Design and Implementation of Multicarrier Modulation Systems Based ...

Sensors & Transducers, Vol. 164, Issue 2, February 2014, pp. 191-198

Sensors & Transducers © 2014 by IFSA Publishing, S. L. http://www.sensorsportal.com

Design and Implementation of Multicarrier Modulation Systems Based on Cosine-modulated Filter Banks Ling Zhuang, * Jing-Yang Qiu and Kai Shao Chongqing University of Posts and Telecommunications (CQUPT), Chongqing Key Lab of Mobile Communication Technology, Chongqing 400065, China Tel.: 13637932347 E-mail: [email protected] Received: 4 November 2013 /Accepted: 28 January 2014 /Published: 28 February 2014 Abstract: Transmultiplexer systems based filter bank have certain advantages compared with existing multicarrier modulation (MCM) systems based discrete Fourier transform (DFT). In this paper, a new cosinemodulated filter bank (CMFB) with the performance of perfect reconstruction was designed and applied to the transmultiplexer based MCM. The prototype filter of the CMFB was designed based on semi-sinusoidal window. Simulation results show that its performance in symbol error rate (SER) and peak to average power ration (PAPR) based on filter bank system are improved, compared to the classical orthogonal frequencydivision multiplexing (OFDM) system. Copyright © 2014 IFSA Publishing, S. L. Keywords: Cosine-modulated filter bank (CMFB), Multicarrier modulation (MCM), Prototype filter, Symbol error rate (SER), Peak to average power ration (PAPR).

1. Introduction Multicarrier modulation (MCM) is an efficient method for transmitting information over the wireless channel. Its basic idea is to divide the channel spectrum into parallel, narrowband subchannels. Orthogonal frequency-division multiplexing (OFDM) can support high data rata and provide high reliability in voice, data, and multimedia communication, and has been widely used for many wireless communication systems due to its robustness and high spectral efficiency to the frequency selective fading channel [1]. In an OFDM system, the inverse fast Fourier transform (IFFT) and fast Fourier transform (FFT) are employed for modulation and demodulation, respectively. In order to maintain orthogonally among subchannels and to meet the requirement of the Nyquist prototype, generally, rectangular window is used in OFDM systems as the

Article number P_1851

prototype filter. However, the level of the side lobe is merely 13 dB and the spectral overlap with neighboring subbands is serious. Another critical drawback of OFDM is the high peak to average power ratio (PAPR) of the output signal [2-5]. To deal with this problem, many solutions have been proposed to improve the frequency characteristics of the OFDM system [6-15]. Windowing method has been introduced to reduce the outband energy [11, 14, 15]. Non-rectangular continuous time pulse shaping filters have been proposed to improve the spectral roll-off of the transmitted signals. However, windowing techniques often mean increasing the number of redundant samples, which will lead to the increase of inter symbol interference (ISI) and amplification of the receiver output noise power. Applying filter bank to MCM was proposed by Vandendorpe [7]. By interchanging the analysis and synthesis filter banks,

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Sensors & Transducers, Vol. 164, Issue 2, February 2014, pp. 191-198 we can obtain a transmultiplexer. However, the main drawback of filter bank based transmultilpexer is that implementations of M transmitting filters and M receiving filters are needed. In order to reduce the complexity, modulated filter banks are employed [10, 12]. In modulated filter banks, analysis and synthesis filters are all obtained by modulating one or two low-pass prototype filter. Therefore, the design complexity of filter banks can be reduced to design the prototype filters. There are two kinds of commonly used modulated filter banks, namely, exponentially-modulated filter bank (EMFB) and cosine-modulated filter banks (CMFB) [16-19]. In general, EMFB is mostly used in the application for complex signal processing like modulation system. Since its subband filters are obtained by exponentially-modulation of prototype filter h0 (n) , the coefficients of hk (n) are generally complex numbers even if h0 (n) has the real coefficients [19]. However, it’s a different case for CMFB. CMFB is usually used in the fields of real signal processing like speech and image processing. Beyond that, the CMFB has advantages like perfect reconstruction, better energy concentration on the low banks and lower PAPR values in the transformdomain signals [20-22]. Such properties are quite advantageous in the MCM system. In this paper, according to the McClellan-Parks algorithm, a linear-phase filter is designed and further employed as the prototype filter of the cosinemodulated transmultiplexer (CMT) and it is applied to the MCM system. The designed modulation system will be compared with the classic OFDM modulation system to show its advantages. The property is evaluated by using parameters like spectral characteristic, symbol error rate (SER) and PAPR. Compared with other optimization algorithms, the CMT system is easily realized and delay of the system is greatly improved. The rest of this paper is organized as follows. In section 2, conventional OFDM system model and CMT model will be introduced. A prototype filter based semi-sinusoidal is designed in section 3. In section 4, the simulation results are presented. Finally, the conclusions are summarized in the last section.

