Patent Publication Number: US-10324068-B2

Title: Temperature compensation in wave-based damage detection systems

Description:
CLAIM OF PRIORITY 
     This application claims priority under 35 U.S.C. § 119(e) to provisional U.S. Patent Application 61/741,434, filed on Jul. 19, 2012, the entire contents of which is hereby incorporated by reference. 
    
    
     BACKGROUND 
     Structural health monitoring systems (SHM) and other damage detection systems use guided wave inspection methods for testing large structures. Guided ultrasonic waves are popular because they travel through the thickness of the structure over long distances with little attenuation. This allows sensors to interrogate large areas all at once. However, guided waves are inherently multi-modal and dispersive in their propagation. Furthermore, structures&#39; boundaries generate reflections and exchange energy between wave modes. These effects make the interpretation of measured data difficult and necessitate the use of baseline measurements with no damage present. 
     Under ideal conditions, SHM techniques easily detect damage as a change in the propagation medium by performing a subtraction or time domain correlation with a baseline signal. The baseline reduces complexity by removing effects from static sources, such as reflecting boundaries. Unfortunately, when the environmental conditions change, these methods are unable to distinguish damage from benign effects. 
     Temperature Compensation 
     One of the most prominent environmental effects to distort signals is temperature fluctuation. Temperature has been shown to modify the velocity of the guided wave modes, which stretches or scales the measured time domain signal. Given an ultrasonic signal measurement x(t), a change in temperature T{⋅} can be modeled as a uniform scale
 
 T{x ( t )}= x (α t ),  (1)
 
where α is an unknown scaling factor. This effect can be attributed to temperature&#39;s influence on the Young&#39;s modulus of the material.
 
     However, this model is an approximation. In the presence of multi-modal and dispersive propagation, temperature&#39;s effects do not perfectly dilate or compress the signal. Even so, the effectiveness of this approximate model has been experimentally demonstrated in references. This implies that, by computing a scale-invariant measure of similarity between the baseline signal and measured data, one can compensate for temperature&#39;s effects on the ultrasonic signals. Two prior art techniques have been proposed for ultrasonic temperature compensation by computing a scale-invariant statistic: (a) optimal signal stretch, and (b) local peak coherence. 
     Optimal Signal Stretch 
     Optimal signal stretch (OSS) is prior art technique for ultrasonic temperature compensation that uses an exhaustive search optimization strategy for finding the scale factor α which minimizes the mean squared error between baseline and measured data. OSS employs a finite library of K stretched baselines {s(α 1  t), s(α 2  t), . . . , s(α K  t)} and computes the mean squared error between the observed signal and each element in the library. The scale factor which minimizes the mean squared error is then declared to be the optimal choice over the given set. As K→∞, OSS computes the optimal scale factor, over all possible scale factors, that minimizes the mean squared error. 
     When the baseline and observed data are normalized to have zero means and equal L 2  norms, minimizing the mean squared error is equivalent to maximizing the sample Pearson product-moment correlation coefficient, or correlation coefficient for short. In the use of OSS, data is normalized so that changes in the signal mean or amplitude, which generally do not correspond with damage, do not bias the results. With normalizing the data, the optimal scale factor α chosen by OSS is defined by 
                     α   =     arg   ⁢       max   k     ⁢       ∫   0   ∞     ⁢           (       x   ⁡     (   t   )       -     μ   x       )     ⁢     (       s   ⁡     (       α   k     ⁢   t     )       -     μ     s   ,   k         )           σ   x     ⁡     (       σ   s     /       α   k         )         ⁢           ⁢   dt             ,     1   &lt;   k   &lt;   K     ,           (   2   )               
where β and σ represent the mean and L 2  norm of each signal,
 
     
       
         
           
             
               
                 
                   
                     
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     The correlation coefficient between the optimally scaled baseline s(α t) and observed data x(t) can then be expressed as 
                       ϕ   xs     =       max   k     ⁢       ∫   0   ∞     ⁢           (       x   ⁡     (   t   )       -     μ   x       )     ⁢     (       s   ⁡     (       α   k     ⁢   t     )       -     μ     s   ,   k         )           σ   x     ⁡     (       σ   s     /       α   k         )         ⁢   dt           ,     1   &lt;   k   &lt;   K     ,           (   4   )               
where value of φ xs  has a range −1≤φ xs ≤1. As K→∞, φ xs  becomes scale-invariant. So if x(t) and s(t) are scaled replicas of one another (i.e., x(t)=s(α t)), then the value of φ xs  is 1. Conversely, if x(t)=−s(α t), then the value of φ xs  is −1. When there is no scale relationship between the two signals, φ xs =0. For finite values of K and situations in which x(t)≈s(α t), φ xs  describes the degree of linear correlation between x(t) and the optimally scaled baseline s(α t).
 
     Since OSS is an optimal technique, assuming the uniform scaling model described in (1), system  110  uses it as a baseline for comparing the techniques described herein and other techniques. Although effective, OSS is computationally inefficient. For a discrete signal of N samples and a library of K stretched baselines, the OSS correlation coefficient can be computed using a matrix-vector multiplication. The computationally complexity of this operation, in big-O notational, is O(K N). Therefore, for sufficiently large signals and a sufficiently dense library, OSS becomes computationally impractical. 
     Local Peak Coherence 
     A second prior art approach to ultrasonic temperature compensation is referred to as the local peak coherence (LPC) technique, which is used to estimate the scale factor from local delay measurements. It has been experimentally shown that diffuse-like signals, cluttered by reflections and multi-path propagation, also exhibit almost perfect signal stretching behavior as temperature varies. Since the diffuse-like condition ensures the measured signal to have a long and continuous duration, the scale factor between the two signals can be estimated from a series of local delay estimates. LPC assumes that, given an observed signal x(t) and uniformly scaled replica x(t)=s(α t), the uniform scaling effect can be approximated as a delay in a small region around t=t 0 ,
 
 x ( t )= s (α t )≈ s ( t −(1+α) t   0 )  as t→t   0 .  (5)
 
     To estimate these delays, a small portion of each signal is windowed around time t=t 0  and the delay between each windowed signal is computed as the argument that maximizes the standard cross-correlation (coherence) function, 
     
       
         
           
             
               
                 
                   
                     
