Abstract:
A method of attenuating spikes in a complex signal comprises examining the complex signal to detect spikes therein and for each detected spike, generating an estimate inverse signal and applying the estimate inverse signal to the complex signal to attenuate the spike associated with the estimate inverse signal.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]    This application claims the benefit of U.S. Provisional Application No. 61/106,949 filed on Oct. 20, 2008 to Zalay entitled “Method for Detection and Iterative Attenuation (Melting) of Large-Amplitude Spikes or Artifacts in Discrete-Time Biological Signals”, the content of which is incorporated herein by reference. 
     
    
     FIELD OF THE INVENTION  
       [0002]    The present invention relates generally to signal processing and in particular, to a method and attenuator for attenuating spikes in complex signals. 
       BACKGROUND OF THE INVENTION 
       [0003]    Analyzing temporally and spectrally complex signals is challenging and requires the use of specialized techniques. Such temporally and spectrally complex signals include for example biological signals (e.g. neural and cardiac signals), noisy or artifact-contaminated communication signals, signals produced by mechanical or electronic measurement or sensing devices (e.g. spectrometers, acoustic transducers etc.) and environmental signals (e.g. atmospheric, oceanic, seismic and astronomic signals). Despite the popularity of Fourier transform-based methods, such methods are ill-suited to deal with non-stationary complex signals. As an alternative, wavelet transform-based methods have proven to be versatile and effective, with improved ability to resolve complex signal features in both the time and frequency domains. 
         [0004]    Large amplitude spikes are common features in temporally and spectrally complex signals. For example, in the case of neural signals, large amplitude spikes may occur spontaneously and/or as a byproduct of stimulation. Unfortunately, large amplitude spikes or artifacts can obscure complex signal components that are relevant to clinical/research diagnosis and/or analysis since their dominant spectral power masks smaller amplitude complex signal features that may be temporally localized at the same regions as the large amplitude spikes. If large amplitude spikes are associated with an underlying rhythm and require removal, then wavelet transform-based methods alone are not capable of eliminating all spike remnants, since spikes with large amplitudes in relation to the rest of the complex signal tend to dominate the coefficients of the wavelet transform across all frequency bands being analyzed. 
         [0005]    For example, electroencephalogram (EEG) artifacts may originate from various physiological and external sources (e.g. eye blinking, muscular movement, cardiac potentials, etc.) and can pose difficulties for EEG analysis, as noted in the publications entitled “Methods for the estimation and removal of artifacts and overlap in ERP waveforms” authored by D. Talsma et al.,  Event - Related Potentials: A Methods Handbook,  MIT Press, 2005, p. 115-148 and “Facts and artifacts in brain electrical activity mapping” authored by K. L. Coburn et al.,  Brain Topology  1, (1):37-45, 1988. In another example, isolation of the baseline complex signal in intracellular neuronal recordings is sometimes warranted due to active research into forms of subthreshold neuronal noise and their role in synaptic function and neuronal communication, making action potential attenuation or removal desirable as noted in the publications entitled “Noise in the nervous system” authored by A. A. Faisal et al.,  Nature Reviews Neuroscience,  9(4):292-303, 2008, “A nonrandom dynamic component in the synaptic noise of a central neuron” authored by P. Faure et al.,  Proceedings of the National Academy of Sciences,  94(12):6506-6511, 1997, and “Subthreshold voltage noise of rat neocortical pyramidal neurons” authored by G. Jacobson et al.,  Journal of Physiology,  564(Pt 1):145-160, 2005. 
         [0006]    Several techniques have been devised to remove large amplitude spikes from complex signals. One approach involves interpolating the baseline potential of the complex signal before and after each spike following its excision, and then lowpass filtering the complex signal. Unfortunately, lowpass filtering the complex signal may remove other high-frequency components in the complex signal not associated with spike artifacts. A cruder method relies on direct bandpass filtering to attenuate dominant-frequency spike components of the complex signal, but this approach usually distorts the underlying complex signal as well. 
         [0007]    As will be appreciated, improvements in complex signal analysis are desired. It is therefore an object of the present invention to provide a novel method and attenuator for attenuating spikes in complex signals. 
       SUMMARY OF THE INVENTION 
       [0008]    Accordingly, in one aspect there is provided a method of attenuating spikes in a complex signal comprising examining the complex signal to detect spikes therein; and for each detected spike, generating an estimate inverse signal and applying the estimate inverse signal to the complex signal to attenuate the spike associated with the estimate inverse signal. 
