Patent Publication Number: US-9838783-B2

Title: Adaptive phase-distortionless magnitude response equalization (MRE) for beamforming applications

Description:
FIELD OF THE DISCLOSURE 
     The instant disclosure relates to magnitude response equalization in multi-sensor systems. More specifically, portions of this disclosure relate to magnitude response equalization of signals from multiple microphone systems using adaptive filtering in the time domain. 
     BACKGROUND 
     Systems containing multiple microphones can detect directional sound by using beam forming techniques where the signals from at least two microphones are compared to observe phase shifts and magnitude differences. Processing signals from two different microphones capturing the same sounds requires equalization because the physical characteristics and magnitude responses may vary between microphones. These variations can exist even between microphones of the same make and model due to minor manufacturing variations. Variations can also be caused by many other factors, such as microphone boots, tube length differences, and other variations. Variations between microphones complicate processing signals from multiple microphone systems because applications, such as beam forming, assume that the differences in the signals measured at each microphone are attributable only to environmental and spacial differences, not differences in how the signals were measured. Accordingly, signal processing in multiple-microphone systems attempts to equalize the raw signals to improve the accuracy of signal processing calculations. 
     One conventional technique for equalizing is off-line calibration during system production. This technique requires manufacturing microphones with extremely low tolerance errors which increases the cost and sensitivity of the microphones. Another conventional technique for equalizing is self-calibration. On-line self-calibration using gain or magnitude response techniques include calculating propagation loss and phase matching. On-line self-calibration using frequency response techniques requires knowing the location of the control stimulus. 
     On-line self-calibration using magnitude response techniques generally operate by transforming the time domain signals for each microphone (e.g., two signals from two separate microphones) into the frequency domain and then calculating an equalization ratio based on the first and second signals across the frequency range. The equalization ratio is then applied to the frequency domain of the second signal in an attempt to match it to the first microphone. The adjusted second signal is then transformed back into the time domain, and further processing, such as beam forming calculations, may be performed with the first and second signals. This technique reduces the error introduced by variations in the two microphones, but introduces additional error in the equalization computations. 
     Manipulating the frequency domain of the second signal using the calculated equalization ratio across all frequencies and then converting back to the time domain introduces error in calculations. The magnitude response of the microphones varies across frequencies such that the calculated equalization ratio only approximates the magnitude differences of the two signals and does not account for varied magnitude responses of the different microphones at different frequencies. Furthermore, the signal generated by the Inverse Fast Fourier Transform (I-FFT) when converting the adjusted second signal from the frequency domain back to the time domain inherently introduces error because of the mathematical limitations of I-FFTs. Such a conventional technique is illustrated in  FIG. 1 , in which the frequency domain signal for x 2 [n] is taken from node  101 , after conversion to frequency domain at block  105 , and equalized at amplifier  102  using the ratio of the frequency domain responses calculated at processing block  103 . The equalized frequency response of x 2 [n] is then transformed in I-FFT block  104 . 
     Shortcomings mentioned here are only representative and are included simply to highlight that a need exists for improved electrical components, particularly for multiple microphone systems employed in consumer-level devices, such as mobile phones. Embodiments described herein address certain shortcomings but not necessarily each and every one described here or known in the art. 
     SUMMARY 
     Magnitude response equalization of multiple sensor systems may be improved by using a time domain impulse response filter that is based on the magnitude responses of the individual signals to equalize the magnitude response of multiple microphones across the desired frequency spectrum. Conventional techniques equalize signals in the frequency domain which creates errors and artifacts that propagate into the time domain representation of an equalized signal when the equalized signal is transformed from the frequency domain into the time domain. The methods and apparatuses described herein reduce or eliminate the signal error introduced by conventional frequency domain equalization techniques by creating a time domain impulse response filter that equalizes signals in the time domain. Thus, avoiding the error and artifacts that are introduced by domain transforms such as the I-FFT. Further, the signal processing is constrained to reduce or prevent introduction of phase differences between input signals. 
     In some embodiments, a time domain impulse response filter is based on the magnitude responses of the individual signals and used to equalize the magnitude response of multiple microphones across the desired frequency spectrum. The magnitude responses for each signal may be calculated in the frequency domain or with other techniques, such as auto-regressive analysis and mathematical signal approximations algorithms like Padé approximations. Applying the time domain impulse response filter based on the magnitude response of the system&#39;s microphones in the time domain to equalize a second microphone with a first microphone avoids the error introduced in prior art systems where equalization of the second signal is done in the frequency domain. 
     According to one embodiment, a method may include receiving, by a processor coupled to a plurality of sensors, at least a first input signal and a second input signal in a time domain from the plurality of sensors; converting, by the processor, the first and second input signals from the time domain to a frequency domain input signal; estimating, by the processor, a magnitude response difference between the first and second input signals based, at least in part, on the frequency domain input signal; converting, by the processor, the magnitude response difference into a time domain impulse response; constraining, by the processor, the time domain impulse response to have a linear phase response; and/or filtering, by the processor, at least one of the first input signal and the second input signal based, at least in part, on the constrained time domain impulse response. 
