Journal of King Saud University - Engineering Sciences (2015) xxx, xxx-xxx

King Saud University Journal of King Saud University - Engineering Sciences

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ORIGINAL ARTICLES

Efficient design of multiplier-less digital channelizers using recombination non-uniform filter banks

Shaeen Kalathil *, Bijili Sravan Kumar, Elizabeth Elias

Department of Electronics and Communication Engineering, National Institute of Technology Calicut, India Received 29 April 2015; accepted 10 November 2015

KEYWORDS

Non-uniform filter banks; Recombination filter banks; Rational sampling factors; Digital channelizers; Software defined radio; Hybrid meta-heuristic algorithms

Abstract A novel approach for the efficient realization of digital channelizers in software defined radios using recombination filter banks is proposed in this paper. Digital channelizer is the core of software defined radio. Computationally efficient design supporting multiple channels with different bandwidths and low complexity are inevitable requirements for the digital channelizers. Recombination filter banks method is used to obtain non-uniform filter banks with rational sampling factors, using a two stage structure. It consists of a uniform filter bank and trans-multiplexer. In this work, the uniform filter bank and trans-multiplexer are designed using cosine modulated filter banks. The prototype filter design is made simple, efficient and fast, using window method. The multiplier-less realization of recombination filter banks in the canonic signed digit space using nature inspired optimization algorithms, results in reduced implementation complexity. © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Software defined radios (SDRs) were initially developed for military applications, but later on, the potential advantages of SDRs are explored and much attention was given for improving their performances. The digital channelizer in an SDR is used to select the desired narrow band channel from the wideband signal. The different wireless standards have different channel spacing or bandwidths. Hence non-uniform

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filter banks are highly suitable in such applications. The equal bandwidth channels can be efficiently extracted using polyphase DFT filter banks (Zangi and Koilpillai, 1999). Only the prototype filter is designed and all other filters are obtained from this filter by modulation. Channels with unequal band-widths are not efficiently extracted with uniform DFT filter banks.

The non-uniform filter bank channelizer using tree structured filter banks is given in Fung and Chan (2002). The tree structured filter bank has a long system delay and the nonuniform decomposition has constraints on bandwidths. The channelizer proposed in Li et al. (2008) obtains the nonuniform subbands by merging the adjacent channels of a uniform cosine modulated filter bank. The number of adjacent channels merged should be an integer multiple of the upper

http://dx.doi.Org/10.1016/j.jksues.2015.11.002

1018-3639 © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

bandedge frequency. However, this method is suitable only for non-uniform channels with integer decimation factors.

The non-uniform filter bank design by merging the adjacent channels of a uniform filter bank result in integer sampling factors (Kumar et al., 2013). A method to design non-uniform filter banks with rational sampling factors was proposed by Cox (1986). It consists of a two stage structure, in which some of the channels of the analysis filters of a uniform filter bank are combined using the synthesis filters of a transmultiplexer (TMUX) with less number of channels. The non-uniform filter banks, thus obtained are called recombination filter banks. Later Xie, Chan and Yuk had extensively analysed the topic and suggested different modifications (Xie, 2004; Chan and Xie, 2006).

In this paper, we propose a new approach for the design of the digital channelizers of software defined radio using recombination filter banks. The uniform filter bank and trans-multiplexer are designed as near perfect reconstruction (NPR) cosine modulated filter banks (CMFBs). Only the prototype filter is required to be designed and all the other analysis and synthesis filters are obtained from this filter by cosine modulation. Compared to perfect reconstruction filter banks, NPR CMFB gives high stopband attenuation and the design requires less number of constraints. The amplitude and aliasing distortions should be within the tolerable limits for the application in hand. Therefore, in this work, a simplified, efficient and fast design technique is given, as the prototype filter is designed using Kaiser window approach (KWA). KWA results in high stopband attenuation. The constraints for spectral inversion and protrusion cancellation of recombination non-uniform filter bank (Chan and Xie, 2006) are satisfied by this method. The coefficients of the prototype filter are represented using minimal signed power of two (SPT) form called canonic signed digit (CSD) representation. The coefficients are synthesized into the CSD representation using different modified meta-heuristic algorithms. The performances are further improved with less number of adders using modified hybrid meta-heuristic algorithm. This results in digital channelizers which are computationally efficient, support multiple channels with different bandwidths and have low implementation complexity.

