresholds are roughly the same in the 2-IFC and 3-IFC paradigms. The results of the 2-IFC and 3-IFC paradigms shown in Fig. 7 are consistent with the hypothesis that subjects base there decision in masking tasks on an intensity cue. The similarities and differences between the results in the two paradigms can be explained by referring to Fig. 6 where Δ I / I is plotted against the probe intensity. If the masker is a pure tone (upper graph in Fig. 6), a just noticeable decrement in intensity is achieved at approximately the same probe level as a just noticeable increment in intensity for the out of phase and in phase addition, respectively. This explains why the in phase and out of phase cases for a pure tone masker in the 3-IFC paradigm are similar. The random phase case for a tone masker has a slightly greater threshold than the fixed phase cases because the correlation (relative phase relationship) between the probe and masker does not always yield a maximal intensity decrement or increment. The similarity in results in the two paradigms when the masker is a noise (bandwidth = 40 and 120 Hz) can be explained in terms of intensity cues with the aid of the lower graph in Fig. 6. The maximum intensity decrement barely exceeds the threshold for detecting a change in intensity (out of phase case represented by the dashed line in lower Fig. 6). As a result, subjects tend to correctly identify the probe interval only if the presence of the probe causes an intensity increment as in the 2-IFC paradigm. Intensity decrements are even less likely to be detected if the masker bandwidth is greater than 40 Hz because the maximum intensity decrement decreases as the bandwidth of the masker increases. Thus the similarity of masked thresholds for tones in noise in the 2-IFC and 3-IFC paradigms is consistent with intensity providing the lowest threshold cue in these masking experiments. The results shown in Figs. 4 - 7 support the hypothesis that JND in intensity and masked threshold, classically discussed as separate and unrelated psycho-acoustic phenomena, are related to the same physical limitations of the auditory system. This concept is further developed into a model of auditory detection which we will test by attempting to predict all the results shown in Fig. 4 from the JND I data. Figure 9 shows the results of these predictions. In this figure, results from Fig. 4 are plotted against the predicted masked thresholds. The in phase and out of phase predictions are computed as described above and illustrated in Fig. 6. Intuitively, one would expect the random relative phase predictions to fall somewhere between the in phase and out of phase predictions. How to compute the random phase predictions is the subject of the next section. 4 Model 4.1 Basic assumptions of the model: The decision variable We now propose a model that is based on the experimental results summarized in Figs. 4, 5, and 7. The equality between the threshold Δ I / I in the masking and the JND I tasks shown in Fig. 5 suggests that the subject uses a decision variable that is a function of the same stimulus parameters in both tasks. The relevant stimulus parameter is closely related to the intensity of the signal. However, the increase in Δ I / I as masker bandwidth increased suggests that the decision variable is a function of stimulus waveform as well since the distribution of stimulus intensity did not change as bandwidth was varied. Figure 8 depicts a model that is consistent with our masking and JNDI data. In this model, the observer bases all their judgments on the output of a hypothetical channel whose value will be referred to as the decision variable. The decision variable is a random function of the input signal. The decision variable function is decomposed into four components which are represented by the four successive processing blocks in Fig. 8. This processing includes a band-pass filter, a non-linear monotonic transformation of the signal, an integrator, followed by the addition of signal dependent internal uncertainty. Since all of the signals used in this paper have sub-critical bandwidths, we can essentially ignore the effects of the filter whose pass-band is wider than the signal's bandwidth. We can summarize the relationship between the SCBW stimulus waveform s(t) and the value of the decision variable N in the model as N = G + e ( G ), where, function λ describes the non-linear transformation of the input signal, T is the duration of the stimulus, and e is a mean zero stochastic process. Random function e represents the internal uncertainty. The uncertainty e is normally distributed with variance that is a function of G : e = Normal (µ = 0,σ 2 G ), In section 5 of this paper we present results which suggest that the non-linear function λ is closely related to the loudness growth function. However, for the masking tasks in this paper, it is sufficient to model λ as a power function [26] ; λ( s ) α s c , where c = 2/3. It will become evident, that for the purpose of relating JND I to masking, the model is robust with respect to the exact form of the function λ. 4.2 Representation of stimuli within the model To the right of the block diagram in Fig. 8 appear three columns of graphs showing signals and distributions at various points in the model for three different inputs. The upper-most row depicts the input acoustic signals. These plots represent snap-shots of a portion of a single presentation of the signal. The remaining rows of graphs are based on multiple presentations of the signal to the system. Hence, the first column represents the case for multiple presentations of a pure tone to the channel; since the pure tone is deterministic, the channel sees the exact same signal during each presentation. The second column represents the case where different realizations of a random SCBW constant spectral level noise have been presented: in this case, the channel sees a different waveform during each presentation. The last column represents the case in which a pure tone has been added to the random SCBW noise used in the middle column; the relative phase between the pure tone and the corresponding component in the SCBW noise is random from trial to trial. 4.2.1 External uncertainty and the distribution of "G" External uncertainty associated with the stimuli is represented by the distribution of G . For any single stimulus presentation, the value of G is given by the integral of the signal after passing through the non-linearity. For multiple stimulus presentations the value of G may be different for each presentation (if the waveform is random) thus yielding a distribution of G values. If the sum of the non-linearities in the auditory system equaled a square law non-linearity (e.g. energy based detector in which c = 2 in Eq. 14) then all individual waveforms in the sample space of constant spectral level noise would yield the exact same value of G . However, because the non-linearity in our model is not a squaring non-linearity, the value of G for any one of these waveforms will be slightly different than the value of G for any of the other waveforms in the sample space. The resulting distribution of G for our noise stimuli is sufficiently close to Gaussian that we can approximate it as G ( noise ) = Normal(µ noise ,σ 2 noise ). The ensemble average value of G over the sample space of noise waveforms, µ noise in Eq. 15, is approximately equal to that of a pure tone of equal intensity; µ noise = E [ G (noise of intensity I )] ≈ G (tone of intensity I ), where E [ G ] is the expected value of G . The variance of G , σ 2 / noise in Eq. 15, depends on the parameters of the noise (e.g. bandwidth of the noise). The distribution of G given by Eq. 15 is for the case of a constant spectral level (constant intensity) noise input. If the input has a random intensity distribution, as is the case for a tone masked by random noise, the distribution of G will not necessarily be Gaussian. Stimulus based uncertainty is depicted in the second (middle) row of graphs in Fig. 8. These graphs illustrate the probability density functions (PDFs) for G that result from many presentations of the stimuli illustrated in the first row. The abscissa corresponds to the value of G while the ordinate represents the frequency of occurrence (probability) of that value of G . For the case of a pure tone (first column in Fig. 8), this distribution is a delta function since identical copies of the signal pass through the non-linearities during each interval. For the case of random constant intensity SCBW noise (middle column in Fig. 8), this distribution will will take the form of a Gaussian (Eq. 15). Finally, for the composite signal consisting of a tone plus the random SCBW noise (last column in Fig. 8), the intensity of the acoustic signal before it enters the channel will vary from trial to trial. This intensity distribution is derived in the Appendix. Hence, the PDF of G depicted in the last column of the second row reflects both the variability of intensity in the composite signal due to the interaction of the tone and noise, as well as the variability introduced by the non-linearities. 