         [0009]    In one embodiment, the examining comprises comparing the amplitude of the complex signal with a threshold value to detect spikes in the complex signal. Prior to the examining, the values of the complex signal are digitally stored in an array. For each value of the complex signal, the value is compared to a plurality of preceding and subsequent values of the complex signal to determine if the value represents a peak and if so, the value of the complex signal is compared to the threshold value. 
         [0010]    In one embodiment, during the generating a scaled mirror image of the detected spike is generated. The generating and applying are performed iteratively until the amplitude of the spike is sufficiently attenuated. 
         [0011]    Following the applying, the attenuated complex signal may be filtered such as for example wavelet or wavelet packet filtered. The complex signal may be a biological signal such as a neural or cardiac signal or one of a noisy and/or an artifact-contaminated communication signal, a mechanical measurement or sensing device signal, an electronic measurement or sensing device signal and an environment signal. 
         [0012]    According to another aspect of the present invention there is provided a method of attenuating spikes in a complex signal comprising subjecting the complex signal to thresholding to detect spikes in the complex signal having amplitudes greater than a first threshold value and subjecting each detected spike to an iterative inverse signal application procedure to attenuate each detected spike so that the amplitude thereof is reduced at least to a second threshold value. 
         [0013]    According to yet another aspect there is provided a computer readable medium embodying a computer program for attenuating spikes in a complex signal, said computer program comprising program code for examining the complex signal to detect spikes therein, program code for generating an estimate inverse signal for each detected spike and program code for applying the estimate inverse signal to the complex signal to attenuate the spike associated with the estimate inverse signal. 
         [0014]    According to yet another aspect there is provided a computer readable medium embodying a computer program for attenuating spikes in a complex signal, said computer program comprising program code for subjecting the complex signal to thresholding to detect spikes in the complex signal having amplitudes greater than a first threshold value and program code for subjecting each detected spike to an iterative inverse signal application procedure to attenuate each detected spike so that the amplitude thereof is reduced at least to a second threshold value. 
         [0015]    According to still yet another aspect there is provided an attenuator for attenuating spikes in a complex signal comprising a spike detector for detecting spikes in the complex signal and a spike attenuator for attenuating each detected spike, said spike attenuator generating, for each detected spike, an estimate inverse signal and applying the estimate inverse signal to the complex signal thereby to attenuate the spike associated with the estimate inverse signal. 
         [0016]    According to still yet another aspect there is provided an attenuator for attenuating spikes in a complex signal comprising a threshold detector for detecting spikes in the complex signal that have amplitudes greater than a first threshold value and an attenuating loop for subjecting each detected spike to an iterative inverse signal application procedure to attenuate each spike to reduce the amplitude thereof to at least a second threshold value. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0017]    Embodiments will now be described, by way of example only, with reference to the accompanying drawings in which: 
           [0018]      FIG. 1A  is a schematic diagram of an attenuator for attenuating spikes in complex signals; 
           [0019]      FIG. 1B  is a schematic diagram of a spike detection loop forming part of the attenuator of  FIG. 1A ; 
           [0020]      FIG. 1C  is a schematic diagram of a spike attenuating loop forming part of the attenuator of  FIG. 1A ; 
           [0021]      FIG. 1D  is a flowchart showing steps performed by the spike detection loop of  FIG. 1B ; 
           [0022]      FIG. 1E  is a flowchart showing steps performed by the spike attenuating loop of  FIG. 1C ; 
           [0023]      FIG. 2  shows a spike in a complex signal and a scaled mirror image of the spike; 
           [0024]      FIG. 3A  shows a time series complex signal having large amplitude spikes; 
           [0025]      FIG. 3B  shows the time series complex signal of  FIG. 3A  after processing by the attenuator of  FIG. 1 ; 
           [0026]      FIG. 4  shows a portion of the time series complex signal of  FIG. 3A  before and after processing by the attenuator of  FIG. 1 ; 
           [0027]      FIG. 5A  shows a complex signal that has been processed by the attenuator of  FIG. 1  to attenuate spikes and then subsequently low-pass filtered; 
           [0028]      FIG. 5B  shows the complex signal of  FIG. 5A  with attenuated large amplitude spikes re-introduced; 
           [0029]      FIG. 6A  shows a composite complex signal with large amplitude spikes formed from the combination of a rhythmic signal and a large amplitude bursting-spike signal; 
           [0030]      FIG. 6B  shows the composite complex signal of  FIG. 6A  after filtering using a wavelet packet filter; 
           [0031]      FIG. 6C  shows the composite complex signal of  FIG. 6A  after processing by the spike attenuating loop of  FIG. 1C ; 
           [0032]      FIG. 6D  shows the composite complex signal of  FIG. 6A  after processing by the spike attenuating loop of  FIG. 1C  followed by filtering using a wavelet packet filter; and 
