If the core idea of Moss' theory 8 is valid for mammalian auditory information processing (as we strongly believe), one has to accept that a mammalian auditory system is capable of recovering an original signal h(t) from the noisy signal f(t) expressed in (1). Guided by Occam's razor, we expect the mechanism of (1) to be generally applicable in the mammalian auditory system. As a first step, it is natural to analyze the energies carried by the signal h(t) and noise e(t). Indeed, in many cases, e.g. 9, the energy addition of the signal and noise is sufficient to explain SR. Moss and his coworkers categorized such SR as Type E (for energy). However, SR has also been observed when the energy addition of the signal and noise is not sufficient to explain the enhancement in sensory perception 10. Moss et al. used the concept of information modulation to explain this observation and categorized such SR as Type I (for information). In other words, the occurrence of of Type I SR relies on characteristics of the signal other than energy. Still, the distinction between Types E and I SR has the disadvantage of requiring evolution of multiple mechanisms for SR in the mammalian auditory system, which would seem less likely than evolution of a single unitary mechanism.

At this point, it is instructional to consider the historical progression of research on signal processing in the latter half of the twentieth century. (We refer the reader to section 1 of 11 for a summary of this history.) Filtering out noise from a noisy signal f(t) as expressed in (1) is a major concern of the community of signal processing, where this task is termed "de-noising". Early researchers developed a substantial number of algorithms for de-noising. However, most of the de-noising algorithms were mathematically proven to be optimal when the characteristics of original signal h(t) could be known to the algorithm in advance. De-noising was thought to require information modulation. In 1994, Donoho and Johnstone 12 dramatically changed the modern understanding of de-noising by proposing wavelet shrinkage. Importantly, wavelet-based algorithms do not require a priori knowledge of the characteristics of the signal (see below) and can be implemented more efficiently than earlier methods such as the fast Fourier transform (FFT).

Since our proposed model employs a recent improvement on analysis of wavelet shrinkage, we will mention some details related to this algorithm. Recall that an original signal is modeled by a function h(t): t ∈ [0,1] ↦ ℝ⁺. In the mammalian auditory system, h(t) necessarily has a certain degree of "smoothness". In the literature on signal processing, this is formulated as a requirement that h(t) belongs to a Hölder class. Recall that a Hölder class Λ^α(M) is a family of functions, which is determined by two parameters α and M as follows: Let ℝ^[0,1] denote the set of all functions defined on [0,1]. For 0 <α ≤ 1, Λ^α(M) def¯¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaadbaqaaiabbsgaKjabbwgaLjabbAgaMbaaaaa@30B0@ {h ∈ ℝ^[0,1]: (∀x₁, x₂ ∈ [0,1]), |h(x₁) - h(x₂)| ≤ M|x₁ - x₂|^α}. For 1 <α, Λ^α(M) def¯¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaadbaqaaiabbsgaKjabbwgaLjabbAgaMbaaaaa@30B0@ {h ∈ ℝ^[0,1]: (∀x ∈ [0,1])|h'(x)| ≤ M, h^⌊α⌋ exists, and (∀x₁, x₂ ∈ [0,1])|h^⌊α⌋ (x₁) - h^⌊α⌋ (x₂)| ≤ M|x₁ - x₂|^{α - ⌊α⌋}}. It is straightforward to see that the concept of Hölder class contains information modulation. For example, a sine wave belongs to a Hölder class with 1 <α; however, the higher the frequency of the wave is, the larger the M must be. Before the advent of wavelet shrinkage, proposed de-noising algorithms required α and M as part of their inputs. Unlike the earlier algorithms, wavelet shrinkage only requires that h(t) belongs to a Hölder class, without further knowledge of α and M. Therefore, wavelet shrinkage provides a universal solution for de-noising. Strikingly, it was mathematically proven that the recovery of a signal by wavelet shrinkage is as good as that obtained by earlier algorithms requiring specific knowledge of α and M 12. Therefore, based on wavelet shrinkage we can propose a model that universally explains SR, including both Types E and I in the same model.