     In certain embodiments, the step of filtering may include equalizing a magnitude response between the first input signal and the second input signal received from the plurality of sensors; the step of estimating the magnitude response difference comprises calculating filter coefficients for an adaptive filter, wherein the step of constraining may include constraining the filter coefficients to be even symmetric and odd length, and wherein the step of filtering comprises applying the adaptive filter with the calculated and constrained filter coefficients. 
     In some embodiments, the method may further include the steps of repeating the steps of receiving, estimating, converting, constraining, and filtering to provide adaptive equalization of the received input signals; delaying at least one of the first input signal and the second input signal that is not filtered based on the constrained time domain impulse response to compensate for a delay introduced by the filtering; the first input signal and the filtered second input signal may be further filtered for spatial recognition; and/or the first input signal and the filtered second input signal may be further filtered for beamforming. 
     According to another embodiment, an apparatus may include a first input node configured to receive a first input signal; a second input node configured to receive a second input signal; and/or a controller coupled to the first input node and coupled to the second input node. The controller may be configured to perform certain steps including receiving the first input signal and the second input signal in a time domain; converting the first and second input signals from the time domain to a frequency domain input signal; estimating a magnitude response difference between the first and second input signals based, at least in part, on the frequency domain input signal; converting the magnitude response difference into a time domain impulse response; constraining the time domain impulse response to have a linear phase response; and/or filtering at least one of the first input signal and the second input signal based, at least in part, on the constrained time domain impulse response. 
     In some embodiments, the controller may perform the step of filtering by equalizing a magnitude response between the first input signal and the second input signal received from the plurality of sensors; and/or may perform the step of estimating the magnitude response difference by calculating filter coefficients for an adaptive filter, wherein the step of constraining comprises constraining the filter coefficients to be even symmetric and odd length, and wherein the step of filtering comprises applying the adaptive filter with the calculated and constrained filter coefficients. 
     In certain embodiments, the controller may also be configured to repeat the steps of receiving, estimating, converting, constraining, and filtering to provide adaptive equalization of the received input signals; and/or configured to delay at least one of the first input signal and the second input signal that is not filtered based on the constrained time domain impulse response to compensate for a delay introduced by the filtering. 
     According to another embodiment, a method may include receiving, by a processor from a plurality of sensors, at least a first input signal and a second input signal in a time domain; computing, by the processor, an auto-regressive (AR) model parameters of the input signals using linear prediction analysis; computing, by the processor, an auto-regressive moving average (ARMA) model parameters corresponding to the magnitude response difference between the two input signals; computing, by the processor, a time domain impulse response corresponding to a magnitude response difference between the first input signal and second input signal where the magnitude response difference is calculated using a Padé approximation based, at least in part, on the auto-regressive model parameters and the auto-regressive moving average model parameters; constraining, by the processor, the time domain impulse response to have a linear phase response; and/or filtering, by the processor, at least one of the first input signal and the second input signal based, at least in part, on the constrained time domain impulse response. 
     In certain embodiments, the step of applying the linear prediction analysis may include generating linear prediction coefficients; and/or the first input signal and the second input signals may include audio information. 
     In yet a further embodiment, an apparatus may include a first input node configured to receive a first audio signal; a second input node configured to receive a second audio signal; and/or a controller coupled to the first input node and coupled to the second input node. The controller may be configured to perform steps including receiving the first input signal and the second input signal in a time domain; computing, by the processor, the auto-regressive (AR) model parameters of the input signals using linear prediction analysis; computing, by the processor, the auto-regressive moving average (ARMA) model parameters corresponding to the magnitude response difference between the two input signals; computing, by the processor, a time domain impulse response corresponding to a magnitude response difference between the first input signal and second input signal where the magnitude response difference is calculated using a Padé approximation based, at least in part, on the auto-regressive model parameters and the auto-regressive moving average model parameters; constraining the time domain impulse response to have a linear phase response; and/or filtering at least one of the first input signal and the second input signal based, at least in part, on the constrained time domain impulse response. 
     In certain embodiments, the controller may be configured to apply the linear prediction analysis by generating linear prediction coefficients; the first input signal and the second input signals may include audio information; and/or the audio information may be audio information received from a first microphone and a second microphone. 
     The foregoing has outlined rather broadly certain features and technical advantages of embodiments of the present invention in order that the detailed description that follows may be better understood. Additional features and advantages will be described hereinafter that form the subject of the claims of the invention. It should be appreciated by those having ordinary skill in the art that the conception and specific embodiment disclosed may be readily utilized as a basis for modifying or designing other structures for carrying out the same or similar purposes. It should also be realized by those having ordinary skill in the art that such equivalent constructions do not depart from the spirit and scope of the invention as set forth in the appended claims. Additional features will be better understood from the following description when considered in connection with the accompanying figures. It is to be expressly understood, however, that each of the figures is provided for the purpose of illustration and description only and is not intended to limit the present invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a more complete understanding of the disclosed system and methods, reference is now made to the following descriptions taken in conjunction with the accompanying drawings. 
