Tight bounds for the median of a gamma distribution.

The median of a standard gamma distribution, as a function of its shape parameter k, has no known representation in terms of elementary functions. In this work we prove the tightest upper and lower bounds of the form 2-1/k(A + k): an upper bound with A = e-γ (with γ being the Euler-Mascheroni constant) and a lower bound with [Formula: see text]. These bounds are valid over the entire domain of k > 0, staying between 48 and 55 percentile. We derive and prove several other new tight bounds in support of the proofs.

On closed-form tight bounds and approximations for the median of a gamma distribution.

The median of a gamma distribution, as a function of its shape parameter k, has no known representation in terms of elementary functions. In this work we use numerical simulations and asymptotic analyses to bound the median, finding bounds of the form 2-1/k(A + Bk), including an upper bound that is tight for low k and a lower bound that is tight for high k. These bounds have closed-form expressions for the constant parameters A and B, and are valid over the entire range of k > 0, staying between 48 and 55 percentile. Furthermore, an interpolation between these bounds yields closed-form expressions that more tightly bound the median, with absolute and relative margins to both upper and lower bounds approaching zero at both low and high values of k. These bound results are not supported with analytical proofs, and hence should be regarded as conjectures. Simple approximation expressions between the bounds are also found, including one in closed form that is exact at k = 1 and stays between 49.97 and 50.03 percentile.

Auditory Measures for the Next Billion Users.

A range of new technologies have the potential to help people, whether traditionally considered hearing impaired or not. These technologies include more sophisticated personal sound amplification products, as well as real-time speech enhancement and speech recognition. They can improve user's communication abilities, but these new approaches require new ways to describe their success and allow engineers to optimize their properties. Speech recognition systems are often optimized using the word-error rate, but when the results are presented in real time, user interface issues become a lot more important than conventional measures of auditory performance. For example, there is a tradeoff between minimizing recognition time (latency) by quickly displaying results versus disturbing the user's cognitive flow by rewriting the results on the screen when the recognizer later needs to change its decisions. This article describes current, new, and future directions for helping billions of people with their hearing. These new technologies bring auditory assistance to new users, especially to those in areas of the world without access to professional medical expertise. In the short term, audio enhancement technologies in inexpensive mobile forms, devices that are quickly becoming necessary to navigate all aspects of our lives, can bring better audio signals to many people. Alternatively, current speech recognition technology may obviate the need for audio amplification or enhancement at all and could be useful for listeners with normal hearing or with hearing loss. With new and dramatically better technology based on deep neural networks, speech enhancement improves the signal to noise ratio, and audio classifiers can recognize sounds in the user's environment. Both use deep neural networks to improve a user's experiences. Longer term, auditory attention decoding is expected to allow our devices to understand where a user is directing their attention and thus allow our devices to respond better to their needs. In all these cases, the technologies turn the hearing assistance problem on its head, and thus require new ways to measure their performance.

Quadratic distortion in a nonlinear cascade model of the human cochlea.

The cascade of asymmetric resonators with fast-acting compression (CARFAC) is a cascade filterbank model that performed well in a comparative study of cochlear models, but exhibited two anomalies in its frequency response and excitation pattern. It is shown here that the underlying reason is CARFAC's inclusion of quadratic distortion, which generates DC and low-frequency components that in a real cochlea would be canceled by reflections at the helicotrema, but since cascade filterbanks lack the reflection mechanism, these low-frequency components cause the observed anomalies. The simulations demonstrate that the anomalies disappear when the model's quadratic distortion parameter is zeroed, while other successful features of the model remain intact.

A 6 µW per channel analog biomimetic cochlear implant processor filterbank architecture with across channels AGC.

A new analog cochlear implant processor filterbank architecture of increased biofidelity, enhanced across-channel contrast and very low power consumption has been designed and prototyped. Each channel implements a biomimetic, asymmetric bandpass-like One-Zero-Gammatone-Filter (OZGF) transfer function, using class-AB log-domain techniques. Each channel's quality factor and suppression are controlled by means of a new low power Automatic Gain Control (AGC) scheme which is coupled across the neighboring channels and emulates lateral inhibition (LI) phenomena in the auditory system. Detailed measurements from a five-channel silicon IC prototype fabricated in a 0.35 µm AMS technology confirm the operation of the coupled AGC scheme and its ability to enhance contrast among channel outputs. The prototype is characterized by an input dynamic range of 92 dB while consuming only 28 µW of power in total ( ∼ 6 µW per channel) under a 1.8 V power supply. The architecture is well-suited for fully-implantable cochlear implants.

Modelling the distortion produced by cochlear compression.

Lyon (J Acoust Soc Am 130:3893-3904, 2011) has described how a cascade of simple asymmetric resonators (CAR) can be used to simulate the filtering of the basilar membrane and how the gain of the resonators can be manipulated by a feedback network to simulate the fast-acting compression (FAC) characteristic of cochlear processing. When the compression is applied to complex tones, each pair of primary components produces both quadratic and cubic distortion tones (DTs), and the cascade architecture of the CAR-FAC system propagates them down to their appropriate place along the basilar membrane, where they combine additively with each other and any primary components at that frequency. This suggests that CAR-FAC systems might be used to study the role of compressive distortion in the perception of complex sounds and that behavioural measurements of cochlear distortion data might be useful when tuning the parameters of CAR-FAC systems.

Cascades of two-pole-two-zero asymmetric resonators are good models of peripheral auditory function.

A cascade of two-pole-two-zero filter stages is a good model of the auditory periphery in two distinct ways. First, in the form of the pole-zero filter cascade, it acts as an auditory filter model that provides an excellent fit to data on human detection of tones in masking noise, with fewer fitting parameters than previously reported filter models such as the roex and gammachirp models. Second, when extended to the form of the cascade of asymmetric resonators with fast-acting compression, it serves as an efficient front-end filterbank for machine-hearing applications, including dynamic nonlinear effects such as fast wide-dynamic-range compression. In their underlying linear approximations, these filters are described by their poles and zeros, that is, by rational transfer functions, which makes them simple to implement in analog or digital domains. Other advantages in these models derive from the close connection of the filter-cascade architecture to wave propagation in the cochlea. These models also reflect the automatic-gain-control function of the auditory system and can maintain approximately constant impulse-response zero-crossing times as the level-dependent parameters change.

Sound retrieval and ranking using sparse auditory representations.

To create systems that understand the sounds that humans are exposed to in everyday life, we need to represent sounds with features that can discriminate among many different sound classes. Here, we use a sound-ranking framework to quantitatively evaluate such representations in a large-scale task. We have adapted a machine-vision method, the passive-aggressive model for image retrieval (PAMIR), which efficiently learns a linear mapping from a very large sparse feature space to a large query-term space. Using this approach, we compare different auditory front ends and different ways of extracting sparse features from high-dimensional auditory images. We tested auditory models that use an adaptive pole-zero filter cascade (PZFC) auditory filter bank and sparse-code feature extraction from stabilized auditory images with multiple vector quantizers. In addition to auditory image models, we compare a family of more conventional mel-frequency cepstral coefficient (MFCC) front ends. The experimental results show a significant advantage for the auditory models over vector-quantized MFCCs. When thousands of sound files with a query vocabulary of thousands of words were ranked, the best precision at top-1 was 73% and the average precision was 35%, reflecting a 18% improvement over the best competing MFCC front end.