The remaining part of the paper is organized as follows: Section 2 gives an introduction of recombination filter bank

and briefly illustrates the design of near perfect reconstruction CMFB. Section 3 explains the design of the proposed digital channelizers for the SDR receivers. Section 4 explains the multiplier-less design of the recombination filter banks using hybrid meta-heuristic algorithm. Section 6 outlines the design and optimization of the CSD coefficient filter bank using various modified meta-heuristic algorithms. Result analysis is given in Section 5 and the conclusion in Section 6.

2. Overview of re-combination filter banks

Recombination non-uniform filter banks (RNUFBs) realize the non-uniform subband decomposition with rational decimation factors. The method involves a uniform filter bank in one stage followed by a uniform transmultiplexer in the required channels. The number of channels of the uniform TMUX will always be less than that of the uniform filter bank. The generic structure of the recombination filter bank is shown in Fig. 1 (Chan and Xie, 2006).

From Fig. 1, it can be seen that ml subbands of the uniform filter bank are combined using the synthesis filters of a ml channel trans-multiplexer. The lth combined channel has a decimation ratio of M/mi. The analysis and synthesis filters will be LTI systems or a cascade of LTI systems with modulation sequence (—1)", provided M and mi are chosen to be co-prime to each other. The co-prime condition enables the interchange of decimators and interpolators. When i is an odd value, the output spectrum will be inverted. The modulation sequence (—1)" eliminates the spectral inversion and also enables to realize a wide range of sampling factors. The frequency shifted versions of Hn+i(z) and Gl;i(z) may overlap and that will result in undesirable or spurious responses. In order to eliminate the spurious responses at the output, the Hn+i and Gii(z) should satisfy the matching conditions as given below (Chan and Xie, 2006).

H(em/x) = Gi(eMx) x e

miM miM

HobÙ \m

ht,(z) - \m

H„+,(.z) \m

Hn+m!-i(z) 1 m

hm- i(Z) \ m

Analysis Filters Synthesis Filters

Figure 1 Generic structure of recombination filter bank (Chan and Xie, 2006).

Here NM, is the filter length of the prototype filter of M channel uniform CMFB and Nmi, is the filter length of the prototype filter of M channel uniform TMUX.

2.1. Design of cosine modulated filter banks

For the design of NPR CMFB, a linear phase FIR filter with good stopband attenuation and which provides flat amplitude distortion function is initially designed. The coefficients of the analysis and synthesis filters are given by Eqs. (2) and (3) respectively (Vaidyanathan, 1993).

hk(n) = 2p0(n)cos(jM (k + 0.5) (n - N) +(-1)kp) (2) fk(n) — 2po(n)cos(M(k + 0.5) in - N) - (-1)kP

k — 0,1,2,..., M - 1 v '

n — 0,1,2,..., N - 1

If the prototype filter has linear phase response, then the overall filter bank will have linear phase response. The adjacent channel aliasing cancellation is inherent in the filter bank design. Remaining is the aliasing between non-adjacent channels. Prototype filter with good stopband attenuation reduces the aliasing between the non-adjacent channels. The 3-dB cutoff frequency of the prototype filter should be at ac 3dB — -M-This condition will reduce the amplitude distortion around the transition frequencies (k+1)p, where k — 0,1,..., M - 1 (Vaidyanathan, 1993).

In this paper, the prototype filter is designed with the window method using the Kaiser window. A closed form design method for the prototype filter using window method is proposed in Bergen and Antoniou (2007), thereby, the direct design of CMFB is possible. The matching condition given by Eq. (1) is satisfied by the prototype filters in order to suppress the spurious responses at the output.

