Efficient implementation of complex exponentially ... - Semantic Scholar
EFFICIENT IMPLEMENTATION OF COMPLEX EXPONENTIALLY-MODULATED FILTER BANKS Juuso Alhava, Ari Wholainen, and Markku Renfors Tampere University of Technology Institute of Communications Engineering P.O. Box 553, FIN-33101 Tampere, FINLAND e-mail:
[email protected], [email protected], [email protected]
ABSTRACT In Exponentially-ModulatedFilter Bank (EMFB) the complex subfilters are generated from a real-valued prototype filter by multiplying the filter impulse response with complex exponential sequences. This gives the possibility to use block transforms with computationally fast algorithms. Our definition for the EMFB is based on Extended Lapped Transform (ELT). Then we may apply Fast ELT based algorithms for cosine-modulatedand sine-modulatedfilter banks (CMFB, SMFB) as basic building blocks in the implementation of this complex filter bank. Alternatively, a modified polyphase structure with 2 branches ( being the sampling rate conversion factor) can be used. The latter structure can be used also with non-perfect reconstruction filter bank designs that allow some additional degrees of freedom in the optimization. In this paper we explore these alternative realization structures for CMFBs and SMFBs.
1. INTRODUCTION The complex exponentially-modulated filter bank we proposed in [1][2] is a tool for subband signal processing of complex signals. The EMFB definition grounds on cosinemodulated filter banks, where the phase of the modulating cosines are as in the extended lapped transform. Then we can use fast algorithms derived for the ELT as a component in EMFB implementations. In addition, we need the ”sister filter bank” of the CMFB, namely the sine-modulated filter bank, that can also be implemented using ELT type of structure. The -band ELT implementation can be divided into prototype filter and modulation parts. The cosine-modulation is done with block transform, Discrete Cosine Transform IV (DCT-IV). If the prototype filter coefficients satisfy the Perfect Reconstruction (PR) condition [3], then we can This research was carried out in the project “Advanced Multicarrier Techniques for Wireless Communications” funded by the Academy of Finland.
0-7803-7761-3/03/$17.00 02003 IEEE
Figure 1: Critically sampled complex filter bank. apply Fast ELT structure with cascaded butterfly sections [4]. When we use nearly-perfect designs [5],the prototype filter can be implemented with 2 polyphase filters, but not with the Fast ELT. These CMFB algorithms require only small changes for SMFB implementations. The relevant parts of the EMFB topic are considered in Section 2. We define the complex subfilters with CMFB and SMFB subfilter equations and represent the subband signal processing system with Lapped Transform (LT) notations. As the EMFB can be computed with two real filter banks, we give the necessary modifications to convert Fast ELT algorithm for SMFB in Section 3. Then in Section 4 we derive the 2 polyphase structure for the SMFB. The same structure for the CMFB is also given for comparison purposes.
2. EXPONENTIALLY-MODULATEDFILTER BANK The cosine- and sine-modulated synthesis filter equations, based on the ELT definitions, are 2 +1 1 k(n) = -h (n)Cos n + 2 k + s - (1)
2 -h
1 +1 T k + - - . (2) 2 The subscript k is the subchannel index. From these real synthesis filters we can construct the exponentially-mod-
k(n) =
(n)sin
n
+
IV-157
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angles 8i . The number of butterfly matrices depends on the overlapping factor as = 0 , 1 , . . . , - 1. The design parameter defines the length of the ELT basis functions: . When we write N =2 = i
COS^^ ,cos81 , . . . ,cos8
= i
{sin801sin81,...,sin8
2-1
2-1
}
(7)
}
(8)
the butterfly matrix is
-
-
(9)
Figure 2: Implementation of the EMFB with LTs. ulated filter bank whose synthesis filters are k
1 = -h 2
-
.
