ALIAS FREE SUBBAND ADAPTIVE FILTERING – A NEW APPROACH

8/2/2019 ALIAS FREE SUBBAND ADAPTIVE FILTERING A NEW APPROACH

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Abstract - In order to overcome the limitations of the fullband

and the previously proposed subband structures a new subband

structure is proposed. This newly proposed subband adaptive

filtering (SAF) structure will converge faster at a lower compu-

tational cost. The interband aliasing is introduced by the

downsampling process, degrades the performance. So the inter-

band aliasing is extracted and removed from the subband sig-

nal with the help of bandwidth increased analysis filter. Here,

an almost alias free SAF structure with critical sampling is pro-

posed to have faster convergence and lower computational com-

plexity. The software used in this paper is mat lab for the simu-lation purpose with a signal processing tool box.

Keywords: Adaptive filtering, interband aliasing, LMS algorithm,

multirate signal processing, and subband adaptive filtering.

I. INTRODUCTIONOne promising method that improves the performance

and reduces the computational cost is subband adaptive

filtering (SAF). It has the potential for a faster conver-

gence and a lower computational complexity than a

fullband structure. However, a subband structure suf-

fers from two deficiencies. First, the interband aliasing

that is introduced by the downsampling process re-quired in reducing the data rate is unavoidable and de-

grades the performance. Second, the filter bank intro-

duces additional computation and system delay. For

these reasons various SAF structures were proposed. In

SAF, signals are decomposed into a number of subband

signals using an analysis filter bank, and the adaptive

filtering is performed on each subband. The result in

each subband is combined into an output using a syn-

thesis filter bank. For a critically sampled SAF with M

subbands, the kth subband error Ek(z), shown in Fig.1,

is

Where X(z), H(z), Hk(z) and Gk(z) represent the z-

transforms of input signal, unknown system, the kth

analysis filter, and the kth adaptive filter to respective-

ly.

Now W

i

m= e

-j2i/M

and the kth analysis filter Hk(z) is abandpass filter .Equation (1) can be written as

Where

Fig.1. Block diagram of the adaptive filtering in the kth sub band.

When the filter bank is real-valued, the terms Hk(z1/M

WkM) and Hk(z1/M Wk+1M) are adjacent to Hk(z

1/M) and

whenHk(z) k(z)is designed with high enough stopband

attenuation, is approximately zero. The second and

third terms on the right-hand side of (2) are the errors

introduced by interband aliasing due to the overlapping

betweenHk(z1/M WkM) andHk(z

1/M) and betweenHk(z1/M

Wk+1M) and Hk(z1/M) . In order to reduce |Ek(z)| to a

value close to zero, the adaptive filter Gk(z) has to

match the two different frequency responses, that is,

bothHk(z1/MWkM) andHk(z1/M) which have considerableoverlap and both Hk(z

1/M Wk+1M) and Hk(z1/M) which

also have considerable overlap between them. Of

course, this is impossible. These mismatches result in

a large |Ek(z)| around = kand (k+1)even after con-

vergence, therefore, the SAF with critical sampling has

a large MSE at the output.

II. ALIAS FREE SAF WITH CRITICAL

SAMPLINGThe interband aliasing components are caused by

downsampling the signal, the downsampling process is

essential in almost all multirate signal processing for

making the overall data rate nearly equivalent to that of

the input. Fig.2 shows the magnitude responses

ALIAS FREE SUBBAND ADAPTIVE FILTERING A NEW APPROACH

1CHAPPIDI ASHOK, 2B.RAMESH BABU1,2Department of Electronics & Communication Engineering,, Dr. Samuel George Institute of Engineering & Technology,

Markapur, Prakasam (Dist.) A.P., INDIA

e-mail: [email protected], [email protected]

