1
Structural SubStructural Sub--band band Decomposition: A New Decomposition: A New
Concept in Digital Signal Concept in Digital Signal ProcessingProcessing
SANJIT K. MITRAMing Hsieh Department of Electrical Engineering
University of Southern CaliforniaLos Angeles, California
2
OutlineOutline• Signal and System Decomposition
– Polyphase decomposition– Structural subband decomposition
• Subband Discrete Transforms- Subband discrete Fourier transform- Subband discrete cosine trasform- Applications
• Subband FIR Filter Design and Implementation
• Subband Adaptive Filtering
3
PolyphasePolyphase DecompositionDecomposition
• In the M-band polyphase decomposition, a sequence {x[n]} is expressed as a sum of M subsequences , obtained by down-sampling {x[n]} by a factor of M with i indicating the phase of the sub-sampling process
]},[{ nxi 10 −≤≤ Mn
,iMnxnxi ][][ +=
4
PolyphasePolyphase DecompositionDecomposition
• For example, for M = 2, for a causal sequence {x[n]}, the two sub-sequences are:
- Even samples of {x[n]}
- Odd samples of {x[n]}
}[6][4][2][0]{]}[{ 0 Lxxxxnx =
}[6][4][2][0]{]}[{ 0 Lxxxxnx =
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PolyphasePolyphase DecompositionDecomposition
• Physical Interpretation – 2-Band Case
z2
2
][nx ][0 nx
][1 nx
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PolyphasePolyphase DecompositionDecomposition
• Likewise, for M = 3, for a causal sequence {x[n]}, the three sub-sequences are:
}[9][6][3][0]{]}[{ 0 Lxxxxnx =
}[10][7][4][1]{]}[{ 1 Lxxxxnx =
}[11][8][5][2]{]}[{ 2 Lxxxxnx =
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PolyphasePolyphase DecompositionDecomposition
• Physical Interpretation – 3-Band Case
z3
3
][nx ][0 nx
][1 nx
3z
][2 nx
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PolyphasePolyphase DecompositionDecomposition• Physical Interpretation – General Case
zM][nx ][0 nx
][1 nx
z][1 nxM −
M
M
M
z][2 nxM −
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PolyphasePolyphase DecompositionDecomposition
• The z-transform X(z) of a finite or infinite length sequence {x[n]} can be expressed as a finite sum of the z-transforms of M subsequences ,
)(zXi]}[{ nxi 1,,1,0 −= Mi K
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PolyphasePolyphase DecompositionDecomposition• The M-band polyphase decomposition of X(z)
is given by
where
• is the i-th polyphase component of X(z)
∑∑−
=
−∞
−∞=
− ==1
0)(][)(
M
i
iMi
n
n zzXznxzX
10,][)( −≤≤+= ∑∞
−∞=
− MiziMnxzXn
ni
)(zXi
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PolyphasePolyphase DecompositionDecomposition• The polyphase decomposition can be
written in matrix form as
where
[ ]⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
−
−−−
)(
)()(
1)(
1
1
0)1(1
MM
M
M
M
zX
zXzX
zzzXM
L
TMzz )()( Xe ⋅=
]1[)( )1(1 −−−= Mzzz Le[ ])()()()( 110 zXzXzXz M −= LX
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PolyphasePolyphase DecompositionDecomposition• Physical interpretation
M
M
M
M
][0 nx
][1 nx
][1 nxM −
1−z
1−z
1−z][nu]1[ −+= Mnx
][nx
][2 nxM −
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PolyphasePolyphase DecompositionDecomposition• Reconstruction of original sequence
][0 nx
][1 nx
][1 nxM −
1−z ]1[ +−= Mnu][nx
][2 nxM −
M
+
+
1−z+
1−z
M
M
M
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PolyphasePolyphase DecompositionDecomposition
• The sequence x[n], i.e., a delayed version of the input sequence u[n], can be developed from the M-sub-sequences by up- sampling each subsequence by a factor of M and then interleaving the outputs of the up- samplers
][nxi
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Structural Structural SubbandSubband DecompositionDecomposition
• The structural subband decomposition of X(z) is given by
where is an nonsingular matrix
[ ]⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
−
−−−
)(
)()(
1)(
1
1
0)1(1
MM
M
M
M
zV
zVzV
zzzXM
L T
][ , jit=T MM ×
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Structural Structural SubbandSubband DecompositionDecomposition
• The structural subband decomposition is thus a generalization of the polyphase decomposition
• The functions are called the structural subband components or generalized polyphase components of X(z)
)(zVk
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Structural Structural SubbandSubband DecompositionDecomposition
• Relation between the polyphase components and the structural sub-band components are given by)(zVi