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     where d w  is the delay estimate for window w, s w (t) and x w (t) are windowed baseline and observed signals, and W is the number of window positions. As the window moves across each signal, the delay value increases linearly according to the scale factor. The scale parameter is then computed by performing a regression analysis on the estimated delays. Using the computed scale factor, the correlation coefficient between the appropriately scaled baseline and measured data is then computed and used as the scale-invariant statistic. 
     As compared with OSS, LPC can be computed very quickly. The delay estimates may be computed using the fast Fourier transform (FFT) algorithm. Assuming, N w  is the number of discrete samples in the window and there are W different window positions, computing every delay requires a computational complexity of O(W N w  log(N w )). In general, N w  will be small. So overall, the computational efficiency is linearly dependent on W, and at a minimum, the scale factor can be estimated using W=2. However, this efficiency comes at the cost of robustness. The technique approximates uniform scaling as a series of delays, which, as shown in equation (5), is true asymptotically as t→t 0 . In general, this may not always be a good approximation. Also, any effect that does not uniformly scale the signal may disrupt a portion of a delay estimates and may significantly alter the scale factor estimate. For this reason, LPC is not robust and may be adversely affected by many effects, including the introduction of damage. 
     Therefore there is still a need for methods for compensation for temperature variations in ultrasonic inspection systems that provide high quality results and do so rapidly. 
     SUMMARY 
     In one aspect of the present disclosure, a method performed by one or more processing devices comprises obtaining first waveform data indicative of traversal of a first signal through a structure at a first time, wherein a first ambient temperature is associated with the structure at the first time; obtaining second waveform data indicative of traversal of a second signal through the structure at a second time, wherein a second ambient temperature is associated with the structure at the second time, the first ambient temperature differs from the second ambient temperature, and a difference between the first ambient temperature and the second ambient temperature causes a distortion of the second signal, relative to the first signal, as the second signal traverses through the structure; applying a scale transform to the first waveform data and the second waveform data; computing, by the processing device and based on applying the scale transform, a scale-cross correlation function that promotes identification of scaling behavior between the first waveform data and the second waveform data; performing one or more of: computing, by the processing device and based on the scale-cross correlation function, a scale factor for the first waveform data and the second waveform data, with the scale factor being indicative of an amount of variation between the first ambient temperature and the second ambient temperature; and computing, by the processing device and based on the scale-cross correlation function, a scale invariant correlation coefficient between the first waveform data and the second waveform data, with the scale invariant correlation coefficient comprising a compensation statistic for compensating for the distortion of the second signal, relative to the first signal, that results from variation in the first ambient temperature and the second ambient temperature. 
     Implementations of the disclosure can include one or more of the following features. In some implementations, the method includes detecting, based on the scale factor and the scale invariant correlation coefficient, one or more areas of structural change in the structure. In other implementations, the structural change comprises a degradation of the structure. In still other implementations, the scale invariant correlation coefficient is indicative of a measure of similarity between the first waveform data and the second waveform data. 
     In some implementations, the scale factor comprises a value that is used as a multiplier in scaling the first waveform data to correspond to the second waveform data. In still other implementations, the processing device is included in a wave-based damage detection system. In yet other implementations, the structure comprises a concrete structure. In still other implementations, the scale-cross correlation function is computed in accordance with: x(t)⋄s α (t)=s −1 { s{x(t)} s{s(t)}}, wherein ⋄ represents a scale cross-correlation operation; wherein the overbar represents a complex conjugation operation; wherein x(t) is the first waveform data; wherein s(t) is the second waveform data; wherein α includes a scaling factor on s(t); wherein s{x(t)} is the scale domain of x(t); wherein s{s(t)} is the scale domain of s(t); and wherein S −1 {⋅} represents an inverse scale transform. 
     In yet other implementations, computing the scale-cross correlation function comprises: computing a product of a scale domain of the first waveform data and a scale domain of the second waveform data. In still other implementations, computing the scale-cross correlation function comprises: resampling the first waveform data; resampling the second waveform data; applying an amplification factor to the resampled first and second waveform data; and cross-correlating the amplified, resampled first and second waveform data. In still other implementations, computing the scale-invariant correlation coefficient comprises: determining an increased value of the scale cross-correlation function in a stretch domain factor, relative to other values of the scale cross-correlation function in the stretch domain factor. 
     In some implementations, one or more of the first signal and the second signal is an ultrasonic wave signal. In still other implementations, the structure comprises one or more of a pipe structure, a heating structure, one or more pipe structures in an oil refinery, one or more pipe structures in a chemical refinery, one or more pipe structures in a gas refinery, one or more natural fuse pipelines, one or more oil pipelines, one or more heating pipe structures, one or more cooling pipe structures, one or more pipe structures in a nuclear power plant, one or more pressure vessels, one or more concrete structures of a bridge, one or more concrete structures of civil infrastructure, one or more portion of an airplace, one or more portions of an aerospace vehicle, one or more portions of a submarine, and one or more metallic structures. In still other implementations, the method includes identifying a maximization of the scale-cross correlation function in the stretch factor domain; wherein computing the scale factor comprises: computing, by the processing device and based on the maximization of the scale-cross correlation function in the stretch factor domain, the scale factor for the first waveform data and the second waveform data; and wherein computing the scale invariant correlation coefficient comprises: computing, by the processing device and based on the maximization of the scale-cross correlation function in the stretch factor domain, the scale invariant correlation coefficient between the first waveform data and the second waveform data. 
     In still other implementations, the method includes identifying a maximization of the scale-cross correlation function in the scale transform domain; wherein computing the scale factor comprises: computing, by the processing device and based on the maximization of the scale-cross correlation function in the scale transform domain, the scale factor for the first waveform data and the second waveform data; and wherein computing the scale invariant correlation coefficient comprises: computing, by the processing device and based on the maximization of the scale-cross correlation function in the scale transform domain, the scale invariant correlation coefficient between the first waveform data and the second waveform data. 
     In still another aspect of the disclosure, one or more machine-readable media are configured to store instructions that are executable by a server to perform operations including obtaining first waveform data indicative of traversal of a first signal through a structure at a first time, wherein a first ambient temperature is associated with the structure at the first time; obtaining second waveform data indicative of traversal of a second signal through the structure at a second time, wherein a second ambient temperature is associated with the structure at the second time, the first ambient temperature differs from the second ambient temperature, and a difference between the first ambient temperature and the second ambient temperature causes a distortion of the second signal, relative to the first signal, as the second signal traverses through the structure; applying a scale transform to the first waveform data and the second waveform data; computing, by the processing device and based on applying the scale transform, a scale-cross correlation function that promotes identification of scaling behavior between the first waveform data and the second waveform data; performing one or more of: computing, by the processing device and based on the scale-cross correlation function, a scale factor for the first waveform data and the second waveform data, with the scale factor being indicative of an amount of variation between the first ambient temperature and the second ambient temperature; and computing, by the processing device and based on the scale-cross correlation function, a scale invariant correlation coefficient between the first waveform data and the second waveform data, with the scale invariant correlation coefficient comprising a compensation statistic for compensating for the distortion of the second signal, relative to the first signal, that results from variation in the first ambient temperature and the second ambient temperature. Implementations of this aspect of the present disclosure can include one or more of the foregoing features. 
     In still another aspect of the disclosure, an electronic system includes a server; and one or more machine-readable media configured to store instructions that are executable by the server to perform operations including: obtaining first waveform data indicative of traversal of a first signal through a structure at a first time, wherein a first ambient temperature is associated with the structure at the first time; obtaining second waveform data indicative of traversal of a second signal through the structure at a second time, wherein a second ambient temperature is associated with the structure at the second time, the first ambient temperature differs from the second ambient temperature, and a difference between the first ambient temperature and the second ambient temperature causes a distortion of the second signal, relative to the first signal, as the second signal traverses through the structure; applying a scale transform to the first waveform data and the second waveform data; computing, by the processing device and based on applying the scale transform, a scale-cross correlation function that promotes identification of scaling behavior between the first waveform data and the second waveform data; performing one or more of: computing, by the processing device and based on the scale-cross correlation function, a scale factor for the first waveform data and the second waveform data, with the scale factor being indicative of an amount of variation between the first ambient temperature and the second ambient temperature; and computing, by the processing device and based on the scale-cross correlation function, a scale invariant correlation coefficient between the first waveform data and the second waveform data, with the scale invariant correlation coefficient comprising a compensation statistic for compensating for the distortion of the second signal, relative to the first signal, that results from variation in the first ambient temperature and the second ambient temperature. Implementations of this aspect of the present disclosure can include one or more of the foregoing features. 
     All or part of the foregoing can be implemented as a computer program product including instructions that are stored on one or more non-transitory machine-readable storage media, and that are executable on one or more processors. All or part of the foregoing can also be implemented as an apparatus, method, or electronic system that can include one or more processors and memory to store executable instructions to implement the stated operations. 
     The details of one or more implementations are set forth in the accompanying drawings and the description below. Other features, objects, and advantages will be apparent from the description and drawings, and from the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  is block diagram of an example environment for generating temperature compensation indicators. 
         FIG. 2  shows a graph indicative of a relationship between a number of samples to resample a uniformly sampled signal and a first-order approximation. 
         FIGS. 3A-3C  show graphs of sampling on a sine wave. 
         FIG. 4  shows a diagram that demonstrates actions in computing the discrete-time exponential-time cross-correlation. 
         FIG. 5  shows curves demonstrating the computational growth rates of OSS (N 2  ln(N)) and the scale cross-correlation (N ln(N) log 2  (N)). 
         FIG. 6  shows a graph of measured ambient temperatures during a simulation. 
         FIGS. 7A-7D  show correlation coefficients for temperature compensation methods. 
         FIGS. 8A-8B  show diagrams of a wideband, modulated sinc waveform that captures a frequency spectrum. 
         FIGS. 9A-9B  show graphs indicative of a wideband response of a thin aluminum plate. 
         FIG. 10  shows correlation coefficients versus scale cross-correlation scale estimates. 
         FIGS. 11A, 11B  are examples of graphs that illustrate the relationships between the standard and compensated correlation coefficients and the scale estimates. 
         FIG. 12  is a flowchart of a process for generating temperature compensation indicators. 
         FIG. 13  is a block diagram of components in an example environment for generating temperature compensation indicators. 
     
    
    
     DETAILED DESCRIPTION 
     A system disclosed herein computes a scale-invariant descriptor based on the Mellin transform, to be used to improve wave-based damage detection systems by compensating for the variation in signals that result from variations in temperature. This compensation statistic is referred to as the scale-invariant correlation coefficient, and the system measures it from the scale cross-correlation function between two signals. 
     Referring to  FIG. 1 , a block diagram is shown of an example environment  100  for generating temperature compensation indicators, e.g., values indicative of an amount to compensate for the variation in signals that result from variations in temperature. There are various types of temperature compensation indictors, including, e.g., a scale invariant coefficient  115 , a scale factor  117 , and so forth. 
     Example environment  100  includes structure  102 , e.g., a concrete structure, a pipe structure, a heating structure, a refinery, and a nuclear power plant and so forth. Example environment  100  also includes system  110  and data repository  114 . In this example, data repository  114  that stores information  109  indicative of various algorithms, including, e.g., algorithms that are represented by and/or in accordance with the equations described herein. In this example, system  110  uses information  109  to compute various temperature compensation indicators. 
     In the example of  FIG. 1 , system  110  is configured to emit signals (e.g., ultrasonic wave signals) to structure  102  at various points in time, e.g., a first time (T 1 ) and a second time (T 2 ). Based on the emitted signals, system  110  records waveform data  104 ,  105  that is indicative of how signals  106 ,  107 , respectively, travel through structure  102  at times T 1 , T 2 , respectively. In this example, system  110  detects structural changes (e.g., damage) in structure  102  based on changes in waveform data  104 ,  105 . 
     In this example, the temperature of structure  102  (e.g., the ambient air surrounding structure  102 ) fluctuates between time T 1  and time T 2 . These fluctuations cause variations in the waveform data  104 ,  105 . To account for these temperature based variations (which are independent from and different from structural damage), system  110  uses the techniques described herein to compute temperature compensation indicators. By application of the temperature compensation indicators to one or more of waveform data  104 ,  105 , system  110  can identify the variations in waveform data  104 ,  105  that are caused by temperature fluctuations and compensate for them, e.g., thereby promote detection of variations in waveform data  104 ,  105  that are caused by structural damage in structure  102 . The example environment  100  may include many thousands of client devices and systems. 
     In this example, one of the algorithms included in information  109  is based on the Mellin transform, which is defined by the integral 
                     ℳ   ⁢     {     x   ⁡     (   t   )       }       =       X   ⁡     (   p   )       =       ∫   0   ∞     ⁢       x   ⁡     (   t   )       ⁢     t     p   -   1       ⁢     dt   .                   (   7   )               
In some examples, the special case for which p=−jc,
 
                     ℳ   ⁢     {     x   ⁡     (   t   )       }       =       X   ⁡     (   jc   )       =       ∫   0   ∞     ⁢       x   ⁡     (   t   )       ⁢     t       -   jc     -   1       ⁢     dt   .                   (   8   )               
is also often referred to as the Mellin transform. To avoid confusion, equation (8) is referred to as the Mellin transform and equation (7) as the generalized Mellin transform. Another case of the generalized Mellin transform is known as the scale transform, which system  110  uses to compute the scale-invariant correlation coefficient. Generally, a scale transform includes a modification (e.g., a transform) between two spaces (e.g., a source space and a destination space) where the axis vectors of the source space are longer or shorter than the corresponding axis vectors in the destination space. This causes objects to stretch or shrink along the axes as they are transformed between the two spaces.
 
     In signal processing and computer vision, the Mellin transform has been utilized for its scale-invariant properties. A time-scale operation on the signal x(t) is expressed as a phase change in the Mellin domain,
 
 M{x (α t )}= X ( jc ) e   jc ln(α).   (9)
 
     The magnitude of the Mellin transform is invariant to changes in the time-scale of a signal. This is analogous to the Fourier magnitude spectrum&#39;s shift invariance property. 
     The generalized Mellin transform is closely related to the Laplace transform. Through algebraic manipulation, the expression for the Mellin transform can be rewritten as 
     
       
         
           
             
               
                 
                   
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     In equation (10), the generalized Mellin transform is equivalent to the Laplace transform, but integrated over logarithmic time. When p=−j c, the Mellin transform becomes equivalent to the Fourier transform integrated over logarithmic time. This implies that, if an input signal x(t) is exponentially skewed by substituting t=e τ , the Mellin transform expression becomes 
                           ℳ   ⁢     {     x   ⁡     (   t   )       }       =       ⁢     X   ⁡     (     j   ⁢           ⁢   c     )                   =       ⁢       ∫   0   ∞     ⁢       x   ⁡     (     e   τ     )       ⁢     e       -           ⁢   jc     ⁢           ⁢   r       ⁢           ⁢   d   ⁢           ⁢   τ                     =       ⁢     ℱ   ⁢     {     x   ⁡     (     e   τ     )       }         ,                 (   11   )               
where F {⋅} represents the Fourier transform. This property is the basis for the fast Mellin transform algorithm.
 