3. Digital channelizers

Fig. 2 shows the generic wide-band receiver for software defined radio. The wideband mixer converts the frequency of the radio frequency signal to the intermediate frequency signal. This wideband signal is digitized by analog to digital converter (ADC). The digital channelizer which follows the ADC, extracts the individual channels of different communication standards. The channelizer is a set of bandpass filters and the output of the bandpass filters after decimations are fed to the baseband processors. Digital filter banks are commonly used as digital channelizers (Zangi and Koilpillai, 1999).

The different wireless standards have different channel bandwidths, hence non-uniform filter banks are the highly appreciated digital channelizer. Different existing approaches of filter bank digital channelizers are given in the introduction part of this paper. It is observed that the digital channelizers using recombination non-uniform filter banks allow rational sampling factors. Hence it is possible to design wideband chan-nelizer with maximum number of constituent subbands.

4. Multiplier-less design of re-combination filter banks

If the filter coefficients are represented using SPT terms, the multipliers can be implemented using shifters and adders. CSD contains minimum number of SPT terms and results in reduced number of shifters and adders (Yu and Lim, 2002). The number of non-zero digits will be minimum. As a result, minimum number of adders and shifters are required for the implementation.

4.1. Prototype filter coefficients in minimal SPT space

A look-up-table approach is used for the fast conversion of the filter coefficients to their corresponding CSD equivalents with prescribed number of non-zero terms (Yu and Lim, 2002). The look-up-table consists of four fields: an index, CSD equivalent, corresponding decimal and number of non-zeros present in the CSD equivalent. The coefficients can be converted to their nearest values in the minimal SPT space with required number of non-zero terms, using the look-up- table.

4.2. Objective function formation

Direct rounding of filter coefficients into the nearest CSD equivalent using look-up-table, may result in intolerable degradation of the filter characteristics. The goal of optimization of the multiplier-less recombination filter bank is to reduce the passband ripple and stopband attenuation, without increasing the total number of non-zeros and thereby adders.

— max I [|P,(ejx) | - 1]| (4)

Here, is the passband edge frequency of the ith prototype filter and Pi(ejx) is the corresponding frequency response. The objective function given in F,1 minimizes the maximum passband ripple.

F, 2 — max iPj I (5)

m>o>si

Here, xi is the stopband edge frequency of the ith prototype filter and the objective function

given in Fi,2 minimizes the maximum error in the stopband of the filter.

Baseband Processing

Baseband Processing

Baseband Processing

Figure 2 Generic wide-band receiver (Zangi and Koilpillai, 1999).

F3 = max(0, g{x)-gb) (6)

Here, g(x) denotes the average number of non-zero terms in the filter and g(x) is the user specified upper limit. When g(x) 6 gb is satisfied, the value of F3 will be zero, otherwise F3 will have a positive value of g(x) — gb.

min / = ai-,iFi,i + y.i,2Fit2 + <^3 (7)

Eq. (7) combines the three objective functions, where ai;1, ai 2 and a3 are the trade-off parameters, which define the relative importance given to each term in the final objective function. The constants ai;1, ai;2 and a3 are chosen by trial and error method. As shown by Eqs. (4)-(6), Fi;1, Fi2 and F3 can be conflicting functions to result in the specified filter performance. Eq. (7) combines these objective functions.

4.3. Optimization of recombination filter bank using modified meta-heuristic algorithms

The different modified meta heuristic algorithms used in this paper are Artificial Bee Colony (ABC) algorithm, Gravitational Search algorithm (GSA) and Harmony Search algorithm (HSA). The ABC algorithm is modified by Manoj and Elias in Manoj and Elias (2012), HSA and GSA algorithms are modified by Manuel and Elias (2012) and Manuel et al. (2012) respectively.

4.4. Optimization of re-combination filter banks using the hybrid algorithms

The hybrid meta-heuristic algorithms, combine the qualities of the constituent algorithms. In this paper, the hybrid optimization algorithm are formed by combining the two algorithms. The steps involved in the hybrid algorithm used in this paper is given as a flowchart in Fig. 3. In this technique the combinations such as HSA and GSA, ABC and HSA, GSA and ABC are run in parallel and the algorithms are coupled at regular intervals.