)(zXi
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
− )(
)()(
)(
)()(
1
10
1
1
10
zX
zXzX
zV
zVzV
MMMM
T
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Structural Structural SubbandSubband DecompositionDecomposition
• If denotes the inverse z-transform of , then it follows that
where is the -th element of• The structural sub-band subsequences
are basically given by a linear combination of the polyphase sub-sequences
)(zVi][nvi
),( li 1−T][nvi
][nxi
10,][][1
0, −≤≤= ∑
−
=Minxtnv
Mii
lll
~
l,it~
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Structural Structural SubbandSubband DecompositionDecomposition
• Physical interpretation
1−z
1−z
1−z][nu]1[ −+= Mnx
][nx M
M
M
M
][0 nv
][1 nv
][1 nvM −
][2 nvM −
1−T
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Structural Structural SubbandSubband DecompositionDecomposition
• Likewise, the polyphase subsequences can be recovered by a linear combination of the structural subband subsequences according to
where is the -th element of T
][nvi
][nxi
10,][][1
0, −≤≤= ∑
−
=Minvtnx
Mii
lll
l,it ),( li
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Structural Structural SubbandSubband DecompositionDecomposition
• A delayed version of the input u[n] can be developed by first up-sampling the M sub- sequences and then generating the subsequences by a linear combination of these up-sampled subsequences, and then interleaving the subsequences
][nvi][nxi^
][nxi^
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Structural Structural SubbandSubband DecompositionDecomposition
• Reconstruction of original sequence
T1−z ]1[ +−= Mnu
][nx+
+
1−z+
1−z
M
][0 nv
][1 nv
][1 nvM −
][2 nvM −
M
M
M
T
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Structural Structural SubbandSubband DecompositionDecomposition
• The digital filter structure generating the structural subband sequences can be considered as an M-channel analysis filter bank, characterized by M transfer functions contained in the vector
TM
T zHzHzHz ])()()([)( 110 −= LHTM zz )( 11)1( −−−−= eT
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Structural Structural SubbandSubband DecompositionDecomposition
• The digital filter structure forming the reconstructed sequence from the structural subband sequences can be considered as an M-channel synthesis filter bank, characterized by M transfer functions contained in the vector
])()()([)( 110 zGzGzGz M −= LG
Te ⋅= )(z
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SubbandSubband MatrixMatrix
• The transfer functions and have bandpass frequency responses for a suitably chosen subband matrix T
• Depending on the application, the matrix T can have various forms
• To be useful in practice, the matrix T should be simple, if possible, both in terms of its elements and its structure
)(zHi )(zGi
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SubbandSubband MatrixMatrix
• Structural simplicity is inherent in the DFT matrix , which can be efficiently implemented using well known FFT methods
• Here, the channel frequency responses have form, providing at least some
frequency selectivity
MW
ωω /sin
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SubbandSubband MatrixMatrix
• However, the elements of are given by
requiring conplex multiplications, choice of could also be advisable if only
very few sub-bands are desired
MWT =
1,0,/2, −≤≤==− MieWt MijiMi l
lll
π
MWT =
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SubbandSubband MatrixMatrix• For example, for M = 4, we have
which do not require any true multiplications
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−−
−−=
jj
jj
111111
111111
T
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−−−−==
jj
jj-
111111
111111
*411 TT
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SubbandSubband MatrixMatrix
• The corresponding magnitude responses are shown below
0 0.5π π 1.5π 2π 0
1
2
3
4
Normalized frequency
Mag
nitu
de
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SubbandSubband MatrixMatrix• Both structural and element-wise
simplicities are inherent in the Hadamard matrix , given by
where is the Hadamard matrix
and is the Kronecker roduct
222 RRRR ⊗⊗⊗= LMMR
⎥⎦⎤
⎢⎣⎡
−= 1111
2R2R
MM ×
22×
⊗
M2 terms
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SubbandSubband MatrixMatrix• From the definition it follows that the order
M of the Hadamard matrix must be a power-of-2, i.e.
