Patent Publication Number: US-7593851-B2

Title: Precision piecewise polynomial approximation for Ephraim-Malah filter

Description:
FIELD 
   Embodiments of the invention relate to the field of speech enhancement; and more specifically, to precision piecewise polynomial approximation for Ephraim-Malah filter. 
   BACKGROUND 
   The problem of enhancing speech degraded by uncorrelated additive noise has recently received much attention. This is due to many potential applications a successful speech enhancement system can have, and because of the available technology which enables the implementation of such intricate algorithms. 
   It has been reported that the noise suppression rule proposed by Ephraim and Malah makes it possible to obtain a significant noise reduction, which leads to an Ephraim-Malah filter weights formula. In one approach, the original Ephraim-Malah filter weights formula has been implemented in a floating-point implementation. Although such implementation provides enough data precision, it lacks efficiency in performance. In another approach, the Ephraim-Malah filter weights formula has been implemented with a fix-point implementation using a traditional curve-fit method, such as polynomial approximation with Taylor&#39;s formula. Although such implementation provides efficiency in performance, it lacks data precision. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The invention may best be understood by referring to the following description and accompanying drawings that are used to illustrate embodiments of the invention. In the drawings: 
       FIG. 1  is block diagram illustrating an exemplary embodiment of a speech enhancement system based on an Ephraim-Malah filter. 
       FIG. 2  is a chart illustrating an exemplary embodiment of a curve analysis. 
       FIG. 3  is a chart illustrating an exemplary embodiment of a curve analysis with band mapping. 
       FIG. 4  is a chart illustrating an exemplary embodiment of an error result of a polynomial approximation process. 
       FIG. 5A  is a block diagram illustrating an exemplary embodiment of a precision piecewise polynomial approximation of an Epharim-Malah filter weights formula. 
       FIG. 5B  is a block diagram illustrating an exemplary embodiment of a data format. 
       FIG. 6  is a block diagram of process logic to perform an enhanced Epharim-Malah filter weights operation. 
       FIG. 7  is a flow diagram illustrating an exemplary embodiment of a process for an enhanced Epharim-Malah filter weights operation. 
       FIG. 8  is a block diagram of an exemplary computer system which may be used to execute an enhanced Epharim-Malah filter weights operation. 
   