           [0033]      FIG. 6E  shows the rhythmic signal forming part of the composite complex signal of  FIG. 6A  after filtering using a wavelet packet filter. 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0034]    In the following, a technique to remove large amplitude spikes from a time series complex signal is described. The technique employs amplitude threshold detection to detect large amplitude spikes in the complex signal and then uses the detected spikes&#39; own rescaled mirror images to iteratively attenuate or “melt” the detected spikes down. Therefore, large amplitude spikes in the complex signal are not filtered out or excised from the complex signal but rather, are transformed so that their prominent high-frequency components and spectral power are greatly diminished. This approach aims to attenuate detected large amplitude spikes in the complex signal to blend the spikes in with their surroundings, rather than remove the spikes completely, so that as much information as possible is preserved in the complex signal before other filtering, such as wavelet packet filtering, is performed. 
         [0035]    Turning now to  FIG. 1A , an attenuator for attenuating spikes in a complex signal is shown and is generally identified by reference numeral  10 . In this embodiment, attenuator  10  is designed to attenuate large amplitude spikes in a biological signal such as for example a neural or cardiac signal. As can be seen, attenuator  10  comprises an adder  12  that receives a recorded complex signal x(k) to be processed as well as the negative of the minimum value of the complex signal −min{x(k)}. The output {tilde over (x)}(k) of adder  12  is applied to a spike detection loop  14 . The spike detection loop  14  communicates with a spike attenuating loop  16 . The output {tilde over (y)}({circumflex over (k)}) of the spike attenuating loop  16  is applied to an adder  18 . Adder  18  also receives the minimum value of the complex signal min{x(k)}, and yields the attenuated complex signal y(k) output of the attenuator  10 . 
         [0036]      FIG. 1B  better illustrates the spike detection loop  14 . As can be seen, spike detection loop  14  comprises a peak detector  30  that receives the output {tilde over (x)}(k) of the adder  12 . The peak detector  30  communicates with a threshold detector  32  and with an indexor  34 . Indexor  34  also communicates with the threshold detector  32  and provides feedback to the peak detector  30 . 
         [0037]      FIG. 1C  better illustrates the spike attenuating loop  16 . As can be seen, the spike attenuating loop  16  comprises an envelope detector  40  that receives output of the spike detection loop  14  and communicates with an inverse signal generator  42 . The output of the inverse signal generator  42  is applied to a multiplier  44  that also receives the output of the spike detection loop  14 . The multiplier  44  communicates with a threshold detector  46 . The threshold detector  46  communicates with the adder  18  and with the envelope detector  40 . 
         [0038]    The general operation of the attenuator  10  will now be described. When a recorded complex signal x(k) is applied to the attenuator  10 , the spike detection loop  14  examines the complex signal and uses amplitude thresholding to detect large amplitude spikes in the complex signal. For each large amplitude spike in the complex signal that is detected, the spike attenuating loop  16  generates an inverse signal that, when combined with the complex signal in an iterative loop, attenuates the associated large amplitude spike so that the prominent high-frequency components and spectral power of the large amplitude spike are greatly diminished. In this manner, large amplitude spikes in the complex signal can be effectively diminished while preserving the integrity of the original complex signal x(k). Further specifics concerning the operation of the attenuator  10  will now be described with particular reference to  FIGS. 1D and 1E . 
         [0039]    Prior to processing a recorded complex signal x(k) to remove large amplitude spikes, the recorded complex signal is examined and the minimum value of the complex signal min{x(k)} is determined by digitally storing the values of the complex signal in an input floating point array and searching the array for the smallest stored value (step  100 ). The recorded complex signal x(k) is then applied to the attenuator  10  and delivered to the adder  12 , which also receives the negative of the minimum value of the complex signal −min{x(k)} determined at step  100 . At adder  12 , the negative of the minimum value of the complex signal −min{x(k)} is added to the values of the complex signal x(k) in the input floating point array, so that all values of the resultant complex signal are non-negative (step  102 ). The complex signal output of the adder  12  is therefore expressed as: 
         [0000]        x ( k )−min{ x ( k )}= {tilde over (x)} ( k ), where  {tilde over (x)} ( k )≧0 for all  k∈Z.    
         [0040]    A global threshold value θ is then computed for the non-negative complex signal (step  104 ). The global threshold value θ may vary depending on the nature of the complex signal being processed. In this embodiment, the global threshold value θ is proportional to the root mean square of the non-negative complex signal and is expressed as: 
         [0000]      θ=C√{square root over (  [{tilde over ( x )}( k )] 2   )} 
         [0000]    where C≧1 and the bar denotes the mean value 
         [0000]    
       