( n ) = k(n)+
k(n)
(n)
n
+
2 +1
T
(3) 1 k+Z
Matrix is the reversing block matrix (dimensions 2 2) with ones on its antidiagonal and the other elements are zlero 1 1 -
-
1
3.2. Fast ELT for Sine-ModulatedFilter Bank 1 = -h 2
(n)
-
where k = O,l,- 1 a n d n = O , l , . . . , N - 1. The scaling factor 2 has been included in the prototype filter. Figure l shows the 2 subchannel filter bank, where the analysis filters are time-reversed and complex conjugated versions of the synthesis filters. It is critically sampled, as we take only the real parts of the analysis filter outputs [6]. This system satisfies the PR condition [ 11. We collect the CMFB and SMFB subfilters in real N lapped transform matrices and [ 1][3]
1 1
Ink =
k(n)
The same ELT angles of the CMFB provide fast algorithm for the SMFB. Figures 3 and 4 show the modified Fast ELT for the SMFB is flowgaphs. The th butterfly matrix
-
-
(1 1)
only the signs of the sine-terms have been reversed. The other necessary change is to replace DCT-IV with Discrete Sine Transform IV (DST-IV). The elements of the DST-IV matrix are
[
(5 )
Ink =
2
-sin
1
"+z
k+-
1
2
- .
(12)
(6) Then we can represent the computational structure of the complex filter bank with lapped transforms as shown in Figure 2. (Note: The subscipt notations in matrix equations (5) and (6) refer to element on nth row and lcth column.) The forward transform matrix T is equivalent to analysis CMFB and is the inverse transform matrix (synthesis CMFB). and are connected similarly with SMFB. We find computationally efficient structure for the EMFB when we implement the LTs with fast algorithms.
There is a simple connection between and . The basis functions of the DST-IV can be obtained from the DCTIV by reversing the columns and changing the sign of odd columns: [ Ink = -1-nk. (13) This means that fast DST-IV transform can be computed in three steps: I) Change the signs of odd elements in input data vector 11) Compute the DCT-IV transform with the fast algorithm [3] 111) Reverse the elements in DCT-IV transformed data vector.
3. FAST EXTENDED LAPPED TRANSFORM
4. 2M POLYPHASE FILTERS FOR MODULATED FILTER BANKS
Ink
= k(n).
3.1. Butterfly Matrices Fast ELT was presented by Malvar in [4]. The algorithm consists of DCT-IV modulation matrix and cascaded butterfly matrix section. Between each butterfly half of the outputs have delays (forward Fast ELT). The elements of the th butterfly matrix are cosines and sines of the ELT
(-V[I
4.1. Matrix Decomposition of The modulating sines for the SMFB can be generated from the DST-IV transform matrix. To show this we decompose the basis functions of the lapped transform in the following manner:
IV-158
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Figure 3: SMFB (analysis) implementation with Fast Direct ELT.
Figure 4: SMFB (synthesis) implementation with Fast Inverse ELT.
. The diag-
First we separate the prototype filter from onal matrix = i
- 1))
{ h (O),h ( l ) , . . . , h (2
Then we write as a decomposition of the prototype filter, matrices and DST-IVs:
-
(14)
[
Ink
=sin
n
1
+k+z 2
-
where n = 0,1,. . . , 2 - 1 and IC = 0 , 1 , . . . , Furthermore we partition the matrices as follows
1
-
contains the prototype filter coefficients and the modulating sines are grouped in . The elements of the modulating matrix are +1
0
-
2
-2
(15) -
1.
(20) This system of matrices describes the synthesis filter bank shown in Figure 5 . The subfilters i( ) are defined as -1
0
0
a(
-1
)=C(-1)h (2+2
)
-2
=O
-1
is a 2 2 matrixand ), . . . , h (2 - 1 22 of the modulating matrix are
Here h (1
[
i
+ 22
i]nk
=sin
+
n
-1
= i { h (22 ), )}. The submatrices
1 2 = i2. , which consist of
+1 + 22 + k+ 2
It can be shown that = - i l and Now we define a 2 matrix 2 2 square submatrices:
- - . (17)
1
-1
-
-
(18)
=E
a(
)
-2
=O
(21) This is not a pure polyphase realization of the prototype filter, because there are 2 subfilters and the interpolation factor is . The analysis bank is constructed similarly. Only now we transpose matrix and multiply it by (- 1) to map 2 subsignals from the polyphase filters to the DST-IV. The analysis bank is shown in Figure 6.