2. System Model 2.1. OFDM OFDM is a special kind of MCM technique which subcarriers keep orthogonal. Fig. 1 illustrates a block diagram of a conventional OFDM system [2-5]. At the transmitter, input signal is arranged into subband signals by a serial-to-parallel (S/P) converter. Then, after signals mapping, the modulation is implemented using IFFT. After IFFT, received signal can be expressed as x ( n) 

(1)

In order to eliminate the influence of the intersymbol-interference (ISI), cyclic prefix （CP）is needed to be inserted before each signal. Then, the output data xt (n) is  x( N  n), n   N g ,, 1 xt (n)   ,  x(n), n  0,1, N  1

(2)

where N g is the length of CP. Finally, the modulated signal and CP are converted to an OFDM symbol by a parallel-to-serial (P/S) converter. At the receiver, firstly, the CP needs to be discarded. Then, demodulation is performed by FFT. When the length of CP is longer than that of the channel impulse response, the interference between consecutive OFDM symbols is eliminated. In this case, the channel can be regarded as a set of independent parallel subchannels, and the received signal is indicated as Y k   H k  X k   wk ,

k  0,  N  1 ,

(3)

where Y  k  means the received signal, X  k  denotes the transmitted signal, H  k  and w  k  are the channel frequency response and the additive Gaussian noise, respectively.

Fig. 1. Block diagram of OFDM.

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1 N 1 j 2 nk )  X (k ) exp( N k 0 N


2.2. M-channel Transmultiplexer Transmultiplexer is an application of filter bank [13]. In general, it can be considered as a MCM system. Compared to the existing discrete Fourier transform (DFT) based MCM (like OFDM), we can adjust the subband overlapping in the transmultiplexer by controlling the prototype filter. A maximally decimated M-channel transmultiplexer is shown in Fig. 2, where Fk ( z ) and H k ( z ) is synthesis and analysis filter, respectively.

At the transmitter, firstly, each input signal is interpolated and filtered by interpolator and synthesis filter. Then, the resulting signals on each subband are added to form a signal for transmission over the wireless channel C ( z ) . At the receiver, an analysis filter bank followed by a set of decimators is used to demodulate and extract the transmitted symbols. If the transmultiplexer is perfect reconstructed, the output signals are equal to the input signals, whereas if the estimated signals receive some interference from the other sources, we would have the nearly perfect reconstruction case [7-13].

Fig. 2. Block diagram for an M-channel transmultiplexer.

In an M-channel transmultiplexer system, the phase distortions can be eliminated if prototype filter has the relationship as h(n)  h( N  1  n) . The amplitude distortion and aliasing distortion can be controlled by the design of the prototype filter. In Fig. 2, the input-output relation between each subband signals in transmultiplexer is given in Eq. (4). 1 Xˆ k ( z )  M

M 1

M 1

i 0

l 0

1

1

 X i ( z )  Fi ( z M W l ) H k ( z M W l ) , (4)

where 0  k  M  1 and W l  e j 2 l M . The synthesis and analysis filters can be combined as M 1

Tki ( z )   Fi ( z l 0

1

M

l

W )H k ( z

1

M

l

W )

i

k

receiving part, Tki ( z ), k  i contributes to needless reconstruction of the input signal X k ( z ) . If T ( z )  0 for k  i , Xˆ ( z ) is independent of ki

k

X i ( z ) . And we get the reconstructed signal 1 Xˆ k ( z )  X k ( z )Tkk ( z ) M

(7)

However, ISI also exists if Tkk ( z ) is not a delay. (5)

According to Eq. (5), Eq. (4) can be rewritten as follows 1 M 1 Xˆ k ( z )   X i ( z )Tki ( z ) M i 0 1 1 M 1 X k ( z )Tkk ( z )    X i ( z )Tki ( z ) M M i 0,i  k

Eq. (6) means that received signal Xˆ k ( z ) is composed of two parts. The first term is related to the input signal X k ( z ) . And the second term is the adjacent channel interference. If Tki ( z )  0 for k  i , Xˆ ( z ) is correlated to X ( z ) . Hence, in the

(6)

When Tkk ( z ) has the form like cz  p , the transmultiplexer is a perfect reconstructed system. Fig. 3 depicts the block diagram of the MCM based M-channel transmultiplexer. Compared with conventional OFDM, we replace IFFT and FFT module with the synthesis and analysis filter bank, respectively. In this way, we can control the size of the stopband attenuation according to the requirement. In section 4, we will give detailed description.

