# Ultrasonic Pulse-Echo Signals for Quantitative Assessment of Reinforced Concrete Anomalies
## 3 Formulation for Accurate Assessment of Void Defects
\[x(t)=\sum_{j=1}^{n}a_{j}(t)\cos\big{[}\varphi_{j}(t)\big{]},\ x(t)=\sum_{j=1}^{n }a_{j}(t)\exp[i\int\omega_{j}(t)dt] \tag{3}\]  
To decompose a signal using the EMD approach, we represent the instantaneous amplitude and instantaneous phase of the \(j\)th component as \(a_{j}(t)\) and \(\varphi_{j}(t)\), respectively, and \(n\) as the number of components. This is achieved through an iterative sifting process, resulting in zero-mean AM-FM components called IMFs. These IMFs must satisfy two requirements: (a) the number of extremes and zero crossings in the IMF must be equal or differ at most by one, and (b) the mean value of the envelopes defined by the local maxima and local minima must be zero at any point. In essence, the signal must be locally symmetrical around the time axis. To find the IMFs for a given signal, we conduct the sifting process to find the IMFs for the signal \(x(t)\), which involves the following steps:  
1. To accurately process the input signal, we must initially identify all local maxima and minima, along with their respective positions and amplitudes. Following this, we can employ cubic spline interpolation to generate an upper envelope consisting of the local maxima and a lower envelope using the local minima. 2. After calculating these envelopes, the envelope mean signal, known as \(m_{1}(t)\), can be determined by taking their mean. Lastly, to finalize the processing, we need to subtract the envelope mean signal from the original input signal (Equation (4)). 3. Check if \(h(t)\) meets the IMF requirements. Treat the data as new data and repeat the process if it does not meet the IMF requirements (Equation (5)). 4. Repeat the sifting procedure k times until the resulting component \(h_{1k}(t)\) is an IMF, which becomes the first IMF (Equation (6)). \[h(t) =x(t)-m_{1}(t)\] (4) \[h_{11}(t) =h_{1}(t)-m_{11}(t)\] (5) \[c_{1}(t) =h_{1k}(t)\] (6)
5. A standard method for extracting and analyzing a signal's underlying components is residual analysis. This process involves subtracting the \(c_{1}(t)\) component from the input signal and defining the resulting remainder as the first residual. Given that the first residual \(r_{1}(t)\) may contain information relating to longer-period components, it is treated as a new data stream. The procedure is repeated for this new signal. This process may be iterated j times, resulting in the generation of \(j\) residuals. By following this approach, it is possible to obtain a refined understanding of the signal and identify the underlying components contributing to its overall structure. \[\left\{ \begin{aligned} r_{1}(t)-c_{2}(t)=r_{2}(t)\\ \vdots\\ r_{n-1}(t)-c_{n}(t)=r_{n}(t)\end{aligned} \right.\] (7) (6) The sifting process is interrupted once either of the two criteria mentioned above is fulfilled: firstly, when the component \(c_{n}(t)\) or the residual \(r_{n}(t)\) is reduced to such a minuscule size that it can be regarded as insignificant, or secondly, when the residual (R) becomes a monotonic function that precludes the extraction of the IMF--the objective IMF can be obtained by adding Equations (4) and (5). The original signal can be expressed as a combination of IMFs and a residual, producing significant implications in signal processing and analysis. This observation has been accepted in the academic community. \[x(t)=\sum_{i=1}^{n}c_{imfi}(t)+r_{n}(t)\] (8)
7. EMD-based denoising, similar to other decomposition-based denoising techniques like wavelet transforms, requires a reliable and robust threshold to distinguish between noise and authentic signal components. In cases where irregularity or noise is present in a time series, the Hurst exponent plays a crucial role in determining the irregularity in the signal. This methodology is more efficient than traditional approaches such as autocorrelation, ANOVA, and spectral analysis in many applications. The Hurst exponent value, whether greater or less than 0.5, indicates the pattern of the non-linearity of the data set.