         
           
             
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         [0000]    The proportionality constant C is greater than or equal to one (1) because it is desirable to detect spikes having amplitudes that are beyond the range of the average complex signal baseline activity. 
         [0041]    The non-negative complex signal {tilde over (x)}(k) stored in the input floating point array is then applied to the peak detector  30 . At the peak detector  30 , the value of the non-negative complex signal {tilde over (x)}(k) at a current index position k o  of the input floating point array is compared to the values of the non-negative complex signal at a plurality of index positions (typically two (2) to five (5) indices) of the input floating point array preceding and subsequent to the current index position k o  to determine if the value of the non-negative complex signal {tilde over (x)}(k) at the current index position k o  represents a local maximum or peak in the non-negative complex signal {tilde over (x)}(k) (steps  106  and  108 ). If the peak detector  30  does not detect a local maximum in the non-negative complex signal {tilde over (x)}(k) at the current index position k o , the value of the non-negative complex signal {tilde over (x)}(k) at the current index position k o , is copied to the corresponding index position of an output floating point array. The peak detector  30  then signals the indexor  34 . The indexor  34  in response examines the current index position k o  to determine if it represents the last index position of the input floating point array (step  110 ). If not, the indexor  34  increments the current index position by one (1) (step  112 ) and the process returns to step  106 . As a result, the peak detector  30  examines the value of the non-negative complex signal {tilde over (x)}(k) at the updated index position to determine if it represents a local maximum or peak in the non-negative complex signal {tilde over (x)}(k). This process continues until the peak detector  30  detects a local maximum in the non-negative complex signal {tilde over (x)}(k) or until the indexor  34  determines that the current index position is equal to the last index position of the input floating point array at step  110 . 
         [0042]    When the peak detector  30  detects a value of the non-negative complex signal {tilde over (x)}(k) that represents a local maximum or peak, the threshold detector  32  compares the value of the non-negative complex signal with the global threshold value θ (step  114 ). If the value of the non-negative complex signal is less than the global threshold value θ (i.e. {tilde over (x)}(k o )&lt;θ), the value of the non-negative complex signal {tilde over (x)}(k) is determined not to be a large amplitude spike. In this case, the value of the non-negative complex signal {tilde over (x)}(k) at the current index position k o  is copied to the corresponding index position of the output floating point array. The threshold detector  32  then signals the indexor  34  and the process reverts to step  110 . The indexor  34  in response examines the current index position to determine if it represents the last index position of the input floating point array. If not, the indexor  34  increments the current index position by one (1) (step  112 ) and the peak detector  30  examines the value of the non-negative complex signal {tilde over (x)}(k) at the updated index position of the input floating point array to determine if it represents a local maximum or peak. 
         [0043]    If the value of the non-negative complex signal {tilde over (x)}(k) is greater than or equal to the global threshold value θ (i.e. {tilde over (x)}(k o )≧θ), the value of the non-negative complex signal is deemed to represent a large amplitude spike. The spike detection loop  14  in turn copies a segment of the non-negative complex signal {tilde over (x)}(k) stored in the input floating point array and outputs the copied segment to the spike attenuating loop  16  (step  116 ). The copied segment of the non-negative complex signal {tilde over (x)}(k) includes the detected large amplitude spike and values of the non-negative complex signal at index positions of the input floating point array that are neighbours to the detected large amplitude spike. The neighbour index positions of the input floating point array are expressed by {circumflex over (k)}∈[k o −ε,k o +ε], which is an interval subset of k defined in the neighborhood e of the identified large amplitude spike. The choice of the neighborhood g is made by inspecting the complex signal visually or by automated measurement before application of the segment to the spike attenuating loop  16  with the condition that the detected large amplitude spike width is less than 2ε. The value for the neighborhood ε, is chosen so that it is large enough to encompass each large amplitude spike or large amplitude spike cluster (if a number of large amplitude spikes are tightly grouped) without overlapping the next large amplitude spike. 