         FIG. 1  is an example block diagram of a system for equalizing a second signal to a first in the frequency domain according to the prior art. 
         FIG. 2  is an example block diagram of an adaptive filter for equalizing a second signal to a first signal in the time domain in which the adaptive filter is based on the magnitude response of the first and second signals. 
         FIG. 3  is an example flow chart of illustrative steps for equalizing a second signal to a first signal in the time domain with an adaptive filter based on the magnitude response of the first and second signals according to one embodiment of the disclosure. 
         FIG. 4  is an example flow chart of illustrative steps for equalizing a second signal to a first signal in the time domain with an adaptive filter based on a magnitude response of the first and second signals that is calculated in the frequency domain according to one embodiment of the disclosure. 
         FIG. 5  is an example block diagram of an adaptive filter for equalizing a second signal to a first signal in the time domain with an adaptive filter based on a magnitude response of the first and second signals that is calculated in the frequency domain according to one embodiment of the disclosure. 
         FIG. 6A  is an example graph illustrating the magnitude response of two microphones without equalization according to one embodiment of the disclosure. 
         FIG. 6B  is an example graph illustrating the magnitude response of two microphones after applying the magnitude response equalization techniques according to one embodiment of the disclosure. 
         FIG. 7  is an example flow chart of illustrative steps for equalizing a second signal to a first signal in the time domain with an adaptive filter based on a magnitude response of the first and second signals that is calculated in the time domain using auto-regressive modeling according to one embodiment of the disclosure. 
         FIG. 8  is an example block diagram of an adaptive filter for equalizing a second signal to a first signal in the time domain in which the adaptive filter is based on a magnitude response of the first and second signals that is calculated in the time domain using auto-regressive modeling according to one embodiment of the disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     An example of the inconsistencies and variations in the magnitude response of different microphones in a multiple microphone system that can be addressed with embodiments of this disclosure are shown in  FIG. 6A . The graph of  FIG. 6A  illustrates a magnitude response of two microphones to a control signal in lines  602  and  604 . Due to microphone mismatches that may arise, for example, during manufacturing, the microphones respond differently to a stimulus at each frequency. Equalizing one microphone&#39;s response to the other microphone&#39;s response may improve the processing of audio captured by the microphones, such as user speech. In some embodiments, a time domain impulse response filter may be applied during equalization of the signals. 
     Referring now to  FIG. 2 , one technique for equalizing magnitude response of two microphones is shown.  FIG. 2  shows such an example system  200  for implementing magnitude response equalization with an adaptive filter according to one embodiment of the disclosure. Input signals x 1 [n] and x 2 [n], such as time domain audio signals from a first and second microphone, are received at input nodes  211  and  212  of the signal processing system  200 . The signals x 1 [n] and x 2 [n] are provided to processing blocks  201  and  202 , which calculate a magnitude response for each time domain signal. The calculated magnitude responses  [n] and  [n] are then used in processing block  203  to calculate a constrained time domain impulse response filter  204 . The constrained time domain impulse response filter, h[n], is then applied to one of the time domain input signals by filter  204  to equalize the first signal x 1 [n] from a first sensor to the second signal x 2 [n] from a second sensor. In one embodiment, a delay block  205  may be inserted after a magnitude response calculation, such as that of processing block  201 , to compensate for delay introduced by filter  204 . 
     Although the signals x 1 [n] and x 2 [n] are described in certain embodiments as being microphone signals, such as those received from digital microelectromechanical systems (MEMS) microphones, any sensor signals may be processed with the systems and methods described herein. The input signals x 1 [n] and x 2 [n] may be digital signals in a time domain representation. Input signals x 1 [n] and x 2 [n] may be received from memory, buffers, or directly from analog-to-digital converters (ADCs) that are coupled to the sensors or microphones. 
     The magnitude response equalization of  FIG. 2  may provide better matching of unmatched microphones because equalization of the second microphone signal to the first signal is performed using a filter in the time domain based on the magnitude response of both microphones. This matching reduces error introduced in prior art systems where equalization of the second microphone is traditionally performed in the frequency domain and then the equalized second microphone signal is transformed from the frequency domain to the time domain.  FIG. 3  is an example signal processing flow for matching magnitude response in the time domain according to one embodiment of the disclosure. Time domain input signals x 1 [n] and x 2 [n] are received at blocks  301  and  302 , respectively, from input nodes. The magnitude response for each of signals x 1 [n] and x 2 [n] is calculated at blocks  303  and  304 , respectively. The magnitude responses of each signal can be estimated in either the time domain or in the frequency domain or a combination of the two. After calculating a magnitude response at blocks  303  and  304 , a time domain impulse response based on the calculated magnitude responses is calculated at block  305 . Because the time domain impulse response might include some phase distortion, the time domain impulse response may be constrained in block  306 . The constrained time domain impulse response is then applied, at block  307 , to one of the input signals, e.g., x 2 [n], to filter the signal and equalize the microphone response of the microphone receiving signal x 2 [n] to the microphone receiving signal x 1 [n]. 
     The constraining of the time domain impulse response results in a minimal or zero introduction of phase distortion to the signals x 1 [n] or x 2 [n]. Beamforming, and other signal processing techniques, calculate parameters based on the time difference of arrival of signals received at the microphones. This time difference of arrival information can be altered if phase information of the microphone signals is distorted by signal processing techniques. By constraining the impulse response, the phase distortion may be reduced or eliminated such that no noticeable effect on the later signal processing occurs. For example, beamforming relies on phase difference information between the microphone signals x 1 [n] and x 2 [n] to form a beam or a null in a particular direction. Constraining the response at block  307  allows the beam forming or null forming to operate with reduced error. 
