Patent Publication Number: US-2023140050-A1

Title: Band filter for filtering a discrete time series signal

Description:
GOVERNMENT LICENSE RIGHTS 
     This invention was made with United States Government support under awards 1755088 awarded by the National Science Foundation. The Government has certain rights in this invention. 
    
    
     TECHNICAL FIELD 
     The embodiments described below relate to filtering signals and, more particularly, to a band filter for filtering a discrete time series signal. 
     BACKGROUND 
     Filtering of signals has broad applications in many fields of science and engineering. Signal processing filters can be generally classified as high-pass, low-pass, band-pass and band-stop. These filters in general can be implemented as infinite impulse response (IIR) or finite impulse response (FIR) filters however there are other techniques such as regression filtering and frequency domain filtering. Due to the desirable characteristics of FIR filters, they are frequently employed. In order to achieve the desired filter characteristics, such as a sharp transition from the pass band to the stop band, long length FIR filters may be required. However, many times, it is desired to filter a short finite duration digital signal. This can cause implementation issues at the beginning and ending of the signal to be filtered such as bias errors. This issue is sometimes referred to as filter warmup. This also affects IIR filters. 
     Another technique used in radar signal processing is frequency domain filtering. The signal to be filtered is transformed to the frequency domain typically using a Discrete Fourier Transform (DFT). In the signal&#39;s frequency domain representation, the undesirable frequencies can be “notched” or set to zero thus eliminating those frequencies. In order to apply a frequency domain filter, the time series signal is usually first multiplied by a window function such as the von Hann (sometimes called Hanning), Blackman or Hamming. These windows attenuate the time series signal thereby, in effect, reducing the number of available independent samples in the windowed data. Reduced independent samples causes the estimate variance of the signal variables to increase. The window function also has the effect of broadening the signal&#39;s spectrum due to the convolution of the unfiltered signal&#39;s frequency spectrum with the frequency spectrum of the window function. This alters some of the characteristics of the signal to be filtered. 
     Accordingly, there is a need for a band filter for filtering a discrete time series signal. For example, there is a need for a band filter for filtering a discrete time series signal that may have a relatively short and finite duration. There is also a need for a band filter for filtering a discrete time series signal without the use of a window function and/or requiring a filter “warm-up” time. 
     SUMMARY 
     A band filter for filtering a discrete time series signal is provided. According to an embodiment, the band filter comprises an input frequency shifter configured to frequency down shift the discrete time series signal to provide a frequency down shifted discrete time series signal, a regression based zero frequency centered band filter communicatively coupled to the input frequency shifter, the regression based zero frequency centered band filter being configured to filter the frequency down shifted discrete time series signal to provide a filtered and frequency down shifted discrete time series signal, and an output frequency shifter communicatively coupled to the regression based zero frequency centered band filter, the output frequency shifter being configured to frequency up shift the filtered and frequency down shifted discrete time series signal. 
     A method of filtering a discrete time series signal with a band filter is provided. According to an embodiment, the method comprises frequency down shifting the discrete time series signal to provide a frequency down shifted discrete time series signal, using a regression based zero frequency centered band filter to filter the frequency down shifted discrete time series signal to provide a filtered and frequency down shifted discrete time series signal, and frequency up shifting the filtered and frequency down shifted discrete time series signal. 
     Aspects 
     According to an aspect, a band filter for filtering a discrete time series signal comprises an input frequency shifter configured to frequency down shift the discrete time series signal to provide a frequency down shifted discrete time series signal, a regression based zero frequency centered band filter communicatively coupled to the input frequency shifter, the regression based zero frequency centered band filter being configured to filter the frequency down shifted discrete time series signal to provide a filtered and frequency down shifted discrete time series signal, and an output frequency shifter communicatively coupled to the regression based zero frequency centered band filter, the output frequency shifter being configured to frequency up shift the filtered and frequency down shifted discrete time series signal. 
     Preferably, the input frequency shifter being configured to frequency down shift the discrete time series signal comprises the input frequency shifter being configured to multiply the discrete time series signal by an exponent of a negatived frequency shift term k 0 . 
     Preferably, the output frequency shifter being configured to frequency up shift the filtered and frequency down shifted discrete time series signal comprises the output frequency shifter being configured to multiply the filtered and frequency down shifted discrete time series signal by an exponent of the frequency shift term k 0 . 
     Preferably, the regression based zero frequency centered band filter comprises a polynomial regression smoothing filter. 
     Preferably, the regression based zero frequency centered band filter comprises a polynomial regression filter communicatively coupled to the input frequency shifter, the polynomial regression filter being configured to receive the frequency down shifted discrete time series signal provided by the input frequency shifter, and smooth the frequency down shifted discrete time series signal to provide a smoothed frequency down shifted discrete time series signal. 
     Preferably, the regression based zero frequency centered band filter further comprises a feed forward loop communicatively coupled to the input frequency shifter, the feed forward loop being configured to receive the frequency down shifted discrete time series signal provided by the input frequency shifter and a summation communicatively coupled to the polynomial regression filter and the feed forward loop, the summation being configured to receive the smoothed frequency down shifted time series signal from the polynomial regression filter, receive the frequency down shifted time series signal from the feed forward loop, and sum the smoothed frequency down shifted discrete time series signal and the frequency down shifted discrete time series signal to provide a filtered and frequency down shifted discrete time series signal. 
     Preferably, the input frequency shifter is further configured to receive the discrete time series signal, the regression based zero frequency centered band filter is further configured to receive the frequency down shifted discrete time series signal, and the output frequency shifter is further configured to receive the filtered and frequency down shifted discrete time series signal. 
     According to an aspect, a method of filtering a discrete time series signal with a band filter comprises frequency down shifting the discrete time series signal to provide a frequency down shifted discrete time series signal, using a regression based zero frequency centered band filter to filter the frequency down shifted discrete time series signal to provide a filtered and frequency down shifted discrete time series signal, and frequency up shifting the filtered and frequency down shifted discrete time series signal. 
     Preferably, frequency down shifting the discrete time series signal comprises multiplying the discrete time series signal by an exponent of a negatived frequency shift term k 0 . 
     Preferably, frequency up shifting the filtered and frequency down shifted discrete time series signal comprises multiplying the filtered and frequency down shifted discrete time series signal by an exponent of the frequency shift term k 0 . 
     Preferably, the regression based zero frequency centered band filter comprises a polynomial regression smoothing filter. 
     Preferably, using the regression based zero frequency centered band filter to filter the frequency down shifted discrete time series signal to provide a filtered and frequency down shifted discrete time series signal comprises using a polynomial regression filter to smooth the frequency down shifted discrete time series signal. 
     Preferably, using the regression based zero frequency centered band filter to filter the frequency down shifted discrete time series signal to provide a filtered and frequency down shifted discrete time series signal further comprises using a summation to sum the smoothed and frequency down shifted discrete time series signal and the frequency down shifted discrete time series signal. 
     Preferably, the method further comprises receiving with a feed forward loop the frequency down shifted discrete time series signal, receiving with a summation the smoothed frequency down shifted time series signal from the polynomial regression filter, receiving with the summation the frequency down shifted time series signal from the feed forward loop, and summing with the summation the smoothed frequency down shifted discrete time series signal and the frequency down shifted discrete time series signal to provide a filtered and frequency down shifted discrete time series signal. 
     Preferably, the method further comprises receiving the discrete time series signal with the input frequency shifter, receiving the frequency down shifted discrete time series signal with the regression based zero frequency centered band filter, and receiving the filtered and frequency down shifted discrete time series signal with the output frequency shifter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The same reference number represents the same element on all drawings. It should be understood that the drawings are not necessarily to scale. 
         FIG.  1    shows a band filter  100  for filtering a discrete time series signal  110 . 
         FIG.  2    shows a band filter  200  for filtering a discrete time series signal  210 . 
         FIG.  3    shows a frequency response graph  300  of a polynomial regression filter. 
         FIGS.  4  through  7    show a first through fourth frequency spectrums  400 - 700  illustrating a filtering of one of two distinct signals to demonstrate an application of the band filter  200  described with reference to  FIG.  2    for filtering a discrete time series signal. 
         FIGS.  8  through  13    illustrates an application of the band filter  200  to SZ(8/64) phase coding. 
         FIG.  14    shows a method  1400  of filtering a discrete time series signal with a band filter. 
     