-'
4.2. Cosine-ModulatedFilter Bank with 2M Polyphase Filters The same decomposition for matrix gives CMFB implementation where polyphase filters a ( ) are identical with the SMFB case in Figure 5 and 6. But the modulation matrix is changed to DCT-IV and is replaced by
I
-I
Now the DST-IV comes into play: We can verify that
-
(22)
I
IV-159
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Figure 5: ELT-based synthesis SMFB with 2 filters.
Figure 7: Forward ELT (analysis CMFB).
polyphase
Thus the EMFB computations can be done with high efficiency. In the continuation we shall consider eficient iniplementation of oversampled filter banks that are obtained by taking the complex (instead of real) subchannel output signals from the analysis bank. Such oversanipled filter banks are very useful in many applications, like channel equa1’1zation in filter bank based transinultiplexer systems. 6. REFERENCES
Figure 6: ELT-based analysis SMFB. polypliase filters is shown in
The analysis CMFB with 2 Figrre 7.
[I] J. Alhasa and M. Renfors, “Coniplex lapped transforms and modulated filter banks,” in Proc. Int. TICSP Wrkshop on Specrral Methods ancl Ariultirare Signal Processing,Tolouse, France, Sept. 7-8, 2002, pp. 8794.
5. CONCLUSIONS
In this paper the implementation structure of the critically sampled coniplex exponentially-modulatedfilter bank uses CMFB and SMFB as building blocks. We have presented a fast computation structure for SMFBs by modifying the cxteiided lapped transform algorithms. When we iniplement these real transforms with ELT-based algorithms. the coniputational coniplexity is of the same order with conventional block transforms. By applying the computational complexity formulas of the Fast ELT [3], we find the number of arithmetic operations per complex input samples for the EMFB. The number ofreal niultiplications and additions (respectively) is
[2] J. Alhava and M. Renfors, “Exponentially-modulated filter bank-based transniultiplexer,” in Proc. 1EE.E Int. JSyrnp.on Circuits
U F ZSwtwzs, ~
Bangkok, Thailand,
May 25-28,2003.
[31 1-1. S. Malvar, Sigrid Processiiig with Lapped Trans,foims.Norwood, MA: Artech House, 1992. [41 H. S. Malvar, “Extended lapped transforins: Properties, applications, and fast algorithms,” IEEE B-uns. SignalProcessing, vol. 40. pp. 2703-2714, Nov. 1992.
[SI A. Viholainen, T. Saramiiki, and M. Renfors, “Nearly
(23) (24)
perfect-reconstiuction cosine-modulated filter bank design for VDSL modems,“ in Proc. h i t . Con/: on Electronics, CirLriirs arid Sjstenis, Pafos, Cyprus, Sept. 5 - 8 , 1999, pp. 373-376.
The complexity is dependent on the decimation factor and the overlapping factor . (Note: TIierc are 2 subchannels in the EMFB.) The number of multiplications have been reduced by scaling sonie buttelfly coeflicieiits to 1 [3]. The complexity of the EMFB with 2 polyphase filters is ( ) = (4 + 2 + 0 , ) (25)
[6] A. Viholainen, T. Hidalgo Stitz, J. Alhava, T. Ihalainen. and M. Renfors. “Complex modulated critically sampled filter banks based on cosinc and sinc modulation:’ in Proc. IEEE h i t . Symp. a n Circvrls wid Systems, Scottsdale, Arizona, USA, May 26-29.2002, vol. 1, pp. 833-836.
( ) =
(2
)
(2
N(
(.
=
=
(4
4-
$ 0 2
+ +
+
)
).
0 2
0 2
).
(26)
rv-160
Authorized licensed use limited to: Tampereen Teknillinen Korkeakoulu. Downloaded on March 24, 2009 at 08:07 from IEEE Xplore. Restrictions apply.
