structural subband decomposition: a new concept in digital ... · concept in digital signal...

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


• 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 =

5


• Physical Interpretation – 2-Band Case

z2

2

][nx ][0 nx

][1 nx

6


• 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 =

7


• Physical Interpretation – 3-Band Case

z3

3

][nx ][0 nx

][1 nx

3z

][2 nx

8

PolyphasePolyphase DecompositionDecomposition• Physical Interpretation – General Case

zM][nx ][0 nx

][1 nx

z][1 nxM −

M

M

M

z][2 nxM −

9


• 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

10

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

11

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

12

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 −

13

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

14


• 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

15

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 ×

16


• 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

17


• 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

18


• 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~

19


• 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

20


• 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

21


• A delayed version of the input u[n] can be developed by first up-sampling the M subsequences and then generating the subsequences by a linear combination of these up-sampled subsequences, and then interleaving the subsequences

][nvi][nxi

^

][nxi^

22


• 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

23


• 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

24


• 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

25

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

26


• 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

27


• 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 Miji

Mi llll

π

MWT =

28

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

29


• The corresponding magnitude responses are shown below

0 0.5π π 1.5π 2π 0

1

2

3

4

Normalized frequency

Mag

nitu

de

30

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 ⊗⊗⊗= LM

MR

⎥⎦⎤

⎢⎣⎡

−= 1111

2R2R

MM ×

22×

⊗

M2 terms

31

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

32

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

33

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

34

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 k

N

NjN eW /2π−=

10 −≤≤ Nk

35


• 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


• 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

37


• 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


• 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

39


• 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

40

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

41

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

42

SubbandSubband DFTDFT• For M = 2 with , we have

• Note: is a lowpass signal, whereas, is a highpass signal

2RT =

210 0],[)1(][)1(][ NkN

kN 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

43

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

NN22 log⋅

)1(log2 2 −NN

1−MR

45


• 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 ≤

46

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

47


• 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

48

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

kNW 2

][0 kV][0 nv][kX

][* kNX −=

180 −≤≤ Nk

~~

][][ kXkX ≈

)1)(1(][ 20

kN

kN WWkV ++⋅=

108

−≤≤ Nk

~

50

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


• 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

53

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

_ _

54

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

55

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

57

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


63Original PEPPERS image


65Standrard DCT, compr 100 Sub-band DCT, compr 100


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


• Parallel IFIR realization

+

][nx

][ny

)(1 zMG −

)(0MzF

)(1MzF

)(2MzF

)(1M

M zF −

)(1 zG

)(0 zG

)(2 zG

70


• 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


• 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 )( L

k zF

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

−−−

)(

)()(

1)(

1

1

0)1(1

LM

L

L

M

zF

zFzF

zzzHM

L T

72


• Corresponding realization

1−z

1−z

1−z

1−T+

][nx

][ny

)(0LzF

)(1LzF

)(2LzF

)(1L

M zF −1−M

1

2

0

73


• 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


• 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


• 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


• 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


• 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


• 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


• 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


• 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


1,,1,0 −= Mk K

84


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


• 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


• 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


• 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


• 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


• 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


• 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


94


• 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


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)(mH

96


• 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 WW

MW MM ×

)(mH)(lkF^

10 −≤≤ Mk

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−+

+⋅⋅⋅= −−−

− ))1((

)()(

)(

)()(

111

1

10

MPH

PHH

F

FF

M

M lMll

lMll

WBT

^^

^^

^^

97


• 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


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


• 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


• 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


• 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


• 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


• Output power spectra for

17.0=ωo

17.0=ωo

114


• 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 subband decomposition: a new concept in digital ... · concept in digital signal...

Documents