    
    
     DETAILED DESCRIPTION 
       FIGS.  1 - 14    and the following description depict specific examples to teach those skilled in the art how to make and use the best mode of embodiments of filtering a discrete time series signal. For the purpose of teaching inventive principles, some conventional aspects have been simplified or omitted. Those skilled in the art will appreciate variations from these examples that fall within the scope of the present description. Those skilled in the art will appreciate that the features described below can be combined in various ways to form multiple variations of filtering a discrete time series signal. As a result, the embodiments described below are not limited to the specific examples described below, but only by the claims and their equivalents. 
     A technique other than FIR and IIR filtering is regression filtering where typically a polynomial is fitted to the time series signal to be filtered. In the case of a polynomial regression-based filter, the polynomial fit can be used to estimate the trend of the time series data or the polynomial fit can be subtracted from the data thus yielding the high frequency content of the signal. This high frequency content is typically called the residuals of the signal. Similar results may be obtained with other regression-based filters. As will be explained below, a regression-based filter, such as a regression based zero frequency centered band filter discussed in the following, may not require the use of a window function. The polynomial regression-based filter may also operate on the entire length of the times series and does not have the issue of filter warmup as do the FIR and IIR filters 
     Frequency Band-Pass Filter 
       FIG.  1    shows a band filter  100  for filtering a discrete time series signal  110 . As shown in  FIG.  1   , a discrete time series signal  110  is provided to an input frequency shifter  120 . The input frequency shifter  120  is communicatively coupled to any device that may provide the discrete time series signal  110 . The input frequency shifter  120  is also communicatively coupled to a regression based zero frequency centered band filter  130 . As shown in  FIG.  1   , an input of the input frequency shifter  120  is communicatively coupled to an output of a device (not shown) configured to provide the discrete time series signal  110  and an output of the input frequency shifter  120  is communicatively coupled to an input of the regression based zero frequency centered band filter  130 . The regression based zero frequency centered band filter  130  is communicatively coupled to the input frequency shifter  120  and an output frequency shifter  140 . As shown in  FIG.  1    the regression based zero frequency centered band filter  130  includes an input and an output, where the input of the regression based zero frequency centered band filter  130  is communicatively coupled to the output of the input frequency shifter  120  and an output of the regression based zero frequency centered band filter  130  is communicatively coupled to an input of the output frequency shifter  140 . 
     The input frequency shifter  120  may be configured to receive the discrete time series signal  110 , perform a frequency down shift of the discrete time series signal  110  to obtain a frequency down shifted discrete time series signal  120   a , and provide the frequency down shifted discrete time series signal  120   a  at the output of the input frequency shifter  120 . The input frequency shifter  120  may be configured to provide the frequency down shifted discrete time series signal  120   a  to the regression based zero frequency centered band filter  130 . The regression based zero frequency centered band filter  130  may be configured to receive and filter the frequency down shifted discrete time series signal  120   a  and provide the filtered and frequency downshifted discrete time series signal  130   a  at the output of the regression based zero frequency centered band filter  130 . The output frequency shifter  140  may be configured to receive the filtered and frequency downshifted discrete time series signal  130   a  from the regression based zero frequency centered band filter  130 , perform a frequency up shift of the filtered and frequency downshifted discrete time series signal  130   a  to obtain a filtered discrete time series signal  150 , and provide the filtered discrete time series signal  150  at the output of the output frequency shifter  140 . 
     The discrete time series signal  110  may be any suitable discrete time series signal. For example, the discrete time series signal  110  may be a series of samples of an analog waveform. The sampling may occur at a constant sampling frequency, although any suitable sampling rate or rates may be employed. For example, some applications may preferably employ a variable sampling rate. The discrete time series signal  110  may represent values in any suitable form. For example, the discrete time series signal  110  may be a series of values indexed by sample number. In an example, the discrete time series signal  110  may be a series of decimal values where each value is indexed from 0 to N, where N+1 is a length of the discrete time series signal  110 . The discrete time series signal  110  may be provided from storage, in real-time from a sampling device, and/or the like. For example, the discrete time series signal  110  may be obtained from a buffer coupled to a communications interface that receives a digitization of an analog signal. 
     The input frequency shifter  120  may be any suitable frequency shifter configured to frequency down shift a discrete time series signal. For example, the input frequency shifter  120  may be configured to receive any suitable discrete time series signal, such as a narrow or broadband signal, a multi-component signal, a plurality of signals, including the narrow or broadband signal and/or multi-component signal, products of the plurality of signals, and/or the like. The input frequency shifter  120  may shift a frequency of the received discrete time series signal down by a frequency shift value of, for example, −k 0 . If the received discrete time series signal is a multi-component signal, then components of the multi-component signal may be frequency down shifted by the frequency down shift amount −k 0 . 