     In engineering, the Mellin transform&#39;s scale invariance property has been used in a variety of applications. These applications include the classification of ships from radar signals and the interpretation of speech waveforms. A variant known as the Fourier-Mellin transform is also popular in several engineering fields. The Fourier-Mellin transform is computed from the Mellin transform of the Fourier magnitude spectrum. As a result, the magnitude of the Fourier-Mellin transform achieves invariance to both shifting and scaling. In two dimensions, the Fourier spectrum is also polar transformed to achieve rotation invariance. The Fourier-Mellin transform has been used in applications for the analysis of speech and Doppler waveforms, as well as for pattern recognition and watermark detection in images. 
     Sometimes, the magnitude of the Mellin transform is analyzed alone to achieve invariance. Unfortunately, this is less than ideal since it requires the removal of phase information, which is necessary for generating complete descriptors of a signal. However, when comparing two signals, the product of the Mellin transformed signals can be used to compute the Mellin cross-correlation function. The maximum of the Mellin cross-correlation is a measure of the scale-invariant similarity between two signals and the location of the maximum can also be used to estimate the scale factor between the signals, as described in further detail below. The Mellin cross-correlation is used in combination with the Fourier-Mellin transform for image registration and pattern recognition applications. 
     Although the Mellin transform has found widespread use in many applications, there are several difficulties involved with implementing it. First, the Mellin transform of a signal is not guaranteed to exist. Second, in discrete-time, efficient computation of the discrete-time Mellin transform requires data to be exponentially sampled in time. This is often impractical to implement. Therefore, the Mellin transform is usually approximated using the fast Mellin transform algorithm, in which uniformly sampled data is exponentially resampled. However, this non-uniform resampling procedure incurs its own Nyquist sampling restrictions and complicates computation of the associate cross-correlation function. 
     In this example, the scale cross-correlation function and scale-invariant correlation coefficient, based on the scale-invariant properties of Mellin transform, provide (and/or are used in) ultrasonic temperature compensation. Below, a case of the generalized Mellin transform known as the scale transform is described. The scale transform exists for all finite energy signals. System  110  derives the continuous-time scale cross-correlation and uses it to derive the scale-invariant correlation coefficient and associated scale factor. System  110  also commutates the scale transform and scale cross-correlation in discrete time, and derives the Mellin resampling theorem to address the numerical difficulties with the fast Mellin transform algorithm. 
     Compared with the temperature compensation methods of the prior art, the techniques for temperature compensation described herein have increased computational speed and robustness (e.g., relative to the computational speed and robustness of the prior art methods). In an example, the continuous-time scale-invariant correlation coefficient described herein is mathematically equivalent to the OSS correlation coefficient as the size of the library K→∞. This suggests the techniques described herein to be optimal in terms of mean squared error. In an example, the scale-invariant correlation coefficient described herein can be computed very efficiently due to its relationship with the Fourier transform. In this example, the scale-invariant correlation coefficient is as effective as OSS, an optimal method, and more robust than LPC. 
     In an example, system  110  implements stretch-based, model-driven temperature compensation algorithms based on the scale transform. In the scale transform domain, system  110  manipulates the stretch factor of signals and compute quantities invariant to changes in that stretch factor. System  110  implements three algorithms for temperature compensation based on these scale domain tools: the scale-invariant correlation (SIC) method, the iterative scale transform (IST) method, and the combined SIC/IST method. 
     The Scale Transform 
     In an example, system  110  uses the scale transform and scale cross-correlation function for computing the scale factor and the scale-invariant correlation coefficient between two signals. In general, the generalized Mellin transform of equation (7) and the Mellin transform of equation (8) of a signal may not converge. Under appropriate conditions, the generalized Mellin transform of a function ƒ(t) converges within some analytic strip a&lt;Re{p}&lt;b, where a and b are real numbers. To promote convergence, system  110  uses the scale transform. The scale transform is a special case of the generalized Mellin transform for which p=−j c+½, 
                     𝒮   ⁢     {     x   ⁡     (   t   )       }       =       X   ⁡     (   c   )       =       ∫   0   ∞     ⁢       x   ⁡     (   t   )       ⁢     t       -   jc     -     1   /   2         ⁢     dt   .                   (   12   )               
where S{⋅} represents the scale transform. The transform is sometimes normalized by 1/(2π) 1/2  to preserve symmetry between it and its inverse. This integral is also sometimes referred to as the Mellin transform or the Mellin transform of square-integrable functions.
 
     The scale transform shares several useful properties with the Fourier transform that are not present in the Mellin transform. For example, the kernel of the inverse scale transform is the complex conjugate of the kernel of the scale transform, as shown in the below equation (13): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             𝒮 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               X 
                               ⁡ 
                               
                                 ( 
                                 c 
                                 ) 
                               
                             
                             } 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           x 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
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                             1 
                             
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                               ⁢ 
                               
                                   
                               
                               ⁢ 
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                           ⁢ 
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 X 
                                 ⁡ 
                                 
                                   ( 
                                   c 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 t 
                                 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     c 
                                   
                                   - 
                                   
                                     1 
                                     ⁢ 
                                     
                                       / 
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 c 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     In equation (13), S −1 {⋅} represents the inverse scale transform. The scale transform also satisfies Parseval&#39;s theorem. This implies that the L 2  norm, or energy, of the signal is conserved between the time and scale domains, in accordance with the below equation (14): 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       0 
                       ∞ 
                     
                     ⁢ 
                     
                       
                         
                            
                           
                             x 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                         
                         2 
                       
                       ⁢ 
                       dt 
                     
                   
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       ∞ 
                     
                     ⁢ 
                     
                       
                         
                            
                           
                             X 
                             ⁡ 
                             
                               ( 
                               c 
                               ) 
                             
                           
                            
                         
                         2 
                       
                       ⁢ 
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         c 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     Therefore, given a finite energy signal, its scale transform and its inverse always exists. Since system  110  is configured to process physical signals that have finite energy, the scale transform may be defined. Parseval&#39;s theorem then also implies that the magnitude of the scale transform exhibits invariance to energy-preserving scaling operations, in accordance with the below equation (15):
 
 S{√{square root over (α)}x (α t )}= X ( c ) e   jc ln(α).   (15)
 
     As with the Mellin transform, the scale transform is closely related to a logarithmic-time Fourier transform. By performing a change of variables, such that t=e τ  and dt=e τ  dτ, the scale transform can be represented in accordance with the below equation (16): 
                           𝒮   ⁢     {     x   ⁡     (   t   )       }       =       ⁢     X   ⁡     (   c   )                   =       ⁢       ∫     -   ∞     ∞     ⁢       x   ⁡     (     e   τ     )       ⁢     e       (       -   jc     +     1   ⁢     /     ⁢   2       )     ⁢   τ       ⁢           ⁢   d   ⁢           ⁢   τ                   =       ⁢     ℱ   ⁢     {       e       (     1   ⁢     /     ⁢   2     )     ⁢   τ       ⁢     x   ⁡     (     e   τ     )         }                       ⁢     ℱ   ⁢       {       x   ~     ⁡     (   τ   )       }     .                     (   16   )               
From this expression, system  110  may also represent the inverse scale transform as indicated in the below equation (17)
 
 s   −1   {X ( c ); t}=e   −(1/2)ln(t)   F   −1   {X ( c );ln( t )}  (17)
 
In an example, system  110  may not compute the inverse scale transform, as performing the substitution of τ=ln(t) has several numerical complications. Instead, system  110  may extract the same information using the inverse Fourier transform.
 
     In equation (16), {tilde over (x)}(τ) is an energy-preserved, exponentially skewed signal, as demonstrated by performing the same change of variables, where t=e τ  and dt=e τ dτ, to show 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       0 
                       ∞ 
                     
                     ⁢ 
                     
                       
                         
                            
                           
                             x 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                         
                         2 
                       
                       ⁢ 
                       dt 
                     
                   
                   = 
                   
                     
                       ∫ 
                       0 
                       ∞ 
                     
                     ⁢ 
                     
                       
                         
                            
                           
                             
                               e 
                               
                                 
                                   ( 
                                   
                                     1 
                                     / 
                                     2 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 τ 
                               
                             
                             ⁢ 
                             
                               x 
                               ⁡ 
                               
                                 ( 
                                 
                                   e 
                                   τ 
                                 
                                 ) 
                               
                             
                           
                            
                         
                         2 
                       
                       ⁢ 
                       
                         dt 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     Therefore, the scale transform is equivalent to the Fourier transform of a signal that has been exponentially time-skewed with constant energy. When performing the inverse scale transform, system  110  can reverse this process
 
 {tilde over (x)} (τ)= F   −1   {X ( c )},  (19)
 
     and then perform an energy-preserving logarithmic skew on {tilde over (x)}(τ), such that x(t)=e (1/2)ln(t) {tilde over (x)}(ln(t)), to return the signal to its original domain. 
     Scale Cross-Correlation Function 
     As with the Mellin transform, system  110  defines a cross-correlation function between two signals and use it to compute the scale factor and scale-invariant descriptor of similarity between them. 
     Optimization in the Stretch Factor Domain α 
     The scale cross-correlation function of two signals x(t) and s(t) is defined by the product of their scale domains, as indicated in the below equation (20):
 
 x ( t )⋄ s   α ( t )= S   −1 {   S{x ( t )}   S{s ( t )}},  (20)
 
     where ⋄ represents the scale cross-correlation operation and the overbar represents a complex conjugation operation. In the time domain, the scale cross-correlation function is defined in accordance with the below equation (21): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             Φ 
                             xs 
                           
                           ⁡ 
                           
                             ( 
                             α 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               x 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             ⁢ 
                             ◇ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 s 
                                 α 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           = 
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 x 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 s 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               dt 
                             
                           
                         
                       
                     
                     
                       
                         α 
                         &gt; 
                         0. 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     In equation (21), if y(t)=x(t)⋄s(t) is the scale cross-correlation of x(t) and s(t), then Φ xs (t) is the inner product of every x(t) and s(α t) pair for values of a greater than zero. 
     In some examples, due to numerical complications, system  110  may not compute the inverse scale transform. To avoid these complications, system  110  further manipulates equation (20) by substituting the inverse scale transform relationship in equation (17) and expressing the scale cross correlation in the ln(a) domain as indicated in the below equation (22): 
                             Φ   xs     ⁡     (   α   )       ⁢       (     =       ⁢   e     )         -     (     1   ⁢     /     ⁢   2     )       ⁢           ⁢     ln   ⁢             ⁢             (   α   )         ⁢     ℱ     -   1       ⁢     {       X   *     (   c   )     ⁢     S   ⁡     (   c   )         ;     ln   ⁢           ⁢     (   α   )         }                 =       ⁢       α       -   1     /   2       ⁢         Ψ   xs     ⁡     (     ln   ⁢           ⁢     (   α   )       )       .                     (   22   )               
The function ψ(ln(α)) is another representation for the scale-cross correlation in the ln(α) domain.
 
Computation of Scale Invariant Correlation (SIC) Method
 
     For Ψ xs (ln(α)), which may be computable by system  110 , the natural logarithm is monotonic and the maximum with respect to α is equivalent to the maximum with respect to ln(α). Therefore, the maximum corresponds to the same maximum in Φ xs (ln(α)). Therefore, system  110  computes the optimal stretch factor (e.g., scale factor) as indicated in the below equation (23): 
                     α   ^     =       exp   (     arg   ⁢           ⁢       max     ln   ⁢           ⁢     (   α   )         ⁢       Ψ   xs     ⁡     (     ln   ⁡     (   α   )       )           )     .             (   23   )               
System  110  defines a scale-invariant correlation coefficient defined generally by the below equation (24):
 
                           ϕ   xs     =       ⁢       max   α     ⁢         α         σ   x     ⁢     σ   s         ⁢       ∫   0   ∞     ⁢     x   *     (   t   )     ⁢     s   ⁡     (     α   ⁢           ⁢   t     )       ⁢           ⁢   dt                       =       ⁢       max   α     ⁢         α         σ   x     ⁢     σ   s         ⁢       Φ   xs     ⁡     (   α   )                         =       ⁢       max     ln   ⁢           ⁢     (   α   )         ⁢       1       σ   x     ⁢     σ   s         ⁢       Ψ   xs     ⁡     (     ln   ⁢           ⁢     (   α   )       )             ,                 (   24   )               
where φ xs  is normalized such that φ xs =1 when x(t) is a stretched replica of s(t). The scale estimate {circumflex over (α)} (e.g., scale factor) specifies how much a signal has scaled or stretch relative to the baseline and the scale-invariant correlation coefficient φ xs  specifies how similar the two signals are, invariant to scaling or stretching.
 