To realize the new model, we must first overcome a mathematical difficulty. Throughout the rest of this paper, we will always denote by h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t) the signal recovered from a noisy signal as expressed in (1). In signal processing, the performance of a de-noising algorithm is mainly judged by the closeness between the recovered signal h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t) and original signal h(t), and this closeness is measured in terms of L₂ norm ||h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t) - h(t)||₂ def¯¯∫01(h˜(t)−h(t))2dt MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaadbaqaaiabbsgaKjabbwgaLjabbAgaMbaadaGcaaqaamaapedabaGaeiikaGIafmiAaGMbaGaacqGGOaakcqWG0baDcqGGPaqkcqGHsislcqWGObaAcqGGOaakcqWG0baDcqGGPaqkcqGGPaqkdaahaaWcbeqaaiabikdaYaaakiabdsgaKjabdsha0bWcbaGaeGimaadabaGaeGymaedaniabgUIiYdaaleqaaaaa@4463@. For SR in the mammalian auditory system, however, when h(t) has few sharp transients which are lost in h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@, one may still have ||h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t) - h(t)||₂ ≈ 0. Of greater concern, for a given original signal h(t), the recovered signal h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t) is random. This is because the noise e(t) is random, and hence, the noisy signal f(t) = h(t) + e(t) is random. While in signal processing, the performance of a de-noising algorithm such as wavelet shrinkage (see 12), is judged by E [||h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t) - h(t)||₂], the average closeness between h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t) and h(t), it would clearly be unacceptable to claim that SR enhances mammalian auditory information processing on the average. Fortunately, the performance of wavelet shrinkage can be judged by sup_1≤t≤1 |h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t) - h(t)| with very high probability 13. That is, the signal recovered by wavelet shrinkage is almost surely (with probability 1) close to an original signal, even when examined in a pointwise fashion. In the next section, we will propose a model for SR based on this new result of Hong and Birget 13. Since in mammalian hearing any part of h(t) may contain crucial information, a necessary condition to recover an original signal h(t) from the noisy signal f(t) = h(t) + e(t) is that the noisy signal be detectable. The proposed model will show that in mammalian hearing, SR occurs if and only if the noisy signal is detectable. In addition, we will demonstrate that the model explains both so-called Types E and I SR in a unitary mechanism.

In the final section, we will indicate how observed physiological structures and functions in mammalian auditory system are consistent with and suggest the proposed model.

Recall that in SR the role of noise e(t) is to sample an original signal h(t) generating a noisy signal f(t) = h(t) + e(t); and that the sampling rate needs to be greater than the frequency of h(t) 7. Mathematically, the sampling of the original signal by noise indicates that SR has a discrete nature. A mammalian auditory system can therefore be viewed as a "device" with the following characteristics. Let n, a large positive integer, denote the sampling rate.

Input: At time instants t = in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@, i = 1, 2, ..., n, an original subthreshold signal h(t), t ∈ [0,1], is sampled by a noise e(t). This results in the noisy samples f(in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@) = h(in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@) + e(in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@).

Output: A recovered signal h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t) obtained by processing the noisy samples f(in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@), i = 1, 2, ..., n.

Since the noise e(t) is intrinsic, i.e., generated within the mammalian auditory system, the intensity is clearly always bounded. That is, the random variable e(t) is bounded. We assume there are two constants 0 ≤ a <b such that e(t) ∈ [a,b]. The criterion for the closeness between the recovered signal h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t) and original signal h(t) is

the smallness of sup ⁡ t ∈ [ 0 , 1 ] | h ˜ ( t ) − h ( t ) |       ( 2 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqG0baDcqqGObaAcqqGLbqzcqqGGaaicqqGZbWCcqqGTbqBcqqGHbqycqqGSbaBcqqGSbaBcqqGUbGBcqqGLbqzcqqGZbWCcqqGZbWCcqqGGaaicqqGVbWBcqqGMbGzdaWfqaqaaiGbcohaZjabcwha1jabcchaWbWcbaGaemiDaqNaeyicI4Saei4waSLaeGimaaJaeiilaWIaeGymaeJaeiyxa0fabeaakiabcYha8jqbdIgaOzaaiaGaeiikaGIaemiDaqNaeiykaKIaeyOeI0IaemiAaGMaeiikaGIaemiDaqNaeiykaKIaeiiFaWNaaCzcaiaaxMaadaqadaqaaiabikdaYaGaayjkaiaawMcaaaaa@5EDF@

where the meaning of the standard mathematical notation "sup" is as follows. Consider all upper bounds for_P(·) where (·) stands for an expression and P is a predicate which the expression must satisfy. Then, sup_P(·) is the smallest possible of all the upper bounds.

The procedure to process the noisy samples follows from the notion of wavelet shrinkage in signal processing. It consists of two linear transforms and one non-linear thresholding. That is, we model a mammalian auditory system as a non-linear system.