     The input frequency shifter  120  may multiply the discrete time series signal  110  by an exponential scaled value of the frequency shift value −k 0  to obtain the frequency down shifted discrete time series signal  120   a . For example, the frequency down shifted discrete time series signal  120   a  may be comprised of all the component discrete time series signals of the frequency down shifted discrete time series signal  120   a  that are frequency down shifted by −k 0 . The frequency down shifted discrete time series signal  120   a  may be received by the regression based zero frequency centered band filter  130 , although any suitable regression based zero frequency centered band filter may be employed. 
     As shown in  FIG.  1   , the regression based zero frequency centered band filter  130  is comprised of a regression filter  132  that is communicatively coupled to the input frequency shifter  120 . The regression filter  132  may be a polynomial regression filter, although any suitable regression filter may be employed such as, for example, Gaussian. Lorentzian, Voigt functions, or other similar or dissimilar functions. For example, in some applications, an inverted logistic sigmoid function (e.g., S-curve) may be used. The regression filter  132  may be configured to smooth the frequency down shifted discrete time series signal  120   a  into a filtered and frequency downshifted discrete time series signal  130   a  and provide the filtered and frequency downshifted discrete time series signal  130   a . The regression filter  132  may be configured to pass any low frequency components of the frequency down shifted discrete time series signal  120   a  received from the input frequency shifter  120 . For example, component discrete time series signals of the frequency down shifted discrete time series signal  120   a  that have frequencies that are within cutoff frequencies of the regression filter  132  may be passed. Accordingly, the regression based zero frequency centered band filter  130  may be any suitable regression based zero frequency centered band filter. The regression based zero frequency centered band filter  130  may be configured to significantly attenuate lower frequency components from the frequency down shifted discrete time series signal  120   a . Accordingly, the regression based zero frequency centered band filter  130  may be configured to pass higher frequency components from the frequency down shifted discrete time series signal  120   a . With more particularity, the regression based zero frequency centered band filter  130  may have pass band outside of which signals are significantly attenuated to obtain the filtered and frequency downshifted discrete time series signal  130   a . Components having frequencies within the pass band of the regression based zero frequency centered band filter  130  may not be attenuated to obtain the filtered and frequency downshifted discrete time series signal  130   a.    
     Accordingly, the filtered and frequency downshifted discrete time series signal  130   a  may be any suitable frequency down shifted and filtered discrete time series signal. For example, as shown in  FIG.  1   , the filtered and frequency downshifted discrete time series signal  130   a  may be comprised of components of the discrete time series signal  110  that, after being frequency down shifted by the input frequency shifter  120 , have frequencies that are within the pass band of the regression filter  132 . The filtered and frequency downshifted discrete time series signal  130   a  may be provided to the output frequency shifter  140 . The filtered and frequency downshifted discrete time series signal  130   a  may be received by the output frequency shifter  140  without any additional processing, although additional intervening devices may be employed in alternative embodiments. 
     The output frequency shifter  140  may be any suitable frequency up shifter. The output frequency shifter  140  may be configured to receive any suitable filtered and frequency downshifted discrete time series signal  130   a , such as a narrow or broadband signal, a multi-component signal, a plurality of signals, including the narrow or broadband signal and/or multi-component signal, products of the plurality of signals, and/or the like. The input frequency shifter  120  may shift a frequency or frequencies of the received filtered and frequency downshifted discrete time series signal  130   a  up by a frequency shift value of, for example, k 0 . For example, if the received filtered and frequency downshifted discrete time series signal  130   a  signal is a multi-component signal, then components of the multi-component signal may be frequency up shifted by the frequency up shift amount k 0 . The output frequency shifter  140  may multiply the filtered and frequency downshifted discrete time series signal  130   a  by an exponential scaled value of the frequency shift value k 0  to obtain the filtered discrete time series signal  150 . The filtered discrete time series signal  150  or xf n  may be any suitable filtered discrete time series signal. For example, the filtered discrete time series signal  150  may be referred to as a band filtered discrete time series signal or, with more particularity, a band-pass filtered discrete time series signal. 
     Frequency Band-Stop Filter 
       FIG.  2    shows a band filter  200  for filtering a discrete time series signal  210 . As shown in  FIG.  2   , the discrete time series signal  210  is provided to an input frequency shifter  220 . The input frequency shifter  220  may be communicatively coupled to any device that may provide the discrete time series signal  210 . The input frequency shifter  220  is communicatively coupled to a regression based zero frequency centered band filter  230 . As shown in  FIG.  1   , an input of the input frequency shifter  220  is communicatively coupled to an output of a device configured to provide the discrete time series signal  210 . An output of the input frequency shifter  220  is communicatively coupled to an input of the regression based zero frequency centered band filter  230 . The regression based zero frequency centered band filter  230  is communicatively coupled to the input frequency shifter  220  and an output frequency shifter  240 . The regression based zero frequency centered band filter  230  includes an input and an output, where the input of the regression based zero frequency centered band filter  230  is communicatively coupled to the output of the input frequency shifter  220  and an output of the regression based zero frequency centered band filter  230  is communicatively coupled to an input of the output frequency shifter  240 . The output frequency shifter  240  includes an input that is communicatively coupled to the output of the regression based zero frequency centered band filter  230 . 
