An Efficient Implementation of Generalized DFT Filter Banks for subband decomposition

An Efficient Implementation of Generalized DFT Filter Banks for subband decomposition

An Efficient Implementation of Generalized

DFT Filter Banks for subband decomposition

1K.N.V.P.S.Rajesh, 1A. Vamsidhar, 1 N.Ganesh, 2K.Raja Rajeswari

1Electronics and Communication Department, D.I.E.T-Anakapalli

2Electronics and Communication Department, Andhra University-Vishakapatnam

Abstract In this paper Generalized DFT filter bank is developed for subband coding. The analysis starts from the signal definition in the analog domain. Simple observations of sampling period changes and matrix decompositions played key role in the development of efficient implementation of the proposed filter bank. We simulated and verified the proposed algorithm using MATLAB software.

Index Terms multirate filter bank, square root raised cosine filter

1. Multirate signal processing is an active research having applications in diverse fields like transmultiplexers, signal and image compression, low complexity adaptive filtering [1, 2, 3] etc.

   Among these signal and image compression gained a lot of attention by researchers [1,2,5]. The idea behind this compression is dividing the given input signal into different subbands and coding each subband separately and transmitting. Number of bits is varying in each subband depending upon the information [1,4,8] in it. At the receiver end these subband signals are combined to generate the signal of interest. Subband decomposition is performed by analysis

   [1,7,8]. Cosine modulated filters also developed from a simple prototype filters [3,8].

   In this paper, we present a generalized DFT filter with reduced implementation complexity. We extended the theory developed into filter banks for subband composition. We consider an analog signal as a sum of subband signals. The sampling rate requirement for subband signals is less compared with the original signal, since frequency content is limited. Number of subbands, analysis filters, and filter length can be varied as designers freedom. By proper matrix formulation of the problem and appropriate decompositions we achieved simplified implementations.

   This paper is organized as follows. Section II presents the analog domain problem formulation and its interpretation in frequency domain. Section III presents the digital domain due to sampling. Section IV implementation of multicarrier filter banks. Simulations are illustrated in section V. Conclusions remarks are given in section VI.

2. The signal xa(t) has to be decomposed into subband signals, sa,m(t) where m=0, 1M-1.We can express xa(t) as sum of modulated versions of subband signals.

   filters at the transmitter side, where as the synthesis filters at the receiver side reconstruct the signal.

   M 1

   x (t) s

   (t )e j mt

   (1)

   a a,m

   Main interest is to reconstruct the signal as close as possible m 0

   to the original signal. Studies have been performed regarding

   the design of the analysis/synthesis filters for perfect reconstruction. Perfect reconstruction is possible by removing two distortions that are usually arising. One is amplitude distortion, other is aliasing distortion. To remove aliasing completely conditions are derived in [1,3,6] as pseudo circulant property.

   Another important issue of interest is reducing the complexity of implementation. Low complexity solutions are obtained

   m denote centre frequency of subband signal sa,m(t).

   Since sa,m(t) is a band limited signal it can be sampled and reconstructed. Let Ts be the sampling period of sa,m(t) and ga,m(t) be the reconstructed filter. Then

   sa,m (t) {sa,m (t) (t lTs )} ga ,m (t) (2)

   l

   from polyphase decomposition. DFT filter bank is one which implementing subband decomposition with sinc filters. Using ploy phase decomposition, it can be implemented with the complexity of a single filter and efficient DFT algorithms

   sa,m (t)

   sa,m (lTs )g(t lTs )

   l

   (3)

   Here ideal impulse sampling is performed and reconstruction is done convolving with ga, m (t).

   Xa ( ) Xa ( )Ha,m ( m ) Xa ( m )Ha,m ( ) (7) Similarly if we are interested to look at the frequency domain of Figure 1

   Sa,m ( )

   s Sa,m (

   k

   k s )

   (8)

   s Sa,m (

   k

   k s )Ga,m ( )

   If sa,m(t) is modulated by exp(jtm) we obtain xa,m(t).Thus

   Fig. 1. Reconstruction of original signal analog representation

   From the original signal xa(t), we can obtain the subband signals with proper frequency shifting and filtering .To obtain sa,m(t), we need to shift frequency m to 0 , then low pass

   X a,m ( )

   s

   k

   Sa,m (

   Sa,m (

   m )

   m k s )Ga,m ( m )

   (9)

   filtering will do. Thus

   Using the equivalent developed between Figure 2 and Figure 3,we can modulated part of Figure 1 towards input side and

   s (t) (x

   (t)e

   j mt ) h

   (t)

   obtain Figure 4

   a,m a a,m

   Fig.2.Decomposition of original signal

   Equation (4) can be expressed as

   (4)

   Fig.4. Modulated version of Fig.1.