Computation of Iterative Scale Transform (IST) Method—Maximization in the Scale Transform Domain c
 
     Instead of computing the estimated scale factor directly from the scale-cross correlation function, system  110  computes it through an optimization in the scale transform domain c, e.g., by maximizing the scale-cross correlation function in the scale transforms domain c. 
     In an example, by applying Parseval&#39;s theorem of 
                 ∫   0   ∞     ⁢     x   *     (   t   )     ⁢     s   ⁡     (   t   )       ⁢           ⁢   dt       =       1     2   ⁢           ⁢   π       ⁢       ∫     -   ∞     ∞     ⁢     X   *     (   c   )     ⁢     S   ⁡     (   c   )       ⁢           ⁢   d   ⁢           ⁢   c               
and a time stretching property in accordance with S{√{square root over (α)}x(αt);c}=X(c)e jcln(α)  to the scale-cross correlation function Φ xs (α), system  110  expresses an optimal stretch estimate in the scale transform domain (e.g., a scale factor) as indicated in the below equation 25:
 
                           α   ^     =       ⁢     arg   ⁢           ⁢       max   o     ⁢         α         σ   x     ⁢     σ   s         ⁢       Φ   xs     ⁡     (   α   )                             ⁢     arg   ⁢           ⁢       max   α     ⁢       1     2   ⁢           ⁢   π   ⁢           ⁢     σ   x     ⁢     σ   s         ⁢       ∫     -   ∞     ∞     ⁢     X   *     (   c   )     ⁢     S   ⁡     (   c   )       ⁢     e     j   ⁢           ⁢   c   ⁢           ⁢   ln   ⁢           ⁢     (   α   )         ⁢           ⁢   d   ⁢           ⁢   c                           (   25   )               
In this example, system  110  modifies the stretch factor α by altering the phase of the scale transform, e.g., by multiplying either X(c) or S(c) by a complex exponential.
 
     The scale-invariant correlation coefficient may then be computed by using from equation (25), in accordance with the below equation (26): 
                     ϕ   xs     ⁢           =       1     2   ⁢           ⁢   π   ⁢           ⁢     σ   x     ⁢     σ   s         ⁢       ∫     -   ∞     ∞     ⁢     X   *     (   c   )     ⁢     S   ⁡     (   c   )       ⁢     e     j   ⁢           ⁢   c   ⁢           ⁢   ln   ⁢           ⁢     (     α   ^     )         ⁢           ⁢   d   ⁢           ⁢     c   .                   (   26   )               
These expressions provide another means of computing the scale factor estimate and scale-invariant correlation coefficient.
 
Performance of Scale Transform Methods
 
     As discussed in further detail below, system  110  implements SIC technique by computing the scale transform of each signal followed by computing the scale cross-correlation function, from which the scale factor and scale-invariant correlation coefficient is extracted. System  110  implements IST by computing the scale transform of each signal and then uses a standard iterative optimization method to compute the scale factor and scale-invariant correlation coefficient directly from the scale transforms. In an example, the SIC technique may be more computationally efficient, e.g., relative to computation efficiency of the IST technique. In still another example, the IST technique is associated with increased resolution in terms of how accurate the results may be, e.g., relative to the accuracy of the SIC technique. 
     Scale-Invariant Correlation (SIC) Method 
     The scale-invariant correlation (SIC) method maximizes Φ xs (ln(α)) directly in the log-stretch factor domain ln(a) as expressed in equation (23). In this example, by sampling x(t) and s(t) in the time domain and truncating the signal to a length of N samples, the scale transform representations, X(c) and S(c), are represented only by a finite number of values. Using equation (23), system  110  computes Φ xs (ln(α)) as the inverse Fourier transform of X*(c)S(c). In this example, system  110  evaluates Φ xs (ln(α)) for a finite, discrete set of stretch factors α. 
     The resolution of the set of stretch factors is defined by the sampling interval of Φ xs (ln(α)). Assuming a unitary sampling period, x(t) is defined over the range 1≤t≤N and x(e T ) is defined for 0≤T≤ln(N). In this example, since x(e T ) is of length N ln(N), the interval between each sample is 1/N. In this example, since Φ xs (ln(α)) is related to x(e T ) and s(e T ) by a Fourier transform followed by an inverse Fourier transform, it also has a sampling interval of 1/N. 
     This implies that the smallest measurable deviation from α=1 is 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     α 
                   
                   = 
                   
                     
                       exp 
                       ⁡ 
                       
                         ( 
                         
                           ± 
                           
                             1 
                             N 
                           
                         
                         ) 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     For sufficiently large values of N, system  110  approximates Δα by a first order Taylor series approximate to obtain 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     α 
                   
                   ≈ 
                   
                     1 
                     ± 
                     
                       
                         1 
                         N 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     In this example, the resolution of SIC is approximately 1/N. Therefore, SIC is limited in resolution. However, system  110  improves this resolution by combining SIC with an iterative optimization approach. 
     System  110  calculates the computational complexity of the SIC method. To compute SIC, system  119  exponentially resamples x(t) and s(t), computes Fourier transforms of x(t) and s(t), compute an inverse Fourier transform, and obtains a maximum of the result (and/or an increased value). Maximizing Φ xs (ln(α)) and exponentially resampling x(t) can both be computed in linear time. In this example, x(e τ ) is of length N ln(N) and the computational complexity of computing its Fourier transform, using the fast Fourier transform algorith, is O(N ln(N)log(N ln(N))), or O(N ln(N)log(N)) after simplifying. Since this is the most computationally expensive operation in SIC, the computational complexity of SIC is also O(N ln(N)log(N)). 
     Iterative Scale Transform (IST) Method 
     The iterative scale transform (IST) method maximizes the scale cross-correlation function Φ xs (α) by phase shifting X*(c) or S(c) in the scale transform domain c as shown in equation (60). Solving this optimization problem iteratively in the scale transform domain c allows IST to have increased precision. In an example, the scale cross-correlation is not (globally) convex, but is locally convex around multiple maxima. 
     To compute the stretch factor estimate {circumflex over (α)} using IST, system  110  computes the scale transforms X*(c) and S(c). As with SIC, the complexity of these operation is O(N ln(N)log(N)). System  110  selects an initial guess for α, multiply S(c) (or X*(c)) by e jcln(α) , and then computes the inner product between X*(c) and S(c)e jcln(α) . Each of these operations has a linear complexity. This process of choosing an α, applying a phase shift, and computing an inner product is then repeated for different values of α by a vortex optimization algorithm until the inner product converges to a maximum value. The complexity of most convex optimization algorithms, neglecting special cases, is O(M 2 ) where M is the number of parameters to optimize across. For this application, system  110  optimizes across one variable α, so M=1 and the complexity is constant. Therefore, the complexity of the optimization procedure is O(N ln(N)), for each iteration, where N ln(N) is the number of samples in the scale transform domain. 
     System  110  also improves the computational speed of IST by taking advantage of the structure of the scale transform. The majority of the energy in a signal is often located early in the scale transform domain. Therefore, system  110  truncates a large portion of the domain with little loss of information. As a result, the cost of the iterative algorith becomes O((ρN ln(N)), where ρ represents the percentage of the scale transform domain retained after truncation. 
     SIC/IST Combination 
     As previously discussed, IST is a very precise estimation strategy but only if the result converges to the global maximum. In contrast, SIC requires no assumption of convexity but has a finite resolution. By combining these two methods, system  110  obtains highly precise estimates and guarantees convergence to the global maximum. This is done by using SIC to generate the initial stretch factor estimate {circumflex over (α)} for IST. In general, the SIC estimate will lie within the locally convexity region around the global maximum of the scale cross-correlation function Φ xs (α). 
     In an example, if N is small enough such that SIC cannot adequately resolve the main lobe of the scale cross-correlation function, then the SIC estimate may not be accurate and IST may not be guaranteed to converge to the globally optimal result. However, for sufficiently large values of N, this is not an issue. 
     Since IST computes the scale transform representations, X(c) and S(c), the only additional step required when combined with SIC is the computation of the inverse Fourier transform in the below equation (29) and maximization over Φ xs (α) in equation (23). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           α 
                           ^ 
                         
                         = 
                           
                         ⁢ 
                         
                           arg 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               max 
                               α 
                             
                             ⁢ 
                             
                               
                                 
                                   α 
                                 
                                 
                                   
                                     σ 
                                     x 
                                   
                                   ⁢ 
                                   
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                                     s 
                                   
                                 
                               
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                                     - 
                                     
                                       ( 
                                       
                                         1 
                                         ⁢ 
                                         
                                           / 
                                         
                                         ⁢ 
                                         2 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ln 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     α 
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 ℱ 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     X 
                                     * 
                                     
                                       ( 
                                       c 
                                       ) 
                                     
                                     ⁢ 
                                     
                                       S 
                                       ⁡ 
                                       
                                         ( 
                                         c 
                                         ) 
                                       
                                     
                                   
                                   ; 
                                   
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                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       α 
                                       ) 
                                     
                                   
                                 
                                 } 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           arg 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               max 
                               α 
                             
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         α 
                                       
                                       
                                         
                                           σ 
                                           x 
                                         
                                         ⁢ 
                                         
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                                           s 
                                         
                                       
                                     
                                     ⁢ 
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                                   ) 
                                 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     X 
                                     * 
                                     
                                       ( 
                                       c 
                                       ) 
                                     
                                     ⁢ 
                                     
                                       S 
                                       ⁡ 
                                       
                                         ( 
                                         c 
                                         ) 
                                       
                                     
                                   
                                   ; 
                                   
                                     ln 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       α 
                                       ) 
                                     
                                   
                                 
                                 } 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           arg 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               max 
                               α 
                             
                             ⁢ 
                             
                               
                                 
                                   α 
                                 
                                 
                                   
                                     σ 
                                     x 
                                   
                                   ⁢ 
                                   
                                     σ 
                                     s 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     Ψ 
                                     
                                       x 
                                       , 
                                       s 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       ln 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         ( 
                                         α 
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     The computational complexity of these operations is O(N ln(N)log(N)), the same as initially computing X(c) and S(c). Therefore, these operations do not change the overall computational complexity of IST and the complexity of SIC/IST is equivalent to the computational complexity of IST. 
     The properties of various temperature compensation techniques are shown in the below Table 1: 
                                         TABLE 1                               Computational   Iterative           Methods   Resolution   Complexity   Complexity                                                        Finite   Search OSS   1/N   O(RN 2 )   —       Resolution   SIC   1/N   O(N ln(N)log(N))   —       Methods       Fine   Search/   —   O(RN 2 )   O(N)       Resolution   Iterative OSS       Methods   SIC/IST   —   O(N ln(N)log(N))   O(pN ln(N))                    
The Discrete-Time Scale Transform
 