International Journal ofSystems , Algorithms &

ApplicationsIIIIJJJJSSSSAAAA AAAA

Volume 2, Issue ICTM 2011, February 2012, ISSN Online: 2277-2677 39

ICTM 2011|June 8-9,2011|Hyderabad|India

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Xkejof the signal that has passed through the kth analy-

sis filter, whose transition bandwidth is /M and its

down sampled version Xdkejfor k = 0, 1,.., M-1. Hence-

forth, only the frequency intervalofk (k+1)will

be considered, since the frequency responses in the other

intervals are just the frequency-shifted and the frequency

-flipped versions of the response in this interval. As

shown in Fig.2 (b), the frequency interval of k

(k+1) is divided into three subintervals as

1 = {/kk+ }

2 = {/k+ (k+1)- }

3 = {/(k+1)- (k+1)}

For 1 the errorEk(ej) can be approximated as

For 2,

finally, For 3, it can be approximated as

Fig.2. Magnitude responses of the signals in the kth subband. (a)

Output of the kth analysis filter. (b) Its downsampled version.

From (4) and (6), the kth adaptive filter Gk(ej) has to

match the desired response H(ej) while simultaneously

matching the aliased response H(ej(-2k)/M) at 1 and

the aliased response H(ej(

-2

(k+1))/M) at 3 inorder tom a k e |Ek( e

j ) z e r o . T h i s i s i m p o s s i b l e .

Fig.3 .Alias-free SAF structure with critical sampling in the kth sub-

band.

In Fig.3, the kth subband portion of the proposed SAF

structure is shown. Hk(ej) is a linear-phase bandwidth-

increased analysis filter of the order 2MNdwhereis an

integer and determines the order ofHk(ej) the interband

aliasing component Xbk(ej) is extracted using the band-

width-increased analysis filterHk(ej) and then its deci-

mated version Xbd

k(ej) is subtracted from the subband

signal Xdk(ej) to obtain the almost alias-free Xnk(e

j).

The spectral dips ofXnk(ej) are reduced using a mini-

mum-phase filter W(ej), and the result Xwk(ej) is used

as an input of the adaptive filter Gk(ej) . The same anal-

ysis is also applied to Dk(ej), which is the kth subband

signal of the desired signal D(ej) and thus the almost

alias-free subband desired signalDwk(ej) with flat spec-

trum is obtained and used as a desired signal of the

adaptive filter Gk(ej). The output ofGk(e

j), Ewk(ej),

passes through 1/W(ej) to negate the effect of W(e

j),

and thenEk(ej) is obtained. In the kth subband,Xdk(e

j)

is given as

Where

The analysis filtersHk(ej) can be designed such that the

overlapping k(ej) between nonadjacent terms can be

much smaller than the overlapping between adjacentones and thus negligible for k= 0, 1 M-1. In order to

eliminate the aliasing term Xdk(ej) from the aliasing

components should be obtained first. To do so, the out-

put signal of the analysis filter in Xk(ej) fig.4(a) is sub-

tracted from the interpolated signal Xik(ej) in Fig. 4(b),

and the result isXik(ej) as

Where M is multiplied toXik(ej) since the amplitude ofXk(e

j) is reduced by M during the downsampling proce-

dure. As shown in Fig.4(c), Xsk(ej) includes only the

interband aliasing components in k/M (k+1)/M

The interband aliasing components are extracted by

passing Xsk(ej) through kth bandwidth-increased analy-

sis filter Hk(ej) , whose magnitude response is almost-

flat from k to (k+1)/Mas shown in fig.4(d); this is the

reason why the term bandwidth-increased is used for

Hk(ej) . In general, the frequency interval where the

magnitude response of the passband of the filter bank is

almost-flat is narrower than that of filter Hk(ej) for a

perfect reconstruction (PR) of the filter bank.

ALIASFREESUBBANDADAPTIVEFILTERINGANEWAPPROACHInternational Journal ofSystems , Algorithms &




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Fig. 4. Magnitude responses of the signals of the proposed SAF

structure. (a) Output signal of the kth analysis filter. (b) Interpolated

version ofXdk(ej). (c) Resulting signal after subtractingXk(e

j)

fromM.Xik(ej). (d) Kth bandwidth-increased analysis filter. (e)Extracted interband aliasing component.