First, a linear transform is carried out to decompose the noisy samples in the cochlea. For simplicity, in accordance with (1) we denote the noisy samples by f. Auditory information carried by the original signal h(t) is encoded by the changes in both amplitude and frequency. Hence, retrieval of the information from h(t) requires its decomposition according to both amplitude and frequency. Now, h(t) is mixed with a noise e(t), generating f(t); and the function of the auditory system is to process the noisy samples f. Thus, a decomposition of f is necessary at the very beginning of the procedure. The principle for such a decomposition is as follows: f is viewed as an element in a function space, usually L²[0,1]; and then, with the choice of a basis of L²[0,1], it finds the projections of f on each component in the basis. Thus, the mathematical quality of the decomposition is determined by the basis. Technically, during mammalian auditory information processing, the noisy samples f are decomposed to allow recovery of h(t). Any basis that is chosen for the decomposition must be sensitive in detecting changes in both amplitude and frequency at the same time. It is mathematically proven that a wavelet basis is the best choice for such a decomposition. While there are many wavelet bases, from the Haar to the Daubechies, we do not specify a particular wavelet basis in the proposed model, except that it is required to be orthonormal.

It must be noted that once a wavelet basis is chosen, the linear transform is constant in the following sense. Recall that in a standard way, a linear transform can be represented by a matrix and vice versa. The matrix representing this linear transform is constant if all entries in the matrix are constants. From a viewpoint of physiology, this indicates that once a mammalian auditory system is developed, it may decompose signals to filter out noise in a fixed manner.

Since the first linear transform decomposes the noisy samples, it is necessary to filter out the noise right after this transform. A non-linear thresholding is applied immediately as the second step in the procedure. It also must be noted that the threshold here is again a constant if the sampling rate n is regarded as fixed. The output from the second step is the decomposed noisy samples with the noise filtered out. Thus, the third (final) step of the procedure is to re-compose the filtered output of the second step. It is carried out by a linear transform, which again is constant in the sense mentioned above for the first step.

Mathematically, we describe the three steps as follows. Two related n × n orthonormal matrices V and V˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGwbGvgaacaaaa@2DF0@ respectively for a discrete wavelet transform (DWT) and its inverse are used for the first and third steps, respectively.

V = ( v 11 v 12 … v 1 n v 21 v 22 … v 2 n … v n 1 v n 2 … v n n ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@57C7@

and

V ˜ = ( v 11 ˜ v 12 ˜ … v 1 n ˜ v 21 ˜ v 22 ˜ … v 2 n ˜ … v n 1 ˜ v n 2 ˜ … v n n ˜ ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5EA8@

For a mammalian auditory system, the two matrices are fixed, i.e. v_ij and vij˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaiaaqaaiabdAha2naaBaaaleaacqWGPbqAcqWGQbGAaeqaaaGccaGLdmaaaaa@31D1@ are fixed during development, and they are used to process any noisy signal entering the system.

A threshold for the second step is defined as

λ n , δ = c ⋅ ( b − a ) ⋅ ( 1 + 2 1 + δ ) ln ⁡ 2 ) log ⁡ 2 n n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF7oaBdaWgaaWcbaGaemOBa4MaeiilaWIae8hTdqgabeaakiabg2da9iabdogaJjabgwSixlabcIcaOiabdkgaIjabgkHiTiabdggaHjabcMcaPiabgwSixpaabmaabaGaeGymaeJaey4kaSIaeGOmaiZaaOaaaeaacqaIXaqmcqGHRaWkcqWF0oazcqGGPaqkcyGGSbaBcqGGUbGBcqaIYaGmaSqabaaakiaawIcacaGLPaaadaGcaaqaamaalaaabaGagiiBaWMaei4Ba8Maei4zaC2aaSbaaSqaaiabikdaYaqabaGccqWGUbGBaeaacqWGUbGBaaaaleqaaaaa@5353@

where c > 0 is a parameter determined according to which wavelet basis is used, and δ > 0 is a parameter related to the accuracy of the auditory information processing. The threshold λ_n,δ is different from the threshold s used by the spike train. Notice that for an auditory system, λ_n,δ is fixed (recall that [a, b] is the range of the intrinsic noise) and is used to process any noisy signal entering the auditory system.

In what follows, we will use some simplified notations. We let h_i denote h(in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@), i = 1,2, ..., n; and let h = (h₁ h₂ ... h_n)^T where (·)^T stands for the transposition of a vector (·). We apply the same notation to e_i, f_i, and h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@_i. Now, we mathematically formulate the three steps mentioned above.