     The input frequency shifter  220  may be configured to receive the discrete time series signal  210 , perform a frequency down shift of the discrete time series signal  210  to obtain a frequency down shifted discrete time series signal  220   a , and provide the frequency down shifted discrete time series signal  220   a  at the output of the input frequency shifter  220 . The input frequency shifter  220  may be configured to provide the frequency down shifted discrete time series signal  220   a  to the regression based zero frequency centered band filter  230 . The regression based zero frequency centered band filter  230  may be configured to receive the frequency down shifted discrete time series signal  220   a , filter the frequency down shifted discrete time series signal  220   a , and provide the filtered and frequency downshifted discrete time series signal  230   a  at the output of the regression based zero frequency centered band filter  230 . The output frequency shifter  240  may be configured to receive the filtered and frequency downshifted discrete time series signal  230   a  from the regression based zero frequency centered band filter  230 , perform a frequency up shift of the filtered and frequency downshifted discrete time series signal  230   a  to obtain and provide a filtered discrete time series signal  250 . 
     For a complex valued discrete time series, a frequency spectrum of a DFT may range from −180 to +180 degrees, or similar, such as π to −π, such that frequency “up” shifting may refer to negative frequencies of the discrete time series signals are frequency down shifted away from zero. As can be appreciated, the input frequency shifters  120 ,  220  and output frequency shifters  140 ,  240  may alternatively be referred to as applied frequency shifters in that a frequency shift is being applied to the discrete time series signal  110 ,  210  by the input frequency shifter  110 ,  210  or the frequency downshifted discrete time series signal  130   a ,  230   a  by the output frequency shifter  140 ,  240 , although any suitable term may be employed. 
     The discrete time series signal  210  may be any suitable discrete time series signal. For example, the discrete time series signal  210  may be a series of samples of an analog waveform. The sampling may occur at a constant sampling frequency, although any suitable sampling rate or rates may be employed. For example, some applications may preferably employ a variable sampling rate. The discrete time series signal  210  may represent values in any suitable form. For example, the discrete time series signal  210  may be a series of values indexed by sample number. In one example, the discrete time series signal  210  may be a series of decimal values where each value is indexed from 0 to n, where n+1 is a length of the discrete time series signal  210 . The discrete time series signal  210  may be provided from storage, in real-time from a sampling device, and/or the like. For example, the discrete time series signal  210  may be obtained from a buffer coupled to a communications interface that receives a digitization of an analog signal. 
     The input frequency shifter  220  may be any suitable frequency shifter that performs a frequency down shift on a discrete time series signal. For example, the input frequency shifter  220  may be configured to receive any suitable discrete time series signal, such as a narrow or broadband signal, a multi-component signal, a plurality of signals, including the narrow or broadband signal and/or multi-component signal, products of the plurality of signals, and/or the like. The input frequency shifter  220  may shift a frequency or frequencies of the received signal down by a frequency shift value of, for example, −k 0 . If the received signal is a multi-component signal, then components of the multi-component signal may be frequency down shifted by the frequency down shift amount −k 0 . 
     The input frequency shifter  220  may multiply the discrete time series signal  210  by an exponential scaled value of the frequency shift value −k 0  to obtain the frequency down shifted discrete time series signal  220   a . Accordingly, the frequency down shifted discrete time series signal  220   a  may be a frequency down shifted discrete time series signal. For example, the frequency down shifted discrete time series signal  220   a  may be comprised of all the component discrete time series signals of the frequency down shifted discrete time series signal  220   a  that are frequency down shifted by −k 0 . The frequency down shifted discrete time series signal  220   a  may be received by the regression based zero frequency centered band filter  230 , although any suitable regression based zero frequency centered band filter may be employed. 
     As shown in  FIG.  2   , the regression based zero frequency centered band filter  230  is comprised of a regression filter  232  and a feed forward loop  234  that is communicatively coupled to the input frequency shifter  220 . The regression filter  232  and the feed forward loop  234  may be configured to receive the frequency down shifted discrete time series signal  220   a . The regression filter  232  may be configured to smooth the frequency down shifted discrete time series signal  220   a  into a smoothed frequency down shifted discrete time series signal  232   a  and provide the smoothed frequency down shifted discrete time series signal  232   a . The feed forward loop  234  may be configured to provide the frequency down shifted discrete time series signal  220   a  without filtering or otherwise altering the frequency down shifted discrete time series signal  220   a . The regression filter  232  and the feed forward loop  234  may be communicatively coupled to a summation  236 . Accordingly, the summation  236  may receive the smoothed frequency down shifted discrete time series signal  232   a  from the regression filter  232  and the frequency down shifted discrete time series signal  220   a  from the feed forward loop  234 . The summation  236  may be configured to subtract the smoothed frequency down shifted discrete time series signal  232   a  from the frequency down shifted discrete time series signal  220   a  to obtain the filtered and frequency downshifted discrete time series signal  230   a.    