In 0

,H

k(e

j)overlaps predominantly with

X(e

j(-2/

M)) when i= k and I = k+1, thusXbk(ej)is given as

which is shown in Fig.4(e), thusXbk(ej). is given as

Where

After down sampling Xbk(ej), an unwanted interband

aliasing occurs due to the overlapping between Xb

k(ej

).in - 0 represented in (11) and Xbk(e

j)in 0

represented in (10).The downsampled interband aliasing

component is

As shown in Fig.4, the kth almost alias-free subband

input signal is given as

by inserting (7) and (13) into (14) and then representingHk(e

j)as e-jNdk(k+1), Xnk(e

j). is given by

WithXk(ej)=Hk(e

j)X(ej),Xnk(ej) at the three subinter-

vals i.e., is given as (16). This analysis also applied to

Dnk(ej) as shown in Fig.4, and thus Dnk(e

j) is given as

(17). IfXnk(ej) andDnk(ej) are used for adaptive filter-ing in subbands instead ofXdk(e

j) andDdk(ej) in the kth

subband, the output of the adaptive filterEk(ej) is given

as (18). This is not the case as in (4) and (6) where Gk

(ej) has to match the desired H(ej/M) while simultane-

ously matching the aliased responses H(ej(-2k)/M) at

1 and H(ej(-2k)/M) at 3 in order to make |Ek(e

j)|

zero, From (18), Gk(ej) needs to match just the desired

response H(ej/M) to make |Ek(ej)|zero. Therefore, the

interband aliasing error can be zero.

Fig.5 (a) Subband input with interband aliasing. (b) Downsampled

version of the extracted interband aliasing component. (c) Almost

alias-free subband input. (d) Filter for the spectral flatness of the

almost alias-free subband signal. (e) Almost alias-free subband input

with the flat spectrum.

As shown in Fig.5c,Xnk(ej) has spectral dips at the sub-

intervals 1 and 3 and so does Dnk(e

j) . The spectral

dips at 1 and 3 can be reduced by using a minimum-phase filter W(ej) that has the following ideal magni-

tude response

The design procedure of is as follows: first, we design a

minimum-phase bandpass filter with passband 2 and

transition bands 1 and 3.The designed minimum-

phase bandpass filter is inverted to give W(ej), and thus

W(ej) is designed to be a minimum-phase filter. Theoutput ofW(e

j) in the kth subbandXwk(ej) is given as

Xwk(ej) = W(ej)Xnk(e

j) -(20)

The kth desired signal Dwk(ej) is obtained using W(e

j)

in the same way as Xwk(ej). Therefore, when Xwk(e

j)





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and Dwk(ej) are used for adaptive filtering in subbands

instead ofXdk(ej) andDdk(e

j) in the kth subband, is giv-

en asEwk(ej) (21). As represented in (21), the kth adap-

tive filter Gk(ej) needs to match onlyH(e

j/M) as in (18)

achieving the spectral flatness. The output ofEk(ej) the

Adaptive filter is obtained by passingEwk(ej) through 1/

W(ej) to negate the effect ofW(ej). Since W(z)is de-

signed to be a minimum-phase filter, 1/W(z) is imple-

mented by inverting W(z). The bandwidth-increased

analysis filters can be designed by cosine modulation of

the prototype filter. The analysis and synthesis filters of

the CMFB are obtained by cosine modulation of the pro-

totype linear-phase lowpass filter. If p[n]is the impulse

response of the FIR prototype filter, the impulse re-

sponses of kth analysis and synthesis filters, hk[n] and, fk[n]are given as

Where L is the order of the FIR prototypefilter, and k=

(-1)k(/4) for k=0,1,...M-1. The term kon the right-hand

side of (22) and (23) is used to cancel the interband ali-

asing component at the output of the filter band and sig-

nificant distortion around frequencies =0 and = re-

spectively to obtain PR. However, the bandwidth-

increased analysis filter band does not need the PR con-

dition, thus kis discarded here. Therefore, if p[n] is

the impulse response of the liner-phase FIR bandwidth-

increased prototype filter, the impulse response of thekth bandwidth-increased analysis filter hk[n] is given as

follows

Where L is the order of the FIR bandwidth-increased

prototype filter for k=0,1,...M-1. In order to design the

magnitude response of the kth bandwidth-increased anal-

ysis filter to be unit from k/M(k+1)/Mthe magni-

tude response of the prototype lowpass filter should be

unit from 0 to /2M Such condition on the magnitude

response of the prototype filter causes amplitude distor-

ton toH0(ej) around =0 and toHM-1(ej)around =but no amplitude distortion for k=1,2...M-2