• It can be shown that
• For M = 4,MMM
RR 11 =−
μ2=M
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−−−−=111111111111
1111T
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SubbandSubband MatrixMatrix• The corresponding magnitude responses are
shown below
• Somewhat higher frequency selectivity of the bandpass responses have been obtained with a slight modified form of the matrix
0 0.5π π 1.5π 2π 0
1
2
3
4
Normalized frequency
Mag
nitu
de
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SubbandSubband Discrete TransformsDiscrete Transforms• An interesting application of the structural
subband decomposition concept is in the approximate, but fast, computation of dominant discrete-transform samples
• Two particular discrete transforms considered here are:(1) Subband discrete Fourier transform,(2) Subband discrete cosine transform
• The concept can be extended to other types of transforms and higher dimensions
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SubbandSubband Discrete Fourier Discrete Fourier TransformTransform
• The N-point DFT X[k] of a length-N sequence x[n] is given by the N samples of its z-transform X(z) evaluated on the unit circle at N equally spaced points,
where
,][)(][1
0∑
−
==
== −N
n
nkNWz WnxzXkX kN
NjN eW
/2π−=
10 −≤≤ Nk
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SubbandSubband Discrete Fourier Discrete Fourier TransformTransform
• From the M-band polyphase decomposition of X(z)
with P = N/M integer, it follows that
[ ]⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
−
−−−
)(
)()(
1)(
1
1
0)1(1
MM
M
M
M
zX
zXzX
zzzXM
L
36
SubbandSubband Discrete Fourier Discrete Fourier TransformTransform
• the DFT samples can alternately be expressed in the form
where and is the P- point DFT of the polyphase component
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
〉〈
〉〈〉〈
=
−
−
][
][][
1][
1
10
)(
PM
PP
kMN
kN
kX
kXkX
WWkXM
L
Pkk P modulo=〉〈 ][kXi][nxi
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SubbandSubband Discrete Fourier Discrete Fourier TransformTransform
• Physical interpretation
][0 nx
][1 nx
][1 nxM −
][2 nxM −
M]1[ −+ Mnx
M
M
M
1−z
1−z
1−z
][nx P-point DFT
P-point DFT
P-point DFT
P-point DFT
+
+
+
][1 pM kX 〉〈−
][2 PM kX 〉〈−
][1 PkX 〉〈
][0 PkX 〉〈][kX
kNW
kNW
kNW
38
SubbandSubband Discrete Fourier Discrete Fourier TransformTransform
• For M = 2, we have
which describes the final twiddle-factor/ butterfly structure of a radix-2, decimation- in-time Cooley-Tukey (CT)-FFT
1T−1−z
(N/2)-point DFT
+
kNW
(N/2)-point DFT
+
1−
][kX
]2[NkX +2
2][nx
]1[ +nx
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SubbandSubband Discrete Fourier Discrete Fourier TransformTransform
• From the M-band structural sub-band decomposition of X(z)
with P = N/M integer, it follows that
[ ]⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
−
−−−
)(
)()(
1)(
1
1
0)1(1
MM
M
M
M
zV
zVzV
zzzXM
L T
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SubbandSubband DFTDFT• the DFT samples can alternately be
expressed in the form
where is the P-point DFT of the i-th structural subband component
• This is the general form of the subband discrete Fourier transform (SB-DFT)
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
〉〈
〉〈〉〈
⋅⋅=
−
−
][
][][
1][
1
10
)(
PM
PP
kMN
kN
kV
kVkV
WWkXM
L T
][kVi][nvi
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SubbandSubband DFTDFT• Physical interpretation
][0 nx
][1 nx
][1 nxM −
][2 nxM −
M
M
M
M
P-point DFT
P-point DFT
P-point DFT
P-point DFT]1[ −+ Mnx
1−z
1−z
1−z
][nx][kX
+
+
+k
NW
kNW
kNW
][1 PM kV 〉〈−
][2 PM kV 〉〈−
][1 PkV 〉〈
][0 PkV 〉〈
1−T T
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SubbandSubband DFTDFT• For M = 2 with , we have
• Note: is a lowpass signal, whereas, is a highpass signal
2RT =
210 0],[)1(][)1(][Nk
NkN kkVWkVWkX ≤≤⋅−+⋅+=
+
kNW
+
1−
][kX
]2[NkX +
(N/2)-point DFT
(N/2)-point DFT2
21−z
][nx
]1[ +nx
][nxL
][nxH
12−R 2R
][0 nv
][1 nv ][ 2/1 NkV 〉〈
][ 2/0 NkV 〉〈
][nxL][nxH
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SubbandSubband DFTDFT
• For , if the decimation by M = 2 is repeated times, a full-band SB-DFT algorithm results
• For , it contains a length-N fast Hadamard transform
ν2=N1−ν
MRT =
44
SubbandSubband DFTDFT• The number of multiplications required is
equal to , same as in the CT-FFT algorithm
• However, there are more additions than that required in the CT-FFT due to the implementation of