5. Results and discussion

All the simulations are done using a Dual Core AMD Opteron processor operating at 2.17 GHz using MATLAB 7.12.0. The prototype filters are linear phase filters with symmetrical coefficients. The optimization variable is obtained initially by concatenating half the number of coefficients of each filter. The algorithms are simulated 10 to 20 times to obtain the best values for the weights ai;1, ai 2 and a3. The main aim of the optimization is to reduce the implementation complexity of the filter, without degrading the performance characteristics of the filter bank. The weights are selected by trial and error method, as a compromise between different objective functions. In the example given in this paper, the stopband attenuation is the worst degraded due to CSD rounding in this example. Hence more weightage is given to ai2, so that the specifications of the filter are met.

5.1. Performance of multiplier-less non-uniform re-combination filter banks using modified meta-heuristic algorithms

Consider a two channel recombination filter bank with rational sampling factors (|, 1). This filter bank requires two prototype filters, one for the three channel filter bank and the other for the two channel trans-multiplexer. To compare the hardware complexity with the existing literature on recombination filter banks, the design in Xie (2004) is chosen. They have designed recombination filter bank with sampling factors (|, 3). The prototype filter length of 3 channel and 2 channel uniform CMFB is 36 and 24 respectively for a stopband attenuation of 40 dB. For the same stopband attenuation, we have designed the prototype filters using KWA (Bergen and Antoniou, 2007) and the prototype filter length obtained is 18 and 12 for the 3 channel and 2 channel uniform CMFB respectively.

For practical applications, high stopband attenuation is required, hence the prototype filters are designed for a stop-band attenuation of 80 dB. The corresponding prototype filter length for 3-channel and 2 channel filters is 39 and 26 respectively. The filter coefficients in the CSD space are optimized for

Table 1 Weightages for each objective function.

Figure 3 Flow chart of hybrid meta-heuristic algorithm used in this work.

«1,1 «1,2 «2,1 «2,2 «3 nb

ABC algorithm 1 6 1 6 1 3

GSA algorithm 0.1 6 0.1 6 0.5 3

HSA algorithm 1 6 1 6 1 3

Table 2 Performance comparisons of CSD coefficient recombination filter banks for different algorithms.

Passband ripple Stopband attn. Passband ripple Stopband attn. Adders Adders

(dB) (2 channel) (dB) (3 channel) (dB) (2 channel) (dB) (3 channel) (SPT terms) (SPT terms)

(2 channel) (3 channel)

Continuous coefficients 1.4 x 10~3 78.3 1.4 x 10~3 78.1

CSD (max. precision) 1.1 x 10~3 78.05 1.3 x 10~3 76.34 35 43

CSD (4 SPT terms) 1.5 x 10~3 65.39 1.8 x 10~3 73.8 30 40

ABC algorithm 5.3 x 10~3 74.6 2.6 x 10~3 74.9 29 37

HSA algorithm 3.9 x 10~3 77.02 2.1 x 10~3 77.6 31 40

GSA algorithm 1.0 x 10~3 78 1.8 x 10~3 78.2 33 42

Hybrid ABC-HSA 0.8 x 10~3 79.98 0.7 x 10~3 78.2 29 39

Hybrid GSA-ABC 1 X 10-3 79.2 2.4 x 10~3 78.7 28 38

Hybrid GSA-HSA 4.7 x 10~3 78.59 1x 10-3 76.9 30 38

---Continuous coefficients

.......CSD Rounded

-Hybrid ABC-HSA

----Hybrid GSA-ABC

\J V » ч ' 1 •

" II I I I "I i

Л/-Л' V « s !

0.4 0.6

Figure 4 Frequency response plots of analysis filters of the recombination filter banks with sampling factors (f, 3).

the combined objective function given in Eq. (7), using meta heuristic algorithms. Table 1 gives the weightage given to each term in the combined objective function for each algorithm. Table 2 compares the performances of the prototype filter in terms of minimum stopband attenuation and maximum passband ripple achieved and the number of adders due to SPT terms. ABC algorithm has got less number of adders due to SPT terms. The three algorithms are combined to form three hybrid meta-heuristic algorithms given by hybrid ABC-HSA, hybrid GSA-ABC and hybrid GSA-HSA. All these algorithms outperform the individual algorithms in terms of performances. The performances of the filter bank are improved and the number of adders due to SPT terms is reduced. Among the three algorithms, hybrid ABC-HSA and hybrid GSA-ABC give good performance with comparatively less number of adders due to SPT terms. Fig. 4 shows the frequency response plots of recombination filter banks with rational sampling factors (|,|) for continuous coefficients, CSD rounded

Table 3 Hardware complexity comparison in terms of required number of LUTs.

