Selection of Appropriate Analysis Method for Low Sampled Data

The employment of sampling rates much below those usually considered necessary in the acquisition process necessitates new knowledge about the selection of an appropri-ate analysis technique which will be used to process the aliased versions of the acoustic signals. This is done by the investigation of different TDE algorithms in the time, fre-quency, and time-frequency (content-based features) domains under the use of low sampling rates. The reason behind this is to find new ways to overcoming the challenges faced when lowering the sampling rate. Such knowledge is useful for specifying a suita-ble domain of analysis for the use of WSNs in ASL and SHM applications.

3.3.2.1 Time Domain Analysis for TDE

Several TDE procedures have been proposed and implemented in the time domain over the years. The basic idea of most of these techniques is based on locating the absolute extremum of the cross-correlation, cross-correlation coefficient function or some other statistic associated with the observed signals. For example, basic cross-correlation (BCC) [180] and generalised cross-correlation (GCC) algorithms [181] search for co-herence among the captured acoustic signals in order to determine the lag at which the cross-correlation function (CCF) has its maximum. This lag then represents the time delay between the two signals, as shown in Table 4. The main difference between the two algorithms is that GCC uses weighting functions (filtering the captured signals) to improve the performance of the TDE approach [181, 182].

As shown in Table 4, the GCC is defined as the inverse Fourier transform of the cross-spectrum of the input signals, , scaled by a weighting function which is summarised in Table 5 for several common GCC methods.

The BCC and GCC approaches have been widely used in different applications, such as sonar, displacement or velocity determination and pattern recognition [183].

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more, they can be applied in flow or strain determination from sequences of images [184]. Time domain approaches have the advantages that they are simple, and therefore not computationally expensive, and more suitable for real-world applications. However, they depend strongly on the coherence among the signals received, which is likely to impact upon their accuracy in precisely estimating the time delay [185], particularly if low sampling rates are used.

Table 4: Basic mathematical expressions for BCC and GCC.

TDE algorithm Mathematical expression Estimated time delay

BCC

∫

GCC ∫

Table 5: Commonly used weighting functions in the GCC method [181].

Method Name Weighting Function

Cross Correlation 1

Roth Impulse Response ⁄

Phase Transform (PHAT) | ⁄ |

Smoothed Coherence Transform (SCOT) √ ⁄

Eckart Filter ⁄

Hannon and Thomson (HT) ⁄ | |

3.3.2.2 Frequency Domain Analysis for TDE

Frequency domain algorithms are based on examining the phase spectrum of two sig-nals. For instance, the GPS is used to compute the time delay between two signals and by estimating the cross-power spectral density (CPSD) of the two signals and

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then computing the phase slope of the CPSD as a function of frequency [185]. Figure 22 shows a typical measured phase spectrum from the test signals, discussed below, in con-junction with an estimate of the actual phase spectrum. As seen in this figure, the signal used is dominated by ambient noise at frequencies below and greater than . Thus, the GPS uses the frequency band to weight the phase in order to improve the TDE which can be described as follows,

Let represent the estimated phase at the frequency from two sensors and with a number of points N; then the time delay is defined as in Equation (3.17) [186]:

∑ ⁄∑ (3.17)

where is a frequency dependent weighting function at frequency .

In this equation, only the frequency band which corresponds to the linear phase gra-dient of CPSD shown in Figure 22 is selected to calculate the time delay.

Figure 22: Selection of the cut-off frequencies ω_0, ω_1, and frequency bandwidth ∆ω.

This GPS algorithm has potential application in radar, sonar and optical imaging sys-tems [187]. It has the advantages over the time domain algorithms that in the frequency domain the time delay is directly estimated, which makes it easier to use filters [188].

This leads to improving the speed of estimation of time delays and simplifying the sig-nal processing algorithms. Furthermore, as shown in Equation (3.17), only a selected frequency range is used in this calculation [185]. This makes the localisation process more robust in a noisy environment, since the CPSD will be weighted according to this

66 range [186].

3.3.2.3 Time-Frequency Domain Analysis for TDE

Content-based feature data analysis algorithms, such as envelope GPS (EGPS) algo-rithm rely on the analysis of the actual contents of the signal, such as the shape or any other information that can be derived from it. This concept has been successfully ap-plied in research fields such as image retrieval applications [189]. In this investigation, this concept will be adopted for the estimation of the time delay between two signals and by first extracting the envelopes or shape of both signals using the Hil-bert function [190] and then, as in frequency domain algorithms, the GPS method is ap-plied to calculate the time delay using the signal envelopes. The reason behind the use of the signal envelope is to smooth the signal shape and to overcome the loss of infor-mation due to the employment of low sampling rates.