   DETAILED DESCRIPTION 
   Precision piecewise polynomial approximation for Ephraim-Malah filter is described herein. In the following description, numerous specific details are set forth. However, it is understood that embodiments of the invention may be practiced without these specific details. In other instances, well-known circuits, structures and techniques have not been shown in detail in order not to obscure the understanding of this description. 
   Some portions of the detailed descriptions which follow are presented in terms of algorithms and symbolic representations of operations on data bits within a computer memory. These algorithmic descriptions and representations are used by those skilled in the data processing arts to most effectively convey the substance of their work to others skilled in the art. An algorithm is here, and generally, conceived to be a self-consistent sequence of operations leading to a desired result. The operations are those requiring physical manipulations of physical quantities. Usually, though not necessarily, these quantities take the form of electrical or magnetic signals capable of being stored, transferred, combined, compared, and otherwise manipulated. It has proven convenient at times, principally for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, or the like. 
   It should be borne in mind, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels applied to these quantities. Unless specifically stated otherwise as apparent from the following discussion, it is appreciated that throughout the description, discussions utilizing terms such as “processing” or “computing” or “calculating” or “determining” or “displaying” or the like, refer to the action and processes of a computer system, or similar data processing device, that manipulates and transforms data represented as physical (e.g. electronic) quantities within the computer system&#39;s registers and memories into other data similarly represented as physical quantities within the computer system memories or registers or other such information storage, transmission or display devices. 
   Embodiments of the present invention also relate to apparatuses for performing the operations described herein. An apparatus may be specially constructed for the required purposes, or it may comprise a general purpose computer selectively activated or reconfigured by a computer program stored in the computer. Such a computer program may be stored in a computer readable storage medium, such as, but is not limited to, any type of disk including floppy disks, optical disks, CD-ROMs, and magnetic-optical disks, read-only memories (ROMs), random access memories (RAMs) such as Dynamic RAM (DRAM), EPROMs, EEPROMs, magnetic or optical cards, or any type of media suitable for storing electronic instructions, and each of the above storage components is coupled to a computer system bus. 
   The algorithms and displays presented herein are not inherently related to any particular computer or other apparatus. Various general purpose systems may be used with programs in accordance with the teachings herein, or it may prove convenient to construct more specialized apparatus to perform the methods. The structure for a variety of these systems will appear from the description below. In addition, embodiments of the present invention are not described with reference to any particular programming language. It will be appreciated that a variety of programming languages may be used to implement the teachings of the embodiments of the invention as described herein. 
   A machine-readable medium includes any mechanism for storing or transmitting information in a form readable by a machine (e.g., a computer). For example, a machine-readable medium includes read only memory (“ROM”); random access memory (“RAM”); magnetic disk storage media; optical storage media; flash memory devices; electrical, optical, acoustical or other form of propagated signals (e.g., carrier waves, infrared signals, digital signals, etc.); etc. 
     FIG. 1  is a block diagram illustrating an exemplary embodiment of an Ephraim-Malah noise suppressor which a precision piecewise polynomial approximation process may be used. In one embodiment, exemplary noise suppressor  100  includes speech data source  101 , time domain to frequency domain (T/F) transform module  102 , noise power spectrum estimation module  103 , speech power spectrum estimation module  104 , filter coefficient computing module  105 , applying filter module  106 , frequency domain to time domain (F/T) transform module  107 , and speech data sink  108 . 
   Referring to  FIG. 1 , according to one embodiment, speech data received from data source  101  may include an input block having the most recently acquired N/2 input samples ζ n  and the previous N/2 input samples ζ n−1  which make up a new input block z n , such as: 
             z   n     =     [           ζ     n   -   1                 ζ   n           ]           
When T/F transform module  102  receives speech data from data source  101 , the input block is multiplied by a square root of a window function. The window function may be constructed such that when its first half is added to its second half, all values add to one. In one embodiment, the window function is a triangular window, which may be defined as follows:
 
             w   ⁡     (   m   )       =     {             m   +   0.5       N   /   2                 for   ⁢           ⁢   m     =   0     ,   …   ⁢           ,       N   /   2     -   1                 1   -     w   ⁡     (     m   -     N   /   2       )                   for   ⁢           ⁢   m     =     N   /   2       ,   …   ⁢           ,     N   -   1                     
The discrete Fourier transform of the input may be calculated as follows:
   Z   n   =F ( z   n   ·√{square root over (w)} ) 
where · denotes point-wise multiplication and √{square root over (w)} denotes a vector containing the square root of the entries of w. F is the Fourier transform matrix with entries of:
   f ( m,n )= e   −j2ππmn/N    
where N is the size of the transform. The discrete Fourier transform can be replaced by FFT (fast Fourier transform), DCT (discrete cosine transform), or DWT (discrete wavelet transform), etc.
 
   The data in frequency domain is then transferred to noise power spectrum estimation module  103  and speech power spectrum estimation module  104 . In noise power spectrum estimation module  103 , the noisy speech magnitude-squared spectral components are averaged to provide an estimate of the noisy speech power spectrum (e.g., power spectral density or PSD). In one embodiment, the estimation may be provided as:
 
 P   n   z ( k )=β n   ·|Z   n ( k )| 2 +(1−β n )· P   n−1   z ( k )
 
wherein adaptive step size β n  is defined as:
 
β n =β min +ρ n−1   y (β max −β min )
 
where β min =0.9, β max =1.0, and ρ n−1   y  is the likelihood of speech presence in frequency bin k. Frequency bin k is an index of coefficients in vector Z n .
 
   An estimation of the clean speech power spectral components is obtained by spectral subtraction and averaging performed by speech power spectrum estimation module  104 . The estimation may be obtained by:
 
 P   n   y ( k )=α n   ·|Ŷ   n−1 ( k )| 2 +(1−α n )·ψ 0 ( P   n   z ( k )− P   n−1   v ( k ))
 
where thresholding operator ψ is defined as
 
               ψ   c     ⁡     (   x   )       =     {           c   ,           x   ≤   c               x   ,           x   &gt;   c                   
where adaptive step size α n  is defined as
 α n =α min +(1−ρ n−1   y )(α max −α min ) 
where α min =0.91, α max =0.95, and ρ n−1   y  is the likelihood of speech presence in frequency bin k. Note that the previous frame&#39;s noise power spectral component is used in this calculation. If the noise floor estimator is independent of the rest of the algorithm, it may be possible to use the current frame&#39;s noise estimate instead.
 