4.2.2 Internal uncertainty and the distribution of "N" The PDF of the decision variable N which includes additional uncertainty contributed by the stochastic nature of the neural representation, the internal uncertainty e . is shown in the lower row of graphs. Let us focus on the case for the pure tone depicted in the first column, for the moment. Since the PDF of G for a tone is a delta function. the JND I for a tone is determined by the variance of e . Thus the variance of e can be inferred from the Δ I / I for a just noticeable difference in intensity of a tone. Since the Δ I / I of a tone depends on intensity [33] , the "near-miss to Weber's law", the variance of e must depend on G (see Eq. 13) [26, 32, 23, 19] . The lower-left graph illustrates the PDF of N for a tone. This normal distribution has mean G and variance σ 2 / G . In the remaining two columns the distribution of G is not a delta function. If the variance of G is not large, the distribution of e will be approximately the same as that for a tone whose G is equal to the expected value of the noise G (Eq. 16). Since the expected value of G is approximately the same in all three cases, the variance of the internal uncertainty e is approximately the same in all three cases. Thus the PDFs for N in the last two columns can be computed by convolving the distribution of e from the tone case with the PDFs for G in the last two columns. The last row of graphs in Fig. 8 illustrate the results of these convolutions. 4.3 Using the model to predict masked threshold from JND I data We hypothesize that the model described in the previous section is predictive of threshold detection tasks requiring the identification of a SCBW probe in the presence of a SCBW masker of equal or greater bandwidth given one caveat: There is no lower threshold information contained in the signal waveform that is not already present in the overall signal intensity. We demonstrated in Experiment III that the stimuli used in this paper fulfill this requirement. However, take the example where the masking stimulus consists of one or two discrete tones whose frequencies are different from the tone probe. In this case, the observer will detect the presence of "beats" (or a change in the beat rate) before detecting a significant change in the overall intensity when the probe is added to the masker - this represents a case where the waveform of the signal contains information in the form of a predictable temporal intensity cue whose perceptual threshold is lower than the change in the overall intensity of the signal. The latter case can not be accounted for using this model. However, we should be able to account for the JND I and the detection thresholds for a tone probe or SCBW noise probe added to a SCBW noise masker using this model. 4.3.1 The "unknowns" in the model Currently, there are two unknowns in the model. The first is the variance of the normally distributed internal uncertainty. This parameter can be inferred from the JND I for a tone. The second is the exact form of the non-linearity, the function λ. In the next section we provide evidence that the function λ is related to the loudness growth function. The exact form of λ is only necessary to predict the JND I for random SCBW noises. Rather than predict the JND I for the noise maskers, we will just use their measured values from Experiment II in this section. 4.3.2 Inferring the PDF of "N" from measured JND I To infer the PDF of N from measurements of JND I , we take advantage of two features in the model. First, since the PDF of N in the JND I task is equal to the convolution of two normal PDFS, the PDF of G and the PDF of e , N must be normally distributed. Second, the expected value of N is approximately linearly related to the cube root of the signal's intensity (see Eq. 14). or equivalently E [ G I +Δ I ] = E [ G I ] + β(( I + Δ I ) 1 3 - I 1 3 ), where β is a constant of proportionality. Thus, according to detection theory the standard deviation of N is related to the Δ I corresponding to the 76% correct level (i.e. d' = 1) in a 2-IFC JND I task as Since the distribution of G is a delta function if measuring the JND I for a tone or a frozen noise, σ N is a direct measure of the variance of the internal uncertainty, e . However, since G is normally distributed in a random noise JND I task, σ N is a measure of the combined internal and external uncertainty. 4.3.3 The PDF of "N" in masking tasks having constant intensity increments Our results show that the masked threshold for a tone probe which is added with a fixed phase relationship to the corresponding component in the random SCBW masker corresponded to the same Δ I / I as observed in the respective JND I experiment (recall Fig. 5). Thus, the intensity increment, regardless of the method by which it is achieved, must yield the same PDF for the decision variable N . It is crucial to emphasize that the intensity of the composite tone plus random SCBW noise signal in this case was constant from trial to trial; in other words, the PDF of Δ I for tone plus noise was a delta function. 4.3.4 The PDF of "N" in masking tasks having random intensity increments When the phase relationship between the tone probe and the random SCBW noise masker is not controlled, the resulting intensity increment randomly varies from trial to trial. The randomness of the intensity increment represents another source of external uncertainty and must be reflected in the distribution of G and, subsequently, N. This case is illustrated in the right-most column of Fig. 8. The PDF for N corresponding to the random intensity increment case can be computed as follows: First determine the PDF for Δ I of the tone plus random SCBW noise; an analytic expression describing this PDF has been derived in the appendix. Then convert this Δ I PDF to a distribution in N -domain using Eq. 18 and convolve it with the PDF for N which was inferred from the measured JND I of the random SCBW noise masker. 4.3.5 Computing estimates of tone threshold masked by random SCBW noise To determine the probability that the subject correctly identifies the interval containing the probe in a 2-IFC task, one has to compute the probability that the value of the decision variable in the tone plus random SCBW noise case is greater than the value of the decision variable in the random SCBW noise alone case: P C = ∫ P ( N masker+probe = n ) · P ( N masker < n ) dn , where P C is the probability of correctly identifying the interval containing the probe, and P(condition) is the probability of realizing "condition" as determined from the PDFs of N. Note that the result of this integral does not depend on the value of β. 4.4 Modeling Results Figure 9 replots the measured masked thresholds (solid lines) from Fig. 4 and compares them to the masked thresholds predicted from the JNDI thresholds (dashed lines) using our model. Predictions for the probe added to the masker with a fixed phase relationship (in phase, or π radians out of phase) are computed by finding the probe tone intensity that corresponds to the same Δ I as found in the JND I experiment using that masker (this procedure was illustrated graphically in Fig. 6). Predictions for the probe added to the masker with a random phase relationship are computed by finding the probe level that corresponded to a probability of 76% correct identification in a 2-IFC task. Specifically, the random phase case was predicted by choosing a probe intensity, computing the intensity distributions within the model as described in the previous section, culminating with the evaluation of Eq. 22. This computation was repeated in an iterative fashion until the probe level corresponding to 76% correct was found. The latter search was terminated when the estimated probe level changed by less than ±0.01 dB from iteration to iteration. The predictions in Fig. 9 were computed using an exponent of c = 2/3 in the non-linearity function λ( s ) = s c . We also found that values for " c " ranging over the decade 0.3 to 3.0 all gave similar predictions (within ±0.2 dB). Since the function λ is used to map from the intensity-domain to the N-domain, and since the values of N are distributed over a very limited neighborhood, these predictions are only sensitive to the behavior of λ over a small range of values. The insensitivity of these predictions to the value of "c" is simply due to the fact that λ can be reasonably approximated by a straight line over the range of interest when its exponent is within the interval [0.3, 3.0]. 