     Signals used to create the magnitude response equalization filter, e.g., the filter at block  307  of  FIG. 3  and filter h[n] of the processing block  204  of  FIG. 2 , can include any signal. In some embodiments, the signal can be processed to create a uniform magnitude across desired frequency ranges, e.g., white noise. However, the magnitude response equalization may be applied at any time with any input signal and does not require a control signal with a uniform magnitude response across frequency ranges. 
     In some embodiments, the magnitude response equalization applied in creating the adaptive filter may be calculated using frequency domain representations of the signals.  FIG. 4  is an example signal processing flow for matching magnitude response in the time domain according to one embodiment of the disclosure in which the adaptive filter is based on the magnitude response of the signals in the frequency domain according to one embodiment of the disclosure. In the example flow of  FIG. 4 , at least two signals from at least two separate sensors are received at blocks  401  and  402 . In some embodiments, the two signals x 1 [n] and x 2 [n] are received in the time domain from a first and second sensor, respectively. Input signals x 1 [n] and x 2 [n] are then converted into the frequency domain at blocks  403  and  404 , respectively. The frequency domain representations of signals x 1 [n] and x 2 [n] are shown as frequency domain representations X 1 (z) and X 2 (z), respectively, at blocks  403  and  405 , but other frequency domain representations may be used in some embodiments. The magnitude response difference between the frequency domain representations X 1 (z) and X 2 (z) is calculated at block  405 . The magnitude response difference includes coefficients that represent the difference in magnitude response for sensor  1  and sensor  2  at several frequencies. The magnitude response difference is then converted into a time domain impulse response filter, h[n], at block  406 . In some embodiments, the filter h[n] is an adaptive filter. In some embodiments, the time domain impulse response filter, h[n], is constrained at block  407  to have a linear phase to prevent phase distortions when applying the filter h[n] to an input signal. The filter h[n] is then applied to one of the input signals, e.g., signal x 2 [n], at block  408 . 
     An adaptive filter may be calculated after frequency domain conversion of the signals as shown in the system of  FIG. 5 . The system of  FIG. 5  receives input signals x 1 [n] and x 2 [n] at nodes  501  and  502  from a first and second sensor, such as two microphones. Nodes  501  and  502  are coupled to respective processing blocks  503  and  504 , where the time domain signals x 1 [n] and x 2 [n] may be buffered, windowed, and/or overlapped. Processing blocks  503  and  504  are coupled to respective Fast Fourier Transform (FFT) processing blocks  505  and  506  where input signals x 1 [n] and x 2 [n] are transformed into the frequency domain. FFT processing block  505  is coupled to magnitude smoothing blocks  507  and  509 , and FFT processing block  506  is coupled to magnitude smoothing blocks  508  and  510 . The magnitude smoothing blocks may estimate the magnitude spectral density (MSD) using any one of the following methods: mean squared displacement (shown in processing blocks  509  and  510 ), the Cepstrum method, running average filtering. Savitzky-Golay smoothing, or other smoothing algorithms. Magnitude smoothing blocks  507 - 510  may perform magnitude smoothing with software or in hardware. Magnitude smoothing in hardware components may be accomplished with, for example, a low-pass or bandpass filter. 
     After this processing in the frequency domain, the signals may be converted back to time domain and used to generate coefficients for adaptive filters blocks  514  and  515 . Magnitude smoothing blocks  507  and  509  are thus coupled to Inverse-Fast Fourier Transform (I-FFT) block  511  and magnitude smoothing blocks  508  and  510  are coupled to I-FFT block  512 . The I-FFT blocks  511  and  512  produce signals {circumflex over (x)} 1 [n] and {circumflex over (x)} 2 [n], respectively, which are time domain representations of the smoothed magnitude spectrums of the microphone signals x 1 [n] and x 2 [n], respectively. I-FFT block  511  is coupled to an error signal processing block  513 , which is coupled to adaptive filter  514 . The adaptive filter  514  is also coupled to I-FFT processing block  512  to receive {circumflex over (x)} 2 [n]. The adaptive filter  514  produces FIR coefficients for the filter h[n] and may be further coupled to the error signal processing block  513  to create a feedback loop where filter h[n] is an input to the error signal processing block  513 . The error signal feedback to the adaptive filter  514  refines the FIR coefficients for the filter h[n] of the adaptive filter to obtain convergence of {circumflex over (x)} 1 [n] and {circumflex over (x)} 2 [n]. The same coefficients can be applied by adaptive filter  515 , which applies the filter to one of the time domain signals x 1 [n] and x 2 [n]. 
     In some embodiments, I-FFT processing block  511  is further coupled to a delay block  518  between the I-FFT block  511  and error signal processing block  513  that imposes a delay, e.g., a simple delay, λ, created by the filter h[n] such that {circumflex over (x)} 1 [n−λ] is the output of delay block  518  and {circumflex over (x)} 1 [n−λ] is synchronized with the {circumflex over (x)} 2 [n] that has passed through adaptive filter  514  when the error signal is calculated in error signal processing block  513 . 
     Referring back to processing blocks  503  and  504 , the blocks  503  and  504  may process the input signals by buffering, overlapping, and/or windowing the signals and then converting to the frequency domain based on the following equation: 
                   X   i     ⁡     [     l   ,   m     ]       =       ∑     n   =   0       N   -   1       ⁢           ⁢       w   ⁡     [   n   ]       ⁢       x   i     ⁡     [     n   ,   m     ]       ⁢     e         -   j     ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢   nl     N             ,     
     ⁢     i   =   1     ,   2   ,     l   =   0     ,   1   ,   …   ⁢           ,     N   -   1     ,         
where w[n] is the windowing function, x i [n,m] is the buffered and overlapped input signal corresponding to the mth superframe, N is the FFT size that can be changed through a tunable parameter, and l is the frequency bin index. The overlap may be fixed at 50%, and the Kaiser-Bessel derived window may be used in this analysis stage. The performance of the magnitude response equalization systems and methods as a whole are not limited by the window function. In some embodiments, a window other than a rectangular window may be applied.
 