   One of the parameters used to compute the Ephraim-Malah suppression rule is the Wiener filter (a different noise suppression rule), which may be performed by filter coefficient module  105 . The Wiener filter weights may be defined as follows: 
               W   n   y     ⁡     (   k   )       =       ψ     W   min       ⁡     (         P   n   y     ⁡     (   k   )             P   n   y     ⁡     (   k   )       +       P     n   -   1     v     ⁡     (   k   )           )             
where W min  may be a threshold similar to the threshold defined by O. Cappe, entitled “Elimination of the Musical Noise Phenomenon with the Ephraim and Malah Noise Suppressor”, IEEE Trans. Speech and Audio Processing., Vol. 2, No. 2, April 1994, pp. 345-349. In Cappe, it was recommended that a lower limit for a priori SNR, which is defined as follows:
 
               ℜ   n   prio     ⁡     (   k   )       =         P   n   y     ⁡     (   k   )           P     n   -   1     v     ⁡     (   k   )               
where    min dB   prio =−15.0 dB may be imposed to avoid musical noise. As a result, which may be transformed to:
 
             W   min     =     1     1   +     10     -       ℜ     min   ⁢           ⁢   dB     prio     10                   
Note that if the Wiener filter is written in terms of the a priori SNR, the Wiener filter calculation may be replaced by a table lookup, which will be described in details further below, according to one embodiment. This approach is particularly useful for processors where divisional operations are expensive.
 
   A posteriori signal to noise ratio (SNR) for each frequency bin may be defined as follows: 
               ℜ   n   post     ⁡     (   k   )       =         P   n   z     ⁡     (   k   )           P     n   -   1     v     ⁡     (   k   )               
The Ephraim-Malah filter weights are given by:
 
               H   n   y     ⁡     (   k   )       =       1       ℜ   n   post     ⁡     (   k   )         ·     M   ⁡     (         W   n   y     ⁡     (   k   )       ⁢       ℜ   n   post     ⁡     (   k   )         )               
where M(·) is a function defined by:
 
   
     
       
         
           
             M 
             ⁡ 
             
               ( 
               θ 
               ) 
             
           
           = 
           
             
               1 
               2 
             
             · 
             
               
                 π 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 θ 
               
             
             · 
             
               
                 ⅇ 
                 
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                         + 
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                       ) 
                     
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   Typically, a noise power spectral estimator may be employed to calculate P n   v (k). Such estimator may be constructed similar to those defined by R. Martin, “Noise Power Spectral Density Estimation Based on Optimal Smoothing and Minimum Statistics,” IEEE  Trans. Speech and Audio , Vol. 9, No. 5, July 2001, pp. 504-512. 
   In general, the probability of speech presence is not calculated directly. Rather, it is roughly approximated by the MMSE (minimum mean-square error) (Wiener) estimator of the overall speech energy, which is defined as follows: 
   
     
       
         
           
             ρ 
             n 
             y 
           
           = 
           
             
               
                 ∑ 
                 
                   k 
                   = 
                   0 
                 
                 
                   N 
                   / 
                   2 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
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                   n 
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                 ⁡ 
                 
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                   k 
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                     0 
                   
                   
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                 ⁢ 
                 
                     
                 
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   The filter coefficients H n   v (k) may be modified to improve perceptual speech quality or reduce perceptible musical tones. For example, to efficiently handle loud, low-pass noise such as those encountered in automotive environments, low-frequency filter coefficients (e.g., below 60 Hz) may be set to zero. Thereafter, filter output may be calculated by applying filter module  106 . The filter output may be defined as follows:
 
 Ŷ   n ( k )= H   n   y ( k )· Z   n ( k )
 
   Finally, time domain filter output is obtained by an inverse FFT, an inverse DFT, or an inverse DWT, etc., to generate final output at speech data sink  108 . The time domain filter output is performed by F/T transform module  107  based on a formula similar to one defined below: 
   