Step 1 Discrete wavelet transform (DWT)

n ( η 1 η 2 ⋮ η n ) ⇐ ( v 11 v 12 … v 1 n v 21 v 22 … v 2 n … v n 1 v n 2 … v n n ) ( h 1 h 2 ⋮ h n ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabd6gaUbWcbeaakmaabmaabaqbaeqabqqaaaaabaacciGae83TdG2aaSbaaSqaaiabigdaXaqabaaakeaacqWF3oaAdaWgaaWcbaGaeGOmaidabeaaaOqaaiabl6Uinbqaaiab=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@715D@

where (η₁ η₂ ... η_n)^T is the decomposed noisy samples.

Step 2 Thresholding

( ζ 1 ζ 2 ⋮ ζ n ) ⇐ ( η 1 η 2 ⋮ η n ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqadaqaauaabeqaeeaaaaqaaGGaciab=z7a6naaBaaaleaacqaIXaqmaeqaaaGcbaGae8NTdO3aaSbaaSqaaiabikdaYaqabaaakeaacqWIUlstaeaacqWF2oGEdaWgaaWcbaGaemOBa4gabeaaaaaakiaawIcacaGLPaaacqGHqhc3daqadaqaauaabeqaeeaaaaqaaiab=D7aOnaaBaaaleaacqaIXaqmaeqaaaGcbaGae83TdG2aaSbaaSqaaiabikdaYaqabaaakeaacqWIUlstaeaacqWF3oaAdaWgaaWcbaGaemOBa4gabeaaaaaakiaawIcacaGLPaaaaaa@4810@

where (ζ₁ ζ₂ ... ζ_n)^T is the result of filtering out the noise from the decomposed noisy samples, which is obtained by

ζ i = { η i i f | η i | ≥ λ n , δ 0 otherwise for  i = 1 , ... , n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF2oGEdaWgaaWcbaGaemyAaKgabeaakiabg2da9maaceqabaqbaeqabeGaaaqaauaabaqaciaaaeaacqWF3oaAdaWgaaWcbaGaemyAaKgabeaaaOqaaGqaaiab+LgaPjab+zgaMjabcYha8jab=D7aOnaaBaaaleaacqWGPbqAaeqaaOGaeiiFaWNaeyyzImRae83UdW2aaSbaaSqaaiabd6gaUjabcYcaSiab=r7aKbqabaaakeaacqaIWaamaeaacqqGVbWBcqqG0baDcqqGObaAcqqGLbqzcqqGYbGCcqqG3bWDcqqGPbqAcqqGZbWCcqqGLbqzaaaabaGaeeOzayMaee4Ba8MaeeOCaiNaeeiiaaIaemyAaKMaeyypa0JaeGymaeJaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaemOBa4gaaaGaay5Eaaaaaa@615A@

Step 3 Inverse of DWT

( h ˜ 1 h ˜ 2 ⋮ h ˜ n ) ⇐ ( v 11 ˜ v 12 ˜ … v 1 n ˜ v 21 ˜ v 22 ˜ … v 2 n ˜ … v n 1 ˜ v n 2 ˜ … v n n ˜ ) ( ζ 1 ζ 2 ⋮ ζ n ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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z7a6naaBaaaleaacqWGUbGBaeqaaaaaaOGaayjkaiaawMcaaaaa@7705@

We refer the reader to chapters 4 and 6 of 14 for details on DWT, thresholding, and the inverse of DWT. This work also describes the nature of simple constant matrices V and V˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGwbGvgaacaaaa@2DF0@, once a wavelet basis is chosen and the sampling n is given. Importantly, the three steps can be carried out within time of an order of n, i.e., the amount of time needed to process the noisy samples is only proportional to the number n of noisy samples. As the sampling rate n can be thought as fixed in a mammalian auditory system, the three steps can process an auditory signal as a stream. In other words, as a linear function of n, the processing rate of the proposed model is as rapid as conceivably possible. In contrast, the processing rate of a FFT is n × log n, so that the delay in processing with an increase in n would preclude online processing.

We set a baseline for the mammalian auditory system. In terms of hearing, this baseline is understood as "absolute silence". Mathematically, the baseline is represented by a constant 0. All signals, including noise, are measured against this baseline and are evaluated in terms of pressure. In this way, h(t), e(t), f(t) and h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t) take positive values. Setting up such a baseline serves the following two purposes:

1. The baseline is fixed, yielding a metric system. For any given mammalian auditory system, our analysis of SR depends on this fixed metric system.