     Referring now to processing blocks  507 ,  508 ,  509 , and  510 , the magnitude spectrum may be computed from the complex frequency spectrum and smoothed using the first order exponential averaging filter based on the following equation:
 
 M   i [ l,m ] =αM   i [ l,m− 1]+(1−α)| X   i [ l,m ]|,
 
where α is a smoothing parameter that can be changed by a user or an algorithm executing on a processor.
 
     The smoothed magnitude spectrum may then be transformed to the time domain using the inverse Fourier transform in blocks  511  and  512  based on the following equation: 
     
       
         
           
             
               
                 
                   
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     The output signal {circumflex over (x)}[n] can be interpreted by assuming that the input signal x i [n] is obtained by filtering a white noise signal by a coloring filter g i [n]. 
     For a wide sense stationary system (WSS),
 
 P   x     i   ( f )=| G   i ( f )| 2   W ( f )
 
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     Thus the signal {circumflex over (x)}i[n] contains only the magnitude response information of the coloring filter g i [n]. The goal of the MRE system and methods is to estimate the magnitude response of the coloring filters and design an equalization filter that matches the magnitude response of one of the coloring filters to the other. The magnitude response of this equalization filter can be: 
     
       
         
           
             
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     The magnitude difference compensation can be implemented in the frequency domain by multiplying the complex spectrum of one of the microphone signals by a real gain function, as is done in the prior art. However, this scaling in frequency domain can introduce artifacts in the synthesized time domain signal. The embodiments described herein instead perform equalization through a time domain filter, e.g., an FIR filter. The filter coefficients are estimated through an adaptive filter that operates on the time domain representation of the smoothed magnitude spectrum of the microphone signals. In some embodiments, the magnitude response equalization block may equalize only for magnitude response differences. Therefore, the coefficients may be updated in such a manner in which the phase response of the filter is constrained to be linear. This linear phase response can translate to introduction of a simple delay at the equalized output. The reference to the adaptive filter is defined as:
 
 x   k =[ {circumflex over (x)}   2 [ k ] {circumflex over (x)}   2 [ k− 1] . . .  {circumflex over (x)}   2 [ k−L+ 1]] T ,
 
where L is the number of filter coefficients that can be tuned through an input parameter. The error signal is then given by the following equation:
 
 e [ k ] ={circumflex over (x)}   1 [ k −λ] −h   k   T   x   k ,
 
where {circumflex over (x)} 1 [k−λ] is the delayed version of the signal whose magnitude spectrum must be matched by the reference signal filtered by the filter coefficients h k . The filter coefficients for an unconstrained adaptive filter may be obtained using the normalized least mean squares (NLMS) recursive update equation as:
 
                 h     k   +   1       =       h   k     +       μ         x   k   T     ⁢     x   k       +   δ       ⁢     e   ⁡     [   k   ]       ⁢     x   k           ,         
where δ is a small regularization factor to prevent division by zero. The linear phase constrained adaptive filter update equation may be obtained by modifying the above equation through exploiting the coefficient symmetry properties of a linear phase FIR filter. The moving average form of a FIR filter may be given by the following equation:
 
 ŷ [ k ] =h   k [0] {circumflex over (x)}   2 [ k ] +h   k [1] {circumflex over (x)}   2 [ k− 1]+ h   k [2] {circumflex over (x)}   2 [ k− 2]+ . . . + h   k [ L− 1] {circumflex over (x)}   2 [ k−L +1].
 
     For a Type I linear phase FIR system, the coefficients may be constrained to be even symmetric and odd length as defined in the following equation:
 
 h [ n ] =h [ L− 1 −n ], 0≦ n≦L− 1,
 
and the delay introduced by this filter may be (L−1)/2 samples. The output of this filter can be defined as
 
     
       
         
           
             
               
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                         x 
                         ^ 
                       
                       2 
                     
                     ⁡ 
                     
                       [ 
                       
                         k 
                         - 
                         
                           
                             L 
                             - 
                             1 
                           
                           2 
                         
                       
                       ] 
                     
                   
                   . 
                 
               
             
           
         
       
     
     Thus, by rearranging the reference buffer, the linear phase FIR filter coefficients can be estimated using the standard NLMS update equation. Specifically, the reference vector and the coefficient vector may be reduced to: 
     
       
         
           
             
               
                 x 
                 k 
                 
                   ( 
                   lp 
                   ) 
                 
               
               = 
               
                 
                   [ 
                   
                     
                       { 
                       
                         
                           
                             
                               x 
                               ^ 
                             
                             2 
                           
                           ⁡ 
                           
                             [ 
                             k 
                             ] 
                           
                         
                         + 
                         
                           
                             
                               x 
                               ^ 
                             
                             2 
                           
                           ⁡ 
                           
                             [ 
                             
                               k 
                               - 
                               L 
                               + 
                               1 
                             
                             ] 
                           
                         
                       
                       } 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             
                               x 
                               ^ 
                             
                             2 
                           
                           ⁡ 
                           
                             [ 
                             
                               k 
                               - 
                               1 
                             
                             ] 
                           
                         
                         + 
                         
                           
                             
                               x 
                               ^ 
                             
                             2 
                           
                           ⁡ 
                           
                             [ 
                             
                               k 
                               - 
                               L 
                               + 
                               2 
                             
                             ] 
                           
                         
                       
                       } 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     … 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           x 
                           ^ 
                         
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           k 
                           - 
                           
                             
                               L 
                               - 
                               1 
                             
                             2 
                           
                         
                         ] 
                       
                     
                   
                   ] 
                 
                 T 
               
             
             , 
             
               
 
             
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 h 
                 k 
                 
                   ( 
                   lp 
                   ) 
                 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               h 
                               k 
                             
                             ⁡ 
                             
                               [ 
                               0 
                               ] 
                             
                           
                         
                         
                           
                             
                               h 
                               k 
                             
                             ⁡ 
                             
                               [ 
                               1 
                               ] 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               h 
                               k 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   L 
                                   - 
                                   1 
                                 
                                 2 
                               
                               ] 
                             
                           
                         
                       
                     
                     ] 
                   
                   T 
                 
                 . 
               
             
           
         
       
     
     The number of unique coefficients in a Type I linear phase filter may be ((L−1)/2+1). In some embodiments, only these unique coefficients may be estimated. The NLMS update equation for a linear phase constrained FIR filter can be modified as shown in the following equation: 
     
       
         
           
             
               h 
               
                 k 
                 + 
                 1 
               
               
                 ( 
                 lp 
                 ) 
               
             
             = 
             
               
                 h 
                 k 
                 
                   ( 
                   lp 
                   ) 
                 
               
               + 
               
                 
                   μ 
                   
                     
                       
                         x 
                         k 
                         
                           
                             ( 
                             lp 
                             ) 
                           
                           ⁢ 
                           T 
                         
                       
                       ⁢ 
                       
                         x 
                         k 
                         
                           ( 
                           lp 
                           ) 
                         
                       
                     
                     + 
                     δ 
                   
                 
                 ⁢ 
                 
                   e 
                   ⁡ 
                   
                     [ 
                     k 
                     ] 
                   
                 
                 ⁢ 
                 
                   
                     x 
                     k 
                     
                       ( 
                       lp 
                       ) 
                     
                   
                   . 
                 
               
             
           
         
       
     
     The delay λ may be set to (L−1)/2 samples to derive the error signal. The adaptation rate may then be tuned through a tunable parameter, selected by a user or determined by a processor. The auto-correlation of the signal {circumflex over (x)} i [n] may be the same as the input signal x i [n], as shown in the following equation:
 
 r   x     i     x     i   [ p ] =r   {circumflex over (x)}     i     {circumflex over (x)}     i   [ p ],
 
where p is the auto-correlation lag index. This relationship means the convergence properties of the adaptive filter that is implemented based on the signals {circumflex over (x)}[n] may be governed by the auto-correlation properties of the original input signals x i [n].
 