     
       
         
           
             
               
                 
                   
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   As mentioned above, the original Ephraim-Malah filter weights formula includes complicated computation which some processors may not be able to offer. The original Ephraim-Malah filter weights formula is defined as follows: 
                     H   n   y     ⁡     (   k   )       =       1       ℜ   n   post     ⁡     (   k   )         ·     M   ⁡     (         W   n   y     ⁡     (   k   )       ⁢       ℜ   n   post     ⁡     (   k   )         )                 (     Eq   .           ⁢   1     )               
where, M (·) is a function defined by:
 
                   M   ⁡     (   θ   )       =       1   2     ·     πθ     ·       ⅇ     -     θ   2         ⁡     [         (     1   +   θ     )     ·       I   0     ⁡     (     θ   2     )         +     θ   ·       I   1     ⁡     (     θ   2     )           ]                 (     Eq   .           ⁢   2     )               
where I 0 (·) and I 1 (·) is order 0 and order 1 of a modified Bessel function of the first kind, which is well known in the art. Further detailed information concerning the modified Bessel function of the first kind can be found at a Web site of:
         http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html
 
W n   y (k) is the Wiener filter defined by:
       
                     W   n   y     ⁡     (   k   )       =       W   min     ⁡     (         P   n   y     ⁡     (   k   )             P   n   y     ⁡     (   k   )       +       P     n   -   1     v     ⁡     (   k   )           )               (     Eq   .           ⁢   3     )               
where W min  is a threshold similar to one defined by O. Cappe, entitled “Elimination of the Musical Noise Phenomenon with the Ephraim and Malah Noise Suppressor,”  IEEE Trans. Speech And Audio Processing , Vol. 2, No. 2, April 1994, pp. 345-349. P n   y (k) is a clean speech PSD (Power Spectral Density) estimation provided by speech power spectrum estimation module  104 . P n   v (k) is a noise PSD estimation provided by noise power spectrum estimation module  103 .
 
   The division operation in equation (1) is a bottleneck for performance of an implementation in software and hardware. 
               Since   ⁢           ⁢     1       ℜ   n   post     ⁡     (   k   )           =         W   n   y     ⁡     (   k   )             W   n   y     ⁡     (   k   )       ⁢       ℜ   n   post     ⁡     (   k   )             ,         
the new
 
Ephraim-Malah filter weights may be transformed into:
   H   n   y ( k )= W   n   y ( k )* M′ ( W   n   y ( k )   n   post ( k )) 
where M′(·) is a function defined by:
 
                     M   ′     ⁡     (   θ   )       =       1   2     ·       π   θ       ·       ⅇ     -     θ   2         ⁡     [         (     1   +   θ     )     ·       I   0     ⁡     (     θ   2     )         +     θ   ·       I   1     ⁡     (     θ   2     )           ]                 (     Eq   .           ⁢   4     )               
where I 0 (·) and I 1 (·) are order 0 and order 1 of a modified Bessel function of the first kind respectively. With the new Ephraim-Malah filter weights formula, the division operation involved in Eq. 1 may be eliminated.
 
     FIG. 2  is graph illustrating an exemplary curve of a M′(·) function. Referring to  FIG. 2 , the dynamic range of curve is large when input value approaches zero. At a point of an input value having zero, the curve approaches ∞. Thus, if M′(·) is implemented via a general piecewise polynomial approximation, the big dynamic range would make the error be substantially large and it would approach ∞ when the input value is approaching zero. Typically, the general piecewise polynomial approximation uses average length band for piecewise polynomial approximation. 
   To solve this problem, a technique for exponential increasing piecewise polynomial approximations is introduced, according to one embodiment. For a fix-point implementation, the input value of M′(·) is represented with a Q22 format. Q format is used to represent a floating-point value using fix-point values. The position of the binary point in a fixed-point number determines how to interpret the scaling of the number. When the hardware performs basic arithmetic such as addition or subtraction, the hardware uses the same logic circuits regardless of the value of the scale factor. The logic circuits have no knowledge of a binary point. They perform signed or unsigned integer arithmetic as if the binary point is at the right of b 0 , b 0  is the location of the least significant (e.g., lowest) bit. For example, according to one embodiment, a 32-bit data may be defined as data format  530  as shown in  FIG. 5B , where MSB is the most significant (e.g., highest) bit and LSB is the least significant (e.g., lowest) bit. 
   In the DSP (digital signal processing) industry, the position of the binary point in the signed and fixed-point data types is expressed in and designated by a Q format notation. This fixed-point notation takes a form of Qm.n, where:
         Q designates that a number is in Q format notation (e.g., the representation for signed fixed-point numbers).   m represents number of bits used to designate the two&#39;s complement integer portion of a number.   n represents number of bits used to designate the two&#39;s complement fractional portion of a number, or number of bits to the right of the binary point.       