     The regression based zero frequency centered band filter  230  may be any suitable regression based zero frequency centered band filter. The regression based zero frequency centered band filter  230  may be configured to attenuate lower frequency components from the frequency down shifted discrete time series signal  220   a . Accordingly, the regression based zero frequency centered band filter  230  may be configured to pass higher frequency components from the frequency down shifted discrete time series signal  220   a . With more particularity, the regression based zero frequency centered band filter  230  may have stop band outside of which components are passed. Within the stop band, which may be defined by a low frequency cutoff and a high frequency cutoff, components having frequencies within the stop band may be significantly attenuated to obtain the filtered and frequency downshifted discrete time series signal  230   a.    
     The regression filter  232  may be the regression filter  132  described above with reference to  FIG.  1   , although any suitable regression based smoothing filter may be employed. The regression filter  232  may be configured to smooth the frequency down shifted discrete time series signal  220   a  into a smoothed frequency down shifted discrete time series signal  232   a . The regression filter  232  may be configured to pass components of the frequency down shifted discrete time series signal  220   a  received from the input frequency shifter  220  that are within a pass band of the regression filter  232 . For example, component discrete time series signals of the frequency down shifted discrete time series signal  220   a  that have frequencies that are within cutoff frequencies of the regression filter  232  may be passed. The pass band of the regression filter  232  may be centered at zero. 
     The smoothed frequency down shifted discrete time series signal  232   a  may be any suitable smoothed and frequency down shifted discrete time series signal that is provided by the regression filter  232 . For example, the smoothed frequency down shifted discrete time series signal  232   a  may be comprised of components of the discrete time series signal  210  that have frequencies that, after being frequency down shifted by the discrete time series signal  210 , are within the pass band of the regression filter  232 . 
     As discussed above, the frequency down shifted discrete time series signal  220   a  is also provided to the feed forward loop  234 . The feed forward loop  234  may be any suitable path that conveys the frequency down shifted discrete time series signal  220   a  to the summation  236 . As shown in  FIG.  2   , the feed forward loop  234  is communicatively coupled to an output of the input frequency shifter  220  and an input of the summation  236 . Accordingly, the frequency down shifted discrete time series signal  220   a  may be conveyed, transmitted, conducted, or the like to the summation  236 . Although no devices are shown in  FIG.  2    as being part of the feed forward loop  234 , alternative feed forward loops may include one or more devices. 
     The summation  236  may be any suitable summation that is configured to sum two discrete time series signals. As shown in  FIG.  2   , the summation  236  is configured to receive the smoothed frequency down shifted discrete time series signal  232   a  and the frequency down shifted discrete time series signal  220   a . The summation  236   a  may sum an amplitude of the smoothed frequency down shifted discrete time series signal  232   a  with an amplitude of the frequency down shifted discrete time series signal  220   a  at a given time index, although any suitable summation may be employed. The summation  236   a  sums the smoothed frequency down shifted discrete time series signal  232   a  and the frequency down shifted discrete time series signal  220   a  to provide the filtered and frequency downshifted discrete time series signal  230   a . The filtered and frequency downshifted discrete time series signal  230   a  may be any suitable filtered and frequency down shifted discrete time series signal provided by the regression based zero frequency centered band filter  230 . 
     The output frequency shifter  240  may be any suitable frequency up shifter. The output frequency shifter  240  may be configured to receive any suitable filtered and frequency downshifted discrete time series signal  230   a , such as a narrow or broadband signal, a multi-component signal, a plurality of signals, including the narrow or broadband signal and/or multi-component signal, products of the plurality of signals, and/or the like. The input frequency shifter  220  may shift a frequency or frequencies of the received filtered and frequency downshifted discrete time series signal  230   a  up by a frequency shift value of, for example, k 0 . For example, if the filtered and frequency downshifted discrete time series signal  230   a  signal is a multi-component signal, then components of the multi-component signal may be frequency up shifted by the frequency up shift amount k 0 . The output frequency shifter  140  may multiply the filtered and frequency downshifted discrete time series signal  230   a  by an exponential scaled value of the frequency shift value k 0  to obtain the filtered discrete time series signal  250 . The filtered discrete time series signal  250  or xf n  may be any suitable filtered discrete time series signal. For example, the filtered discrete time series signal  250  may be referred to as a band filtered discrete time series signal or, with more particularity, a band-stop filtered discrete time series signal. The filtered discrete time series signal  250  or xf n  may be a filtered discrete time series signal. For example, the filtered discrete time series signal  250  may be referred to as band filtered discrete time series signal or, with more particularity, a band-stop filtered discrete time series signal. 
     The band filters  100 ,  200  may be referred to or described as a regression frequency shift filter (RFSF), although any suitable descriptive term may be employed. As will be described in more detail below with reference to  FIGS.  3 - 7   , the band filter  200  described with reference to  FIG.  2    may separate two signals by attenuating one of the two signals. Accordingly, the band filter  200  may be used filter out or attenuate a frequency component of a discrete time series signal, as is described in more detail in the following. 
     The discrete time series signal  210  may be frequency shifted by a frequency shift value of, for example, −k 0  by the input frequency shifter  220  so that a selected center frequency of component discrete time series signal in the discrete time series signal  210  is in a zero frequency location of the regression filter  232 . This may be accomplished with the well-known general Fourier frequency shift transform pair property, 
         x ( t ) e   (jω     0     t)   ⇔X (ω−ω 0 ),  [1]
 