II. SIMULATIONSFour simulation experiments were conducted.

A .White Noise Input - The simulations using the pro-

posed SAF were conducted by varying the number of

subbands, that is, forM= 2, 4, 8, and 16.The MSE evo-

lutions of the proposed SAF and the fullband NLMS

algorithm are shown in Fig.6. The figure shown that the

convergence rates of both the proposed SAF and the

fullband NLMS algorithms were similar for a white

noise input, however, the fullband reached lower steady

state MSE than the proposed after convergence.

Fig.6. MSE performance for the proposed SAF algorithm for differ-

ent values of M and the fullband NLMS algorithm for a white noise

input.

B. Colored Input- The simulation using a colored input

was conducted. The MSE evolution is shown in Fig.7.

The proposed SAF algorithm performed better than the

fullband NLMS algorithm. Increasing the number of

subbands led to better convergence rate, but larger MSE

after convergence

Fig.7.MSE performance for the proposed SAF algorithm for different

values of M and the fullband NLMS algorithm for a colored input.

C. Speech - An acoustic echo cancellation (AEC) exper-

iment was conducted with real speech sampled at 8kHz. The residual echoes using the proposed eight-

channel SAF algorithm and the fullband NLMS algo-

rithm are shown in Fig.8. It shows the proposed SAF

algorithm cancelled the echo better than the fullband

NLMS algorithm.

Fig.8. Residual echo signals using the proposed eight-channel alias-

free SAF algorithm and the fullband NLMS algorithm for speech

input.

D. Comparison of Performances - The performance of

the proposed SAF algorithm was compared to those of

the SAF algorithms when M = 4. For a colored noise of

length 200000, simulation result shown in Fig.9 was

obtained. The proposed SAF achieved faster conver-

gence rate and lower MSE than the other algorithms.





[ ] [ ] ( ) )22(2

5.0cos2

+

+= kk

Lnk

Mnpnh

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5.0cos2

+= kk Lnk

Mnpnf

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Lnk

Mnpnh k


5/5

Fig.9. Comparison of the performances of the proposed alias-free

SAF algorithm and the conventional SAF algorithms for a colored

input when M = 4.

III. CONCLUSIONIn this paper, a structure with critical sampling that is

virtually alias-free is proposed. The interband aliasing is

extracted in each subband using the bandwidth-increased

FIR linear-phase analysis filters and then subtracted

from each subband signal. The almost alias-free sub-

band signals have spectral dips, so the spectral dips are

reduced using a filter for the spectral flatness and then

the outputs are used for adaptive filtering in each sub-

band. Simulations results show that the proposed sub-

band structure achieves similar convergence rate to the

fullband structure for white noise input and better con-

vergence rate than both the equivalent fullband at lower

computational complexity and the conventional SAF

structures for colored input.

IV. REFERENCES[1] Gilloire and M. Vetterli, Adaptive filtering in subbands, in

Proc.IEEE Int. Conf. Acoust., Speech, Signal Process., Apr. 1988,

pp. 15721575.

[2] S. S. Pradhan and V. U. Reddy, A new approach to subband

adaptive filtering, IEEE Trans. Signal Process., vol. 47, no. 8, pp.

655664, Aug. 1999.

[3] P. P. Vaidyanathan , Multirate Systems and Filter banks. Eng-

lewood Cliffs, NJ: Prentice-Hall, 1993.

[4] R. D. Koilpillai and P. P. Vaidyanathan, Cosine-modulated FIR

filter banks satisfying perfect reconstruction, IEEE Trans. Signal

Process, vol. 40, no. 4, pp. 770783, Apr. 1992.

[5] S. Haykin , Adaptive Filter Theory. Englewood Cliffs, NJ: Pren-tice- Hall, 1996.





ALIAS FREE SUBBAND ADAPTIVE FILTERING – A NEW APPROACH

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