   In a Q format, the most significant bit is designated as a sign bit. Representing a signed fixed-point data type in a Q format requires m+n+1 bits to account for the sign. For example, Q15 is a signed 32-bit number with n=15 bits to the right of the binary point which is defined as Q16.15. In this notation, there is (1 sign bit)+(m=16 integer bits)+(n=15 fractional bits)=32 bits total in the data type. In a Q format notation, when Q16.15 is indicated the data type fixed on 32-bit, m=32−n−1 is often implied. As a result Q15 is used to represent Q16.15 instead.
         According to one embodiment, from θ=2 7  to 2 31 , the range is divided into 24 bands, each band is defined as [2 i ,2 i+1 ),i=7 . . . 30. Each band is mapped to equal length cell to analyze the curve, as shown in  FIG. 3 . As shown in  FIG. 3 , exponential increasing piecewise approach limits the dynamic range and provides high precision for fix-point implementation. In the i th  band [2 i ,2 i+1 ), a two-order polynomial approximation may be used to calculate the output result. In one embodiment, the two-order polynomial approximation may be defined as follows:
 
 f ( x )= P 0 +P 1 *x+P 2 *x   2   (Eq. 5)
       

   In general, a fixed Q value, such as Q31, Q15, is used for fix-point implementation. To achieve a high-precision output, a dynamic Q Value of parameters is designed. Referring to Eq. 5, since P1 and P2 change greatly in different band, dynamic Q value may be designed for parameter P1 and P2 to maintain high precision. In one embodiment, the Q value of P1 is (i+5) and the Q value of P2 is (i−4), where i is an index of the corresponding band (i from 0 to 23). The representation of P0 is defined as a Q22 format for all segments. 
   In one embodiment, P0 may be defined as follows: 
   
     
       
         
             
             
             
             
             
             
           
             
                 
             
           
          
             
               P0[24] = { 
                 
                 
                 
                 
                 
             
             
                669498645, 
               473414302, 
               334764744, 
               236728959, 
               167413213, 
               118408092, 
             
             
                83768274, 
               59291231, 
               42007360, 
               29819668, 
               21249233, 
               15255427, 
             
             
                11108711, 
               8299601, 
               6470760, 
               5361107, 
               4756698, 
               4463049, 
             
             
                4325745, 
               4259445, 
               4226742, 
               4210491, 
               4202389, 
               4198345 
             
             
               }; 
             
             
                 
             
          
         
       
     
       
       
         
           In one embodiment, P1 may be defined as follows: 
         
       
     
  
   
     
       
         
             
             
             
             
             
             
           
             
                 
             
           
          
             
               P1[24] = { 
                 
                 
                 
                 
                 
             
             
                72453962, 
               51231813, 
               36225125, 
               25613282, 
               18108852, 
               12801395, 
             
             
                9047010, 
               6390220, 
               4508716, 
               3174276, 
               2225121, 
               1546422, 
             
             
                1056723, 
               698929, 
               435261, 
               245271, 
               121987, 
               56781, 
             
             
                27183, 
               13368, 
               6635, 
               3306, 
               1650, 
               824 
             
             
               }; 
             
             
                 
             
          
         
       
     
   
   In one embodiment, P2 may be defined as follows: 
   
     
       
         
             
             
             
             
             
             
           
             
                 
             
           
          
             
               P2[24] = { 
                 
                 
                 
                 
                 
             
             
                1576642499, 
               557423223, 
               197075987, 
               69674844, 
               24632335, 
               8707825, 
             