     where x(t) is the time series signal X(ω) is the Fourier transform of x(t), ω is frequency, to is the frequency shift, j is the square root of −1, and t is time. For a discrete time series signal, such as, for example, the discrete time series signal  210  described above, equation [1] can be equivalent expressed, 
     
       
         
           
             
               
                 
                   
                     
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     where   indicates a Fourier transform, n is the time index, k is the frequency index, k 0  is the integer frequency shift parameter, and N is the length of the discrete time series signal. 
     An algorithm for band-stop filtering, such as the band filter  200  described above, at an arbitrary frequency may be as follows: 1. select the center frequency, k 0 , for the regression filter, 2. frequency shift the time series signal according to Equation [2](using −k 0 ), 3. apply a regression filter to the frequency shifted signal, 
     
       
         
           
             
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     Frequency shift the filtered signal back to the original frequency locations using 
     
       
         
           
             
               
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     where xs n  is the frequency shifted and filtered time series and xf n  is a final band-stop filtered discrete time series signal, such as, for example, the filtered discrete time series signal  250 . The regression filter stop-band bandwidth may, for polynomial regression filters, a function of the time series length and the polynomial order. The stop bandwidth can be chosen from the regression filter&#39;s frequency response plot, as will be described in more detail in the following with reference to  FIGS.  3 - 7   . 
       FIG.  3    shows a frequency response graph  300  of a regression filter. As shown in  FIG.  3   , the frequency response graph  300  includes a frequency axis  310  and a power axis  320 . The frequency axis  310  ranges from −25 to 25 m s −1  (i.e., a frequency index of ‘m’, which may be at a sampling rate of, for example, Ts=0.001 s) which can correspond to +/−180 degrees, and the power axis  320  ranges from −50 to 0 dB, although any suitable ranges may be employed. For example, other frequency response graphs may include a frequency axis that range from +/−π radians, or other similar ranges. The frequency response graph  300  also includes a plurality of frequency response plots  330 . Each of the frequency response plots  330  is a frequency response of a polynomial regression filter having a particular order. 
     As shown in  FIG.  3   , the frequency response plots  330  are for a polynomial regression filter for 64 data points for odd-numbered polynomial orders 1 through 21. As can be seen, the frequency response plots  330  include stop bands  330   a , which are centered as zero frequency. The higher the order of the polynomial, the wider a bandwidth of a given stop band of the stop bands  330   a  in the regression filter. 
       FIGS.  4  through  7    show a first through fourth frequency spectrums  400 - 700  illustrating a filtering of one of two distinct signals to demonstrate an application of the band filter  200  described with reference to  FIG.  2    for filtering a discrete time series signal. As shown in  FIGS.  4 - 7   , the first through fourth frequency spectrums  400 - 700  have a discrete bandwidth of 64 data points, although any suitable bandwidth may be employed. Accordingly, as shown in  FIGS.  4 - 7   , the first through fourth frequency spectrums include a frequency axis  410  and a power axis  420 . The first through fourth frequency spectrums  400 - 700  illustrate a dual signal frequency spectrum  430 , a frequency shifted dual signal frequency spectrum  530 , a filtered frequency shifted dual signal frequency spectrum  630 , and a filtered dual signal frequency spectrum  730 . 
     In the example shown in  FIGS.  4 - 7   , two signals with separate distinct frequency spectra are separated via the band filter  100  as an illustration of the technique. The power spectrum is shown in  FIG.  4   . The wider spectrum width signal, centered at about frequency index  52 , is now shifted to zero velocity via Eq. [ 2 ]. The resulting frequency sifted signal is shown in  FIG.  5   . Now that the desired signal is centered at zero frequency, a regression filter is applied to eliminate the power at those frequencies around zero. This is shown in  FIG.  6   . Finally, the data in  FIG.  6    are frequency shifted back to their original frequency locations. This is shown in  FIG.  7   . Thus, band-stop filtering has been accomplished without the use of finite impulse response (FIR) or infinite impulse response (IIR) techniques and without using a time domain window function. 
     As shown in  FIG.  4   , the dual signal frequency spectrum  430  includes the two distinct signals, which are referenced as a first signal  430   a  and a second signal  430   b.    
     The first signal  430   a  has a bandwidth that is narrower than a bandwidth of the second signal  430   b . The bandwidth of the first signal  430   a  is centered at about the frequency index of ‘0’ and the bandwidth of the second signal  430   b  is centered at about the frequency index of ‘52’, although any suitable number of signals and/or values of center frequencies of signal bandwidths may be employed. 
     As shown in  FIG.  5   , the frequency shifted dual signal frequency spectrum  530  includes a frequency shifted first signal  530   a  and a frequency shifted second signal  530   b  that respectively correspond to the first and second signal  430   a ,  430   b  of  FIG.  4   . As can be appreciated, the frequency shifted dual signal frequency spectrum  530  is a frequency shifted version of the dual signal frequency spectrum  430  shown in  FIG.  4   . 
     As shown in  FIG.  6   , the filtered frequency shifted dual signal frequency spectrum  630  includes a frequency shifted first signal  630   a  that is essentially the same as the frequency shifted first signal  530   a  of  FIG.  5   . However, the filtered frequency shifted dual signal frequency spectrum  630  includes an attenuated frequency shifted second signal  630   b  that results from attenuating the frequency shifted second signal  530   b  of  FIG.  5   . More specifically, the attenuated frequency shifted second signal  630   b  of the frequency shifted dual signal frequency spectrum  530  may result from an attenuation of the frequency shifted second signal  530   b  by the band filter  200  described above with reference to  FIG.  2   . 