5 Modeling the non-linearity In the previous section we developed a model that accurately predicts masked thresholds from the JND I of the masking signals; we will now outline a model that can predict the JND I of the masking signals from the JND I of a tone. The combination of these two models will enable us to predict random signal JND I and masked thresholds from a single estimate of the internal uncertainty, the tone JND I . Although only a approximation of λ was necessary to predict masked thresholds from the JND I measurements, a precise description of λ is necessary to predict the JND I of noise signals given the tone JND I . The non-linearity determines how the waveform uncertainty is converted into a perceptual intensity uncertainty. Since the instantaneous intensity of the waveform varies from zero to some large value, we need a precise description of λ over this entire range rather than an approximate description over a narrow range as was the case for predicting masked thresholds from JND I . λ logical choice for the non-linearity is the loudness growth function. The decision variable in our model is solely a function of signal intensity, and because the decision variable represents the subjective ranking of stimulus intensity, the loudness growth function relating signal intensity (or pressure) to loudness is consistent with this model. We adopted Fletcher's algebraic approximation to the loudness growth function [10] . λ( s ) = loudness( s ) = s s 40 2 3 1 = 0.005/ s s 40 ) 2 , where the loudness units are "sones", and s 40 is the amplitude of a sine wave whose loudness is 40 phons. To predict the noise JND I , we ran several thousand realizations of the constant spectral level noise stimuli through the model (Fig. 8) with Eq. 23 as the non-linearity to build an estimate of the PDF of G , the external uncertainty. A separate PDF of G was estimated for each bandwidth of noise. The internal uncertainty estimated from the tone JND I was then convolved with the PDF of G to arrive at the decision variable distribution, the PDF of N . Finally, the Δ N necessary for 76% correct discrimination was computed and converted to an equivalent Δ I using Eq. 23. Figure 10 displays the constant spectral level noise JND I predictions along with their measured values for subject #1. Two separate predictions appear in Fig. 10: (1) predictions using Fletcher's loudness growth function (Eq. 23) as the non-linearity, and (2) predictions using the energy model (Eq. 14 with c = 2) as the non-linearity. Quantitatively, the predictions using the loudness growth function as the non-linearity are all within the 95% confidence limits of the measured results. Qualitatively, the predicted JND I as a function of bandwidth follow the same pattern as the measured JND I . The energy model, on the other hand, completely fails to predict the noise JND I . The JND I predicted using the energy model for the noise are slightly greater than that for the tone due to the small amount of energy variability introduced by the 5 msec on/off ramps used to gate the random noise. 6 Discussion 6.1 Masking is an intensity discrimination task In this paper we set out to explain the dependence of tone masked threshold on the bandwidth of the masking stimulus (i.e. tone versus sub-critical bandwidth random noise maskers) and to develop a quantitative model that predicts tone masked thresholds. We reported the results of several masking and JND I experiments which In Fig. 5 we demonstrated that the relative intensity increment Δ I / I at masked threshold caused by the addition of a tone to a random narrow band masker is equal to the JND I of the masker. The masking experiment in Fig. 5 was designed such that the addition of a constant amplitude probe tone caused a constant (deterministic) increment in the intensity of the stimulus from trial to trial even though the waveform of the masking noise was changing from interval to interval. The equality between the threshold intensity increment in the masking and JND I task of Fig. 5 is a quantitative demonstration of Miller's [27] hypothesis that masking and JND I tasks are fundamentally related. Further, the equality in Δ I / I between the masking and the JND I task demonstrates that the decision variable upon which the observer bases their decision is a function of the same stimulus attribute, the stimulus intensity. However, the 2-IFC paradigm used to collect these data did not allow us to rule out that an additional waveform cue might also be present but not utilized. We verified that stimulus intensity is sufficient to account for the lowest absolute threshold of detectability by repeating the masking experiment using a 3-IFC paradigm. Although the subjects had at their disposal both a temporal waveform cue and an overall intensity cue to identify the "different" interval in the 3-IFC task, a comparison of the 3-IFC to the 2-IFC results in Fig. 7 demonstrates that the waveform cue does not have a lower threshold than the intensity cue. Incrementing the intensity of the SCBW noise masker by adding an in phase tone (tone masked by noise) and incrementing the intensity by adding a scaled copy of the masker (masker JNDI) represents two extreme methods for changing the intensity of the masker without changing its bandwidth. In one case, we changed the amplitude of only one component in the masker's spectrum and in the other case we changed the amplitude of all components in the masker's spectrum. However, the Δ I / I at detection threshold was the same in these two cases. This observation implies that any SCBW probe whose bandwidth is less than or equal to the masker and causes a constant intensity change when added to the masker (i.e. having a fixed correlation with the masker) will have the same threshold Δ I / I . When we did not control the correlation (phase relationship) between the tone probe and masker, the experiment was analogous to the standard tone in noise masking task. In this situation the Δ I varied from trial to trial depending on the particular realization of the noise masker (Eq. 5). This additional uncertainty in the intensity increment is reflected in the distribution of the decision variable. The distribution of the decision variable in this masker plus probe case can no longer be approximated by a linear shift of the masker alone distribution. The distribution of the decision variable will be determined not only by the neural representational uncertainty (the additive Gaussian uncertainty in Fig. 8) and uncertainty contributed by passing a signal with random waveform through a non-linear system, but by the uncertainty in the Δ I as well. The effect of intensity increment uncertainty on the decision variable distribution is illustrated in the last column of Fig. 8. The Δ I uncertainty is yet another source that contributes to elevating the masked threshold. This study was initially motivated by the need to explain the large difference in masked threshold for a tone probe masked by another tone versus an equal intensity narrow band random noise (recall Fig. 2). We can now explain the basis for this difference and, more importantly, we can predict this difference quantitatively. There are three factors that contribute to increase the threshold when the masker is a SCBW random noise rather than a tone. We can break down the significance of each of these factors with the aid of Fig. 11. Figure 11 replots the results for subject #1 from Experiment II (these are the same data as previously shown in Fig. 4). A large circle has been drawn around two data points on this graph that correspond to the masked threshold for a tone masked by a 70dB SPL tone (threshold = 46 dB SPL) and a tone masked by a 70 dB SPL 40Hz wide random noise (threshold = 67 dB SPL). A horizontal dotted line has been drawn at the probe level corresponding to masked threshold in the presence of these two maskers. There is a 21 dB difference between these masked thresholds. Two additional horizontal dotted lines have been drawn at the level corresponding to the threshold JND I of the 40Hz wide noise masker (50 dB SPL) and the masked threshold for a tone added in phase to the same noise masker (57 dB SPL). The areas bounded by the four dotted lines have been numbered 1 through 3. These three areas (increments) show the relative contribution of three factors responsible for the difference between masked threshold for tone versus noise maskers. Increment #1 shows that there is a 4 dB increase in threshold due to the waveform uncertainty associated with the random constant spectral level noise masker. Increment #2 shows that there is a 7 dB increase due to the algebraic consequence of converting the JND I Δ I for a noise probe to an equivalent Δ I for a tone probe added in phase to the masker. Finally, increment #3 shows that there is a 10 dB increase due to the Δ I uncertainty associated with adding a tone to random noise with no correlation control. 