             
                3077959, 
               1087711, 
               384201, 
               135577, 
               47749, 
               16748, 
             
             
                5823, 
               1986, 
               648, 
               193, 
               49, 
               11, 
             
             
                2, 
               0, 
               0, 
               0, 
               0, 
               0 
             
             
               }; 
             
             
                 
             
          
         
       
     
   
     FIG. 4  is an error result of exponential increasing piecewise two-order polynomial approximations, according to one embodiment. Referring to  FIG. 4 , graph  402  represents the maximum absolute error of bands and graph  401  represents errors in percentage. As shown in  FIG. 4 , the maximum error percentage is less than 1%. The error of traditional curve-fit approach may reach nearly 50% when input value is approaching zero. 
   According to one embodiment, when the input value (Q22 format) of M′(·) is in a range of (2 7 ,2 31 ), M′(·) is determined by exponential increasing piecewise two-order polynomial approximations with 24 bands, as described above. When the input value (Q22 format) of M′(·) is small, such as, for example, in a [0,2 7 ) range, it is not suitable to be used in a curve-fit method because the one-order differential coefficient and the two-order differential coefficient are changed greatly at different bands. As a result, according to one embodiment, a table is used for the small input value to achieve high precision. According to one embodiment, when a threshold is set as 2 7 , a table may be designed to have 129 values. It would be appreciated that other thresholds may be defined. Higher threshold would lead to higher performance since less computation is involved. However, data table associated with the threshold may be increased and more memory is needed. Therefore, a balance of resources may be required. In one embodiment, an exemplary data table may be defined as follows: 
   
     
       
         
             
           
             
                 
             
           
          
             
               DIRECT_VALUE[129]= 
             
             
               { 
             
             
               2147483647, 
             
             
               1815, 1283, 1048, 907, 812, 741, 686, 642, 605, 574, 547, 524, 503, 485, 
             
             
               469, 454, 440, 428, 416, 406, 396, 387, 378, 370, 363, 356, 349, 343, 
             
             
               337, 331, 326, 321, 316, 311, 307, 303, 298, 294, 291, 287, 283, 280, 
             
             
               277, 274, 271, 268, 265, 262, 259, 257, 254, 252, 249, 247, 245, 243, 
             
             
               240, 238, 236, 234, 232, 231, 229, 227, 225, 223, 222, 220, 219, 217, 
             
             
               215, 214, 212, 211, 210, 208, 207, 206, 204, 203, 202, 200, 199, 198, 
             
             
               197, 196, 195, 193, 192, 191, 190, 189, 188, 187, 186, 185, 184, 183, 
             
             
               182, 182, 181, 180, 179, 178, 177, 176, 175, 175, 174, 173, 172, 172, 
             
             
               171, 170, 169, 169, 168, 167, 166, 166, 165, 164, 164, 163, 162, 162, 
             
             
               161, 160 
             
             
               }; 
             
             
                 
             
          
         
       
     
   
     FIG. 5A  is a block diagram illustrating exemplary embodiment of operations for high-precision implement algorithm of Ephraim-Malah filter weights formula. The operations may be performed by hardware (e.g., circuitry, dedicated logic, etc.), software (such as programs run on a general purpose computer or a dedicated machine), or a combination of both. Referring to  FIG. 5A , at processing block  501 , input parameters of function Ephraim_Mala( ) θ, which is a product of Wiener filter W n   y (k) and posterior SNR    n   post (k), is received by process logic, where k represents an index of frequency point. 
   For a fix-point implementation, according to one embodiment, W n   y (k) is implemented using Q31 format and    n   post (k) is implemented using Q15 format. At processing block  502 , according to one embodiment, since θ is implemented in a Q22 format, the 0 may be obtained by process logic via following transformation:
 
θ =W   n   y ( k )×   n   post ( k )&gt;&gt;(31+15−22)  (Eq. 6)
 
where &gt;&gt;represents a shift operation. In one embodiment, θ is a 32-bit value which is suitable for a 32-bit processor. It would be appreciated that θ may be implemented in other forms for other types of processor, such as 64-bit processors, etc.
 