     As shown in  FIG.  7   , the filtered dual signal frequency spectrum  730  includes a first signal  730   a  that corresponds to the first signal  430   a  shown in  FIG.  4    and an attenuated second signal  630   b  that may result from frequency shifting the attenuated frequency shifted second signal  630   b  shown in  FIG.  6   . More specifically, the filtered dual signal frequency spectrum  730  may result from frequency up shifting the filtered frequency shifted dual signal frequency spectrum  630  by a shift frequency, which may be the same as the shift frequency used to frequency down shift the dual signal frequency spectrum  430 . 
     Summarizing, the dual signal frequency spectrum  430  may be down shifted by a frequency shift value of −k 0  to obtain the frequency shifted dual signal frequency spectrum  530  shown in  FIG.  5   . As discussed above, the frequency shifted dual signal frequency spectrum  530  may be filtered by the band filter  200  to obtain the filtered frequency shifted dual signal frequency spectrum  630  shown in  FIG.  6   . The filtered frequency shifted dual signal frequency spectrum  630  show in  FIG.  6    may be frequency up shifted by the same amount to obtain the filtered dual signal frequency spectrum  730 . For example, the filtered frequency shifted dual signal frequency spectrum  630  may be up shifted by a frequency shift value of +k 0  to obtain the filtered dual signal frequency spectrum  730 . 
       FIGS.  8  through  13    illustrates an application of the band filter  200  to SZ(8/64) phase coding. The SZ(8/64) phase coding is a technique to mitigate range-velocity ambiguities in the nation&#39;s NEXRAD weather radar data (Sachidananda and Zrnic, 1999). SZ phase coding provides for the separation of multiple trip overlaid weather echoes. This is accomplished by applying a phase code to the individual radar transmit pulses. Upon reception of the return signal, the first trip echo will be coded with one set of phases while the second trip echo is phase coded with a different set of phases. Assuming that the first trip echo is the strongest echo, the first trip echo is made coherent by cohering the signal using the conjugate of the known transmit phases. At this point, the second trip echo (the weak trip) possesses phases that are termed the modulation code that distributes the weak trip echo across the entire frequency range making the second trip echo behave like white noise. 
     Prior the development of the band filter  200  discussed above, to separate the strong trip (ST) and weak trip (WT) echoes, first a window function is applied to the ST cohered time series. The spectrum of the signal is calculated and the center velocity of the ST is found. Usually, a ¾ spectrum width notch is applied centered at the ST calculated velocity. At this point the ST trip signal has been eliminated. The spectrum is then transformed back to the time domain where the WT is recohered using the conjugate of the modulation code. The WT signal power and velocity can then be estimated, but with some measurement error. 
     The band filter  200  described above may now be used to eliminate the ST echo illustrated via an example. The time series window function will not be required so that the WT power and velocity can be recovered with reduced measurement error, as will be described in the following with reference to  FIGS.  8 - 13   .  FIGS.  8 - 13    show SZ phase coding graphs  800 - 1300  that include a frequency index axis  810 - 1310  and a magnitude axis  820 - 1320 . The SZ phase coding graphs  800 - 1300  are shown as including frequency spectrums. As shown in  FIG.  8   , the SZ phase coding graph  800  shows an ST signal frequency spectrum  830 . The SZ phase coding graph  900  of  FIG.  9    shows a WT signal frequency spectrum  930 . Also shown in  FIGS.  8  and  9    is a zero or “0” velocity line. The zero-velocity line is at a frequency index of about 33. 
     As shown in  FIG.  8   , the ST signal frequency spectrum  830  is a frequency spectrum of an ST signal. As can be appreciated from the ST signal frequency spectrum  830 , the ST signal may have a center frequency that is at a frequency index of about 38. As shown in  FIG.  9   , the WT signal frequency spectrum  930  is a frequency spectrum of a WT signal. The WT signal frequency spectrum  930  is centered at a frequency index of about 38. 
     In  FIG.  10   , the SZ phase coding graph  1000  illustrates a modulated WT signal frequency spectrum  1030 . More specifically, the modulated WT signal frequency spectrum  1030  is a spectrum of a WT signal after the SZ (8/64) modulation code is applied. Due to the SZ (8/64) modulation code being applied, eight replicas of the WT signal frequency spectrum  930  are produced resulting in the modulated WT signal frequency spectrum  1030 . As can be seen in  FIG.  10   , the eight replicas are distributed equally across the frequency spectrum in the modulated WT signal frequency spectrum  1030 . 
     In  FIG.  11   , the SZ phase coding graph  1100  includes a combined ST/WT signal frequency spectrum  1130 . More specifically, the ST signal illustrated in  FIG.  8    as the ST signal frequency spectrum  830  is combined with the modulated WT signal frequency spectrum  1030  by adding point-by-point in the time domain the ST signal and the modulated WT signal and frequency shifting the resulting signal to be centered as zero velocity (illustrated by the “0” line). More specifically, for a given time-index, a magnitude value of the ST signal is added with a magnitude value of the modulated WT signal and the resulting signal is frequency shifted to be about the “0” velocity line. As a result, the combined ST/WT signal frequency spectrum  1130  includes a ST signal frequency spectrum portion  1130   a  and a modulated WT signal spectrum portion  1130   b.    