2. While our analysis of SR is not concerned with energy, we will be able to compute energy based on the metric system, so as to to show how the proposed model naturally covers all types of SR in the mammalian auditory system.

We can assume that the threshold for a stimulus to be detected by the system is s > 0. Here, s is a constant against the baseline, i.e., the threshold is fixed (see Figure 1). Recall that within time interval [0,1], noise e(t) samples an original subthreshold signal h(t), generating the noisy samples f_i = h_i + e_i where f_i = f(in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@), h_i = h(in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@), e_i = e(in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@), i = 1, 2, ..., n. Since in mammalian auditory information processing, any h_i may contain a critical part of the information carried by h, a necessary condition for SR to occur is

f_i = h_i + e_i ≥ s for all 1 ≤ i ≤ n     (3)

That is, all noisy samples must be detectable. However, the detectability of the samples does not at all automatically imply that the information carried by h(t) is retrievable, since at the moment of acquisition, the samples are a mixture of signal h(t) and noise e(t).

Our goal is to use the proposed model to prove that the necessary condition expressed in (3) is also a sufficient condition for SR to occur. In precise mathematical terms, if all the noisy samples are detectable, then the information carried by h(t) can be retrieved almost surely.

Recall that an original signal is represented by a function h(t): t ∈ [0,1] ↦ ℝ⁺ in a Hölder class Λ^α(M). It is straightforward to see that for all t₁, t₂ ∈ [0,1]

|h(t₁) - h(t₂)| ≤ M|t₁ - t₂|^α

which implies that for all 1 ≤ i ≤ n and for all t∈[i−1n,in] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG0baDcqGHiiIZdaWadaqaamaalaaabaGaemyAaKMaeyOeI0IaeGymaedabaGaemOBa4gaaiabcYcaSmaalaaabaGaemyAaKgabaGaemOBa4gaaaGaay5waiaaw2faaaaa@39F0@

| h ( t ) − h i | ≤ M ( 1 n ) α       ( 4 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGG8baFcqWGObaAcqGGOaakcqWG0baDcqGGPaqkcqGHsislcqWGObaAdaWgaaWcbaGaemyAaKgabeaakiabcYha8jabgsMiJkabd2eannaabmaabaWaaSaaaeaacqaIXaqmaeaacqWGUbGBaaaacaGLOaGaayzkaaWaaWbaaSqabeaaiiGacqWFXoqyaaGccaWLjaGaaCzcamaabmaabaGaeGinaqdacaGLOaGaayzkaaaaaa@4465@

In physiological terms, (4) indicates that a mammalian auditory system uses its intrinsic noise to sample the original signal at a high rate so that information loss between two consecutive samples is negligible. On the other hand, (4) implies that the capability of a mammalian auditory system is limited; it has to lose some information between two consecutive noisy samples. Mathematically, with (4) we need to focus only on h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@ = (h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@₁ h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@₂ ... h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@_n), the recovery from the noisy samples obtained in Step 3.

In signal processing, the criterion to judge the quality of the recovery h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@ is the average squared error E[Qavg(n)] def¯¯ E[1n∑i=1n(hi−h˜i)2] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGfbqrcqGGBbWwcqWGrbqudaWgaaWcbaGaeeyyaeMaeeODayNaee4zaCgabeaakiabcIcaOiabd6gaUjabcMcaPiabc2faDjabbccaGmaameaabaGaeeizaqMaeeyzauMaeeOzaygaaiabbccaGiabbweafnaadmaabaWaaSaaaeaacqaIXaqmaeaacqWGUbGBaaWaaabmaeaacqGGOaakcqWGObaAdaWgaaWcbaGaemyAaKgabeaakiabgkHiTiqbdIgaOzaaiaWaaSbaaSqaaiabdMgaPbqabaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaaaeaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGUbGBa0GaeyyeIuoaaOGaay5waiaaw2faaaaa@5418@, yet, as discussed in the previous section, this criterion is not acceptable for mammalian auditory information processing.