6.2 Relationship between the masker and JND I We found that the JND I for constant spectral level SCBW random noise is greater than that for an equal intensity tone (Fig. 5) at the intensities tested. In section 5 we showed that the larger Δ I for the random noise is the result of a "broadening" of the decision variable distribution due to passing a signal whose waveform varies from interval to interval through a non-linear system. Broadening of the decision variable distribution is reflected as an increase in the Δ I required to be able to discriminate an intensity increment in the stimulus. If the input signal's energy is constant amongst intervals as is the case for our flat spectrum noise, the integral of the signal after the non-linearity will not necessarily be constant. Only a system which has a squaring non-linearity followed by an integrator, such as Green's [14] or de Boer's [6] "energy detector" models, would predict a constant value for the decision variable if presented with our constant energy SCBW random noise. We demonstrated in Fig. 10 that the energy detector model fails to predict the increase in JND I as the bandwidth of the signal is increased from zero. However, using a loudness growth function as the non-linearity does predict the increase in JND I for noise. Loudness growth functions vary amongst individuals depending on amount and type of hearing loss [36] . Therefore, since the loudness growth function partially determines a subject's noise JND I , one would expect variability amongst individuals for noise JND I as we observed. We did not measure loudness growth functions in our experiments. However. subject #4. who had significant presbyacusis hearing loss, did not show a significant change in JND I as a function of signal bandwidth as the other subjects (see Fig. 4). The observation is consistent with subject #4 having a loudness growth function that is significantly different from the other three subjects. The result further suggests that subject #4's loudness growth function is similar to a squaring non-linearity (i.e. a true "energy detector"). It is interesting to contrast this subject dependent variability in noise JND I which strongly depends on the form of the non-linearity to the relative lack of variability in masked thresholds for tones in noise which depends on the statistics of the random correlation between masker and probe and only partially on the non-linearity. Our model also implies that if we had used band-passed Gaussian noise rather than a constant energy SCBW noise, the JND I would have been even larger. A larger JND I for narrow band Gaussian noise would be expected because the energy of the signal would vary amongst intervals in addition to the waveform variations. especially for very narrow band signals where energy variance is greatest. If the stimulus is deterministic (i.e. repeatable waveform) as is the case for a pure tone, passage through the non-linearity will yield the same output signal from interval to interval and thus not contribute additional uncertainty to the signal's level. Our model predicts that the JND I of all deterministic sub-critical bandwidth signals should be the same. In other words, a "frozen" SCBW noise stimulus should have the same JND I as an equal intensity tone because both signals present the same waveform to the system non-linearities from interval to interval. 6.3 Relation of the model to previous work 6.3.1 Comparing optimal detectors to the human auditory system The psychoacoustic performance of the auditory system in 2-IFC tasks has been compared to two types of optimal detectors: (1) A detector that compares the energy (or, equivalently, the intensity) in the two intervals [14, 6] , and (2) a detector that compares the signal waveforms in the two intervals [21] . If the task is to identify which of two intervals contains a tone, where one interval contains random narrow band Gaussian noise plus the tone and the other interval contains the random noise alone, these two detectors yield identical performance. Peterson et al. [29] analyzed the performance of an optimal detector designed to discriminate between a masker alone and masker plus probe based on the resulting waveform change for the case of a band-limited, time-truncated Gaussian noise masker. If the probe is a random narrow band Gaussian noise of the same bandwidth and duration as the noise masker (i.e. a noise JND I task), Peterson et al.'s results are identical to those obtained by Green using a detector whose criterion is the intensity change [14] . Peterson et al. [29] also derived the discriminability of a waveform change resulting from adding a finite duration sinusoid to the random narrow band Gaussian noise masker. The derivation assumed that the sinusoid's frequency was within the pass band of the random noise and that its phase was unknown to the observer. We have derived the detectability of the resulting intensity increment under the same conditions and obtained the same analytic result as that obtained for Peterson's optimal detector based on waveform changes. From this identity we conclude that there is no additional information contained in the waveform of the signal (such as the envelope, or peaks, etc.) that is not already present in the intensity of the signal when discriminating between the Gaussian noise alone and noise plus tone case. One can get an intuitive feel for these theoretical results by viewing Gaussian noise as a signal whose waveform and spectrum are being "roved" thereby eliminating or minimizing the details of the waveform information. For example, a bandlimited, time-truncated Gaussian noise has a spectrum whose component amplitudes are Rayleigh distributed. Adding a low level tone to this noise will change the amplitude of one of the components in the noise spectrum, however, this amplitude change may not be distinguishable from the inherent random variability of the component's amplitude. The situation is analogous to that for the intensity of the noise which is also random. In summary, these theoretical results demonstrate that an optimal detector whose decision variable is a function of the signal waveform will not yield better performance than a detector whose decision variable is a function of only the signal intensity in two specific detection tasks relevant to this paper: (1) Detection of a tone in random narrow band Gaussian noise, and (2) detection of an intensity increment in random narrow band Gaussian noise. These conclusions apply to a detector that is optimal. However, attempts to quantitatively predict human psychophysical performance using optimal detection models [14, 6, 21] have generally failed. These failures suggest that the human observer performs sub-optimally in auditory detection tasks. Sub-optimal performance in humans is due to both (1) the non-linear distortion of the information contained in the signal waveform as it is transmitted through the auditory system to the site where discrimination actually takes place as well as (2) the limited spectral resolution of the auditory system. In any case, the successes of optimal detection models in qualitatively predicting human performance suggests that these models provide useful insight into psycho-acoustic performance but we must be particularly careful when interpreting the quantitative predictions. Although the constant spectral level noise used in this paper does not have the random spectrum of a true Gaussian noise, the results of Experiment III demonstrate that the waveform cue associated with adding a tone to this noise does not have a lower threshold than the intensity cue. If the human auditory system had the capacity to resolve the individual frequency components of a SCBW noise, perhaps the spectral (waveform) cue would provide a