     In  FIG.  12   , the SZ phase coding graph  1200  shows a filtered ST/WT signal frequency spectrum  1230 . The filtered ST/WT signal frequency spectrum  1230  includes an attenuated portion  1230   a . As can be seen, the attenuated portion  1230   a  has a magnitude of less than −100 dB magnitude whereas the remainder of the filtered ST/WT signal frequency spectrum  1230  has a magnitude of approximate −10 dB. The filtered ST/WT signal frequency spectrum  1230  may be obtained by filtering the combined ST/WT signal with the band filter  200  described above and frequency shifting the resulting signal to the center frequency of the ST signal as shown in  FIG.  8   . As a result, a center frequency of the stop band is centered at a frequency index of about 38. 
       FIG.  13    shows a recombined ST/WT signal frequency spectrum  1330 . As shown in  FIG.  13   , the recombined ST/WT signal frequency spectrum  1330  includes a ST signal frequency spectrum portion  1330   a  and a WT signal frequency spectrum portion  1330   b . The recombined ST/WT signal frequency spectrum  1330  can be used to effectively estimate a power and a velocity of the WT signal without the use of a window function. As a result, the power and velocity are recovered with smaller measurement error when compared to the measurement error associated with the use of a Hanning window. 
       FIG.  14    shows a method  1400  of filtering a discrete time series signal with a band filter. In step  1410 , the method  1400  may frequency down shift a discrete time series signal. In step  1420 , the method  1400  may use a regression based zero frequency centered band filter to filter the frequency down shifted discrete time series signal. The method  1400 , in step  1430 , may frequency up shift the filtered and frequency down shifted discrete time series signal. 
     The discrete time series signal to be down shifted may be the discrete time series signal  110 ,  210  described above, although any suitable discrete time series signal may be employed. The discrete time series signal may be down shifted by, for example, the input frequency shifters  120 ,  220  described above, although any suitable input frequency shifter may be employed. The step  1410  may frequency down shift the discrete time series signal to provide a frequency down shifted discrete time series signal. 
     In step  1420 , the frequency down shifted signal may be filtered by using a regression based zero frequency centered band filter to filter the frequency down shifted discrete time series signal to provide a filtered and frequency down shifted discrete time series signal. The regression based zero frequency centered band filter may be the regression based zero frequency centered band filters  130 ,  230  described above, although any suitable regression based zero frequency centered band filter may be employed. 
     As discussed above with reference to  FIG.  1   , the regression based zero frequency centered band filter  130  may be comprised of a regression filter  132  that is a time domain filter that passes a discrete time series signal within two cutoff frequencies. As discussed above with reference to  FIG.  2   , the regression based zero frequency centered band filter  230  may be comprised of a regression filter  232  similar to the regression filter  132  of  FIG.  1    and a feed forward loop  234  that is communicatively coupled to the input frequency shifter  220 . 
     Referring again to  FIG.  14   , the step  1430  may frequency up shift the filtered and frequency down shifted discrete time series signal to provide a filtered discrete time series signal. The filtered discrete time series signal obtained from step  1430  may be the filtered discrete time series signals  150 ,  250  described above, although any suitable filtered discrete time series signal may be employed. 
     Frequency down shifting the discrete time series signal may comprise multiplying the discrete time series signal by an exponent of a negatived frequency shift term k 0  and wherein frequency up shifting the filtered and frequency down shifted discrete time series signal comprises multiplying the filtered and frequency down shifted discrete time series signal by an exponent of the frequency shift term k 0 . 
     The regression based zero frequency centered band filter comprises a polynomial regression smoothing filter. The regression based zero frequency centered band filter to filter the frequency down shifted discrete time series signal to provide a filtered and frequency down shifted discrete time series signal may comprise using a polynomial regression filter to smooth the frequency down shifted discrete time series signal and using a summation to sum the smoothed and frequency down shifted discrete time series signal and the frequency down shifted discrete time series signal. 
     The band filters  100 ,  200  and method  1400  may filter a discrete time series signal. In particular, the band filters  100 ,  200  and the method  1400  employ a regression based zero frequency centered band filter  130 ,  230  to ensure that filtering discrete time series signal with relatively short and finite duration and without the use of a window function and/or requiring a filter “warm-up” time. More specifically, the regression based zero frequency centered band filter  130 ,  230  both employ the regression filter  132 ,  232  that passes low frequency components of a frequency down shifted discrete time series signal. Accordingly, the benefits of discussed above may be realized. 
     The detailed descriptions of the above embodiments are not exhaustive descriptions of all embodiments contemplated by the inventors to be within the scope of the present description. Indeed, persons skilled in the art will recognize that certain elements of the above-described embodiments may variously be combined or eliminated to create further embodiments, and such further embodiments fall within the scope and teachings of the present description. It will also be apparent to those of ordinary skill in the art that the above-described embodiments may be combined, in whole or in part, to create additional embodiments within the scope and teachings of the present description. 
     Thus, although specific embodiments are described herein for illustrative purposes, various equivalent modifications are possible within the scope of the present description, as those skilled in the relevant art will recognize. The teachings provided herein can be applied to other band filters for filtering a discrete time series signal and not just to the embodiments described above and shown in the accompanying figures. Accordingly, the scope of the embodiments described above should be determined from the following claims.