     When the equalization filter coefficients is estimated from the time domain equivalent of the magnitude spectrum, the filter may be separately applied on the raw input signal x 2 [n]. Specifically, the equalized output may be defined by the following equation: 
     
       
         
           
             
               
                 y 
                 2 
               
               ⁡ 
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     0 
                   
                   
                     
                       
                         L 
                         - 
                         1 
                       
                       2 
                     
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       h 
                       k 
                       
                         ( 
                         lp 
                         ) 
                       
                     
                     ⁡ 
                     
                       [ 
                       i 
                       ] 
                     
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           x 
                           2 
                         
                         ⁡ 
                         
                           [ 
                           
                             k 
                             - 
                             i 
                           
                           ] 
                         
                       
                       + 
                       
                         
                           x 
                           2 
                         
                         ⁡ 
                         
                           [ 
                           
                             k 
                             - 
                             L 
                             + 
                             1 
                             + 
                             i 
                           
                           ] 
                         
                       
                     
                     } 
                   
                 
               
               + 
               
                 
                   
                     h 
                     k 
                     
                       ( 
                       lp 
                       ) 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         L 
                         - 
                         1 
                       
                       2 
                     
                     ] 
                   
                 
                 ⁢ 
                 
                   
                     
                       x 
                       2 
                     
                     ⁡ 
                     
                       [ 
                       
                         k 
                         - 
                         
                           
                             L 
                             - 
                             1 
                           
                           2 
                         
                       
                       ] 
                     
                   
                   . 
                 
               
             
           
         
       
     
     The unequalized input may be delayed to compensate for the delay introduced by the linear phase FIR filter given by the following equation: 
     
       
         
           
             
               
                 y 
                 1 
               
               ⁡ 
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                   x 
                   1 
                 
                 ⁡ 
                 
                   [ 
                   
                     k 
                     - 
                     
                       
                         L 
                         - 
                         1 
                       
                       2 
                     
                   
                   ] 
                 
               
               . 
             
           
         
       
     
     The output of delay block  518  may be y 1 [n], and the output of adaptive filter  514  may be y 2 [n]. The signals y 1 [n] and y 2 [n] may be further filtered for beamforming applications (e.g., beamforming or spatial filtering). For example, beamforming with y 1 [n] and y 2 [n] may include filtering the signals x 1 [n] and x 2 [n]. Filtering the signals x 1 [n] and x 2 [n] to alter the phase or magnitude of at least one of the signals x 1 [n] and x 2 [n] may be used to amplify or nullify signals within the signals x 1 [n] and x 2 [n]. In some embodiments, beamform filtering using y 1 [n] and y 2 [n] may be used to detect the location of the signal source by calculating, for example, magnitude and phase shift differences between y 1 [n] and y 2 [n] caused, at least in part, by the spatial relationship between the sensors that produce signals x 1 [n] and x 2 [n]. 
       FIG. 6A  illustrates the spectral plot of two input sensors, labelled Mic 1  and Mic 2 . This figure highlights the problem addressed by the systems and methods disclosed herein. Signal comparisons between the data of two sensors with different spectral responses, such as Mic 1  and Mic 2  in  FIG. 6A , must be equalized to perform further processing on the signals, such as beam forming.  FIG. 6B  illustrates the spectral plot of the raw data of Mic 1  plotted with the spectral plot of the filtered raw data from Mic 2 , where the Mic 2  raw data has been filtered using one embodiment of the systems and methods described herein. As shown in  FIG. 6B , the filtered Mic 2  signal  606  is equalized across the relevant frequency spectrum to match the magnitude response of Mic 1  signal  608 . Comparative signal analysis between Mic 1  and Mic 2  may be enhanced by the embodiments herein by removing signal processing errors that would otherwise be caused by the inherent or environmental differences in the first and second microphones producing signals Mic 1  and Mic 2 . 
     In some environments, the input signal consists of noise and speech and the relative magnitude spectrum of the speech and the noise can be very different. In such scenarios, matching the magnitude spectrum at all times may cause undesirable results. Accordingly, some embodiments further include an adaptive enable input signal that controls the time instances in which the smoothed magnitude spectrum estimation is enabled. The adaptive filter may be updated only when the adaptive enable input signal control signal is true because the input signals {circumflex over (x)}[n] changes only when the smoothed magnitude spectrum estimation is enabled. 
     In some embodiments, the magnitude response used in creating the adaptive filter for equalizing signals in the time domain may be calculated using statistical approximations of the time domain representations of the signals.  FIG. 7  is an example signal processing flow for matching magnitude response in the time domain according to one embodiment of the disclosure. The adaptive filter may be, for example, based on the magnitude response of the signals found using auto-regressive techniques and a Padé approximation.  FIG. 7  illustrates an embodiment of the methods of the present invention where input signals x 1 [n] and x 2 [n] are received at blocks  701  and  702 , respectively. Block  703  calculates an estimate of Auto-Regressive (AR) model parameters of signal x 1 [n], and block  704  calculates an estimate of the AR model parameters of signal x 2 [n]. Next, the Auto-Regressive Moving Average (ARMA) model parameters are calculated at block  705  to correspond to the magnitude response difference between signals x 1 [n] and x 2 [n]. The ARMA model parameters may then be used at block  706  to estimate a time domain impulse response that corresponds to a magnitude response difference between input signals x 1 [n] and x 2 [n]. The estimated time domain impulse response may be constrained to create a time domain impulse response filter with a linear phase. One of the signals, such as x 2 [n], is then filtered at block  708  using the constrained time domain impulse response calculated at blocks  706  and  707 . Additionally, the unfiltered signal, x 1 [n], may be delayed in some embodiments to compensate for delay caused by the time domain impulse response filter applied at block  708 . 
     An adaptive filter may be calculated using time domain approximations as shown in the system of  FIG. 8 . The system of  FIG. 8  receives input signals x 1 [n] and x 2 [n] at nodes  801  and  802  from a first and second sensor, respectively. Nodes  801  and  802  are coupled to respective processing blocks  803  and  804 . Processing blocks  803  and  805  calculate Linear Prediction Coefficients (LPC) for input signal x 1 [n], and processing blocks  804  and  806  calculate LPCs for input signal x 2 [n]. In some embodiments, the LPCs are estimated using auto-regressive (AR) model parameters. 
     Processing block  803  is coupled to processing block  805 , and processing block  804  is coupled to processing block  806 . Processing blocks  805  and  806  receive the LPCs for input signals x 1 [n] and x 2 [n], respectively, and calculate a magnitude response difference between using the auto-regressive, moving average (ARMA) system coefficients of input signals x 1 [n] and x 2 [n]. In some embodiments, processing blocks  805  and  806  then perform a Padé approximation using the ARMA coefficients to approximate a time domain impulse response that corresponds to the magnitude difference between input signals x 1 [n] and x 2 [n]. In some embodiments, processing block  805  is further coupled to processing block  807 , and processing block  806  is further coupled to processing block  808 . Processing blocks  807  and  808  perform smoothing similar to that described with respect to blocks  507 - 510  of  FIG. 5 . In some embodiments, processing blocks  807  and  808  may also constrain the estimated time domain impulse response calculated in processing blocks  805  and  806  such that the time domain impulse response has a linear phase response. The constraining of the estimated time domain impulse response may be performed as described below by, for example, applying a filtering delay. 
     Processing block  807  is coupled to error signal processing block  809  where an error signal is calculated. Processing block  808  is coupled to adaptive filter  810  where the time domain impulse response coefficients are used to create an adaptive filter. The adaptive filter  810  is further coupled to error signal processing block  809  through a feedback loop. The adaptive filter  810  creates filter h[n] that is applied in processing block  811  to the original input signal x 2 [n]. Some embodiments may include a delay block  812  coupled between processing block  807  and error signal processing block  809  to calculate the delay of the time domain impulse response. The calculated delay from delay block  812  may be applied to the unfiltered signal (not shown in  FIG. 8 ), x 1 [n], to keep input signals x 1 [n] and x 2 [n] synchronized after x 2 [n] is filtered by adaptive filter h[n] in processing block  811 . 
     For example, in one embodiment, processing blocks  803  and  804  may calculate linear prediction coefficients (LPCs) based on the following equation:
 
