**1. Introduction**

Correlation is one of the most common and useful statistical concepts. It measures the strength and direction of a linear relationship between two variables. **Figure 1** shows some examples of correlations between pairs of variables, including white noise signals with different phases, pure tones with the same frequency and phase, pure tones with different frequencies, human

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

voice signals and time-delayed versions of the same signal, environmental noise signals and time-delayed versions of the same signal, and environmental noise signals obtained at the left and right ears. The correlation coefficient ranges between −1 and 1, and characterizes the strength of the relationships between the two variables.

**Figure 1.** Relationship between two variables. (a) White noise signals with different phases, (b) pure tones with the same frequency, (c) pure tones with different frequencies, (d) human voice signals and time-delayed versions of the same signal, (e) environmental noise and time-delayed versions of the same signal, and (f) environmental noise signals obtained at the left and right ears.

When a signal is represented as a time series, it is characterized by periodicity or randomness as a function of time. **Figure 2** shows some examples of relationships between a signal and the time-delayed version of that signal. The signals included in the figure are white noise, pure tones, a human voice, and train noise. The way in which correlation coefficients change as a function of time can be evaluated using an autocorrelation function (ACF). An ACF is a set of correlation coefficients that characterize the relations between the points in a series and time-delayed version of the same set. In other words, the ACF is a time-domain function that measures how much a waveform resembles the delayed version of itself. While the values of an ACF can extend beyond −1 and 1, the normalized ACF (NACF) for a signal, *φ*(*τ*), is defined by

Psychophysiological Evidence of an Autocorrelation Mechanism in the Human Auditory System http://dx.doi.org/10.5772/66198 189

$$\phi(\tau) = \frac{\Phi(\tau)}{\Phi(0)}\tag{1}$$

**Figure 2.** Relationships between a signal and a time-delayed version of the same signal. (a) White noise, (b) pure tones, (c) a human voice, and (d) train noise.

where

voice signals and time-delayed versions of the same signal, environmental noise signals and time-delayed versions of the same signal, and environmental noise signals obtained at the left and right ears. The correlation coefficient ranges between −1 and 1, and characterizes the

**Figure 1.** Relationship between two variables. (a) White noise signals with different phases, (b) pure tones with the same frequency, (c) pure tones with different frequencies, (d) human voice signals and time-delayed versions of the same signal, (e) environmental noise and time-delayed versions of the same signal, and (f) environmental noise signals

When a signal is represented as a time series, it is characterized by periodicity or randomness as a function of time. **Figure 2** shows some examples of relationships between a signal and the time-delayed version of that signal. The signals included in the figure are white noise, pure tones, a human voice, and train noise. The way in which correlation coefficients change as a function of time can be evaluated using an autocorrelation function (ACF). An ACF is a set of correlation coefficients that characterize the relations between the points in a series and time-delayed version of the same set. In other words, the ACF is a time-domain function that measures how much a waveform resembles the delayed version of itself. While the values of an ACF can extend beyond −1 and 1, the normalized ACF (NACF) for a signal,

strength of the relationships between the two variables.

188 Advances in Clinical Audiology

obtained at the left and right ears.

*φ*(*τ*), is defined by

$$\Phi(\tau) = \frac{1}{2T} \int\_{-T}^{+T} p(t)p(t+\tau)dt\tag{2}$$

That is, the ACF is normalized by the maximum value of the ACF at the point with zero delay, Φ(0), thus restricting the values to fit the range between −1 and 1. **Figure 3** shows some examples of the NACF. As white noise is random, the ACF is close to zero. As pure tones are completely periodic, the ACF is also periodic and the maximum and minimum values are 1 and −1, respectively. The human voice and environmental noise have periodic components, so the ACF values for these stimuli are high at the dominant frequency.