         [0044]    Before the non-negative complex signal segment is processed by the spike attenuating loop  16 , a local threshold value {circumflex over (θ)} based on the values of the non-negative complex signal segment is computed (step  120 ). In this embodiment, the local threshold value {circumflex over (θ)} is equal to: 
         [0000]      {circumflex over (θ)}={circumflex over ( C )}√{square root over (  [{tilde over ( x )}({circumflex over ( k )})] 2   )} 
         [0000]    where (0&lt;Ĉ≦1). The proportionality constant Ĉ is approximately one (1) or less so that the amplitude of the large amplitude spike is attenuated to within a range equivalent to or less than that of the average complex signal baseline activity in its neighborhood. 
         [0045]    With the local threshold value {circumflex over (θ)} determined, the envelope detector  40  fits a sigmoidal envelope around the large amplitude spike in the non-negative complex signal segment (step  122 ) such that: 
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         [0000]    where ξ 1  and ξ 2  are parameters adjusting the edge slant and relative width of the sigmoidal envelope, respectively. The properties of the sigmoidal envelope as controlled by ξ 1  and ξ 2  affect both the isolation of the large amplitude spike from its surroundings, and preservation of the complex signal baseline activity from the effects of the spike attenuating transformation. As such, ξ 1  is chosen to be large enough to enable sharp delineation of the large amplitude spike from its neighborhood, and ξ 2  is chosen to be some fraction in relation to the neighborhood radius, ε. For example, parameter choices that are adequate for most time series complex signals are ξ 1 =100 and ξ 2 =0.2. 
         [0046]    With the sigmoidal envelope fitted around the large amplitude spike, the inverse signal generator  42  generates an inverse scaled image of the large amplitude spike (step  124 ) according to: 
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         [0000]    with scaling 0&lt;&lt;η&lt;1 (i.e. arbitrarily close to 1). A parameter choice η=0.999 has been shown to be satisfactory. 
         [0047]    The inverse scaled image of the large amplitude spike is then applied to the multiplier  44  which also receives the non-negative complex signal segment output by the threshold detector  32  (step  126 ). The product output z({circumflex over (k)}) of the multiplier  44  which is expressed as: 
         [0000]        z ( {circumflex over (k)} )=μ( {circumflex over (k)} )· {tilde over (x)} ( {circumflex over (k)} ) 
         [0000]    is used to update the values of the non-negative complex signal segment. The updated values of the non-negative complex signal segment are then applied to the threshold detector  46 . The threshold detector  46  compares the updated values of the non-negative complex signal segment (i.e. the resultant product output z({circumflex over (k)})) with the local threshold value {circumflex over (θ)} (steps  128  and  130 ). If the amplitude of any of the updated values of the non-negative complex signal segment is greater than or equal to the local threshold value {circumflex over (θ)} (i.e. z({circumflex over (k)})≧{circumflex over (θ)} for any {circumflex over (k)}), steps  122 ,  124  and  126  as described above are performed again using the updated values of the non-negative complex signal segment. This iterative process continues until the updated values of the non-negative complex signal segment (i.e. the product output z({circumflex over (k)}) of the multiplier  44 ) are less than the local threshold value {circumflex over (θ)}, or the number of permitted iterations has reached an allowed limit. 