[email protected], [email protected], [email protected]
ABSTRACT In Exponentially-ModulatedFilter Bank (EMFB) the complex subfilters are generated from a real-valued prototype filter by multiplying the filter impulse response with complex exponential sequences. This gives the possibility to use block transforms with computationally fast algorithms. Our definition for the EMFB is based on Extended Lapped Transform (ELT). Then we may apply Fast ELT based algorithms for cosine-modulatedand sine-modulatedfilter banks (CMFB, SMFB) as basic building blocks in the implementation of this complex filter bank. Alternatively, a modified polyphase structure with 2 branches ( being the sampling rate conversion factor) can be used. The latter structure can be used also with non-perfect reconstruction filter bank designs that allow some additional degrees of freedom in the optimization. In this paper we explore these alternative realization structures for CMFBs and SMFBs.
1. INTRODUCTION The complex exponentially-modulated filter bank we proposed in [1][2] is a tool for subband signal processing of complex signals. The EMFB definition grounds on cosinemodulated filter banks, where the phase of the modulating cosines are as in the extended lapped transform. Then we can use fast algorithms derived for the ELT as a component in EMFB implementations. In addition, we need the ”sister filter bank” of the CMFB, namely the sine-modulated filter bank, that can also be implemented using ELT type of structure. The -band ELT implementation can be divided into prototype filter and modulation parts. The cosine-modulation is done with block transform, Discrete Cosine Transform IV (DCT-IV). If the prototype filter coefficients satisfy the Perfect Reconstruction (PR) condition [3], then we can This research was carried out in the project “Advanced Multicarrier Techniques for Wireless Communications” funded by the Academy of Finland.
0-7803-7761-3/03/$17.00 02003 IEEE
Figure 1: Critically sampled complex filter bank. apply Fast ELT structure with cascaded butterfly sections [4]. When we use nearly-perfect designs [5],the prototype filter can be implemented with 2 polyphase filters, but not with the Fast ELT. These CMFB algorithms require only small changes for SMFB implementations. The relevant parts of the EMFB topic are considered in Section 2. We define the complex subfilters with CMFB and SMFB subfilter equations and represent the subband signal processing system with Lapped Transform (LT) notations. As the EMFB can be computed with two real filter banks, we give the necessary modifications to convert Fast ELT algorithm for SMFB in Section 3. Then in Section 4 we derive the 2 polyphase structure for the SMFB. The same structure for the CMFB is also given for comparison purposes.
2. EXPONENTIALLY-MODULATEDFILTER BANK The cosine- and sine-modulated synthesis filter equations, based on the ELT definitions, are 2 +1 1 k(n) = -h (n)Cos n + 2 k + s - (1)
2 -h
1 +1 T k + - - . (2) 2 The subscript k is the subchannel index. From these real synthesis filters we can construct the exponentially-mod-
k(n) =
(n)sin
n
+
IV-157
Authorized licensed use limited to: Tampereen Teknillinen Korkeakoulu. Downloaded on March 24, 2009 at 08:07 from IEEE Xplore. Restrictions apply.
angles 8i . The number of butterfly matrices depends on the overlapping factor as = 0 , 1 , . . . , - 1. The design parameter defines the length of the ELT basis functions: . When we write N =2 = i
COS^^ ,cos81 , . . . ,cos8
= i
{sin801sin81,...,sin8
2-1
2-1
}
(7)
}
(8)
the butterfly matrix is
-
-
(9)
Figure 2: Implementation of the EMFB with LTs. ulated filter bank whose synthesis filters are k
1 = -h 2
-
.