     In additional to processing continuous-time signals, system  110  is also configured to process discrete time signals that are sampled. As implied from equations (11) and (18), a discrete-time Mellin transform or discrete-time scale transform (DTST) can be computed by exponentially sampling a continuous-time signal and taking its discrete-time Fourier transform (DTFT). Unfortunately, uniform sampling is usually performed. This presents several challenges in implementing the Mellin or scale transforms. 
     If the samples are uniformly spaced, then the discrete-time Mellin or scale transforms must be approximated. One option is to replace the continuous-time signals with discrete-time signals and the integrals with summations. However, this provides a poor approximation of the transform. In practice, this “direct′” approach has been shown to be less effective than other approximations and is slow, requiring O(N 2 ) complexity. More often, the Mellin and scale transforms are computed by exponentially resampling the uniformly sampled signal and computing its Fourier transform with the FFT. This is known as the fast Mellin transform (FMT) algorithm. 
     Compared to the direct Mellin transform approximation, the fast approximation is more accurate and computationally efficient. The exponential resampling operation is implemented by interpolating unknown points on an exponential grid. According to the Nyquist-Shannon sampling theorem, a band-limited signal may be perfectly reconstructed by sinc interpolation. However, performing sinc interpolation on an exponential grid is impractical and computationally expensive. Instead, cubic spline interpolation is commonly used. Cubic spline interpolation provides a close approximation to sinc interpolation and may be computed with O(M) complexity, where M is the number of points on the exponential grid. The DTFT of the resampled signal then is computed using the FFT algorithm, which requires O(M log(M)) complexity. 
     Mellin Resampling Theorem 
       FIGS. 3A-3C  illustrate examples of a sinusoid sampled on uniform and an exponential grid. Referring to  FIG. 3A , graph  300  illustrates a continuous-time sine wave. Referring to  FIG. 3B , graph  302  illustrates a uniformly sampled sine wave. Referring to  FIG. 3C , graph  304  illustrates an exponentially sampled sine wave. 
     In generating an exponentially resampled signal, system  110  uses an exponential grid. To generate the exponential grid, system  110  determines 1) where the first sample in the exponential grid is located in time; and 2) a number of samples to be included in the exponential grid. In this example, system  110  defines the first sample&#39;s location and, from that result, derives an exact solution for the necessary number of samples. 
     In an example, signals may begin at time t=0, which is the scaling origin. In another example, on an exponential grid, t=0 translates to τ=ln(0)=−∞. So as the first sample approaches t=0, the sampling interval approaches 0 and the number of necessary samples approaches infinity. To define the location of the first sample, system  110  enforces causality in the exponential-time domain. A causal signal is nonzero for time τ≥0. In this example, causality is in exponential time, the signal x(t) will be nonzero for values ln(t)≥0 or t≥1. 
     When exponential-time causality is assumed, the scale transform becomes 
     
       
         
           
             
               
                 
                   
                     
                       ℱ 
                       ⁢ 
                       
                         { 
                         
                           
                             e 
                             
                               
                                 ( 
                                 
                                   1 
                                   / 
                                   2 
                                 
                                 ) 
                               
                               ⁢ 
                               τ 
                             
                           
                           ⁢ 
                           
                             x 
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 τ 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                     = 
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                       ⁢ 
                       
                         
                           e 
                           
                             
                               ( 
                               
                                 1 
                                 / 
                                 2 
                               
                               ) 
                             
                             ⁢ 
                             τ 
                           
                         
                         ⁢ 
                         
                           x 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               τ 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           e 
                           
                             - 
                             jc 
                           
                         
                         ⁢ 
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         τ 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       𝒮 
                       ⁢ 
                       
                         { 
                         
                           x 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         } 
                       
                     
                     = 
                     
                       
                         ∫ 
                         1 
                         ∞ 
                       
                       ⁢ 
                       
                         
                           x 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         ⁢ 
                         
                           t 
                           
                             
                               - 
                               jc 
                             
                             - 
                             
                               1 
                               / 
                               2 
                             
                           
                         
                         ⁢ 
                         
                           dt 
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     Exponential-time causality implies that the time signal x(t) begins at time t=1. In continuous time, this is not advantageous since it removes data. However, for a uniformly sampled discrete-time signal, this assumption is very practical. In discrete-time, t=nT, where T is the sampling interval. The sampling interval T scales the signal x[nT], which translates to a phase shift in the scale domain,
 
 S{x [ nT ]}= T   −1/2   S{x [ n ]} e   jc ln(T) .  (31)
 
     Therefore, when system  110  computes the DTST, system  110  determines (or accesses information specifying that) the signal has a sampling rate of 1. Under a unitary sampling rate, t=1 occurs when n=1. So in discrete-time, the causality condition allows us to exponentially resample starting from n=1 and neglect the asymptotic conditions around n=0, the scaling origin. This may require us to remove a single sample, assuming the first sample is located on the scaling origin. However, when analyzing the scaling behavior of a signal, the origin contains no information since it is not affected by scaling, x[α 0]=x[0], and the scale transform of values at the origin are not well defined. For example, the scale transform of a Dirac impulse δ(t) is a division by zero 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           𝒮 
                           ⁢ 
                           
                             { 
                             
                               δ 
                               ⁢ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             } 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                           ⁢ 
                           
                             
                               δ 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             ⁢ 
                             
                               t 
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   c 
                                 
                                 - 
                                 
                                   1 
                                   ⁢ 
                                   
                                     / 
                                   
                                   ⁢ 
                                   2 
                                 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             dt 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               
                                 ( 
                                 0 
                                 ) 
                               
                               
                                 
                                   - 
                                   jc 
                                 
                                 - 
                                 
                                   1 
                                   ⁢ 
                                   
                                     / 
                                   
                                   ⁢ 
                                   2 
                                 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     For these reasons, it is practical for us to assume the first sample is n=1 and the exponential-time domain is causal. 
     By enforcing causality, the exponentially resampled signal can be computed with a finite number of samples. To properly resample the signal, however, system  110  ensures that no information (or a reduced amount of information) is lost. This condition requires that the minimum sampling rate satisfies the Nyquist sampling criteria,
 
 {tilde over (F)}   s ≥2 f   max ,  (33)
 
     where {tilde over (F)} s  is the minimum sampling rate on the exponential grid and f max  is the largest non-zero frequency in the signal. For example, if the uniform signal is critically sampled, where the sampling rate F s =2f max , then the exponentially resampled signal will require more samples than the uniform signal. 
     Mellin Resampling Theorem. 
     In this example, {tilde over (x)}[m] includes a causal, exponentially resampled signal generated from the uniformly sampled signal x[n], where x[n] is sampled at the Nyquist rate F s , and exists for all integers in the domain 1≤n≤N. Assume {tilde over (x)}[m] to have a minimum sampling rate of {tilde over (F)} s  and include M samples. To promote maintenance of the Nyquist criteria {tilde over (F)} s ≥F s  is maintained, system  110  determines M in accordance with the below equation (34): 
     
       
         
           
             
               
                 
                   M 
                   ≥ 
                   
                     
                       
                         ln 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             / 
                             N 
                           
                           ) 
                         
                       
                       
                         ln 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             - 
                             
                               1 
                               / 
                               N 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     1. 
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     In this example, without following the integer constraint, an exponentially resampled signal {tilde over (x)}[ñ] would exist across the domain 0≤ñ≤ln(N). To satisfy the integer constraint, system  110  determines ñ in accordance with the equation shown in the below equation (35): 
     
       
         
           
             
               
                 
                   
                     
                       n 
                       ~ 
                     
                     = 
                     
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Λ 
                       
                       = 
                       
                         m 
                         ⁡ 
                         
                           ( 
                           
                             
                               ln 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 N 
                                 ) 
                               
                             
                             
                               M 
                               - 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     0 
                     ≤ 
                     m 
                     ≤ 
                     
                       M 
                       - 
                       1. 
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     The constant Λ represents the sampling interval for the exponentially resampled signal. Λ is referred to as the Mellin interval and its inverse Λ −1  as the Mellin rate. To then maintain the Nyquist criteria, the minimum sampling rate in {tilde over (x)}[m] must be greater than or equal to the Nyquist rate. This is the same as stating that maximum sampling interval, which lies between the last two samples as illustrated in  FIG. 2 , in {tilde over (x)}[m] must be less than or equal to one uniform sampling interval. Following that logic, system  110  makes a determination in accordance with the below equation (36): 
     
       
         
           
             
               
                 
                   
                     1 
                     ≥ 
                     
                       N 
                       - 
                       
                         exp 
                         ⁡ 
                         
                           ( 
                           
                             
                               ( 
                               
                                 M 
                                 - 
                                 2 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 ln 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   N 
                                   ) 
                                 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     1 
                     ≥ 
                     
                       N 
                       - 
                       
                         N 
                         
                           
                             M 
                             - 
                             2 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       N 
                       
                         
                           M 
                           - 
                           3 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                     
                     ≥ 
                     
                       N 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             M 
                             - 
                             2 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ln 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         N 
                         ) 
                       
                     
                     ≥ 
                     
                       ln 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           N 
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       M 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             ln 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               N 
                               ) 
                             
                           
                           - 
                           
                             ln 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 N 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     ≥ 
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ln 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           N 
                           ) 
                         
                       
                       - 
                       
                         ln 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             N 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     M 
                     ≥ 
                     
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             N 
                             ) 
                           
                         
                         - 
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             
                               N 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             N 
                             ) 
                           
                         
                         - 
                         
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               N 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     M 
                     ≥ 
                     
                       
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             N 
                             ) 
                           
                         
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             
                               N 
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       1 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     M 
                     ≥ 
                     
                       
                         
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               / 
                               N 
                             
                             ) 
                           
                         
                         
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 1 
                                 / 
                                 N 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       1. 
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     The Mellin Resampling Theorem provides an exact bound on the number of samples necessary to exponentially resample a signal, starting from n=1, while maintaining the Nyquist criteria. In practice, M is an integer, so a ceiling operation would be applied to the result. When a signal is critically resampled in accordance with the below equation (37): 
     
       
         
           
             
               
                 
                   
                     M 
                     = 
                     
                       
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               / 
                               N 
                             
                             ) 
                           
                         
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 1 
                                 / 
                                 N 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       1 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     the Mellin interval from equation (35) becomes
 
Λ=−ln(1−1 /N ).  (38)
 
     In this example, M increases with N. In this example, ln(1−1/N) is replaced with its first order Taylor series expansion −1/N. After making that substitution, the first-order approximation of equation (34)(34) becomes 
     
       
         
           
             
               
                 
                   
                     
                       
                         ln 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             / 
                             N 
                           
                           ) 
                         
                       
                       
                         ln 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             - 
                             
                               1 
                               / 
                               N 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     1 
                   
                   ≈ 
                   
                     
                       N 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ln 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         N 
                         ) 
                       
                     
                     + 
                     1. 
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     The error function for this approximation is then defined by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             g 
                             ⁡ 
                             
                               ( 
                               N 
                               ) 
                             
                           
                           = 
                             
                           ⁢ 
                           
                             
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     / 
                                     N 
                                   
                                   ) 
                                 
                               
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       1 
                                       / 
                                       N 
                                     
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               N 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ln 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 N 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             - 
                             
                               ln 
                               ⁡ 
                               
                                 ( 
                                 N 
                                 ) 
                               
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   1 
                                   + 
                                   
                                     N 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ln 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           
                                             1 
                                             / 
                                             N 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 
                                   ln 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         1 
                                         / 
                                         N 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                   ⁢ 
                   