lower threshold cue. 6.3.2 Physiological correlates The development of the model and perspectives presented in this paper have been influenced not only by the successes of previously published psychoacoustic models but by recent advances in the understanding of neurophysiological signal processing in the central nervous system (CNS). However, the model has been presented as a phenomenological construction due to the difficulty in assigning, with defensible certainty, specific physiological mechanisms to the model elements. The filter represents the transduction of motion at a fixed point on the basilar membrane to a neural representation of the signal amplitude. This element represents a filter since each point on the basilar membrane is particularly sensitive to a narrow range of frequencies (the critical band). The non-linearity represents not only the mechanical non-linearities present in the basilar membrane motion but, also, non-linearities contributed by the transduction of membrane motion to auditory nerve impulses by the inner hair cells, and the subsequent non-linearities contributed by neural processing and transmission across synapses to higher order neurons. Hence the stimulus level estimated at the site of cognition will be a function of these system non-linearities [46] . Recent neurophysiological evidence suggests that the representation of acoustic stimulus envelope in the CNS is enhanced in the output of the principal cells of the ventral cochlear nucleus (VCN) relative to the auditory nerve [40, 41] . This suggests that the auditory system may be particularly concerned with preserving, or enhancing, information about the stimulus envelope. Further, it is evident that the neural representation of stimulus envelope in the output of VCN principal cells is a non-linear function of the acoustic stimulus [40, 41] . Neurophysiological evidence also suggests that information about the stimulus envelope is represented in the output of auditory cortical cells, however, this representation appears to be low pass filtered with respect to that in the output of the VCN (see [24] for review). These properties of the central representation of acoustic stimuli are consistent with the notion that information about the stimulus envelope is available to higher (cognitive) centers in the brain, thus providing a foundation for performing comparisons of stimulus envelope over time and comparisons of integrated level. 6.3.3 Psychophysical correlates The additive internal Gaussian distributed uncertainty is simply an approximate way of accounting for the inherent stochastic nature of the neural encoding. Many psychophysical models have explicitly accounted for this by including an internal source that behaves as as an independent additive uncertainty. Most published models assume that the internal uncertainty is Gaussian distributed but they differ in how they model the dependence of the variance on the stimulus. For example, it has been proposed that this variance is proportional to: stimulus intensity [30, 32] , the expected value of the decision variable [26, 23, 19] , and the square of the stimulus intensity [6] or combinations of the these [47, 32] . It is well known that the statistics of auditory nerve fibers with best frequencies greater than 3k-5k Hz can be viewed as a Poisson process that is modified by a refractory dead-time to a first approximation [38, 42, 25] . Thus, if the discrimination process was based on counts of events in the auditory nerve as Fletcher and Munson originally proposed [12] , it would be reasonable to assume that the internal variance was related to the expected value of the decision variable as some have suggested [26, 23, 19] . However, as opposed to the auditory nerve, the statistics of neural discharge patterns of principal cells in the auditory CNS can be quite regular as well [3, 43, 1] . Since it is ambiguous as to where the actual discrimination precisely occurs within the auditory system, it is not possible to predict which statistical properties should be assumed for modeling discrimination processes. The success of several conceptually similar models in predicting tone JND I [26, 23, 19] and loudness growth [23, 19] suggests that the internal uncertainty distribution can be modeled as a Gaussian process whose mean-to-variance ratio is approximately constant over a broad range of intensities. Since discriminating between different stimuli is a conscious task, it is assumed that the assignment of a value to the decision variable occurs at a cortical level. We model this process as an integration. The transformed and noisy representation of the signal waveform is integrated over its duration to arrive at a value that represents the subjective ranking of the input stimulus, the decision variable. This process can be equivalently viewed as counting the number of action potentials elicited by the stimulus [12, 46] . Temporal details of the signal are ignored by an integrator; however, our experimental results suggest that there is no additional information contained in the temporal details of the waveform if the task is to detect a tone in SCBW noise. The success of the "loudness model" (Sec. 5) in predicting the JND I of constant spectral level random noise suggests a correspondence between the perceived loudness of the signal and the value of the decision variable in our model. It is well known that loudness is a monotonically increasing function of stimulus duration up to several hundred milliseconds [28, 31, 34] . Thus the integration element in our model should ultimately incorporate a time weighting function that represents the duration over which the subject can integrate the signal level (i.e. a leaky integrator). This "leaky" integrator can be equivalently viewed as a low-pass filter with a time constant on the order of hundreds of milliseconds. We assume that our 100 millisecond stimuli were short enough to not be affected by the integration limits of the auditory system and, therefore, did not incorporate this feature into the model. The low-pass filter represented by the integrator in our model is distinct from and dominates the low-pass filtering contributed by brainstem neural processing which has a time constant on the order of milliseconds. This low-pass filtering which is inherent to all neurons probably limits the detectability of amplitude modulations in the signal. It has been shown that the perceived loudness of a pure tone and a narrow band noise (BW < CB) of equal intensity (i.e. equal RMS SPL) is the same [9] . This relationship probably is not strictly true since the filters are not rectangular, however, departures from this relationship do not exceed the error in published loudness matching results. Hence, the non-linearity in our model must fulfill this requirement (see Eq. 16). 6.4 Conclusions Historically, masking and just noticeable differences in intensity have been addressed as separate phenomena. Several papers have hypothesized a connection between the two phenomena. Our results demonstrate this connection quantitatively. Masking and JND I are both related to the same fundamental Δ I limitations of the auditory system. Overall intensity change is the lowest threshold cue for detection of tones in noise. Masked thresholds for arbitrary probes can be accurately predicted given the JND I of the masking stimulus and the correlation between the probe and masker. Non-linearities in the auditory system contribute to the difference between tone and random noise JND I . If the non-linearities in the auditory system are known, it is possible to predict masked thresholds for probes in the presence of any sub-critical bandwidth masker given only the JND I for a pure tone (which specifies the distribution of the additive internal uncertainty). The difference in masking efficacy of tone versus random noise maskers is due to (1). The increase in the JND I due to the random waveform of the masker, and (2) the uncertainty in the intensity increment resulting from adding the probe to the random masker. 7 Acknowledgments We would like to thank D.A. Berkley, O. Ghitia, J.L. Hall, and M.M. Sondhi ("The Hearing Group" at Bell Laboratories) for the many thoughtful discussions regarding the material presented in the paper. Also, special thanks to subjects #1 thru #6. 8 Appendix 8.1 Amplitude distribution of the sum of two sinusoids with random relative phase Derivation of the amplitude distribution for the sum of two sinusoidal signals with random relative phase: Let a ( t ) = A cos(ω 0 +  a ), b ( t ) = B cos(ω 0 +  b ),  =  a -  b , and (i.e.  is uniformly distributed over the interval [0, π].) Find the distribution of Y , f Y ( Y ), using the relationship f Y ( Y ) = f  ()/ dY d where Combining Equations 31-33 and the relationship sin( x ) = ± 1 - cos 2 ( x ) , we get f Y ( Y ) = Y π AB {1 - [ Y 2 -( A 2 + B 2 ) 2 AB ] 2 } . 