   The frequency domain relation from Figure 4 is given as follows

   s (t)

   x ( )e

   j m h (t )d

   Sa,m ( )

   s Sa,m (

   k

   k s )

   a,m a a,m

   (5)

   s Sa,m (

   m k s )

   (10)

   s (t)

   e j mt

   x ( )h (t

   )e j m (t ) d k

   a,m a a,m

   Now Figure 2 can be modified as Figure 3

   s Sa,m (

   k

   m k s )Ga,m ( m )

   Clearly (9), (10) give same final result.

3. Fig.3.Modified version of figure 2

   Now looking at the frequency domain Figure 2 implies

   In the previous discussion is based on analog implementation. With reference with the original signal sampling period, we develop the digital equivalent analysis. We choose T, as the sampling rate of xa(t) for proper reconstruction based on Nyquist criteria. Subband sampling

   Xa ( ) Xa (

   Figure 3 implies

   m ) Xa (

   m )Ha,m ( ) (6)

   period Ts is larger compared to T. The relationship between T and Ts is given by T= Ts/N, for some positive integer N. If we sample xa(t), it result xa(nT)=x(n).

   For subband signals

   sa,m (nTs )

   sa,m (nNT )

   sm (nN)

   (11)

   The same relation can be concluded from Figure 5.

   Fig.5 (a). Upsampling process

   Fig.5 (b). Downsampling process

   Equation (1) after samples t=nT

   j nT

   M 1

   Fig.7 (a). Decomposing of the signal by downsampling

   Fig.7 (b). Modulated version of the Fig.7 (a).

   xa nT sa,m (nT )e m

   m 0

   Using (12) its equivalent to

   (12)

   Now we look at mathematical analysis of Figures .From Figure 6(b)

   M 1

   { s (lT )g

   (nT lT )}e j (

   mT )n

   (13)

   ((s

   (n) )e j

   mn ) (g

   (n)e j

   mn )

   m 0 l

   a,m s a,m s

   m N m

   Using T =NT, T=

   (s (n)e j

   mnN ) (g

   (n)e j

   mn )

   s

   M 1

   x(n)

   m m

   s (Nl)g

   (n Nl)e j m n

   (14)

   m

   Let gm (n)e

   j mn

   N

   bm (n) , sm (n)e

   m

   j m Nn

   ym (n)

   m m

   m 0 l

   xm (n)

   k

   ym (k )bm (n kN )

   (15)

   From Figure 7(b) s

   (n) ((x(n) * h

   (n)e j mn )e

   j mn )

   m m N

   Let h (n)e j mn a (n) then

   m m

   s (n)

   e j mnN

   x(k )a

   (nN k )

   (16)

   Fig.6 (a). Reconstruction of the signal by upsampling

   Fig.6 (b) Modulated version Fig.6 (a).

   Using (14) Figure 6 can be developed. It is easy to note that sing Figure 1 and Figure 5(a) one can develop Figure 6(a). Simply by applying result in Figure 5(a) on Figure 4 we obtain Figure 6(b). Similarly by applying 5(b) on Figure 2, 3 we obtain Figures 7(a), 7(b).

   m m

   k

   Equation (16) given decomposition of x (n) into subbands sm(n). Where (15) with x(n) =x0(n) +x1(n) +xM-1(n), gives synthesis of original signal from subband signals. It is easy to combine Figure7 (b) followed by 6(b)

   Fig.8. The reformed digital implementation of generalized multicarrier filter banks

   f 0 b 0

   b 0 ….

   b 0 y k

4. IMPLEMENTATION OF MULTICARRIER FILTER BANKS

   k 0 1

   M 1 0

   To simplify the implementation task, all the filters can be fk 1

   b 1 b

   1 …. b 1 y k

   M 1

   developed from a prototype filter g(n). We choose it be the length L=2kN+1 for group delay k. We know that

   0 1

   : : : …. :

   1

   :

   (20)

   j mn

   f L 1

   k

   b L 1 b L 1 ….

   b L 1 yM 1 k

   Analysis filter

   am n hm n e

   L 1 0 1

   M 1 L M M 1

   Synthesis filter b n g n e j mn

   B b n g n e j 2 / M (m (M 1/2))(n (L 1/2))

   (21)

   m m

   To make all filters casual a common shift of (L-1)/2 samples is introduced.

   n,m m

   BT T I 1 M 1 I

   am

   n g L 1

   n e j 2 / M (m (M 1/2))(n ( L 1/2)) m m

   b L 1

   n a

   (n)

   (17)

   I2M

   I2M

   …..