Continuous Max. precision Hybrid Hybrid

coefficients (8 SPT terms) ABC-GSA ABC-HSA

No: of multipliers 33 0 0 0

No: of adders 66 144 132 134

No: of LUTs required 3267 1152 1056 1072

Table 4 Common world wide wireless technologies.

Service Technology Launch year Channel spacing (MHz)

1G cellular AMPS 1983 0.03

2G cellular GSM 1991 0.2

3G cellular CDMA2000 1xEV-DO Rev.A 2002 1.25

3G cellular CDMA2000 1xEV-DO Rev.B 2010 3.75, 5, 7.5, 10, 11.5, 15, 18.75

3G cellular WCDMA FDD 2001 5, 10

3.5G cellular HSDPA 2007 5

4G MBWA i Burst HC-SDMA 2005 0.625

Fixed WiMAX IEEE802.16d 2004 1.75, 3.5, 7, 20

4G cellular LTE 2009 1.4, 3, 5, 10, 15, 20

Digital radio DAB 1995 1.715

Digital TV 2007 8

Digital cable TV 6

Personal area networks IEEE 802.15.4a (Zigbee) 2003 2, 5

Personal area networks ANT 2006 1

Personal area networks IEEE 802.15.1 (Bluetooth) 2002 1

— CDMA2000 4x (5MHz)

— WCDMA FDD (10MHz) ...... HSDPA (5MHz)

— IEEE 806.16d (3.5MHz)

— LTE (3Mhz)

— Digital TV (8MHz)

.— Digital Cable TV (6MHz)

...... IEEE 802.15.4a (5MHz)

-.- IEEE 802.15.1 (1MHz)

— IEEE 802.15.4a (2MHz)

Figure 5 Proposed digital channelizers for various wireless standards.

coefficients, hybrid ABC-HSA and hybrid ABC-GSA optimized.

The hardware complexity comparison for the prototype filters is shown in Table 3. It can be observed that a tremendous reduction in hardware complexity is obtained by the multiplier-less realization using hybrid meta-heuristic algorithms.

5.2. Digital channelizers for SDR implementations using the proposed re-combination filter banks

transmultiplexers are designed as cosine modulated filter bank, in which the prototype filter is designed using window method. The constraints for spectral inversion and spurious response cancellation are satisfied by this method. The implementation complexity is reduced by synthesizing the filter coefficients in CSD space using hybrid meta-heuristic algorithms. The flexibility in implementing a considerable number of distinct wireless standards in a single SDR is also demonstrated.

The proposed non-uniform sharp transition width CMFB can be applied for efficiently implementing the digital channelizer block of SDR. Table 4, shows the common world-wide wireless technologies and their corresponding channel bandwidths. It is clear from the table that different communication standards have different channel spacing and hence non-uniform filter banks are highly preferred. In order to demonstrate the adaptability of RNUFB for implementing the SDR channelizers a design example is given. Initially a 97-channel uniform CMFB is designed. As 97 is a prime number, the co-prime condition will be naturally satisfied for all integer values of mi. Hence the uniform trans-multiplexer can choose any number of channels. In the same manner, mi can be chosen as any odd number, if M is taken as a power of two number. The different channels of 97-channel uniform CMFB are suitably combined using different uniform trans-multiplexers. Fig. 5 shows the frequency response of the recombination filter bank designed for various wireless standards. It is evident from Fig. 5 that, the non-uniform subbands with different band-widths for a large number of channels are possible with recombination filter banks.

6. Conclusion

The realization of digital channelizers in the software defined radio using recombination non-uniform filter bank is investigated in this paper. The uniform filter bank and

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