8.2 Intensity distribution when a flat spectrum noise masker is added to a random phase tone Here we derive the intensity distribution of the masker + probe signal when the masker is a flat spectrum NBN of the type discussed in the Methods section and when the probe is a sinusoidal signal with the same frequency as one of the components of the SCBW noise. We have already discussed the case where the probe is added with some fixed phase relationship to the corresponding SCBW noise component; this case was trivial since the resultant intensity distribution is a constant. We now assume that the phase relationship between the probe and SCBW noise component is uniformly distributed and use Equation 34 to derive an analytic expression for the intensity distribution as a function of the masker bandwidth and intensity, and probe intensity. The distribution is expressed in decibels relative to the masker intensity. Assume that when the masker is of zero bandwidth (a pure tone) its amplitude is A 0 bw . Then, as one increases the bandwidth of the masker while maintaining the same RMS, each sinusoidal component of the masker will have an amplitude of A comp = A 0bw / N comp , where N comp is the number of components. If we define B to be the amplitude of the tone probe, then the intensity of the masker + probe signal, Z dB is given by Z dB = 10log[( N comp - 1) A 2 comp - Y 2 ], where Y is the amplitude of the sinusoidal signal resulting from adding the probe to the corresponding component of the masker, Y = ∥ a comp + b ∥. Our objective is to derive an analytic expression for the distribution of Z dB relative to the masker alone intensity. The distribution of Z dB can be determined from Eqs. 35 and 34 using f Z ( Z ) = f Y ( Y )/ dZ dY , where f Y ( Y ) is given by Eq. 34, and dZ dY = 20 Y ln 10[( N comp - 1) A 2 comp + Y 2 ] . After some substitutions, we find that f z ( Z ) = ln 10 10 Z /10 20π A comp B 1 - ( 10 Z /10 - A 2 - B 2 2 A comp B ) 2 . References [1] Carol C. Blackburn and Murray B. Sachs. Classification of unit types in the anteroventral cochlear nucleus: PST histograms and regularity analysis. J. Neurophysiol., 62(6):1303-29, December 1989. [2] C. E. Bos and E. de Boer. Masking and discrimination. J. Acoust. Soc. Am., 39(4):708-15, 1966. [3] T. R. Bourk. Electrical Responses of Neural Units in the Anteroventral Cochlear Nucleus of the Cat. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, 1976. [4] Soren Buus. Level discrimination of frozen and random noise. J. Acoust. Soc. Am., 87(6):2643-54, June 1990. [5] E. de Boer. Note on the critical bandwidth. J. Acoust. Soc. Am., 34(7):985-986, July 1962. [6] E. de Boer. Intensity discrimination of fluctuating signals. J. Acoust. Soc. Am., 40(3):552-60, 1966. [7] James P. Egan and Harold W. Hake. On the masking patterns of a simple auditory stimulus. J. Acoust. Soc. Am., 22(5):622-30, September 1950. [8] Richard H. Ehmer. Masking by tones vs noise bands. J. Acoust. Soc. Am., 31(9):1253-6, September 1959. [9] R. Feldkeller and E. Zwicker. Das Ohr als Nachrichtenempfänger. S. Hirzel Verlag, Stuttgart, 1956. [10] Harvey Fletcher. Speech and Hearing in Communication. Robert E. Krieger Publishing Co., Huntington, NY, 1972 (1953). [11] Harvey Fletcher and R.H. Galt. Perception of speech and its relation to telephony. J. Acoust. Soc. Am., 22:89-151, 1950. [12] Harvey Fletcher and W.A. Munson. Loudness, its definition, measurement, and calculation. J. Acoust. Soc. Am., 5:82-108, 1933. [13] Harvey Fletcher and W.A. Munson. Relation between loudness and masking. J. Acoust. Soc. Am. , 9:1-10, 1937. [14] David M. Green. Auditory detection of a noise signal. J. Acoust. Soc. Am., 32(1):121-31, January 1960. [15] David M. Green and John A. Swets. Signal Detection Theory and Psychophysics. John Wiley and Sons, Inc.. New York, 1966. [16] Donald D. Greenwood. Auditory masking and the critical band. J. Acoust. Soc. Am., 33(4):484-502, April 1961a. [17] Donald D. Greenwood. The intensity DL of tones: Dependence of signal/masker ratio on tone level and on spectrum of added noise. Hear. Res. , 65:1-39, 1993. [18] Joe L. Hall. Hybrid adaptive procedure for estimation of psychometric functions. J. Acoust. Soc. Am., 69(6):1763-9, June 1981. [19] William S. Hellman and Rhona P. Hellman. Intensity discrimination as the driving force force for loudness. application to pure tones in quiet. J. Acoust. Soc. Am., 87( 3 ):1255-65, Starch 1990. [20] I. J. Hirsh, W. A. Rosenblith, and W.D. Ward. The masking of clicks by pure tones and bands of noise. J. Acoust. Soc. Am., 22(5):631-7, September 1950. [21] Lloyd A. Jeffress. Stimulus-oriented approach to detection. J. Acoust. Soc. Am., 36(4):766-74, April 1964. [22] James D. Johnston and Karlheinz Brandenburg. Wideband coding - perceptual considerations for speech and music. In Sadaoki Furui and M. Mohan Sondhi, editors, Advances in Speech Signal Processing, pages 109-140. Marcel Dekker, Inc., New York, 1992. [23] Gerard Lachs, Radhi Al-Shaikh, Rosalie A. Saia, and Malvin C. Teich. A neural-counting model based on physiological characteristics of the peripheral auditory system. v. application to loudness estimation and intensity discrimination. IEEE Trans. Syst., Man, Cybern., SMC-14(16):819-36, November 1984. [24] Gerald Langner. Periodicity coding in the auditory system. Hear. Res., 60:115-42, 1992. [25] June Li: Estimation of the recovery discharge probability in cat auditory nerve spike trains and computer simulations . PhD thesis, The Johns Hopkins University, Baltimore, MD, May 1991. [26] W. J. McGill and J. P. Goldberg. Pure-tone intensity discrimination and energy detection. J. Acoust. Soc. Am., 44(3):576-81, 1968. [27] George A. Miller. Sensitivity to changes in the intensity of white noise and its relation to masking and loudness. J. Acoust. Soc. Am., 19(4):609-19, July 1947. [28] W. A. Munson. The growth of auditory sensation. J. Acoust. Soc. Am., 19(4):584-91, July 1947. [29] W. W. Peterson, T. G. Birdsall, and W. C. Fox. The theory of signal detectability. Inst. Radio Engrs. Trans. Grp, Inf, Theory, 4:171-212, 1954. [30] Sheila M. Pfafflin and M. V. Mathews. Energy-detection model of monaural auditory detection. J. Acoust. Soc. Am. , 34(12):1842-53, December 1962. [31] E. Port. Ueber die Lautstärke einzelner kurzer Schallimpulse. Acustica, 13:212-23, 1963. [32] David H. Raab and Israel A. Goldberg. Auditory intensity discrimination with bursts of reproducible noise. J. Acoust. Soc. Am. , 57(2):437-47, February 1975. [33] R.R. Riesz. The differential sensitivity of the ear for pure tones. Phys. Rev., 31:867-75, 1928. [34] Bertram Scharf. Loudness. In Edward C. Carterette and Morton P. Friedman, editors, Handbook of Perception, volume IV Hearing, pages 187-242. Academic Press, NY, 1978. [35] M. R. Schroeder, B. S. Atal, and J. L. Hall. Optimizing digital speech coders by exploiting masking properties of the human ear. J. Acoust. Soc. Am., 66(6):1647-52, December 1979. [36] John C. Steinberg and Mark B. Gardner. On the auditory significance of the term hearing loss. J. Acoust. Soc. Am., 11:270-7, January 1940. [37] S.S. Stevens and H. Davis. Hearing, Its Psychology and Physiology. Reprinted by the American Inst. Physics for the Acoust. Soc. Am., 1983, 1983 (1938). [38] M. C. Teich and S. M. Khanna. Pulse-number distribution for the neural spike train in the cat's auditory nerve. J. Acoust. Soc. Am., 77:1110-28, 1985. [39] Raymond N. J. Veldhuis. Bit rates in audio source coding. IEEE J. Select. Areas Commun., 10(1):86-96, January 1992. [40] Xiaoqin Wang and Murray B. Sachs. Neural encoding of single-formant stimuli in the cat i. responses of auditory nerve fibers. J. Neurophysiol., 70(3):1054-75, September 1993. [41] Xiaoqin Wang and Murray B. Sachs. Neural encoding of single-formant stimuli in the cat ii. responses of anteroventral cochlear nucleus units. J. Neurophysiol., (in press), 1994. [42] E. D. Young and Patrick E. Barta. Rate responses of auditory nerve fibers to tones in noise near masked threshold. J. Acoust. Soc. Am., 79(2):426-42, February 1986. [43] Eric D. Young, Jeanne-Marie Robert, and William P. Shofner. Regularity and latency of units in the ventral cochlear nucleus: Implications for unit classification and generation of response properties. J. Neurophysiol., 60(1):1-29, July 1988. [44] I. M. Young and C. H. Wenner. Masking