 {circumflex over (x)} ( n )=α 1   (i)   x   i ( n− 1)+α 2   (i)   x   i ( n− 2)+ . . . α L   (i)   x   i ( n−L ),
 
with
 
 {circumflex over (X)}   i ( z )=(α 1   (i)   z   −1 + . . . +α L   (i)   z   −L ) X   i ( z ),
 
and
 
 {circumflex over (X)}   i ( z )= A   i ( z ) X   i ( z ).
 
     The parameters α 1   (i)  . . . α L   (i)  may be estimated using the Levinson&#39;s-Durbin algorithm through estimating the auto-correlation sequence based on the following equation: 
     
       
         
           
             
               
                 
                   γ 
                   
                     
                       x 
                       i 
                     
                     ⁢ 
                     
                       x 
                       i 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   m 
                   ) 
                 
               
               = 
               
                 
                   1 
                   N 
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         x 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         n 
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         x 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         
                           n 
                           - 
                           m 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     m 
                     ⁢ 
                     
                       : 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
             
             , 
             
               … 
               ⁢ 
               
                   
               
               ⁢ 
               
                 L 
                 . 
               
             
           
         
       
     
     In some embodiments, the magnitude response difference calculated in processing blocks  805  and  806  may be defined as: 
     
       
         
           
             
               H 
               ⁡ 
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 
                   
                     A 
                     2 
                   
                   ⁡ 
                   
                     ( 
                     z 
                     ) 
                   
                 
                 
                   
                     A 
                     1 
                   
                   ⁡ 
                   
                     ( 
                     z 
                     ) 
                   
                 
               
               . 
             
           
         
       
     
     In some embodiments that calculated the LPC coefficients described above, the adaptive filter may be defined by the following equation: 
     
       
         
           
             
               H 
               ⁡ 
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 1 
                 + 
                 
                   
                     a 
                     1 
                     
                       ( 
                       2 
                       ) 
                     
                   
                   ⁢ 
                   
                     z 
                     
                       - 
                       1 
                     
                   
                 
                 + 
                 
                   
                     a 
                     2 
                     
                       ( 
                       2 
                       ) 
                     
                   
                   ⁢ 
                   
                     z 
                     
                       - 
                       2 
                     
                   
                 
                 + 
                 
                   … 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     a 
                     L 
                     
                       ( 
                       2 
                       ) 
                     
                   
                   ⁢ 
                   
                     z 
                     
                       - 
                       L 
                     
                   
                 
               
               
                 1 
                 + 
                 
                   
                     a 
                     1 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   ⁢ 
                   
                     z 
                     
                       - 
                       1 
                     
                   
                 
                 + 
                 
                   
                     a 
                     2 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   ⁢ 
                   
                     z 
                     
                       - 
                       2 
                     
                   
                 
                 + 
                 
                   … 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     a 
                     L 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   ⁢ 
                   
                     z 
                     
                       - 
                       L 
                     
                   
                 
               
             
           
         
       
     
     When processing blocks  805  and  806  apply a Pade approximation, the auto-regressive, moving average system (ARMA), represented by H(z), and the moving average, represented by, Ĥ(z) may be approximately equal as defined in the following equation:
 
 H ( z )≈ Ĥ ( z ),
 
The approximation may then be expanded and represented as defined by the following equation:
 
                   1   +       a   1     (   2   )       ⁢     z     -   1         +       a   2     (   2   )       ⁢     z     -   2         +     …   ⁢           ⁢     a   L     (   2   )       ⁢     z     -   L             1   +       a   1     (   1   )       ⁢     z     -   1         +       a   2     (   1   )       ⁢     z     -   2         +     …   ⁢           ⁢     a   L     (   1   )       ⁢     z     -   L             =       b   0     +       b   1     ⁢     z     -   1         +       b   2     ⁢     z     -   1         +     …   ⁢           ⁢     b   M     ⁢     z     -   M             ,         
when M&gt;&gt;L. The coefficients b 0  . . . b M  may then be solved by carrying the denominator on the left over to the right and equating the coefficients to create a linear system of equations.
 