• In the general case with a different sub-band matrix T, additional multiplications may arise
NN 22 log⋅
)1(log2 2 −NN
1−MR
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SubbandSubband DFTDFT
• If the signal is a priori band-limited to a cut- off frequency , it may be simply down-sampled by a factor of M, and only N/M values feed a shorter FFT: the polyphase approach is then applicable
• If, however, the signal is not strictly band- limited, aliasing occurs
Mc /πω ≤
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SubbandSubband DFTDFT• In the subband approach aliasing effects are
reduced by the pre-filters
• Then, if the reduced aliasing is acceptable, branches can be dropped by pruning the SB- DFT and obtain approximate values of the dominant DFT samples
)(zHi
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SubbandSubband DFTDFT
• For example, if 1 band in an M-band subband decomposition is dominant, branches can be dropped and calculate a standard CT-FFT of length N/M of one decimated signal
• For an 8-band analysis, only 40% of the CT-FFT computer time is needed
)1( −M
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SubbandSubband DFTDFT• Approximate SB-DFT calculation with M =
4, , and dropping of 3 out of 4 bands4RT =
kNW
][nx
(N/4)-point DFT
1−z 1−z
1−z 1−z
1−z 1−z
4
4
4
4
+ +
14−R
kNW2
][0 kV][0 nv
][kX][* kNX −=
180 −≤≤Nk
~~
][][ kXkX ≈)1)(1(][ 20
kN
kN WWkV ++⋅=
108
−≤≤ Nk
~
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SubbandSubband DFTDFT• Adaptive band selection in the case of
Hadamard transform based sub-band DFT• Based on averaged (signs of) differences
between and in the 2-band DFT computation scheme shown below
][0 nv ][1 nv
+
kNW
+
1−
][kX
]2[NkX +
(N/2)-point DFT
(N/2)-point DFT2
21−z
][nx
]1[ +nx
][nxL
][nxH
12−R 2R
][0 nv
][1 nv ][ 2/1 NkV 〉〈
][ 2/0 NkV 〉〈
51
SubbandSubband DFTDFT
• In the general case of M > 2, the method is based on averaged (signs of) differences between corresponding subband component pairs
• The online estimation causes only a minor loss of computational advantage gained by the subband calculation
52
SubbandSubband Discrete Cosine Discrete Cosine TransformTransform
• The structural subband decomposition concept has also been applied to the approximate, but efficient, computation of the dominant samples of the DCT
• One of the most common forms of the DCT of a length-N sequence x[n], with N even, is given by
∑ −≤≤⎟⎠⎞
⎜⎝⎛ += 10,
2)12(cos][2][ Nk
NknnxkC π
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SubbandSubband DCTDCT• By applying the subband processing to x[n]
we can write
1−z
][nx
]1[ +nx
][nxL
][nxH
][0 nv
][1 nv2
2
+
+
1−
],[2
sin2][2
cos2][ 00 kSNkkC
NkkC ⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛= ππ
10 −≤≤ Nk
_ _
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SubbandSubband DCTDCTwhere
with denoting the (N/2)-point DCT (discrete cosine transform) of
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
−≤≤+−−
=
−≤≤
=
112
],[2
,0
12
0],[
][
0
0
0
NkNkNC
Nk
NkkC
kC_
][0 kC][0 nv
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SubbandSubband DCTDCTand
with denoting the (N/2)-point DST (discrete sine transform) of
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
−≤≤+−
=−
−≤≤
= ∑−
=
112
],[
2,][)1(2
12
0],[
][
1
2/)2(
01
1
1
NkNkNS
Nknx
NkkS
kSN
n
n_
][1 kS][1 nv
56
SubbandSubband DCTDCT• The computation of the N-point DCT C[k]
requiring the computation of an (N/2)-point DCT and an (N/2)-point DST has been referred to as the subband DCT
• The above process can be continued to decompose the sub-sequences and
, provided N/2 is an even integer• The process terminates when the final
subsequences are of length 2
][1 kS][0 kC
][1 nv][0 nv
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SubbandSubband DCTDCT
• By exploiting the spectral contents of the subsequences, an efficient DCT algorithm can be developed
• For example, if x[n] is known to have most of its energy in the low frequencies, a reasonable approximation to C[k] can be obtained by discarding terms associated with high frequencies
58
SubbandSubband DCTDCT
• The resulting approximation is given by
• The SB-DCT concept can be extended to higher dimensions
⎪⎩
⎪⎨⎧ −≤≤⎟
⎠⎞
⎜⎝⎛
≅otherwise,0
12
0][2
cos2][ 0NkkC
Nk
kCπ
60Original BABOON image
Image Compression ApplicationImage Compression Application