of white noise by pure tone, frequency-modulated tone, and narrow-band noise. J. Acoust. Soc. Am., 41(3):700-6, November 1966. [45] Eberhard Zwicker and Bertram Scharf. A model of loudness summation. Psychological Review, 72(1):3-26, 1965. [46] Josef J. Zwislocki. Temporal summation of loudness: An analysis. J. Acoust. Soc. Am., 46(2):431-441, 1969. [47] Josef J. Zwislocki and Herbert N. Jordan. On the relations of intensity jnds to loudness and neural noise. J. Acoust. Soc. Am., 79(3):772-80, March 1986. Abstract A classical problem in auditory theory is the relation between the loudness L ( 1 ) and the intensity JND Δ I . The intensity JND is frequently expressed in terms of the Weber fraction delined by J( I ) ≡ Δ I / I because it is expected that this ratio should be a constant (i.e., Weber's Law). Somewhat frustrating is the observation that J ( I ) is not a constant for the most elementary case of the pure tone JND. Equally frustrating is the fact that it remains unexplained why Weber's law holds as well as it does. We define the loudness JND Δ L ( I ) as the change in loudness corresponding to Δ I ( I ). Using the Hellman and Hellman (JASA, 1990) theory, we shall compare the Riesz 1928 JND data to the Fletcher and Munson (JASA, 1933) loudness growth data. Based on this comparison we will argue that Δ L ≈ L , and we shall show that Fechner's Law does not hold for loudness. The implication of this result is that over a large range of levels (0 to 70 dB SL) and frequencies (0.25 to 16 kHz) the statistics of the single-trial loudness are Poisson-like, namely where L ≡ ε and σ 2 ( ) ≡ ε( - L ) 2 . This is consistent with the idea that the pure tone loudness code is based on neural discharge rate, at least for this range of frequencies and levels. Definitions: Intensity JND I is the signal intensity ∫( s ( t )) 2 dt The intensity JND is defined by the threshold change in intensity Δ I , where I + Δ I = ∫( s ( t ) + α s ( t )) 2 dt = I + 2α I + α 2 I , where α is the just noticable change in signal gain. Thus Δ I = I (2α + α 2 ), The Weber Fraction is defined as J ≡ Δ I I = 2α + α 2 Definitions: Loudness JND ( I ), the single-trial loudness, is a random variable. L ( I ) ≡ ε ( I ), the loudness, is the ensemble mean of the single-trial loudness. σ 2 ( ) ≡ ε( - L ) 2 is the variance of the single-trial loudness. Taylor series expansion for the (mean) loudness L about intensity I is L ( I + Δ I ) = L ( I ) + Δ I dL dI + HOT Ignoring higher order terms (HOT), we define the loudness JND Δ L as: Δ L ( I ) ≡ L ( I + Δ I ) - L ( I ) = Δ I dL dI | , This says that the loudness JND ΔL ( I ) is related to the intensity JND ΔI ( I ) by the slope of the loudness function, evaluated at I. We call this the small-JND approximation. How do we count JND's? Method: Starting from the Taylor series small-JND approximation Δ L = Δ I dL dI , separate variables and integrate to get the number of JNDs. The total number of intensity JNDs N 12 ≡ N ( I 1 , I 2 ) between I 1 and I 2 is (Riesz. 1928; Fletcher 1929, pages 160,226) The total number of loudness JNDs N 12 ≡ N ( L 1 , L 2 ) between L 1 and L 2 is The number of JND must be the same no matter how they are counted. What is Weber's Law? (1846) Weber's Law says the Weber fraction is a constant ( J ≡ Δ I / I = J 0 ) From Weber's Law and the intensity JND counting function Eq. 2: When Webers's Law holds (e.g., wide band noise), the number of intensity JNDs is proportional to the log() of the intensity ratio. What is Fechner's Hypothesis (1860)? Fechner's hypothesis (or postulate) was that the loudness JND Δ L ( I ) is constant (i.e., independent of I ) Fechner assumed "that the total change in loudness between two intensities I 1 and I 2 may be found by counting the number of JNDs." From Fechner's hypothesis and the JND counting formula Eq. 3: What is Fechner's Law? We obtain Fechner's Law by setting the number of intensity JNDs, assuming Weber's Law holds (Eq. 7), equal to the number of loudness JNDs, assuming Fechner's hypothesis holds (Eq. 9). This equality gives ( L 2 - L 1 ) Δ L = 1 J 0 log I 2 / I 1 . Perhaps it should be called the Weber-Fechner Law because it depends on both Fechner's hypothesis and Weber's Law to be simultaneously true. (This is what Fletcher called it in his 1929 book.) What we would like to show: Fechner's hypothesis (Δ L is constant) is not true Therefore: Fechner's law can not be true. Conclusions: We see directly from the raw data of Riesz and Fletcher-Munson that N 12 = C ( L 2 - L 1 ), where C is almost independent of frequency and intensity, and is close to I. It follows that Δ L = 2 C L ( I ) since by direct integration of Eq. 3 (treating C as a constant) which is the experimental result. Fechner's Hypothesis (constant Δ L ) cannot be true because it leads to (Eq. 9) N 12 = ( L 2 - L 1 )/Δ L . This equation is in serious disagreement with the data, summarized by Eq. 11. DISCUSSION: What does it mean that the loudness JND is proportional to the square root of the loudness (i.e., Δ L ≈ L ( I ) )? ANSWER: From the above analysis: Δ L ≈ L ( I ) ) From signal detection theory: Eliminating Δ L , we conclude that where L ≡ ε and σ 2 ( ) ≡ ε( - L) 2 ). In summary: THE MEAN OF THE SINGLE-TRIAL LOUDNESS IS PROPORTIONAL TO THE VARIANCE OF THE SINGLE-TRIAL LOUDNESS We need a model to explore this further. MODEL ASSUMPTIONS: IT the single-trial loudness ( I ) were Poisson below 70 dB SPL, this would satisfy our experimental observations. If were limited by the refractory nature of pulse propagation at high levels (dead time), this might account for effects above 70 dB. Comments The loudness would be Poisson, as we assume, if the single-trial loudness ( I ) is the total count of spikes due to the stimulus. Based on loudness additivity experiments, it is reasonable to assume that where ( I,t,x ) is a Poisson random variable called the single-trial instantaneous partial loudness, which describes the neural spike rate at intensity I , time t, and place x , and x L is the length of the basilar membrane. This assumption will not be used in the argument but is here to motivate the definitions. For such a dead-time modified Poisson process, r ( )σ 2 ( ) = L ( I ), where σ 2 ( ) ≡ ε( - L) 2 and L ≡ ε . The mean to variance ratio r is called the Fano factor. For levels below the refractory limit, r = 1 (the Poisson process). Due to the refractory nature of action potential propagation, we assume that r ( ) depends on ( I,t,x ). Derivation of the intensity JND counting formula From the small-JND approximation Eq. 1 Δ L = Δ I d L dI . Separate variables, integrate, and from signal detection theory Δ L = d' σ The left side is the definition of N 12 and on the right side we use the Poisson assumption N 12 = ∫ dL d' L ( I )/ r ) d' If we can ignore the level dependence of r and bring it out of the integral, we find N 12 = 2 r d ' ( L 2 - L 1 ) which agrees with Eq. 11. Derivation of Weber fraction given the loudness From the small-JND approximation Eq. 1 Δ L = Δ I dL dI . Solve for J ( I ) ≡ Δ I / I J ( I ) = Δ L /( I dL dI ). From SDT assumption From the Poisson assumption which allows us to calculate J ( I )) given L ( I ) (Hellman and Hellman 1990). What is the relation between the JND and Masking? For the case of the JND: I + Δ I = ∫( s ( t ) + αs ( t )) 2 dt, where s ( t ) is the signal and α is a scale factor. Thus J ≡ Δ I I = 2α + α 2 . For the case of masking: I + Δ I = ∫( m ( t ) + α p ( t )) 2 dt , where rn ( t ) is the masker and α is a scale factor applied to the probe signal p ( t ). We define the intensity of p to be equal to that of rn . Thus J ≡ Δ I I = 2α ∫ rn ( t ) p ( t ) dt I + α 2 . We have found that the interaction term is very important in masking (Sydorenko and Allen 1995) and may be interpreted as the direction cosine between rn and p (i.e., cos () ≡ ( rn , p )/∥ rn ∥) 2 . References Flewcher, H. (1929). Spreech and Hearing . D. Van Nostrand Company, Inc., New York. Fletcher, H. and Munson, W. (1933). "Loudness, its definition, measurement, and calculatiun." Journal of the Acoustical Society of America 5 :82-108. Hellman, W. and Hellman, R. (1990). "Intensity discrimination as the driving force for loudness. Application to pure tones in quiet." Journal of the Acoustical Society of America 87 (3): 1255-1271. Riesz. R. (1928). "Differential intensity sensitivity of the car fur pure tones." Physical Review 31 (2):867-875. Sydorenko. M. and Allen, J. (1995). "The relationship between the just noticalbe difference (jnd) in intensity and the masked threshold of sub-critical bandwidth signals," in Midwinter research meeting . ARO. Talk 218.