                     ( 
                     40 
                     ) 
                   
                 
               
             
           
         
       
     
     Assuming N&gt;1, ln(1−1/N)&lt;0 and N ln(1−1/N)&lt;−1. From these inequalities, system  110  determines that the approximation error g(N)&lt;0. In this example, under a first order approximation,
 
 M&gt;N  ln( N )+1.  (41)
 
     Therefore, M increases with N at a rate of approximately N ln(N). Referring to  FIG. 2 , diagram  200  shows the first-order approximation error curve g(N). In general, the approximation is very close, deviating by approximately 1.15 samples for every power of 10 samples N. This also demonstrates that the Mellin resampling theorem is consistent with previous results, which demonstrated that, to maintain the Nyquist criteria, M&gt;N ln(N). 
     Discrete-Time Scale Transform 
     Using the Mellin resampling theorem in equation (34), the DTST is defined as implemented by the fast Mellin transform algorithm. In this example, {tilde over (x)}[mΛ] includes the exponentially resampled and amplified signal generated from a uniformly sampled signal with a unitary sampling rate x[n],
 
 {tilde over (x)} [ m Λ]= e   (1/2)mΛ   x [ e   mΛ ].  (42)
 
     The Mellin interval Λ is defined in equation (35). The DTST is then defined as 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           𝒮 
                           ⁢ 
                           
                             { 
                             
                               x 
                               ⁡ 
                               
                                 [ 
                                 n 
                                 ] 
                               
                             
                             } 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           X 
                           ⁡ 
                           
                             ( 
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 c 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               o 
                             
                             ∞ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               x 
                               ⁡ 
                               
                                 [ 
                                 
                                   m 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Λ 
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             ℱ 
                             ⁢ 
                             
                               { 
                               
                                 
                                   x 
                                   ~ 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     m 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Λ 
                                   
                                   ] 
                                 
                               
                               } 
                             
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
         
         
           
             where F now represents the DTFT of a discrete-time signal. The inverse transform can be computed by taking the inverse DTFT 
           
         
       
    
     
       
         
           
             
               
                 
                   
                     
                       
                         x 
                         ~ 
                       
                       ⁡ 
                       
                         [ 
                         
                           m 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Λ 
                         
                         ] 
                       
                     
                     = 
                     
                       
                         
                           ℱ 
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           { 
                           
                             X 
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 jc 
                               
                               ) 
                             
                           
                           } 
                         
                       
                       = 
                       
                         
                           ∫ 
                           
                             - 
                             π 
                           
                           π 
                         
                         ⁢ 
                         
                           
                             X 
                             ⁡ 
                             
                               ( 
                               
                                 e 
                                 jc 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             e 
                             jct 
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           c 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
         
         
           
             and logarithmically resampling and attenuating back to the original domain. 
           
         
       
    
     In another example, system  110  uses a perfect discrete-time interpolator. In this example, {tilde over (x)}[m] represents the energy-preserving time-skewed replica of the original uniformly sampled signal, such that 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         ∞ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                            
                           
                             x 
                             ⁡ 
                             
                               [ 
                               n 
                               ] 
                             
                           
                            
                         
                         2 
                       
                     
                     = 
                     Λ 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         
                           - 
                           ∞ 
                         
                       
                       ∞ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                            
                           
                             x 
                             ⁡ 
                             
                               [ 
                               
                                 Λ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 m 
                               
                               ] 
                             
                           
                            
                         
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     In this example, the actual energy in the exponentially resampled signal will depend on the interpolation process used. 
     As in the continuous case, a scaling operation on a discrete-time signal translates to an energy-normalizing phase shift in the scale domain. For a uniformly sampled signal, a scaling operation x[βn] becomes a shift after the exponential resampling process. Therefore, in the scale domain, the scaling operation becomes 
                           𝒮   ⁢     {     x   ⁡     [     β   ⁢           ⁢   n     ]       }       =       ⁢     ℱ   ⁢     {       e       (       -   1     ⁢     /     ⁢   2     )     ⁢           ⁢     ln   ⁡     (   β   )           ⁢       x   ~     ⁡     [       m   ⁢           ⁢   Λ     +     ln   ⁡     (   β   )         ]         }                   =       ⁢       β       -   1     ⁢     /     ⁢   2       ⁢     X   ⁡     (   c   )       ⁢       e       -   j     ⁢           ⁢   c   ⁢           ⁢   ln   ⁢           ⁢     (   β   )         .                     (   46   )               
Discrete-Time Scale Cross-Correlation Function
 
     In this example, x[nT] and s[nT] include uniformly sampled signals with sampling interval T. The discrete-time Mellin cross-correlation between x[nT] and s[nT] is defined as 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               Φ 
                               xs 
                             
                             ⁡ 
                             
                               [ 
                               α 
                               ] 
                             
                           
                           = 
                             
                           ⁢ 
                           
                             
                               x 
                               ⁡ 
                               
                                 [ 
                                 nT 
                                 ] 
                               
                             
                             ⁢ 
                             
                               
                                 ◇s 
                                 α 
                               
                               ⁡ 
                               
                                 [ 
                                 nT 
                                 ] 
                               
                             
                           
                         
                         , 
                         
                           α 
                           &gt; 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               ∞ 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 x 
                                 ⁡ 
                                 
                                   [ 
                                   nT 
                                   ] 
                                 
                               
                               ⁢ 
                               
                                 s 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     α 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     nT 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         , 
                         
                           α 
                           &gt; 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             ( 
                             
                               = 
                                 
                               ⁢ 
                               𝒮 
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             { 
                             
                               
                                 
                                   𝒮 
                                   ⁢ 
                                   
                                     { 
                                     
                                       x 
                                       ⁡ 
                                       
                                         [ 
                                         nT 
                                         ] 
                                       
                                     
                                     } 
                                   
                                 
                                 _ 
                               
                               ⁢ 
                               𝒮 
                               ⁢ 
                               
                                 { 
                                 
                                   s 
                                   ⁡ 
                                   
                                     [ 
                                     nT 
                                     ] 
                                   
                                 
                                 } 
                               
                             
                             } 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
     Since T scales the signal, it translates to a phase shift in the scale domain so that 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             Φ 
                             xs 
                           
                           ⁡ 
                           
                             [ 
                             α 
                             ] 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             𝒮 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   ℒ 
                                   ⁢ 
                                   
                                     { 
                                     
                                       x 
                                       ⁡ 
                                       
                                         [ 
                                         nT 
                                         ] 
                                       
                                     
                                     } 
                                   
                                 
                                 _ 
                               
                               ⁢ 
                               
                                 T 
                                 
                                   
                                     - 
                                     1 
                                   
                                   ⁢ 
                                   
                                     / 
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ln 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     T 
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               𝒮 
                               ⁢ 
                               
                                 { 
                                 
                                   s 
                                   ⁡ 
                                   
                                     [ 
                                     n 
                                     ] 
                                   
                                 
                                 } 
                               
                               ⁢ 
                               
                                 T 
                                 
                                   
                                     - 
                                     1 
                                   
                                   ⁢ 
                                   
                                     / 
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ln 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     T 
                                     ) 
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             T 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             𝒮 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               { 
                               
                                 
                                   
                                     𝒮 
                                     ⁢ 
                                     
                                       { 
                                       
                                         x 
                                         ⁡ 
                                         
                                           [ 
                                           nT 
                                           ] 
                                         
                                       
                                       } 
                                     
                                   
                                   _ 
                                 
                                 ⁢ 
                                 𝒮 
                                 ⁢ 
                                 
                                   { 
                                   
                                     s 
                                     ⁡ 
                                     
                                       [ 
                                       n 
                                       ] 
                                     
                                   
                                   } 
                                 
                               
                               } 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
     In an example, discrete-time scale cross-correlation is defined from the pair of exponentially resampled and amplified signals, in accordance with the below equations:
 
 {tilde over (x)} [ m Λ]= e   (1/2)mΛ   x [ e   mΛ]   (49)
 
 {tilde over (s)} [ m Λ]= e   (1/2)mΛ   s [ e   mΛ],   (50)
         where Λ is the Mellin interval defined in (38). From these signals, system  110  determines the exponentially-skewed scale cross-correlation function, in accordance with the below equation (51):       

     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             Φ 
                             xs 
                           
                           ⁡ 
                           
                             [ 
                             
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Λ 
                             
                             ] 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             T 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             e 
                             
                               
                                 - 
                                 
                                   ( 
                                   
                                     1 
                                     ⁢ 
                                     
                                       / 
                                     
                                     ⁢ 
                                     2 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Λ 
                             
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               ∞ 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   x 
                                   ~ 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     m 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Λ 
                                   
                                   ] 
                                 
                               
                               ⁢ 
                               
                                 
                                   s 
                                   ~ 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       m 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       Λ 
                                     
                                     + 
                                     
                                       k 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       Λ 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             T 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             e 
                             
                               
                                 - 
                                 
                                   ( 
                                   
                                     1 
                                     ⁢ 
                                     
                                       / 
                                     
                                     ⁢ 
                                     2 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Λ 
                             
                           
                           ⁢ 
                           
                             F 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   
                                     ℱ 
                                     ⁢ 
                                     
                                       
                                         x 
                                         ~ 
                                       
                                       ⁡ 
                                       
                                         [ 
                                         
                                           m 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           Λ 
                                         
                                         ] 
                                       
                                     
                                   
                                   } 
                                 
                                 _ 
                               
                               ⁢ 
                               ℱ 
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     s 
                                     ~ 
                                   
                                   ⁡ 
                                   
                                     [ 
                                     
                                       m 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       Λ 
                                     
                                     ] 
                                   
                                 
                                 } 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             T 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             e 
                             
                               
                                 - 
                                 
                                   ( 
                                   
                                     1 
                                     ⁢ 
                                     
                                       / 
                                     
                                     ⁢ 
                                     2 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Λ 
                             
                           
                           ⁢ 
                           
                             F 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   𝒮 
                                   ⁢ 
                                   
                                     { 
                                     
                                       x 
                                       ⁡ 
                                       
                                         [ 
                                         n 
                                         ] 
                                       
                                     
                                     } 
                                   
                                 
                                 _ 
                               
                               ⁢ 
                               𝒮 
                               ⁢ 
                               
                                 { 
                                 
                                   s 
                                   ⁡ 
                                   
                                     [ 
                                     n 
                                     ] 
                                   
                                 
                                 } 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               e 
                               
                                 
                                   - 
                                   
                                     ( 
                                     
                                       1 
                                       ⁢ 
                                       
                                         / 
                                       
                                       ⁢ 
                                       2 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Λ 
                               
                             
                             ⁢ 
                             
                               
                                 Ψ 
                                 xs 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   k 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Λ 
                                 
                                 ] 
                               
                             
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
         
         
           
             where kΛ=ln(α). System  110  defines Ψ xs [kΛ] to be the exponential-time cross-correlation function in discrete time,
 
Ψ xs [ k Λ]= F   −1 { S { x [ nT ]} S{s [ nT ]}}.  (52)
 
           
         
       
    