     The coefficients may be constrained to the linear phase, such as by applying a filtering delay. For example the approximation may then be expanded and represented by the following equation: 
                 1   +       a   1     (   2   )       ⁢     z     -   1         +       a   2     (   2   )       ⁢     z     -   2         +     …   ⁢           ⁢     a   L     (   2   )       ⁢     z     -   L             1   +       a   1     (   1   )       ⁢     z     -   1         +       a   2     (   1   )       ⁢     z     -   2         +     …   ⁢           ⁢     a   L     (   1   )       ⁢     z     -   L             =         b   M     ⁢     z       -   D     +   M         +     …   ⁢           ⁢     b   2     ⁢     z       -   D     +   2         +       b   1     ⁢     z       -   D     +   1         +       b   0     ⁢     z     -   D         +       b   1     ⁢     z       -   D     -   1         +       b   2     ⁢     z       -   D     -   2         +     …   ⁢           ⁢     b   M     ⁢       z       -   D     -   M       .               
A set of linear equations may similarly be formulated from this equation to equate polynomials, as discussed above to create set of constrained coefficients to be used in filter h[n] in processing blocks  810  and  811 . The linear system of equations can then be solved to obtain the coefficients b 0 , . . . , b h .
 
     In some embodiments, the error signal calculated in error signal processing block  809  may be calculated based on the following equation:
 
 e ( n )= x   i ( n )− {circumflex over (x)}   i ( n ).
 
     In some environments, the input signal consists of noise and speech and the relative magnitude spectrum of the speech and the noise can be very different. In such scenarios, matching the magnitude spectrum at all times may cause undesirable results. Accordingly, some embodiments further include an adaptive enable input signal that controls the time instances in which any of the magnitude equalization processing blocks  803 - 811  are enabled. The adaptive filter h[n] in processing block  811  may be updated only when the adaptive enable input signal control signal is true because the input signals X[n] changes only when the magnitude equalization processing blocks are enabled. 
     The time domain adaptive filter and other components and methods described above may be implemented in an audio controller of a device, such as a mobile device, to process signals received from near and/or far microphones of the mobile device. The mobile device may be, for example, a mobile phone, a tablet computer, a laptop computer, or a wireless earpiece. A processor of the mobile device, such as the device&#39;s application processor, may implement a processing technique, such as those described above with reference to  FIG. 2 ,  FIG. 3 ,  FIG. 4 ,  FIG. 5 ,  FIG. 7 , and/or  FIG. 8 , or other circuitry for processing. Alternatively, the mobile device may include specific hardware for performing these functions, such as a digital signal processor (DSP). The controller may include the processor, digital signal processor (DSP), and/or other circuitry related to signal processing. In some embodiments, the controller may be integrated into an audio coder/decoder (CODEC) chip along with other audio processing circuitry, such as adaptive echo cancellation (AEC), adaptive noise cancellation (ANC), pulse width modulators (PWM), and/or audio amplifiers. 
     The schematic flow chart diagrams of  FIG. 3 ,  FIG. 4 ,  FIG. 5 ,  FIG. 7 , and  FIG. 8  are generally set forth as a logical flow chart diagram. As such, the depicted order and labeled steps are indicative of aspects of the disclosed method. Other steps and methods may be conceived that are equivalent in function, logic, or effect to one or more steps, or portions thereof, of the illustrated method. Additionally, the format and symbols employed are provided to explain the logical steps of the method and are understood not to limit the scope of the method. Although various arrow types and line types may be employed in the flow chart diagram, they are understood not to limit the scope of the corresponding method. Indeed, some arrows or other connectors may be used to indicate only the logical flow of the method. For instance, an arrow may indicate a waiting or monitoring period of unspecified duration between enumerated steps of the depicted method. Additionally, the order in which a particular method occurs may or may not strictly adhere to the order of the corresponding steps shown. 
     If implemented in firmware and/or software, functions described above may be stored as one or more instructions or code on a computer-readable medium. Examples include non-transitory computer-readable media encoded with a data structure and computer-readable media encoded with a computer program. Computer-readable media includes physical computer storage media. A storage medium may be any available medium that can be accessed by a computer. By way of example, and not limitation, such computer-readable media can include random access memory (RAM), read-only memory (ROM), electrically-erasable programmable read-only memory (EEPROM), compact disc read-only memory (CD-ROM) or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium that can be used to store desired program code in the form of instructions or data structures and that can be accessed by a computer. Disk and disc includes compact discs (CD), laser discs, optical discs, digital versatile discs (DVD), floppy disks and Blu-ray discs. Generally, disks reproduce data magnetically, and discs reproduce data optically. Combinations of the above should also be included within the scope of computer-readable media. 
     In addition to storage on computer readable medium, instructions and/or data may be provided as signals on transmission media included in a communication apparatus. For example, a communication apparatus may include a transceiver having signals indicative of instructions and data. The instructions and data are configured to cause one or more processors to implement the functions outlined in the claims. 
     Although the present disclosure and certain representative advantages have been described in detail, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the disclosure as defined by the appended claims. Moreover, the scope of the present application is not intended to be limited to the particular embodiments of the process, machine, manufacture, composition of matter, means, methods and steps described in the specification. As one of ordinary skill in the art will readily appreciate from the present disclosure, processes, machines, manufacture, compositions of matter, means, methods, or steps, presently existing or later to be developed that perform substantially the same function or achieve substantially the same result as the corresponding embodiments described herein may be utilized. Accordingly, the appended claims are intended to include within their scope such processes, machines, manufacture, compositions of matter, means, methods, or steps.