( n ) = k(n)+
k(n)
(n)
n
+
2 +1
T
(3) 1 k+Z
Matrix is the reversing block matrix (dimensions 2 2) with ones on its antidiagonal and the other elements are zlero 1 1 -
-
1
3.2. Fast ELT for Sine-ModulatedFilter Bank 1 = -h 2
(n)
-
where k = O,l,- 1 a n d n = O , l , . . . , N - 1. The scaling factor 2 has been included in the prototype filter. Figure l shows the 2 subchannel filter bank, where the analysis filters are time-reversed and complex conjugated versions of the synthesis filters. It is critically sampled, as we take only the real parts of the analysis filter outputs [6]. This system satisfies the PR condition [ 11. We collect the CMFB and SMFB subfilters in real N lapped transform matrices and [ 1][3]
1 1
Ink =
k(n)
The same ELT angles of the CMFB provide fast algorithm for the SMFB. Figures 3 and 4 show the modified Fast ELT for the SMFB is flowgaphs. The th butterfly matrix
-
-
(1 1)
only the signs of the sine-terms have been reversed. The other necessary change is to replace DCT-IV with Discrete Sine Transform IV (DST-IV). The elements of the DST-IV matrix are
[
(5 )
Ink =
2
-sin
1
"+z
k+-
1
2
- .
(12)
(6) Then we can represent the computational structure of the complex filter bank with lapped transforms as shown in Figure 2. (Note: The subscipt notations in matrix equations (5) and (6) refer to element on nth row and lcth column.) The forward transform matrix T is equivalent to analysis CMFB and is the inverse transform matrix (synthesis CMFB). and are connected similarly with SMFB. We find computationally efficient structure for the EMFB when we implement the LTs with fast algorithms.
There is a simple connection between and . The basis functions of the DST-IV can be obtained from the DCTIV by reversing the columns and changing the sign of odd columns: [ Ink = -1-nk. (13) This means that fast DST-IV transform can be computed in three steps: I) Change the signs of odd elements in input data vector 11) Compute the DCT-IV transform with the fast algorithm [3] 111) Reverse the elements in DCT-IV transformed data vector.
3. FAST EXTENDED LAPPED TRANSFORM
4. 2M POLYPHASE FILTERS FOR MODULATED FILTER BANKS
Ink
= k(n).
3.1. Butterfly Matrices Fast ELT was presented by Malvar in [4]. The algorithm consists of DCT-IV modulation matrix and cascaded butterfly matrix section. Between each butterfly half of the outputs have delays (forward Fast ELT). The elements of the th butterfly matrix are cosines and sines of the ELT
(-V[I
4.1. Matrix Decomposition of The modulating sines for the SMFB can be generated from the DST-IV transform matrix. To show this we decompose the basis functions of the lapped transform in the following manner:
IV-158
Authorized licensed use limited to: Tampereen Teknillinen Korkeakoulu. Downloaded on March 24, 2009 at 08:07 from IEEE Xplore. Restrictions apply.
Figure 3: SMFB (analysis) implementation with Fast Direct ELT.
Figure 4: SMFB (synthesis) implementation with Fast Inverse ELT.
. The diag-
First we separate the prototype filter from onal matrix = i
- 1))
{ h (O),h ( l ) , . . . , h (2
Then we write as a decomposition of the prototype filter, matrices and DST-IVs:
-
(14)
[
Ink
=sin
n
1
+k+z 2
-
where n = 0,1,. . . , 2 - 1 and IC = 0 , 1 , . . . , Furthermore we partition the matrices as follows
1
-
contains the prototype filter coefficients and the modulating sines are grouped in . The elements of the modulating matrix are +1
0
-
2
-2
(15) -
1.
(20) This system of matrices describes the synthesis filter bank shown in Figure 5 . The subfilters i( ) are defined as -1
0
0
a(
-1
)=C(-1)h (2+2
)
-2
=O
-1
is a 2 2 matrixand ), . . . , h (2 - 1 22 of the modulating matrix are
Here h (1
[
i
+ 22
i]nk
=sin
+
n
-1
= i { h (22 ), )}. The submatrices
1 2 = i2. , which consist of
+1 + 22 + k+ 2
It can be shown that = - i l and Now we define a 2 matrix 2 2 square submatrices:
- - . (17)
1
-1
-
-
(18)
=E
a(
)
-2
=O
(21) This is not a pure polyphase realization of the prototype filter, because there are 2 subfilters and the interpolation factor is . The analysis bank is constructed similarly. Only now we transpose matrix and multiply it by (- 1) to map 2 subsignals from the polyphase filters to the DST-IV. The analysis bank is shown in Figure 6.