     If [mΛ] and [mΛ] are nonzero within the range 0≤m≤M−1, then Φxs[kΛ] and Ψxs[kΛ] will be nonzero over −(M−1)≤k≤M−1. Direct computation of the scale cross-correlation Φxs[α] requires implementation of the inverse scale transform, and therefore a logarithmic decimation of Φxs[kΛ]. However, Ψxs[kΛ] is not a causal signal, an assumption made by the Mellin resampling theorem. In an example, system  110  computes the scale factor and scale-invariant correlation coefficient between two signals directly from Ψxs[kΛ]. Referring to  FIG. 4 , process  400  illustrates actions in the computation of the Ψxs[kΛ] from two windowed sinusoids, which vary by a scale factor of 2. In this example, system  110  receives ( 402 ) sinusoids  403   a ,  403   b . Sinusoid  403   a  oscillates at 20 Hz. Sinusoid  403   b  oscillates at 40 Hz. System  110  exponentially resamples ( 404 ) sinusoids  403   a ,  403   b , resulting in signals  405   a ,  405   b , respectively. System  110  multiples ( 406 ) the resampled sinusoids by an exponential amplification factor, resulting in signals  407   a ,  407   b , respectively. System  110  also cross-correlates ( 408 ) the multiplied, resampled sinusoids with each other, resulting in signal  409 . 
     Discrete-Time Scale Factor Computation 
     In this example, x[n] and s[n] include scaled replicas of each other such that x[n]=s[βn]. After exponentially resampling and amplifying the signals, the scale becomes a shift
 
 {tilde over (s)} [ m Λ]= e   (1/2)mΛ   s [ e   mΛ] 
 
 {tilde over (x)} [ m Λ]=(β) −1/2   {tilde over (s)} [ m Λ+ln(β)]  (53)
 
     Under these circumstances, system  110  applies the Cauchy-Schwarz inequality to the exponential-time cross-correlation function Ψ xs [kΛ] in equation (52), 
     
       
         
           
             
               
                 
                   
                     
                       
                         1 
                         
                           
                             T 
                             2 
                           
                           ⁢ 
                           β 
                         
                       
                       ⁢ 
                       
                         
                            
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               ∞ 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   s 
                                   ~ 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       m 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       Λ 
                                     
                                     + 
                                     
                                       ln 
                                       ⁡ 
                                       
                                         ( 
                                         β 
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               ⁢ 
                               
                                 
                                   s 
                                   ~ 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       ( 
                                       
                                         m 
                                         + 
                                         k 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     Λ 
                                   
                                   ] 
                                 
                               
                             
                           
                            
                         
                         2 
                       
                     
                     ≤ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         1 
                         
                           
                             T 
                             2 
                           
                           ⁢ 
                           β 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             
                               - 
                               ∞ 
                             
                           
                           ∞ 
                         
                         ⁢ 
                         
                           
                             
                                
                               
                                 
                                   s 
                                   ~ 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       m 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       Λ 
                                     
                                     + 
                                     
                                       ln 
                                       ⁡ 
                                       
                                         ( 
                                         β 
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                                
                             
                             2 
                           
                           · 
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               ∞ 
                             
                             ⁢ 
                             
                               
                                  
                                 
                                   
                                     s 
                                     ~ 
                                   
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         ( 
                                         
                                           m 
                                           + 
                                           k 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       Λ 
                                     
                                     ] 
                                   
                                 
                                  
                               
                               2 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   = 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       1 
                       
                         
                           T 
                           2 
                         
                         ⁢ 
                         β 
                       
                     
                     ⁢ 
                     
                       
                         
                            
                           
                             
                               1 
                               Λ 
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   
                                     - 
                                     ∞ 
                                   
                                 
                                 ∞ 
                               
                               ⁢ 
                               
                                 
                                    
                                   
                                     
                                       s 
                                       ~ 
                                     
                                     [ 
                                     m 
                                     ) 
                                   
                                    
                                 
                                 2 
                               
                             
                           
                            
                         
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
           
         
       
     
     Assuming s[mΛ] is not a periodic function, the expressions in equation (54) will be equal when kΛ=ln(β). This implies that, for non-periodic functions, the scale cross-correlation has a unique maximum when 
     
       
         
           
             
               
                 
                   
                     arg 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         max 
                         k 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           Ψ 
                           xs 
                         
                         ⁡ 
                         
                           [ 
                           
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Λ 
                           
                           ] 
                         
                       
                     
                   
                   = 
                   
                     
                       ln 
                       ⁡ 
                       
                         ( 
                         β 
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         Λ 
                         
                           - 
                           1 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
           
         
       
     
     Through algebraic manipulation, this concludes that the scale factor is equal to 
     
       
         
           
             
               
                 
                   β 
                   = 
                   
                     
                       exp 
                       ⁡ 
                       
                         ( 
                         
                           Λ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           arg 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               max 
                               k 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 Ψ 
                                 xs 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   k 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Λ 
                                 
                                 ] 
                               
                             
                           
                         
                         ) 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
     The function Ψ xs [kΛ] is generally not convex. Therefore, when implementing, care must be taken to compute the maximum. 
     Discrete-Time Scale-Invariant Correlation Coefficient 
     For simplicity, assume both s[nT] and x[nT] are zero-mean signals so that their exponentially resampled versions {tilde over (s)}[mΛ] and {tilde over (x)}[mΛ] are also zero-mean. From the Cauchy-Schwarz inequality in equation (52), system  110  determines the maximum value of the exponential-time cross-correlation Ψ xs [kΛ] in accordance with the below equation (55): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             max 
                             k 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               Ψ 
                               xs 
                             
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Λ 
                               
                               ) 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             
                               T 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Λ 
                               ⁢ 
                               
                                 β 
                               
                             
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 
                                   - 
                                   ∞ 
                                 
                               
                               ∞ 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                  
                                 
                                   
                                     s 
                                     ~ 
                                   
                                   ⁡ 
                                   
                                     [ 
                                     m 
                                     ] 
                                   
                                 
                                  
                               
                               2 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             T 
                           
                           ⁢ 
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     
                                       - 
                                       ∞ 
                                     
                                   
                                   ∞ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                      
                                     
                                       
                                         s 
                                         ~ 
                                       
                                       ⁡ 
                                       
                                         [ 
                                         
                                           m 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           Λ 
                                         
                                         ] 
                                       
                                     
                                      
                                   
                                   2 
                                 
                               
                             
                             · 
                             
                               
                                 
                                   ∑ 
                                   
                                     m 
                                     = 
                                     
                                       - 
                                       ∞ 
                                     
                                   
                                   ∞ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                      
                                     
                                       
                                         x 
                                         ~ 
                                       
                                       ⁡ 
                                       
                                         [ 
                                         
                                           m 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           Λ 
                                         
                                         ] 
                                       
                                     
                                      
                                   
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             T 
                           
                           ⁢ 
                           
                             σ 
                             
                               s 
                               ~ 
                             
                           
                           ⁢ 
                           
                             σ 
                             
                               x 
                               ~ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   57 
                   ) 
                 
               
             
           
         
       
     
     System  110  computes the discrete-time scale-invariant correlation coefficient the in accordance with the below equation (58): 
     
       
         
           
             
               
                 
                   
                     
                       ψ 
                       xs 
                     
                     = 
                     
                       
                         T 
                         
                           
                             σ 
                             
                               s 
                               ~ 
                             
                           
                           ⁢ 
                           
                             σ 
                             
                               x 
                               ~ 
                             
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             max 
                             k 
                           
                           ⁢ 
                           
                             
                               Ψ 
                               xs 
                             
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Λ 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
           
         
       