61Standard DCT, compr 50 Sub-band DCT, compr 50
Image Compression ApplicationImage Compression Application
62Standard DCT, compr 100 Sub-band DCT, compr 100
Image Compression ApplicationImage Compression Application
63Original PEPPERS image
Image Compression ApplicationImage Compression Application
64Standard DCT, compr 50 Sub-band DCT, compr 50
Image Compression ApplicationImage Compression Application
65Standrard DCT, compr 100 Sub-band DCT, compr 100
Image Compression ApplicationImage Compression Application
66
Efficient FIR Filter Design and Efficient FIR Filter Design and ImplementationImplementation
• Consider an FIR filter H(z) with an impulse response{h[n]} of length
• By applying the structural subband decomposition to H(z) we arrive at
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=
−
−−−
)(
)()(
1)(
1
1
0)1(1
MM
M
M
M
zF
zFzF
zzzHM
L T
MPN ×=
67
Efficient FIR Filter Design and Efficient FIR Filter Design and ImplementationImplementation
• The M-band structural subband decomposition of H(z) can be alternately expressed as
where is given by
∑−
==
1
0)()()(
M
k
Mkk zFzGzH
10,)(1
0, −≤≤= ∑
−
=
− MkztzGM j
kkl
l
)(zGk
68
Efficient FIR Filter Efficient FIR Filter ImplementationImplementation
• Realizations of H(z) based on the structural subband decomposition are as follows:
1−z
1−z
1−z
1−T+
][nx
][ny
)(0MzF
)(1MzF
)(2MzF
)(1M
M zF −
69
Efficient FIR Filter Efficient FIR Filter ImplementationImplementation
• Parallel IFIR realization
+
][nx
][ny
)(1 zMG −
)(0MzF
)(1MzF
)(2MzF
)(1M
M zF −
)(1 zG
)(0 zG
)(2 zG
70
Efficient FIR Filter Efficient FIR Filter ImplementationImplementation
• Thus the second realization can be considered as a generalization of the interpolated FIR (IFIR) structure, where
is the interpolator and , the shaping filter, is of length P = N/M
• Note: Delays in the implementation of the sub-filters in both realizations can be shared leading to a canonic realization of the overall structure
)(zFk
)( Mk zF
)(zGk
71
Efficient FIR Filter Efficient FIR Filter ImplementationImplementation
• Further generalization obtained by choosing the number of bands M (i.e. the sub-band transform size) different from the sparsity factor L of the subfilters )( Lk zF
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=
−
−−−
)(
)()(
1)(
1
1
0)1(1
LM
L
L
M
zF
zFzF
zzzHM
L T
72
Efficient FIR Filter Efficient FIR Filter ImplementationImplementation
• Corresponding realization
1−z
1−z
1−z
1−T+
][nx
][ny
)(0LzF
)(1LzF
)(2LzF
)(1L
M zF −1−M
1
2
0
73
Efficient FIR Filter Efficient FIR Filter ImplementationImplementation
• For , the modified structure can realize any FIR transfer function H(z) of length up to , where P is the length of
• Coefficients of are no longer unique, resulting in an infinite number of realizations for a given H(z) with fixed L and M
• For L < M, there is an increase in the number of multipliers
ML ≤
MLPN +−= )1()(zFk
)(zFk
74
Efficient FIR Filter Efficient FIR Filter ImplementationImplementation
• Computational complexity of the overall structure can be reduced by choosing “simple” invertible transform matrices Tsuch as the Hadamard matrix
• Each interpolator section is a cascade of μ basic interpolators of the form
75
Efficient FIR Filter Efficient FIR Filter ImplementationImplementation
• For an M-branch decomposition, the interpolator has a lowpass magnitude response given by
• The interpolator has a highpass magnitude response given by
• The remaining interpolators with have each a bandpass magnitude response
)(0 zG
)2/sin()2/sin()(0 ω
ωω MeG j =)(1 zG
]2/)sin[(]2/)(sin[)(1 ωπ
ωπω−
−= MeG j
)(zGk 1,0≠k
76
Efficient FIR Filter Efficient FIR Filter ImplementationImplementation
• Each of the branches thus contributes to the overall response essentially within a “subband” associated with the corresponding interpolator
• For a narrow-band FIR filter, it may be possible to drop branches from the overall structure if these branches do not contribute significantly to the filter’s frequency response, thus leading to a computationally efficient realization
77
Efficient FIR Filter Efficient FIR Filter ImplementationImplementation
• For L = M, the coefficients of the subfilters can be expressed in terms of the coefficients {h[n]} of the overall filter H(z):
• Each subfilter has, in general, P non-zero coefficients
][nfk)(zFk
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+