Claims: A method of encoding an audio signal (x(i)), the audio signal having a corresponding frequency domain representation, the frequency domain representation comprising one or more sets of frequency domain coefficient signals (S m ), and one or more of said coefficient signals being grouped into at least one frequency band, the method comprising the steps of: computing (10) a first loudness measure (L k ) of a portion of said audio signal associated with the frequency band; based on at least the first loudness measure, quantizing (80) one set of the frequency domain coefficient signals corresponding to the frequency band to generate an encoded audio signal; CHARACTERIZED BY synthesizing (120, 130) the portion of the audio signal in response to the encoded audio signal; computing (30) a second loudness measure (L and k ) of the synthesized portion of the audio signal (x and(i)); generating (20) a differential measure indicative of a difference between the first loudness measure and the second loudness measure; and based on at least a metric value d' k (i) which is a function of a measure of a degree of variability in loudness, and the differential measure, re-quantizing (80) the set of frequency domain coefficient signals to generate a re-quantized encoded audio signal. A method as claimed in claim 1 wherein the measure of a degree of variability in loudness is a function of the first loudness measure. A method as claimed in claim 1 wherein the metric value is a ratio of the differential measure to the measure of a degree of variability in loudness. A method as claimed in claim 1 comprising the step of comparing said metric value with at least a first predetermined threshold. A method as claimed in claim 4 wherein the re-quantizing step comprises the step of adjusting (83) values of the frequency domain coefficient signals in the set to change a phase of the portion of the audio signal when said metric value exceeds the first predetermined threshold, the adjusted versions of the frequency domain coefficients being re-quantized. A method as claimed in claim 5 wherein the adjusting step comprises the step of generating a plurality of random values, each random value corresponding to a respective one of the frequency domain coefficient signals in the set, each coefficient signal value in the set being adjusted by the corresponding random value. A method as claimed in claim 4 wherein said metric value is also compared with a second predetermined threshold in the comparing step, said second predetermined threshold being lower than said first predetermined threshold. A method as claimed in claim 7 wherein the set of frequency domain coefficient signals are initially quantized in accordance with a first quantization level, and the set of frequency domain coefficient signals are re-quantized in accordance with a second, different quantization level when said metric value is below the second predetermined threshold. A method as claimed in claim I wherein the step of computing the first loudness measure comprises the steps of: transforming (11-k) said portion of said audio signal into first subband signals representative thereof, each first subband signal being associated with a respective one of a plurality of predetermined subbands; and associating (81) selected ones of said predetermined subbands with the frequency band, the first loudness measure comprising an individual loudness measure of each of those first subband signals which are associated with the selected subbands, the set of the frequency domain coefficient signals being quantized based on the individual loudness measure of each of those first subbands signals; and wherein the step of computing the second loudness measure comprises the step of transforming (30) the synthesized portion of the audio signal into second subband signals representative thereof, each of said second subband signals being associated with a respective one of the plurality of predetermined subbands, the second loudness measure comprising a loudness measure of each of those second subband signals associated with the selected subbands. A method as claimed in claim 9 wherein the predetermined subbands model a basilar membrane of a cochlea for a bandwidth of the portion of the audio signal. A method as claimed in claim 9 wherein the measure of a degree of variability in loudness varies with the first loudness measure in accordance with a non-homogeneous Poisson process. A method as claimed in claim 11 wherein the measure of a degree of variability in loudness is proportional to a square root of the first loudness measure. A method as claimed in claim 9 wherein the first loudness measure and the second loudness measure are each computed using a nonlinear process. A method as claimed in claim 13 wherein said nonlinear process comprises a squaring function (12-k) and a cube-root function (14-k). Apparatus for encoding an audio signal (x(i)), the audio signal having a corresponding frequency domain representation, the frequency domain representation comprising one or more sets of frequency domain coefficient signals (S m ), and one or more of said coefficient signals being grouped into at least one frequency band, the apparatus comprising: a processor (10, 80, 120, 130, 30, 20) for computing a first loudness measure (L k ) of a portion of said audio signal associated with the frequency band; based on at least the first loudness measure, quantizing one set of the frequency domain coefficient signals corresponding to the frequency band to generate an encoded audio signal; CHARACTERIZED in that the processor comprises means for synthesizing the portion of the audio signal in response to the encoded audio signal; means for computing a second loudness measure (L and k ) of the synthesized portion of the audio signal (x and(i)) and means for generating a differential measure indicative of a difference between the first loudness measure and the second loudness measure; and based on at least a metric value d' k (i) which is a function of a measure of a degree of variability in loudness, and the differential measure, re-quantizing the set of frequency domain coefficient signals to generate a re-quantized encoded audio signal. Apparatus as claimed in claim 15 wherein the measure of a degree of variability in loudness is a function of the first loudness measure. Apparatus as claimed in claim 15 wherein the metric value is a ratio of said differential measure to the measure of a degree of variability in loudness. Apparatus as claimed in claim 15 wherein said metric value is compared with at least a first predetermined threshold. Apparatus as claimed in claim 18 wherein the processor comprises an adjuster (83) for adjusting values of the set of frequency domain coefficient signals to change a phase of the portion of the audio signal when said metric value exceeds the first predetermined threshold, the adjusted versions of the frequency domain coefficients being re-quantized. Apparatus as claimed in claim 19 wherein the adjuster comprises a generator for generating a plurality of random values, each random value corresponding to a respective one of the frequency domain coefficient signals in the set, each coefficient signal value in the set being adjusted by the corresponding random value. Apparatus as claimed in claim 18 wherein said metric value is also compared with a second predetermined threshold, said second predetermined threshold being lower than said first predetermined threshold. Apparatus as claimed in claim 21 wherein the set of frequency domain coefficient signals are initially quantized in accordance with a first quantization level, and the set of frequency domain coefficient signals are re-quantized in accordance with a second, different quantization level when said metric value is below the second predetermined threshold. Apparatus as claimed in claim 15 comprising: a first filter arrangement (11-k) for transforming said portion of said audio signal into first subband signals representative thereof, each first subband signal being associated with a respective one of a plurality of predetermined subbands; a mapper (81) for associating selected ones of said predetermined subbands with the frequency band, the first loudness measure comprising an individual loudness measure of each of those first subband signals which are associated with the selected subbands, the set of frequency domain coefficient signals being quantized based on the individual loudness measure of each of those first subband signals; and a second filter arrangement (30) for transforming the synthesized portion of the audio signal into second subband signals representative thereof, each of said second subband signals being associated with a respective one of the plurality of predetermined subbands, the second loudness measure comprising a loudness measure of each of those second subband signals associated with the selected subbands. Apparatus as claimed in claim 23 wherein the predetermined subbands model a basilar membrane of a cochlea for a bandwidth of the portion of the audio signal. Apparatus as claimed in claim 23 wherein the measure of a degree of variability in loudness varies with the first loudness measure in accordance with a non-homogeneous Poisson process. Apparatus as claimed in claim 25 wherein the measure of a degree of variability in loudness is proportional to a square root of the first loudness measure. Apparatus as claimed in claim 23 wherein the processor comprises at least one sub-processor (12-k, 14-k) for computing each of the first loudness measure and the second loudness measure, in accordance with a nonlinear process. Apparatus as claimed in claim 27 wherein the at least one sub-processor is capable of performing a squaring function and a cube-root function.

Inventors: ALLEN JONATHAN B, SINHA DEEPEN, SYDORENKO MARK R, ALLEN, JONATHAN B., SINHA, DEEPEN, SYDORENKO, MARK R.
Applicants: AT & T CORP, AT&T IPM CORP.
Application year: 1996
Main classification: H04B1
IPC codes: H04B1, G10L19, G11B20, H03M7