It was mathematically proven that for a Hölder class Λ^α(M) the best that any de-noising algorithm can achieve is

E [ Q avg ( n ) ] ≤ C ⋅ ( log ⁡ 2 n n ) 2 α 1 + 2 α       ( 5 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieaacqWFfbqrcqGGBbWwcqWGrbqudaWgaaWcbaGaeeyyaeMaeeODayNaee4zaCgabeaakiabcIcaOiabd6gaUjabcMcaPiabc2faDjabgsMiJkabdoeadjabgwSixpaabmaabaWaaSaaaeaacyGGSbaBcqGGVbWBcqGGNbWzdaWgaaWcbaGaeGOmaidabeaakiabd6gaUbqaaiabd6gaUbaaaiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaeGOmaidcciGae4xSdegabaGaeGymaeJaey4kaSIaeGOmaiJae4xSdegaaaaakiaaxMaacaWLjaWaaeWaaeaacqaI1aqnaiaawIcacaGLPaaaaaa@5284@

where C is a constant (see section 2 of 11). One would expect that we can sharpen (5) by considering the probabilistic behavior of

Q max ⁡ ( n ) def ¯ ¯ max ⁡ 1 ≤ i ≤ n { | h i − h ˜ i | }       ( 6 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaGagiyBa0MaeiyyaeMaeiiEaGhabeaakiabcIcaOiabd6gaUjabcMcaPmaameaabaGaeeizaqMaeeyzauMaeeOzaygaamaaxababaGagiyBa0MaeiyyaeMaeiiEaGhaleaacqaIXaqmcqGHKjYOcqWGPbqAcqGHKjYOcqWGUbGBaeqaaOWaaiWabeaacqGG8baFcqWGObaAdaWgaaWcbaGaemyAaKgabeaakiabgkHiTiqbdIgaOzaaiaWaaSbaaSqaaiabdMgaPbqabaGccqGG8baFaiaawUhacaGL9baacaWLjaGaaCzcamaabmaabaGaeGOnaydacaGLOaGaayzkaaaaaa@549A@

For the moment, suppose that we can prove that Qmax⁡(n)≤C⋅(log⁡2nn)2α1+2α MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaGagiyBa0MaeiyyaeMaeiiEaGhabeaakiabcIcaOiabd6gaUjabcMcaPiabgsMiJkabdoeadjabgwSixpaabmaabaWaaSaaaeaacyGGSbaBcqGGVbWBcqGGNbWzdaWgaaWcbaGaeGOmaidabeaakiabd6gaUbqaaiabd6gaUbaaaiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaeGOmaidcciGae8xSdegabaGaeGymaeJaey4kaSIaeGOmaiJae8xSdegaaaaaaaa@4B34@ almost surely. This will not violate (5); but will strengthen it so that we can use (6) in the analysis of SR. However, there is a technical flaw in (6), which is an issue commonly overlooked in the current analysis on SR.

First, (5) and (6) can be used in signal processing since noise is always considered as a random variable with zero mean. However, in a mammalian auditory system, noise is a random variable with positive mean. For a mammalian auditory system, the baseline can logically be set at absolute silence and mathematically fixed at 0. When noise is used to sample an original signal, then it must be measured above the baseline, and hence, must have a positive mean. Accordingly, noise with a positive mean was used in 9. Thus, we would expect h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@_i > h_i which in mathematical terms can be expressed as

h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@_i = h_i + W MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFwe=vaaa@384D@_i

Obviously, W MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFwe=vaaa@384D@_i must be a constant; for otherwise, h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@_i, the recovery from the noisy samples, will be skewed, which may cause a severe loss of information carried by h_i (the original signal). On the other hand, W MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFwe=vaaa@384D@_i are from the noise. Recall that we assumed that the noise is represented by a bounded random variable e(t) with 0 ≤ a ≤ e(t) ≤ b where a <b are constants. Without loss of generality, we let the mean of this random variable be m = a+b2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdggaHjabgUcaRiabdkgaIbqaaiabikdaYaaaaaa@3128@ > 0. The best scenario that one can expect is W MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFwe=vaaa@384D@_i = m almost surely. For the moment, suppose this can be proven. Then, we can rewrite (6) as