+⋅⋅=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
− )]1([
][][
1
]1[
]1[]0[
PMkh
Mkhkh
MPf
ff
M
k
kk
MMR
79
Efficient FIR Filter Efficient FIR Filter ImplementationImplementation
• Simpler realizations are obtained in the case of linear-phase FIR filters
• The 4-branch realization of a length-8 type 2 FIR filter is shown below
+ y[n]x[n]
]0[0f
]0[3f
]0[5f
]0[6f
11 −+ z
11 −− z
11 −− z
11 −+ z
21 −− z
21 −− z
21 −+ z
21 −+ z
41 −+ z
41 −+ z
41 −− z
41 −− z
80
Efficient FIR Filter DesignEfficient FIR Filter Design
• The structural subband decomposition of an FIR transfer function H(z) simplifies considerably the filter design process
• To this end, two different design approaches have been advanced
81
Efficient FIR Filter DesignEfficient FIR Filter Design
• In one approach, each branch is designed one-at-a-time using either a least-squares minimization method or a minimax optimization method
• In the other approach, each subfilter is designed using a frequency sampling method
82
Efficient FIR Filter DesignEfficient FIR Filter Design
• Let H(ω) denote the amplitude function of a linear-phase frequency response
• For the parallel IFIR structure we then have
where and are the amplitude functions of the k-th interpolator and the k-th sub-filter, respectively
∑−
=ωω=ω
1
0)()()(
M
kkk MFGH
)(ωkG )( Mk ωF
o
83
• Filter design problem - Determine the N/2M coefficients of each sparse subfilter for to approximate a specified )(ωH
)( Mk zF
Efficient FIR Filter DesignEfficient FIR Filter Design
1,,1,0 −= Mk K
84
Efficient FIR Filter DesignEfficient FIR Filter Design
Least-squares optimization -• By taking the samples of the respective
amplitude functions at D suitably chosen discrete frequency points in the interval
, we can writeπω ≤≤0
∑−
==
1
0
M
kkkfGh
~~~
85
Efficient FIR Filter DesignEfficient FIR Filter Design
• where- a vector representing the discretized version of- a diagonal matrix with diagonal elements given by samples of- a column vector containing samples of
)(ωH
)(ωkG
)( Mk ωF
h~
kG~
kf~
86
Efficient FIR Filter DesignEfficient FIR Filter Design
• If denotes the desired amplitude response samples of the parallel IFIR structure, the approximation error is then given by
• Design objective - Minimize the -norm of e separately with respect to each of the sub-filters
2L
∑−
=−=−=
1
0
M
kkkdd fGhhhe
~~~ ~ ~
dh~
87
Efficient FIR Filter DesignEfficient FIR Filter Design
• The minimization procedure results in the determination of the coefficients of all sub-filters from which the impulse response samples of the overall filter can be obtained
• The computational complexity of the modified least-squares method is smaller by a factor of 1/M compared to that of the direct least-squares method
][nfk
88
Efficient FIR Filter DesignEfficient FIR Filter Design
• Example - Design a linear-phase lowpass FIR filter of length 128 using an 8-band decomposition
• Filter specifications: passband edge at 0.02π and stopband edge at 0.04π
•
The gain response of the filter designed using the least-squares approach is shown on the next slide
89
Efficient FIR Filter DesignEfficient FIR Filter Design• Gain response
90
Efficient FIR Filter DesignEfficient FIR Filter DesignMinimax optimization -• Here, the weighted error of approximation
for a linear-phase filter design is given by
where is the desired amplitude response and is a weighting function
∑−
=−=
1
0
M
kkkd MFGHWE )]()()()[()( ωωωωω
)(ωdH)(ωW
91
Efficient FIR Filter DesignEfficient FIR Filter Design
• The optimization is carried out over one subfilter at a time using the Remez method
• The computational complexity of the structural subband based method is smaller by a factor of 1/M compared to that of the Parks-McClellan method
92
Efficient FIR Filter DesignEfficient FIR Filter Design
• Example - Design a bandpass FIR filter of length with passband edges at 0.15π
and
0.16π, and stopband edges at 0.1π
and 0.21π, respectively
• Passband and stopband ripples are assumed to have equal weights
• Assume a 8-band decomposition
93
Efficient FIR Filter DesignEfficient FIR Filter Design• Gain response
94
Efficient FIR Filter DesignEfficient FIR Filter Design
• It is possible to design a nearly optimum FIR filter, based on a 2-band Hadamard- matrix based structural subband decomposition, by applying the minimax routine to each of the two smaller size subfilters without repeated iterations and combining the paths