**Figure 3.** Examples of the NACF for (a) white noise, (b) pure tones, (c) the human voice, and (d) train noise.

Mathematically, the ACF contains the same information as the power spectrum of a given signal. For characterization of auditory signals, five factors are extracted from the ACF [1]. The first factor is the energy at the point with zero delay, given by Φ(0), which corresponds to the equivalent continuous sound pressure level (SPL). The second and third factors are the amplitude and delay time of the first maximum peak of the NACF, *φ*1 and *τ*1, which are related to the perceived pitch strength and pitch [2, 3]. The fourth factor is the effective duration of the envelope of the NACF, *τe*, which is defined by the 10th percentile delay. It represents a repetitive feature containing the auditory signal itself and is related to the preferred condition for the temporal factors of a sound field, such as reverberation time and the delay time of the first reflection [3, 4]. The fifth factor is the width of the amplitude of the NACF around the origin of the delay time, *Wφ(0)*, which is defined as having a value of 0.5. It corresponds to the spectral centroid [1]. The definitions of the ACF factors are depicted in **Figure 4**.

Psychophysiological Evidence of an Autocorrelation Mechanism in the Human Auditory System http://dx.doi.org/10.5772/66198 191

**Figure 4.** Definitions of the ACF factors, *φ*1, *τ*1, *τe*, and *Wφ*(0).

**Figure 3.** Examples of the NACF for (a) white noise, (b) pure tones, (c) the human voice, and (d) train noise.

190 Advances in Clinical Audiology

spectral centroid [1]. The definitions of the ACF factors are depicted in **Figure 4**.

Mathematically, the ACF contains the same information as the power spectrum of a given signal. For characterization of auditory signals, five factors are extracted from the ACF [1]. The first factor is the energy at the point with zero delay, given by Φ(0), which corresponds to the equivalent continuous sound pressure level (SPL). The second and third factors are the amplitude and delay time of the first maximum peak of the NACF, *φ*1 and *τ*1, which are related to the perceived pitch strength and pitch [2, 3]. The fourth factor is the effective duration of the envelope of the NACF, *τe*, which is defined by the 10th percentile delay. It represents a repetitive feature containing the auditory signal itself and is related to the preferred condition for the temporal factors of a sound field, such as reverberation time and the delay time of the first reflection [3, 4]. The fifth factor is the width of the amplitude of the NACF around the origin of the delay time, *Wφ(0)*, which is defined as having a value of 0.5. It corresponds to the The ACF is one of the most famous models for describing the perception of pitch and pitch strength. Pitch is thought to be extracted by the ACF in the temporal model of pitch perception [e.g., 5–7] and pitch strength corresponds to *φ*1, which represents the degree of temporal regularity of a sound [e.g., 1–3, 6]. It is possible to systematically manipulate the values of *φ*<sup>1</sup> using iterated rippled noise (IRN). IRN is produced by adding a delayed version of a noise signal to the original signal, and then repeating this delay and addition process [2]. Increasing the number of iterations increases the periodicity and *φ*1 value.

Physiologically, IRN elicits signals in auditory nerve fibers [8, 9] and cochlear nucleus neurons [10–12], indicating that the pitch of IRN is represented in the firing patterns of action potentials locked to either the temporal fine structure or the envelope periodicity. That is, autocorrelationlike behavior in the fine structure of the neural firing patterns suggests that the pitch of IRN is based on an ACF mechanism. Indeed, the pooled interspike interval distributions of auditory nerve discharge patterns in response to complex sounds are similar to the ACF of the stimulus waveform, and *φ*1 of the ACF corresponds to pitch strength [13, 14].

Therefore, to find the physiological counterparts of an ACF mechanism in the human auditory cortex, we used magnetoencephalography (MEG) to investigate the auditory evoked magnetic field (AEF) elicited by IRN and bandpass filtered noise (BPN). The *φ*1 value can be manipulated systematically by changing the bandwidth of the BPN. A narrower bandwidth produces a higher *φ*1. In MEG, the measured signals are generated by synchronized neuronal activity in the human brain. The time resolution is in the range of milliseconds. Thus, this technique can be used to examine rapid changes in cortical activity that reflects ongoing signal processing in the brain; electrical events in single neurons typically last from one to several tens of milliseconds [15]. With respect to the psychological aspect of sound perception, we evaluated the effects of the other ACF factor, i.e., *τe*, on loudness and annoyance because it can explain changes in loudness even when SPL conditions are unchanged.