-'
4.2. Cosine-ModulatedFilter Bank with 2M Polyphase Filters The same decomposition for matrix gives CMFB implementation where polyphase filters a ( ) are identical with the SMFB case in Figure 5 and 6. But the modulation matrix is changed to DCT-IV and is replaced by
I
-I
Now the DST-IV comes into play: We can verify that
-
(22)
I
IV-159
Authorized licensed use limited to: Tampereen Teknillinen Korkeakoulu. Downloaded on March 24, 2009 at 08:07 from IEEE Xplore. Restrictions apply.
Figure 5: ELT-based synthesis SMFB with 2 filters.
Figure 7: Forward ELT (analysis CMFB).
polyphase
Thus the EMFB computations can be done with high efficiency. In the continuation we shall consider eficient iniplementation of oversampled filter banks that are obtained by taking the complex (instead of real) subchannel output signals from the analysis bank. Such oversanipled filter banks are very useful in many applications, like channel equa1’1zation in filter bank based transinultiplexer systems. 6. REFERENCES
Figure 6: ELT-based analysis SMFB. polypliase filters is shown in
The analysis CMFB with 2 Figrre 7.
[I] J. Alhasa and M. Renfors, “Coniplex lapped transforms and modulated filter banks,” in Proc. Int. TICSP Wrkshop on Specrral Methods ancl Ariultirare Signal Processing,Tolouse, France, Sept. 7-8, 2002, pp. 8794.
5. CONCLUSIONS
In this paper the implementation structure of the critically sampled coniplex exponentially-modulatedfilter bank uses CMFB and SMFB as building blocks. We have presented a fast computation structure for SMFBs by modifying the cxteiided lapped transform algorithms. When we iniplement these real transforms with ELT-based algorithms. the coniputational coniplexity is of the same order with conventional block transforms. By applying the computational complexity formulas of the Fast ELT [3], we find the number of arithmetic operations per complex input samples for the EMFB. The number ofreal niultiplications and additions (respectively) is
[2] J. Alhava and M. Renfors, “Exponentially-modulated filter bank-based transniultiplexer,” in Proc. 1EE.E Int. JSyrnp.on Circuits
U F ZSwtwzs, ~
Bangkok, Thailand,
May 25-28,2003.
[31 1-1. S. Malvar, Sigrid Processiiig with Lapped Trans,foims.Norwood, MA: Artech House, 1992. [41 H. S. Malvar, “Extended lapped transforins: Properties, applications, and fast algorithms,” IEEE B-uns. SignalProcessing, vol. 40. pp. 2703-2714, Nov. 1992.
[SI A. Viholainen, T. Saramiiki, and M. Renfors, “Nearly
(23) (24)
perfect-reconstiuction cosine-modulated filter bank design for VDSL modems,“ in Proc. h i t . Con/: on Electronics, CirLriirs arid Sjstenis, Pafos, Cyprus, Sept. 5 - 8 , 1999, pp. 373-376.
The complexity is dependent on the decimation factor and the overlapping factor . (Note: TIierc are 2 subchannels in the EMFB.) The number of multiplications have been reduced by scaling sonie buttelfly coeflicieiits to 1 [3]. The complexity of the EMFB with 2 polyphase filters is ( ) = (4 + 2 + 0 , ) (25)
[6] A. Viholainen, T. Hidalgo Stitz, J. Alhava, T. Ihalainen. and M. Renfors. “Complex modulated critically sampled filter banks based on cosinc and sinc modulation:’ in Proc. IEEE h i t . Symp. a n Circvrls wid Systems, Scottsdale, Arizona, USA, May 26-29.2002, vol. 1, pp. 833-836.
( ) =
(2
)
(2
N(
(.
=
=
(4
4-
$ 0 2
+ +
+
)
).
0 2
0 2
).
(26)
rv-160
Authorized licensed use limited to: Tampereen Teknillinen Korkeakoulu. Downloaded on March 24, 2009 at 08:07 from IEEE Xplore. Restrictions apply.