95
Efficient FIR Filter DesignEfficient FIR Filter Design
Frequency-sampling approach• Here, simple analytical expressions for the
passband, transition band, and the stopband are first sampled at equally-spaced points on the unit circle to arrive at the original frequency samples, , , of the overall parallel IFIR structure
10 −≤≤ Nm)(mĤ
96
Efficient FIR Filter DesignEfficient FIR Filter Design
• From the desired frequency samples of the subfilters, , ,
, are then determined using
where B and is an DFT matrix
10 −≤≤ Pl
][diag )( ll L 11 −= MNN WWMW MM ×
)(mĤ)(lkF
^
10 −≤≤ Mk
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
−+
+⋅⋅⋅= −−−
− ))1((
)()(
)(
)()(
111
1
10
MPH
PHH
F
FF
M
M lMll
lMll
WBT
^^
^^
^^
97
Efficient FIR Filter DesignEfficient FIR Filter Design
• An IDFT of the vector of the frequency samples of each subfilter yields its impulse response samples
• Example - Design a half-band FIR filter with a passband ripple of and a stopband ripple of using a 4-band decomposition
0013.0=δ p001.0=δs
98
Efficient FIR Filter DesignEfficient FIR Filter Design• Gain response
99
Efficient Decimator and Efficient Decimator and Interpolator StructuresInterpolator Structures
• Structural sub-band decomposition-based structure can be computationally more efficient than the conventional polyphase decomposition-based structure in realizing decimators and interpolators employing linear-phase Nyquist filters
• To this end, it is necessary to use transform matrices that transfer the filter coefficient symmetry to the sub-filters
100
Efficient Decimator and Efficient Decimator and Interpolator StructuresInterpolator Structures
• A factor-of-4 interpolator structure
1−z 1−z
+
+
++
+
+
+
4
4
4
4
+
+
1−z
1−z
1−z4R
][10f ][00f
][02f][13f ][03f
][01f
101
SubbandSubband Adaptive FilteringAdaptive Filtering• Based on the generalized structural sub-
band realization
1−z
1−z
1−z][nx
T+ ][ny
)( LzF0
)( LzF1
)( LM zF 2−)( LM zF 1−
Adaptation algorithm + ][nd][ne +
_
][nv0
][nv1
][nvM 2−
][nvM 1−
102
SubbandSubband Adaptive FilteringAdaptive Filtering
• Here, the input signal x[n] is first processed by a fixed unitary transform T, generating the signals , which are then filtered by the sparse adaptive sub-filters
MM ×][nvi
)( Li zF
103
SubbandSubband Adaptive FilteringAdaptive Filtering
• For large values of M, recursive DFT or DCT algorithms are computationally more efficient to implement the transform T than the FFT-type algorithms
• For small values of M, dedicated fast non- recursive algorithms are preferred to implement the transform T
104
SubbandSubband Adaptive FilteringAdaptive Filtering• The output y[n] can be expressed as
where v[n]
is the vector of transformed inputs, andf
is the subfilter coefficient vector containing the -th coefficient of each sub-filter
∑−
=⋅−=
1
0
MT nLnny
lll ][f][v][
[ ]TM nvnvnv ][][][ 110 −= L
[ ] TM nfnfnfn ][][][][ ,,, llll L 110 −=
l
105
SubbandSubband Adaptive FilteringAdaptive FilteringNormalized LMS Algorithm -• The subfilter coefficient vector update
equation is given by
• where μ is the adaptation step size, and is an diagonal matrix containing the power estimates of
],[*][][][ Lnnenfnf lll −Λ+=+ v221 μ
2ΛMM ×
110 −= P,,, Kl
][nvi
106
SubbandSubband Adaptive FilteringAdaptive Filtering
• For M = L = N, i.e., P =1 (in which case each of the sub-filters consists of a single coefficient), the proposed method reduces to the transform-domain LMS algorithm
• For M = L = 1, and T = 1, the proposed method reduces to the conventional time- domain LMS algorithm
107
SubbandSubband Adaptive FilteringAdaptive Filtering• The sub-band adaptive filter structure offers
additional flexibility in the choice of the number of sub-bands M and the sparsity factor L
• This feature is attractive in the case of higher- order adaptive filters, as it provides a reduction in the computational complexity compared to the transform-domain algorithm and improved convergence performance compared to the LMS algorithm
108
SubbandSubband Adaptive FilteringAdaptive Filtering
• Choice of a transform T with good frequency selection decreases the correlation among the transformed signals, which can be used to obtain a significant improvement in the convergence speed of the LMS algorithm for colored input signals
• In these cases, the DFT or DCT have been found to be useful