     
     As in continuous time, ψ xs  is 1 when x[nT]=s[βnT], ψ xs  is −1 when x[nT]=−s[βnT], and there is no scale relationship between the two signals when ψ xs =0. System  110  uses the values between the extrema to determine the degree scale-invariant linear correlation between the two signals. 
     Simulation Setup and Methodology 
     In an example, system  110  is configured to compare the scale cross-correlation, LPC, and OSS compensation techniques using ultrasonic data from a thin plate under variable temperature conditions. To produce guided waves, system  110  uses a pair of synchronized lead zirconate titanate (PZT) piezoelectric transducers permanently bonded to the surface of a 9.8 cm wide by 30.5 cm long by 0.1 cm aluminum plate. Signals were generated and measured at a sampling rate of 1 MHz using a National Instruments PXI data acquisition equipment. For each measurement, the transducers were driven using a wide-band impulse-like sinc excitation and its response recorded for 10 ms. Referring to  FIGS. 8A, 8B , diagrams  800 ,  802  show the excitation of a time domain signal and a power spectral density, respectively. Referring to  FIGS. 9A, 9B , diagrams  900 ,  902  show the response (to excitation) of a time domain signal and a power spectral density, respectively. 
     The experiment ran for 36.3 hours between 11:30 AM and 11:50 PM of the following day. During this time, the aluminum plate was cooled and warmed in a refrigerator while guided waves were generated and automatically recorded by the transducers at 2 minute intervals. The ambient temperature within the refrigerator was also measured automatically at 1 minute intervals. Referring to  FIG. 6 , graph  600  shows the measured ambient temperature over time. Starting at 11:30 AM on day one, the temperature was gradually reduced from room temperature, 20.5° C., to 3.37° C. until 2:51 PM. Between 2:51 PM and 7:33 PM, the temperature was then gradually increased back to room temperature, 20.4° C. After 7:33 PM, the ambient temperature decreased and then remained relatively constant, at around 3.89° C., between midnight and 10:36 AM the next day. Between 10:36 AM and 5:56 PM, the temperature was again gradually increased to 20.94° C. and then gradually decreased to 4.17° C. until 11:50 PM, when the experiment ended. 
     At 6:18 PM on the first day, a cylindrical, steel, greased-coupled mass with a diameter of 3.8 cm and a length of 4.5 cm, was placed on top of the aluminum plate to simulate damage. At 4:04 PM on the second day, the mass was removed from the plate. The mass was removed to confirm that the changes in the wave propagation were a result of the mass and not an artifact introduced by the placement procedure. The timing also allowed us to measure both an increase and decrease in temperature with and without the mass present. Using a mass in place of damage allows easy repeatability of the experiment. While the mass may not perfectly simulate damage, it changes the propagation environment in ways that temperature will not. In this way, the mass acts similar to damage. 
     An impulsive signal was chosen in order to drive the PZT transducers to excite as wide a frequency band as possible. Signal scaling occurs in both the time and frequency domains. Therefore, limiting the observable frequencies will reduce the accuracy of the uniform scaling model used by the temperature compensation strategies. 
     After recording each guided wave record, the signals are high-pass filtered with a 3 kHz cutoff frequency and advanced 0.5 ms. The high-pass filtering operation in done to remove systematic low frequency noise in the system. This noise arises from various sources, such as mechanical vibrations and the pyroelectric effect. The time shift operation is necessary to remove group delay introduced by the excitation signal. Failing to account for group delay will introduce errors into the compensation strategies since the scaling origin will then not be located at t=0. 
     Correlation Analysis 
     System  110  compares five different descriptors used for detecting the mass: the correlation coefficient, the OSS correlation coefficient, two variations on the LPC correlation coefficient, and the scale-invariant correlation coefficient. The correlation coefficient is a standard measure of similarity but is very sensitive to changes in scale. In contrast, the OSS, LPC, and scale-invariant correlation coefficients are scale-invariant measures and more robust to changes in temperature. 
     For each LPC method, system  110  utilizes a different form of linear regression. The first uses a standard least-squares linear regression to estimate the scale factor. The second LPC method uses a robust linear regression method using MATLAB&#39;s robustfit function. This function performs an iteratively reweighted least-squares regression, which is more robust to outlier data. In implementing each LPC method, system  110  uses a local window size of 1000 samples with a hop size of 1 sample between each cross-correlation operation. A larger hop size would increase the computational speed of the technique, but the hop size was kept to a minimum to maximize the accuracy of the linear regression. In computing the OSS correlation coefficient, system  110  accesses a library of 200 baselines over a scale range of 0.99 to 1.01, which is reasonable for this application. The library was designed to have the same resolution as the scale cross-correlation. 
     Referring to  FIGS. 7A, 7B , graphs  700 ,  702  show the correlation coefficients computed over the course of the experiment for two different baseline signals, taken at 20.5° C. and 12.4° C. respectively. The first baseline is near the highest temperature measured and the second is approximately the median temperature. Two baselines illustrate how each method is affected by different ranges of temperature change. In both plots, the standard correlation coefficient, labeled “no compensation,” is significantly affected by temperature. As the temperature deviates from the baseline temperature, the correlation coefficient decreases significantly. However, when the mass is added or removed, the correlation coefficient shows minor changes. 
     The introduction of the mass causes significant changes for each of the compensated correlation coefficients. However, the LPC correlation coefficients are significantly affected by other effects as well. In  FIG. 7A  and  FIG. 7B , the LPC correlation coefficients are almost equal to the OSS and scale-invariant correlation coefficients when the temperature is close to the baseline temperature. However, when the temperature deviates too far from the baseline, the LPC correlation coefficients drop and vary erratically. When the mass is on the plate, this erratic behavior becomes even more prevalent. 
     The OSS and scale-invariant correlation coefficients of vary significantly at the moment the mass is introduced. The two techniques are almost equal to each other throughout the experiment. However, both descriptors, as well as LPC, do vary weakly with temperature. This indicates that it may be difficult to distinguish the mass from a sharp change in temperature using the compensated correlation coefficients alone. By utilizing both these scale-invariant descriptors as well as an estimate of the scale change, system  110  has an improved ability to detect the mass. 
     Temperature and Scale 
     Referring to  FIGS. 7C and 7D , graphs  704 ,  706  show the scale factor estimates for each observed signal relative to two baseline signals. Graph  704  illustrates scale estimates with baseline at 20.5° C. Graph  706  illustrates scale estimates with baseline at 13.1° C. 
     Each temperature compensation method is associated with an estimate. Comparison of  FIGS. 7A-7D  with  FIG. 6  shows that the scale estimates from OSS and the scale cross-correlation follow a similar trend as the ambient temperature. The LPC technique estimate, however, is a poor predictor of temperature when the current temperature is far from the baseline temperature and when the mass is on the specimen. As with the correlation results, the OSS and scale cross-correlation produce almost identical results, and the scale estimates are not largely affected by the addition or removal of the mass. 
     Referring to  FIG. 10 , graph  1000  shows the relationship between scale and temperature to be approximately linear, regardless of the state of the mass. The data, however, does exhibit a hysteresis, in which the scale value transitions slower than the temperature. This phenomenon is a result of the temperature sensor recording the ambient temperature rather than the temperature of the specimen. The temperature of the metal specimen changes at a slower rate than the air surrounding it. Therefore the speed of the guided waves, and scale change in the signal, will change slower than the temperature as recorded by the sensor. 
     Mass Classification 
     As illustrated in the  FIGS. 7A-7B , the scale-invariant correlation coefficient is not perfectly invariant to temperature. However, in  FIGS. 7C-7D , the scale estimate is affected by temperature. Therefore, the combination of these two descriptors can be used to provide a more complete description of the system than either one alone. 
       FIGS. 11A, 11B  illustrate the relationships between the standard and compensated correlation coefficients and the scale estimates, using a baseline signal at 13.1° C. Referring to  FIG. 11A , graph  1100  shows the relationship between scale, estimated using the scale cross-correlation, and the standard correlation coefficient without temperature compensation. In this plot, there are little separation between the data taken with the mass present and the data taken without the mass. Referring to  FIG. 11B , graph  1102  illustrates the relationship between the scale-invariant correlation coefficient and its associated scale estimate. In this plot, there is a clear separation between the data taken with and without the mass. Given a linear discriminator, the two conditions can be classified with one hundred percent accuracy. Using both the scale-invariant correlation coefficient and scale estimate allows us to better differentiate model errors caused by temperature and more significant deviations caused by damage or other effects. In this example, the two curves may eventually meet. This implies that, as the temperature deviates further from the baseline, the mass will become increasingly more difficult to detect. 
     System  110  implements a computationally efficient technique, the scale cross-correlation function, for analyzing the scaling behavior between two signals. Through analytical and experimental comparison with the optimal signal stretch method, the scale cross-correlation is an optimal (minimum mean squared error) temperature compensation method, for a uniform scaling model for temperature. Given a baseline and observed signal, system  110  computes the scale factor and the scale-invariant correlation coefficient between them. The computation is presented for continuous-time and discrete time systems. In discrete time, system  110  derives the Mellin resampling theorem, which states the minimum number of samples necessary to exponentially resample a uniformly sampled signal.  FIG. 5  shows curves in graph  500  demonstrating the computational growth rates of OSS (N2 ln(N)) and the scale cross-correlation (N ln(N) log 2(N)). 
     The estimate of the scale factor is a robust, linear predictor of temperature in the system. The scale-invariant correlation coefficient was shown to be approximately invariant to variations in temperature. Using both the scale factor and scale-invariant correlation coefficient, a mass, simulating damage, could be detected with a linear discriminator. 
     Since the scale cross-correlation requires no dense baseline library, it is a potentially very flexible tool for temperature compensation. Furthermore, its close relationship with the Fourier transform demonstrates it to be a potentially powerful tool as well. 
     Referring to  FIG. 12 , system  110  executes process  1200  in computing temperature compensation indicators, e.g., a scale factor, a scale invariant correlation coefficient, and so forth. In operation, system  110  records ( 1202 ) a signal at a first point in time. The record signaled may include waveform data. The record signal is a signal that has travelled through a structure. At a second point in time, system  110  records ( 1204 ) another signal that has travelled through the structure. Between the first point in time and the second point in time, the structure has experienced one or more structure changes (e.g., damage or other structural modifications). 
     Using the recorded signals, system  110  applies ( 1206 ) one or more time compensation algorithms, including, e.g., an algorithm for computing the scale cross-correlation function (from the scale transform), an algorithm for computing the scale factor (e.g., from the scale cross-correlation function), an algorithm for computing the scale-invariant correlation coefficient (e.g., from the scale cross-correlation function), and so forth. System  110  computes ( 1208 ) time compensation indicators, e.g., using the time compensation algorithms. System  110  computes various types of time compensation indicators, including, e.g., a scale factor, a scale-invariant correlation coefficient, and so forth. Using the time compensation indicators, system  110  detects ( 1210 ) damage in the structure. For example, system  110  applies the computed time compensation indicators to the record signals to correct and/or identify variations in the signals that are caused by temperature fluctuations between times T 1 , T 2 . After identifying the variations that are caused by temperature fluctuations, system  110  identifies other variations in the signals, with these other variations being indicative of structural changes in the structure. 
     Referring to  FIG. 13 , components  1300  of an environment (e.g., environment  100 ) for generating temperature compensation indicators. Client device  101  can be any sort of computing device capable of taking input from a user and communicating over a network (not shown) with system  110  and/or with other client devices. In an example, client device  101  is a device associated with structure  102  ( FIG. 1 ) for recording and/or tracking traversal of a signal through structure  102 . For example, client device  101  can be a mobile device, a desktop computer, a laptop, a personal digital assistant (“PDA”), a server, an embedded computing system, a mobile device and so forth. Client device  101  can include monitor  1108 , which renders visual representations of interface  1106 . 
     System  110  can be any of a variety of computing devices capable of receiving data, such as a server, a distributed computing system, a desktop computer, a laptop, a cell phone, a rack-mounted server, and so forth. System  110  may be a single server or a group of servers that are at a same location or at different locations. 
     System  110  can receive data from client device  101  via interfaces  1106 , including, e.g., graphical user interfaces. Interfaces  1106  can be any type of interface capable of receiving data over a network, such as an Ethernet interface, a wireless networking interface, a fiber-optic networking interface, a modem, and so forth. System  110  also includes a processor  1002  and memory  1004 . A bus system (not shown), including, for example, a data bus and a motherboard, can be used to establish and to control data communication between the components of server  110 . In the example of  FIG. 13 , memory  1004  includes data engine  1302  for executing the techniques and operations described herein. 
     Processor  1002  may include one or more microprocessors. Generally, processor  1002  may include any appropriate processor and/or logic that is capable of receiving and storing data, and of communicating over a network (not shown). Memory  1004  can include a hard drive and a random access memory storage device, such as a dynamic random access memory, machine-readable media, or other types of non-transitory machine-readable storage devices. Components  1300  also include data repository  114 , which is configured to store data collected through system  110  and generated by system  110 . 
     Embodiments can be implemented in digital electronic circuitry, or in computer hardware, firmware, software, or in combinations thereof. Apparatus of the invention can be implemented in a computer program product tangibly embodied or stored in a machine-readable storage device and/or machine readable media for execution by a programmable processor; and method actions can be performed by a programmable processor executing a program of instructions to perform functions and operations of the invention by operating on input data and generating output. 
     The techniques described herein can be implemented advantageously in one or more computer programs that are executable on a programmable system including at least one programmable processor coupled to receive data and instructions from, and to transmit data and instructions to, a data storage system, at least one input device, and at least one output device. Each computer program can be implemented in a high-level procedural or object oriented programming language, or in assembly or machine language if desired; and in any case, the language can be a compiled or interpreted language. 
     Suitable processors include, by way of example, both general and special purpose microprocessors. Generally, a processor will receive instructions and data from a read-only memory and/or a random access memory. Generally, a computer will include one or more mass storage devices for storing data files; such devices include magnetic disks, such as internal hard disks and removable disks; magneto-optical disks; and optical disks. Computer readable storage media are storage devices suitable for tangibly embodying computer program instructions and data include all forms of volatile memory such as RAM and non-volatile memory, including by way of example semiconductor memory devices, such as EPROM, EEPROM, and flash memory devices; magnetic disks such as internal hard disks and removable disks; magneto-optical disks; and CD-ROM disks. Any of the foregoing can be supplemented by, or incorporated in, ASICs (application-specific integrated circuits). 
     Other embodiments are within the scope and spirit of the description claims. In another example, due to the nature of software, functions described above can be implemented using software, hardware, firmware, hardwiring, or combinations of any of these. Features implementing functions may also be physically located at various positions, including being distributed such that portions of functions are implemented at different physical locations. 
     A number of embodiments have been described. Nevertheless, it will be understood that various modifications can be made without departing from the spirit and scope of the processes and techniques described herein. In addition, the logic flows depicted in the figures do not require the particular order shown, or sequential order, to achieve desirable results. In addition, other steps can be provided, or steps can be eliminated, from the described flows, and other components can be added to, or removed from, the described systems. Accordingly, other embodiments are within the scope of the following claims.