   I2M

   (22)

   I

   2M ,L

   m m

   Implementation of synthesis/analysis filter bank algorithm as

   diag g 0

   g 1 …..

   g L 1

   follows.

   Where T W H

   , Wm= normalized DFT matrix

   1 M 2

   1. Synthesis Filter Bank:

      1 , 2 diagonal matrix with equations

      1 k ,k

      e j (k (M 1/2))( L 1)/ M

      Now we will discuss development of synthesis filter bank. The transmitted signal can be expressed as

      2 k ,k

      e j k (M 1)/ M

      xm n

      k

      ym k bm

      n Nk

      (18)

      The relationship is fk

      Byk

      ym n sm

      n e j mnN

      1. The Synthesis filter bank is implemented by below algorithm

         This can be depicts as follows

         Step 1) Set the initial value of m and k: k=0, n=0. Set an L- point sequence: d (l) =0, l=0, 1L-1.

         Step 2) Perform M-point IFFT on M subband inputs ym(k),

         with the transform matrix T W H

         .Gives the result

         Fig.9. Synthesis filter bank representation

         f

         m

         k

         M 1

         , m=0, 1, 2.M-1.

         1 M 2

         k

         fk n ym

         k bm n

         Step3) Expand M-point sequence to 2M-point sequence

         m 0 (19)

         M 1

         fk m f m

         0 m M 1

         fk n kN ym

         m 0

         k bm

         n kN

         1 M 1 f m

         M m 2M 1

         k

         Where f n

         [b n b n

         ……

         b n ]

         y0 k

         k

         y1 k

         And then expand L-point l=0, 1.L-1.

         fk l f

         l sequence,

         2M

         k 0 1

         M 1 :

         Step 4) Calculate fk

         l g l fk

         l l=0,1,.L-1.

         n=0, 1.L-1 k=0, 1 .

         yM 1 k

         Step 5)Calculate

         d (l)

         d (l)

         fk (l) l=0,1,.L-1

         Step 6) Output the first N samples of the L point sequence

         d(l) : f (n l)

         sequence d(l) as

         d (l)

         l=0,1,.N-1. Update the L-point

         d (l)

         d (l N ) 0

         l L N 1

         Step 5) perform FFT on M-point sequence xn (m)

         with

         0 L N l L 1

         transform matrix T W to get reconstructed

         Step 7) Set k k 1, n n N and go to step 2.

         ym (n) m=0,1..2M-1.

         1 M 2

         Step 6) set n=n+1 and go to step 2.

   2. Analysis Filter Bank:

   Now we will discuss development of Analysis filter bank.

   L 1

   ym n am k x nN k

5. Based on the analysis done so far, simulations are performed

   k 0 using MATLAB to justify the effectiveness of the algorithm.

   L 1

   am L 1

   k x nN L 1 k

   The prototype filter is chooses to be a square root raised cosine filter of length L=433.The impulse response, frequency

   k 0 response are plotted in below

   L 1

   m

   (b

   k ) x nN L 1 k

   (23)

   k 0

   This can be depicts as follows

   Fig. 10. Analysis filter bank representation

   x nN L 1

   y0 n

   Fig.11. Square root raised cosine filter

   Let xn

   x Nn L 2

   :

   y y1 n

   n :

   x nN

   yM 1 n

   n n

   y BH x (24)

   This expression decomposes input signal xn into subbands yn.Thus analysis filter bank can be implemented as follows.

   1. The Analysis filter bank is implemented by below algorithm

   Step 1) set the initial value of n: n= (L-1)/N

   Step2) Take an L-point sequence from input signal: xn (k) x(Nn L 1 k) , k=0, 1,L-1.

   Fig.12.FFT filter response

   We are using the prototype filter and various modulations on it and from the original signal different bands are extracted the original signal spectrum and the extracted subband spectrum are illustrated in below figures.

   Step 3) calculate xn (l)

   g(l)xn (l) , k=0,1,L-1

   Qm

   Step 4) calculate xn (m)

   xn (2Mq m) ,m=0, 1…2M-1

   q 0

   and Qm is an integer no more than (L-1-m)/M .calculate

   xn (m)

   xn (m) ( 1)

   M 1

   xn (m M )

   m=0,1..2M-1.

   Fig.13 (a). Spectrum consists of five bands

   Fig.13 (b) Decomposition of band 1

   Fig.13 (c) Decomposition of band 2

   Fig.13 (d) Decomposition of band 3

   Fig.13 (e) Decomposition of band 4

   Fig.13 (f) Decomposition of band 5

6. CONCLUSION

In this paper we presented algorithms to implement generalized DFT filter bank. From a single prototype filter various subband filters are developed using the generalized DFT matrix operation. Factorization of generalized DFT matrix played the major role in developing efficient implementation algorithm.

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