Q max ⁡ ˜ ( n ) def ¯ ¯ max ⁡ 1 ≤ i ≤ n { | ( h i + m ) − h ˜ i | } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaiaaqaaiabdgfarnaaBaaaleaacyGGTbqBcqGGHbqycqGG4baEaeqaaaGccaGLdmaacqGGOaakcqWGUbGBcqGGPaqkdaadbaqaaiabbsgaKjabbwgaLjabbAgaMbaadaWfqaqaaiGbc2gaTjabcggaHjabcIha4bWcbaGaeGymaeJaeyizImQaemyAaKMaeyizImQaemOBa4gabeaakmaacmqabaGaeiiFaWNaeiikaGIaemiAaG2aaSbaaSqaaiabdMgaPbqabaGccqGHRaWkcqWGTbqBcqGGPaqkcqGHsislcuWGObaAgaacamaaBaaaleaacqWGPbqAaeqaaOGaeiiFaWhacaGL7bGaayzFaaaaaa@558C@

Hong and Birget 13 showed that with the threshold λ_n,δ by Steps 1, 2 and 3 we have, for all n ≥ 512

Pr ⁡ { Q max ⁡ ˜ ( n ) ≤ ( c 1 + c 2 δ ) ( log ⁡ 2 n n ) α 1 + 2 α } ≥ 1 − 9 n 1 + δ       ( 7 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacyGGqbaucqGGYbGCdaGadeqaamaaGaaabaGaemyuae1aaSbaaSqaaiGbc2gaTjabcggaHjabcIha4bqabaaakiaawoWaaiabcIcaOiabd6gaUjabcMcaPiabgsMiJkabcIcaOiabdogaJnaaBaaaleaacqaIXaqmaeqaaOGaey4kaSIaem4yam2aaSbaaSqaaiabikdaYaqabaacciGccqWF0oazcqGGPaqkdaqadaqaamaalaaabaGagiiBaWMaei4Ba8Maei4zaC2aaSbaaSqaaiabikdaYaqabaGccqWGUbGBaeaacqWGUbGBaaaacaGLOaGaayzkaaWaaWbaaSqabeaadaWcaaqaaiab=f7aHbqaaiabigdaXiabgUcaRiabikdaYiab=f7aHbaaaaaakiaawUhacaGL9baacqGHLjYScqaIXaqmcqGHsisldaWcaaqaaiabiMda5aqaaiabd6gaUnaaCaaaleqabaGaeGymaeJaey4kaSIae8hTdqgaaaaakiaaxMaacaWLjaWaaeWaaeaacqaI3aWnaiaawIcacaGLPaaaaaa@6327@

where c₁ and c₂ depend only on (b - a), M, and α. (We note the following. In 13 the mean m of the random variable e(t) was supposed to be zero; however, with a trivial modification, all proofs can be applied when m > 0.) Since n is large and δ > 0, we have 1−9n1+δ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqaIXaqmcqGHsisldaWcaaqaaiabiMda5aqaaiabd6gaUnaaCaaaleqabaGaeGymaeJaey4kaSccciGae8hTdqgaaaaaaaa@34A9@ and (log⁡2nn)α1+2α MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqadaqaamaalaaabaGagiiBaWMaei4Ba8Maei4zaC2aaSbaaSqaaiabikdaYaqabaGccqWGUbGBaeaacqWGUbGBaaaacaGLOaGaayzkaaWaaWbaaSqabeaadaWcaaqaaGGaciab=f7aHbqaaiabigdaXiabgUcaRiabikdaYiab=f7aHbaaaaaaaa@3C96@ which are extremely close to 1 and 0, respectively. Thus, (7) indicates that the error Qmax⁡˜(n) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaiaaqaaiabdgfarnaaBaaaleaacyGGTbqBcqGGHbqycqGG4baEaeqaaaGccaGLdmaacqGGOaakcqWGUbGBcqGGPaqkaaa@360C@ is almost surely close to 0. A key step in proving (7) was to apply a deep result in measure concentration 15, a recently developed field in probability.

Summarizing all discussed thus far in this subsection, with the proposed model we can conclude that a mammalian auditory system processes an original subthreshold signal h(t) <s as follows. At time instants t = in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@, i = 1,2, ..., n, the intrinsic noise e(t) with mean m > 0 is employed to sample the original signal, generating the detectable noisy samples f(in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@) = h(in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@) + e(in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@) ≥ s. Then, by the Step 1, 2 and 3, the system recovers the noisy samples, obtaining h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(in MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdMgaPbqaaiabd6gaUbaaaaa@2F7C@). (7) indicates that almost surely