   If θ is greater than a predetermined threshold, such as 2 7 , at processing block  504 , an index value and a mantissa value are extracted from θ, as shown as 32-bit number  550  in  FIG. 5B  according to an embodiment. Referring to  FIGS. 5A and 5B , at processing block  504 , function Evaluate (θ) extracts the index=24−n  551 , mantissa  552 . n is the number of leading zero bits at 32-bit number  550 . For the example, as shown in  FIG. 5B , index=24−6=18, mantissa  552 =1110000001111100110 in binary, which is 459750 in decimal. In one embodiment, n  551  and mantissa  552  can be extracted through an instruction of a processor, such as, for example, Intel Xscale microprocessor with a CLZ instruction which is available from Intel Corporation. 
   At processing block  505 , since X, which is a mantissa, such as mantissa  552 , is implemented in a Q22 format. P0[i] is implemented in a Q22 format. P1[i] is implemented in a dynamic Q value, such as (5+i). P2[i] is implemented in dynamic Q value (i−4). Result M′(θ) is implemented in a Q22 format. In one embodiment, processing block  505  may be implemented in one or more major operations by process logic. 
     FIG. 6  is a block diagram illustrating exemplary embodiment of operations in a fix-point implementation. Referring to  FIG. 6 , exemplary operation  600  includes a first operation  601  and a second operation  602 . Operations  601  and  602  may be performed by hardware (e.g., circuitry, dedicated logic, etc.), software (such as programs run on a general purpose computer or a dedicated machine), or a combination of both. For example, blocks  601  and  602  may represent two circuits having individual components, such as multiplier, shifter, and adder, etc. Alternatively, blocks  601  and  602  may be embedded in a processor, such as a microprocessor. Furthermore, operations involved in blocks  601  and  602  may be implemented as an instruction recognized and executable by a processor, such as a CLZ instruction of Intel Xscale microprocessor. Other components apparent to those with ordinary skills in the art may be included. 
   According to one embodiment, processes involved in first operation  601  may be defined as follows: 
                     TEMP   =       ⁢     (       (     P   ⁢           ⁢     2   ⁡     [   i   ]       ×   X     )     &gt;&gt;       (   22     ⁢           -     (       (     i   +   5     )     -     (     i   -   4     )       )         )       )     -     P   ⁢           ⁢     1   ⁡     [   i   ]                     =       ⁢       (       (     P   ⁢           ⁢     2   ⁡     [   i   ]       ×   X     )     &gt;&gt;   13     )     -     P   ⁢           ⁢     1   ⁡     [   i   ]                       
In a particular embodiment, first operation  601  includes a multiplier  603 , a shifter  604 , and an adder  605 . Multiplier  603  multiplies P2 and X (mantissa) and generates a first intermediate value at an output of multiplier  603 . Shifter  604  receives the first intermediate value from the output of multiplier  603  and shifts the intermediate value by a value of 22, resulting in a second intermediate value. Adder  605  adds the second intermediate value with P1 and generate an output Temp, as described above, of first operation  601 .
 
   According to one embodiment, processes involved in second operation  602  may be defined as follows:
 
 M′ (θ)=(( X×TEMP )&gt;&gt;( i+ 5))+ P 0 [i] 
 
During second operation  602 , multiplier  606  multiplies output Temp from the first operation  601  with mantissa X and generates a third intermediate value. Shifter  607  receives the third intermediate value and shifts a value of (i+5), where i is the index, and generates a fourth intermediate value. Adder  608  adds the fourth intermediate value with P0 and generates a final output representing M′(θ) described above. All processes described above do not invoke any mathematical division operations.
 