         [0048]    When the threshold detector  46  determines that the product output z({circumflex over (k)}) of the multiplier  44  is less than the local threshold value {circumflex over (θ)}, signifying that the large amplitude spike has been sufficiently attenuated, the updated values of the non-negative complex signal segment are copied to corresponding index positions of the output floating point array. The threshold detector  46  then signals the indexor  34  to update the current index position so that it is equal to the last index position of the non-negative complex signal segment that was copied to the output floating point array (step  132 ). The process then reverts to step  110  where the indexor  34  examines the current index position to determine if it represents the last index position of the input floating point array. If not, the indexor  34  increments the current index position by one (1) and above process is performed on the non-negative complex signal {circumflex over (x)}(k) at the next index position of the input floating point array. After the non-negative complex signal {circumflex over (x)}(k) has been processed as described above, the processed non-negative complex signal ŷ(k) stored in the output floating point array is applied to the adder  18 . The adder  18  which also receives the minimum value of the complex signal min{x(k)} adds the minimum value of the complex signal to the values of the complex signal ŷ(k) in the output floating point array (step  134 ) and outputs the resultant attenuated complex signal y(k). 
         [0049]      FIG. 2  shows an example of a large amplitude spike  200  in a complex signal that is detected together with a scaled mirror image  204  of the large amplitude spike as generated by the inverse signal generator  42 . In this example, the large amplitude spike has a peak amplitude of H and is located at index position k o . The sigmoidal envelope fitted around the large amplitude spike  200  is shown by dotted line  202 . As described above, a scaled inversion transformation is carried out to obtain the scaled mirror image  204  of the large amplitude spike μ({circumflex over (k)}), which is used to attenuate the large amplitude spike through repeated multiplication. During the iterative process, the most prominent features of the large amplitude spike  200  are reduced the fastest. The baseline complex signal is left virtually untouched. 
         [0050]      FIGS. 3A and 3B  shows a time series complex signal containing large amplitude spikes before and after processing by the attenuator  10 .  FIG. 4  is an enlarged view of a portion of the time series complex signal of  FIG. 3A  showing one large amplitude spike. The dotted line  210  represents the transformed spike once it has been attenuated. 
         [0051]    If information about spike amplitude and phase is important, as in the case with action potentials in an intracellular neuronal recording, as information concerning the phases and amplitudes of removed large amplitude spikes is not lost, large amplitude spikes that have been removed from a complex signal can be re-introduced to a post processed complex signal as shown in  FIGS. 5A and 5B . 
         [0052]    If the time series complex signal contains biphasic large amplitude spikes or large amplitude spike components of negative polarity, the input time series can be inverted such that [x(k)] inverted =−x(k) prior to processing to eliminate the negative large amplitude spike components. Alternatively, the threshold definitions for detection and attenuation can be reconfigured to detect and attenuate large amplitude spikes of negative polarity. 
         [0053]    In order to effectively remove large amplitude spikes or extract their underlying rhythms from complex signals such as neural recordings, the complex signals should be applied to the attenuator  10  in advance of other filtering methods, such as for example the wavelet packet rhythm extraction method described in co-pending U.S. application Ser. No. ______ to Zalay et al. filed on even date herewith and entitled “Method and System for Detecting and Isolating Rhythmic Signal Features from an Input Signal Using the Wavelet Packet Transform” and assigned to Neurochip Corporation of Ontario, Canada, assignee of the subject application, the content of which is incorporated herein by reference. In  FIG. 6A , a composite complex signal (X 3 ) made up of an intracellularly recorded baseline rhythmic signal (X 1 ) and a large amplitude bursting-spike signal (X 2 ) is shown that was simulated using coupled oscillators. Composite complex signal X 3  is similar in form to interictal bursting activity recorded intracellularly from neurons. As is typically the case with neuronal recordings, the subthreshold activity and the spiking activity are both independently rhythmic. 
         [0054]    To quantify the signals (X 1 ) and (X 3 ), each signal is subdivided into time segments. In this example, each signal is divided into six equal time segments and the sample average for the mean, variance, skewness and kurtosis are calculated as set out in Table 1 below. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Statistical moments (sample mean ± std. err. mean) ♦  of unprocessed, spike attenuated 
               