| ( h ( i n ) + m ) − h ˜ ( i n ) | ≈ 0 for all  i = 1 , 2 , ... , n       ( 8 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaWaaqWaaeaadaqadaqaaiabdIgaOjabcIcaOmaalaaabaGaemyAaKgabaGaemOBa4gaaiabcMcaPiabgUcaRiabd2gaTbGaayjkaiaawMcaaiabgkHiTiqbdIgaOzaaiaGaeiikaGYaaSaaaeaacqWGPbqAaeaacqWGUbGBaaGaeiykaKcacaGLhWUaayjcSdGaeyisISRaeGimaadabaGaeeOzayMaee4Ba8MaeeOCaiNaeeiiaaIaeeyyaeMaeeiBaWMaeeiBaWMaeeiiaaIaemyAaKMaeyypa0JaeGymaeJaeiilaWIaeGOmaiJaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaemOBa4gaaiaaxMaacaWLjaWaaeWaaeaacqaI4aaoaiaawIcacaGLPaaaaaa@5B71@

This means that the system amplifies the original subthreshold signal by simply translating it up with the mean m of the intrinsic noise. Figure 1 illustrates this. Some remarks need to made.

• Our analysis of SR takes an approach that differs substantially from the current view of SR as applied to sensory physiology. However, our approach does follow from the core idea by Moss 8 that noise enhances hearing by sampling the subthreshold signal. Using recent deep results in signal processing (1513), our analysis further provides a strong statement that a necessary and sufficient condition for SR to occur in a mammalian auditory system is that all the samples by the noise are detectable.

• Our model and analysis do not involve energy and information modulation (as was also apparent in Moss' original description of SR in sensory systems 8). However, we formulate this idea in a rigorous and concrete way: using noise to sample an original subthreshold signal, a mammalian auditory system processes the noisy samples to translate the original signal up (in amplitude) by the mean of the noise.

• A new insight that our model and analysis adds to SR is as follows: When a mammalian auditory system processes the noisy samples, it may deposit energy into the recovered signal, and this added energy is expended in the recovery process. As a consequence, our result suggests that information modulation is not a likely mechanism for SR, as discussed below.

Recall that in the analysis of a mammalian auditory system above, all signals and noise are evaluated in terms of their pressure against a fixed baseline. Thus, we can compute energies of signals and noise. The energy carried by an original subthreshold signal is

E signal = ∫ 0 1 h ( t ) 2 d t MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFfbqrdaWgaaWcbaGaee4CamNaeeyAaKMaee4zaCMaeeOBa4MaeeyyaeMaeeiBaWgabeaakiabg2da9maapedabaGaemiAaGMaeiikaGIaemiDaqNaeiykaKYaaWbaaSqabeaacqaIYaGmaaaabaGaeGimaadabaGaeGymaedaniabgUIiYdGccqWGKbazcqWG0baDaaa@438A@

Since the sample rate n is large, we can interpolate h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@_i, i = 1, 2, ..., n, by segments to have a function h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t), t ∈ [0,1]. This is equivalent to taking the Haar wavelet as the basis, and thus, will not violate our analysis presented above. Our analysis above showed h˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGObaAgaacaaaa@2E14@(t) ≡ h(t) + m almost surely. Hence, we can write the energy carried by the recovered signal as

E recovery = ∫ 0 1 h ˜ ( t ) 2 d t = ∫ 0 1 ( h ( t ) + m ) 2 d t MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFfbqrdaWgaaWcbaGaeeOCaiNaeeyzauMaee4yamMaee4Ba8MaeeODayNaeeyzauMaeeOCaiNaeeyEaKhabeaakiabg2da9maapedabaGafmiAaGMbaGaacqGGOaakcqWG0baDcqGGPaqkdaahaaWcbeqaaiabikdaYaaaaeaacqaIWaamaeaacqaIXaqma0Gaey4kIipakiabdsgaKjabdsha0jabg2da9maapedabaGaeiikaGIaemiAaGMaeiikaGIaemiDaqNaeiykaKIaey4kaSIaemyBa0MaeiykaKYaaWbaaSqabeaacqaIYaGmaaaabaGaeGimaadabaGaeGymaedaniabgUIiYdGccqWGKbazcqWG0baDaaa@57E9@

and hence,

E recovery = ∫ 0 1 h ( t ) 2 + 2 m ∫ 0 1 h ( t ) d t + m 2 = E signal + 2 m ∫ 0 1 h ( t ) d t + m 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7216@

Recall that the energy carried by a random variable equals the deviation of the random variable. Thus, we can see that the energy carried by the intrinsic noise as a random variable is

E_noise = λm² for 0 <λ < 1

where for a given noise λ is a constant. Therefore,

E recovery = [ E signal + E noise ]