     FIG. 7  is a flow diagram illustrating an exemplary embodiment of a process for generating Ephraim-Malah filter coefficients. The process may be performed by hardware (e.g., circuitry, dedicated logic, etc.), software (such as programs run on a general purpose computer or a dedicated machine), or a combination of both. In one embodiment, exemplary process  700  includes computing a first parameter based on Wiener filter weights and posterior signal-to-noise (SNR) via a polynomial approximation mechanism without using a mathematical division operation, and generating Ephrain-Malah filter coefficients based on the first parameter. 
   Referring to  FIG. 7 , at block  701 , Wiener filter weights (e.g., W n   y ( k )) and posterior SNR (e.g.,    n   post (k)) are received. Wiener filter weights and posterior SNR may be obtained based on speech and noise power spectrum estimations which may be performed by speech and noise power spectrum estimation modules  104  and  103  of  FIG. 1  respectively. At block  702 , a first parameter (e.g., θ) is computed based on the Wiener filter weights and posterior SNR. In one embodiment, the first parameter is a 32-bit value. At block  703 , if the first parameter is equal to or less than a threshold, a second parameter (e.g., M′(θ)) is retrieved from a database based on the first parameter, at block  707 . In one embodiment, the threshold is defined as 2 7 . According to one embodiment, the database includes one or more data tables that store the second parameter corresponding to the first parameter. Thereafter, at block  706 , Ephraim-Malah filter coefficients are computed based on the second parameter. 
   If the first parameter is greater than the threshold, at block  704 , an index and a mantissa are determined based on the first parameter. In one embodiment, the index is determined based on the number of the leading zero of the first parameter and the mantissa is determined based in part on the remaining portion of the first parameter, such as for example, parameter  550  shown in  FIG. 5B . At block  705 , a second parameter (e.g., M′(θ)) is computed based on the index and mantissa using a polynomial approximation mechanism without invoking a mathematical division operation. In one embodiment, the polynomial approximation mechanism includes a two-order polynomial approximation operation, which may be defined as follows:
 
 f ( x )= P 0 +P 1 *x+P 2 *x   2  
 
In one embodiment, P0 is in a Q22 format. P1 is determined based on a dynamic Q value of (5+i), where i is an index value. P2 is determined based on a dynamic Q value of (i−4), where i is an index value. At block  706 , Ephraim-Malah filter coefficients are computed based on the second parameter.
 
     FIG. 8  shows a block diagram of an exemplary computer which may be used with an embodiment of the invention. For example, system  800  shown in  FIG. 8  may include hardware, software, or the both, to perform the above discussed processes shown in  FIGS. 5A ,  6 , and  7 . Note that while  FIG. 8  illustrates various components of a computer system, it is not intended to represent any particular architecture or manner of interconnecting the components, as such details are not germane to the present invention. It will also be appreciated that network computers, handheld computers, cell phones, and other data processing systems which have fewer components or perhaps more components may also be used with the present invention. 
   As shown in  FIG. 8 , the computer system  800 , which is a form of a data processing system, includes a bus  802  which is coupled to a microprocessor  803  and a ROM  807 , a volatile RAM  805 , and a non-volatile memory  806 . The microprocessor  803 , which may be a Pentium processor from Intel Corporation, is coupled to cache memory  804  as shown in the example of  FIG. 8 . The bus  802  interconnects these various components together and also interconnects these components  803 ,  807 ,  805 , and  806  to a display controller and display device  808 , as well as to input/output (I/O) devices  810 , which may be mice, keyboards, modems, network interfaces, printers, and other devices which are well-known in the art. Typically, the input/output devices  810  are coupled to the system through input/output controllers  809 . The volatile RAM  805  is typically implemented as dynamic RAM (DRAM) which requires power continuously in order to refresh or maintain the data in the memory. The non-volatile memory  806  is typically a magnetic hard drive, a magnetic optical drive, an optical drive, or a DVD RAM or other type of memory system which maintains data even after power is removed from the system. Typically the non-volatile memory will also be a random access memory, although this is not required. While  FIG. 8  shows that the non-volatile memory is a local device coupled directly to the rest of the components in the data processing system, it will be appreciated that the present invention may utilize a non-volatile memory which is remote from the system, such as a network storage device which is coupled to the data processing system through a network interface such as a modem or Ethernet interface. The bus  802  may include one or more buses connected to each other through various bridges, controllers, and/or adapters, as is well-known in the art. In one embodiment, the I/O controller  809  includes a USB (Universal Serial Bus) adapter for controlling USB peripherals. 
   Precision piecewise polynomial approximation for Ephraim-Malah filter has been described herein. In the foregoing specification, the invention has been described with reference to specific exemplary embodiments thereof. It will be evident that various modifications may be made thereto without departing from the broader spirit and scope of the invention as set forth in the following claims. The specification and drawings are, accordingly, to be regarded in an illustrative sense rather than a restrictive sense.