               
                 (SA) and wavelet packet filtered (WPF) signals pertaining to FIG. 2. 
               
             
          
           
               
                   
                   
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                 Signal 
                 Process 
                 Threshold 
                 Mean 
                 Variance 
                 Skewness 
                 Kurtosis 
               
               
                   
               
               
                 X1 
                 WPF 
                 0.0010 
                 −0.34 ± 0.42 
                  6.54 ± 0.23 
                 0.031 ± 0.037 
                 2.810 ± 0.042 
               
               
                   
                 — 
                 — 
                 −0.39 ± 0.32 
                 12.5 ± 1.5 
                 0.173 ± 0.071 
                 2.787 ± 0.083 
               
               
                 X3 
                 — 
                 — 
                  7.19 ± 0.31 
                 181 ± 12 
                 2.69 ± 0.11 
                 11.05 ± 0.74  
               
               
                   
                 WPF 
                 0.0055 
                 −0.25 ± 0.27 
                 12.5 ± 1.0 
                 0.94 ± 0.19 
                 10.2 ± 1.6  
               
               
                   
                 SA 
                 — 
                  3.70 ± 0.24 
                 26.0 ± 3.1 
                 0.394 ± 0.091 
                 2.82 ± 0.12 
               
               
                   
                 SA + WPF 
                 0.0030 
                  3.66 ± 0.36 
                  6.45 ± 0.40 
                 0.153 ± 0.041 
                 2.829 ± 0.088 
               
               
                   
               
               
                   ♦ n = 6 sequential time series intervals of 2000 pts. 
               
             
          
         
       
     
         [0055]    For signals filtered using a wavelet packet filter technique such as the wavelet packet rhythm extraction method described in the above-incorporated Zalay et al. PCT application, the initial normalized interpeak interval variance threshold (θ λ ) and power significance level (β P ) are set to 0.001 and 0.01, respectively. The subsequent pruning value of the interpeak interval variance threshold θ λ  determining the final tree is selected by minimizing the sum of squared errors between rhythmic signal X 1  after wavelet packet filtering and the given signal, based on values of their four statistical moments. This is done to maximize similarity of the signals while at the same time avoiding an arbitrary choice of threshold, for purposes of comparison. 
         [0056]    A two-sided paired t-test and a Wilcoxon signed ranked test are performed on the individual and combined statistical moments, excluding the mean, to determine whether the difference between rhythmic signal X 1  after wavelet packet filtering and the various versions of composite complex signal X 3  is statistically significant in each case. The mean is excluded because the initial addition of large amplitude bursting-spike signal X 2  to rhythmic signal X 1  introduces a non-rhythmic offset that is not removed by neural rhythm extraction; furthermore, this offset can easily be subtracted to correct for the difference in the means. The results of the hypothesis tests set out in Table 2 below confirm that complex composite signal X 3  when processed initially using attenuator  10  and then subsequently wavelet packet filtered as shown in  FIG. 6D  does not differ significantly from rhythmic signal X 1  after wavelet packet filtering for all categories tested, thereby confirming that the attenuator  10  is best suited to be used as a pre-processing stage to wavelet packet filtering when dealing with complex signals containing large amplitude spikes. 
         [0000]    
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Two-sided paired t-test (TT) and Wilcoxon signed rank test (WT) p             
               
             
          
           
               
                   
                 Variance 
                 Skewness 
                 Kurtosis 
                 Combined 
               
             
          
           
               
                 Signal 
                 Process 
                 TT 
                 WT 
                 TT 
                 WT 
                 TT 
                 WT 
                 TT 
                 WT 
               
               
                   
               
               
                 X1 
                 — 
                 0.011  
                 0.031 
                 
                   0.16 
                 
                   0.31   
                 
                   0.75 
                 
                   0.68   
                 0.025 
                 0.014  
               
               
                 X3 
                 — 
                 2.7 × 10 −5   
                 0.031 
                 3.7 × 10 −6   
                 0.031 
                   0.00011 
                 0.031 
                  0.0060 
                 0.00020 
               
               
                   
                 WPF 
                 0.0013 
                 0.031 
                  0.0051 
                 0.031 
                  0.0051 
                 0.031 
                 4.9 × 10 −5   
                 0.00020 
               
               
                   
                 SA 
                 0.0013 
                 0.031 
                  0.013 
                 0.031 
                 
                   0.93 
                 
                   0.84   
                 0.013 
                 0.0038  
               
               
                   
                 SA + WPF 
                   0.85   
                   0.84   
                 
                   0.13 
                 
                   0.22   
                 
                   0.87 
                 
                   1     
                   0.89   
                 0.37   
               
               
                   
               
               
                 Comparison is against X1 WPF with respect to statistical moments (excluding the mean). 
               
               
                             values that do not reject the null hypothesis at 5% significance are bolded. 
               