109
SubbandSubband Adaptive FilteringAdaptive Filtering• The contribution of each sub-filter is mainly
restricted to a frequency sub-band, which can be used advantageously to increase the speed of convergence of the adaptive algorithm
• The structure also has the flexibility of allowing sub-bands not contributing greatly to the overall frequency response to be removed, reducing the number of operations needed for the filter implementation
110
SubbandSubband Adaptive FilteringAdaptive Filtering
• Example - We examine the behavior of the subband adaptive line enhancer (ALE)
• Input consists of a single sinusoid of unit amplitude plus white Gaussian noise with a variance 0.25 (SNR = 3 dB)
• We choose N = 128, M = 8, P = 16• For a DCT transform matrix we choose L = 8• For a DFT transform matrix we choose L = 4
111
SubbandSubband Adaptive FilteringAdaptive Filtering• The coefficients were updated using the
LMS algorithm• The output power spectra estimated using
averaged periodograms of 16 data blocks of length 512 for the different ALE structures
• In the DCT structure, 2 bands out of 8 were kept
• In the DFT structure, 1 band out of 8 were kept
112
SubbandSubband Adaptive FilteringAdaptive Filtering• In both DCT and DFT cases, the number of
operations required for the ALE implementation was about 1/4-th of those required in the conventional ALE implementation
• Further savings in the number of operations in the subband ALE approach results when a frequency estimate of the input sinusoid is required
113
SubbandSubband Adaptive FilteringAdaptive Filtering
• Output power spectra for
17.0=ωo
17.0=ωo
114
SubbandSubband Adaptive FilteringAdaptive Filtering
• Output power spectra for the subband ALE structures show some minor peaks due to band removals which may be acceptable in most applications
• Subband ALE approch has been used in acoustic echo cancellation and adaptive channel equalization
Structural Sub-band Decomposition: A New Concept in Digital Signal ProcessingOutlinePolyphase DecompositionPolyphase DecompositionPolyphase DecompositionPolyphase DecompositionPolyphase DecompositionPolyphase DecompositionPolyphase DecompositionPolyphase DecompositionPolyphase DecompositionPolyphase DecompositionPolyphase DecompositionPolyphase DecompositionStructural Subband DecompositionStructural Subband DecompositionStructural Subband DecompositionStructural Subband DecompositionStructural Subband DecompositionStructural Subband DecompositionStructural Subband DecompositionStructural Subband DecompositionStructural Subband DecompositionStructural Subband DecompositionSubband MatrixSubband MatrixSubband MatrixSubband MatrixSubband MatrixSubband MatrixSubband MatrixSubband MatrixSubband Discrete TransformsSubband Discrete Fourier TransformSubband Discrete Fourier TransformSubband Discrete Fourier TransformSubband Discrete Fourier TransformSubband Discrete Fourier TransformSubband Discrete Fourier TransformSubband DFTSubband DFTSubband DFTSubband DFTSubband DFTSubband DFTSubband DFTSubband DFTSubband DFTSubband DFTSubband DFTSubband Discrete Cosine TransformSubband DCTSubband DCTSubband DCTSubband DCTSubband DCTSubband DCTImage Compression ApplicationImage Compression ApplicationImage Compression ApplicationImage Compression ApplicationImage Compression ApplicationImage Compression ApplicationEfficient FIR Filter Design and ImplementationEfficient FIR Filter Design and ImplementationEfficient FIR Filter ImplementationEfficient FIR Filter ImplementationEfficient FIR Filter ImplementationEfficient FIR Filter ImplementationEfficient FIR Filter ImplementationEfficient FIR Filter ImplementationEfficient FIR Filter ImplementationEfficient FIR Filter ImplementationEfficient FIR Filter ImplementationEfficient FIR Filter ImplementationEfficient FIR Filter ImplementationEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient FIR Filter DesignEfficient Decimator and Interpolator StructuresEfficient Decimator and Interpolator StructuresSubband Adaptive FilteringSubband Adaptive FilteringSubband Adaptive FilteringSubband Adaptive FilteringSubband Adaptive FilteringSubband Adaptive FilteringSubband Adaptive FilteringSubband Adaptive FilteringSubband Adaptive FilteringSubband Adaptive FilteringSubband Adaptive FilteringSubband Adaptive FilteringSubband Adaptive FilteringSubband Adaptive Filtering