             
          
         
       
     
         [0057]      FIG. 6B  shows that composite complex signal X 3  after wavelet packet filtering without attenuating reduces much of the signature of the large amplitude spike bursts, but at the expense of removing too much of the complex signal from the time-frequency plane. The spike attenuating loop  16  alone also removes the high-frequency components of the large amplitude spike bursts, but does not eliminate the low amplitude baseline rhythm associated with them as shown in  FIG. 6C . 
         [0058]    Large amplitude spikes or artifacts interfere with the extraction and separation of baseline rhythms because their dominant spectral power masks other smaller-amplitude features. Besides the utilitarian reasons for dispensing with large amplitude spikes/artifacts, isolation of the non-rhythmic baseline complex signal may itself be of interest, due to active research into forms of (subthreshold) neuronal noise and their role in synaptic function and neuronal communication. The performance of wavelet packet filtering in the presence of large amplitude spikes is comparable to the performance of conventional filtering, in that they are both inadequate. To address this shortcoming, the attenuator  10  works well as a preprocessing stage to wavelet packet filtering. In contrast to previous techniques of large amplitude spike removal, attenuator  10  does not rely on excision or filtering; rather, it iteratively updates a scaled mirror image of the target large amplitude spike which it uses to selectively attenuate or “melt” the large amplitude spike away, thereby eliminating the large amplitude spike&#39;s broad spectral footprint while preserving the baseline complex signal. As a result, the attenuator  10  minimizes signal distortion and information loss, which is important for subsequent wavelet packet filtering. 
         [0059]    The attenuator  10  can be implemented in software that is run on a general purpose computing device such as for example a personal computer (PC) or other digital electronic hardware or can be implemented on a logic processing device such as for example a field programmable gate array (FPGA) or application specific intergrated circuit (ASIC), for either offline or online processing of complex signals. The general purpose computing device comprises, for example, a processing unit, system memory (volatile and/or non-volatile memory), other removable or non-removable memory (hard drive, RAM, ROM, EEPROM, CD-ROM, DVD, flash memory, etc.), and a system bus coupling various components to the processing unit. In the software implementation, the processing unit runs program modules including but not limited to routines, programs, object components, data structures etc. that may be embodied as computer readable program code stored on a computer readable medium. A computer readable medium is any data storage device that can store data, which can thereafter be read by a computer system. Examples of computer readable medium include for example read-only memory, random-access memory, flash memory, CD-ROMs, magnetic tape, optical data storage devices and other storage media. The computer readable program code can also be distributed over a network including coupled computer systems so that the computer readable program code is stored and executed in a distributed fashion or copied over a network for local execution. 
         [0060]    Although the attenuator  10  is described above with specific reference to the processing of a biological signal and in particular to a recorded neural signal, those of skill in the art will appreciate that the attenuator  10  may be employed in virtually any environment where it is desired to attenuate large amplitude spikes in a complex signal. For example, attenuation of large amplitude spikes or artifacts may be desirable in wireless/wired communications signals, measurement or detector device signals (e.g. spectrometer signals, radar signals, acoustic transducer signals, etc.), environmental signals (atmospheric signals, oceanic signals, seismic signals, astronomic signals etc.) or as a pre-processing step to other signal processing applications such as filtering, transforms, sub-space methods, blind-channel identification, unsupervised learning (e.g. clustering), classification and/or tracking. 
         [0061]    In addition, the attenuator  10  is not limited to the processing of recorded complex signals. Incoming complex signals may also processed generally in “real-time” or “online”. In this case, to adapt the attenuator  10  for online processing of complex signals, a memory buffer is employed to store prior samples of the complex signal corresponding to a sufficiently large moving time window (interval), allowing for a running update to the thresholds and computation of the inverse scaled images and sigmoidal envelopes. 
         [0062]    If desired, as a pre-processing spike attenuating loop step, the minimum of the non-negative complex signal segment min{{tilde over (x)}({circumflex over (k)})} may be subtracted from the values of the non-negative complex signal segment before the non-negative complex signal segment {tilde over (x)}({circumflex over (k)}) is applied to the spike attenuating loop  16 . In this case, the minimum of the non-negative complex signal segment is added back to the attenuated non-negative complex signal segment before the attenuated complex signal is applied to adder  18 . 
         [0063]    Those of skill in the art will also appreciate that other variations and modifications from those described may be made without departing from the scope and spirit of the invention, as defined by the appended claims.