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Fig. 3. Block diagram of multicarrier modulation system based on CMT.

3. Cosine-modulated Filter Bank CMFB is a specially designed filter bank, in which all the analysis and synthesis filter bank are derived from a linear phase FIR prototype filter by cosine modulation. Therefore, during the CMFB, only the prototype filter is needed to be designed. In an M-channel CMFB, the analysis and synthesis filters can be expressed as  N 1   hk (n)  2h(n) cos  (2k  1) (n  )  (1) k  , 2M 2 4 

(8)

 N 1   f k (n)  2h(n) cos  (2k  1) (n  )  (1)k  , 2 M 2 4 

(9)

where 0  k  M  1 , 0  n  N  1 , and h(n) is the N order prototype filter. In general, the following results are typically desired in a CMFB [16-18]: 1) Aliasing errors: Spectral characteristic is desired as

 

H e j  0 , for    M

2)

(10)

Distortion function: T  z  is exactly or

nearly a delay. In particular, the magnitude response T e j is required to be flat.

 

 

   H e 

T e j  1 , where T e j 

2 M 1 k 0

j  k  M 



2

(11)

3) Linear phase property of prototype filter: Prototype filter should be satisfied , where N is the filter length. h n  h  N 1 n 4) Filter length: In order to minimize the delay, the length of prototype filter should be attenuated as much as possible.

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In this paper, we choose Parks-McClellan algorithm to design the prototype filter. Then, we will briefly introduce Parks-McClellan algorithm. There are four parameters to be determined, that is, the passband edge  p , the stopband edge s , passbandstopband error ratio  , and filter length N . According to Eq. (10), the first prerequisite for perfect reconstruction can be achieved by assigning the stopping edge s   M . Then the second parameter should be considered. Observing the Eq. (11), the T  e j  is periodic with period  M and

we should only consider T  e j  in the interval

 0, 

M.

 

T e j can

s   M ,

With

be

approximated by

 

 

T e j   H e j

2



H e

j  -  M 





let   max H  e j   H e 2



2

, for 0     M

j   M 



2

1

(12)

0  M

If filter length N and passband-stopband error ratio  are given, the only parameter needed to be optimized is the passband edge  p . And the method is optimized to minimize  . According to the above four constraints and Parks-McClellan algorithm, we choose semi 1   sinusoidal window h(n)  sin  n   as the 2  2M   prototype filter, where the length of the prototype filter N is fixed as 2M , M is the number of channels. If the channel is ideal, the filter bank is perfect reconstructed. Then, we will present the data processing of the designed filter bank.


Analysis filter bank can be decomposed as h  (h 0 h1 ) , and the synthesis filter bank can be expressed as f  (f0 f1 ) where h0 (iM  1)  h0 (iM )  h (iM ) h1 (iM  1)  hi    1     hM 1 (iM ) hM 1 (iM  1)  f 0 (iM )  f (iM )  fi    1    f M 1 (iM )



h0 (iM  M  1)  h1 (iM  M  1)        hM 1 (iM  M  1)  

 f 0 (iM  1)



f1 (iM  1)



  f M 1 (iM  1) 

f 0 (iM  M  1)  f1 (iM  M  1)      f M 1 (iM  M  1) 

(13)

(14) Fig. 4. Amplitude-frequency characteristic of rectangular window and semi-sinusoidal window.

Firstly, we should add zeros before the first frame input signal x(1) to compose the

 x(1)

x(0)  . T

Then, we have the first frame output signal  x(1)  y (0)  (h 0 h1 )  .  x(0)  After repeating the above steps,  x(2)  y (1)  (h 0 h1 )   can be obtained. According to  x(1)  y (0) and y (1) , we can obtain the first frame reconstruction signal xˆ (1) as

xˆ (1)  f0

 f0h 0

 x(2)  0    x(1)  h1     x(0)   x(2)    f0h1  f1h 0 f1h1   x(1)   x(0)   

h f1   0 0

h1T h1  h 0T h 0

(17)

where N  2mM is the length of the prototype filter, M is the number of the channels. Prototype filter can be expressed as polyphase decomposition 2M1 m1

H z  h j  2iM z 

 j2iM

j0 i0

2M1

 

  z j Ej z2M , j0

(18)

m 1

(15)

E j  z    h  j  2iM  z  i

(19)

i 0

According to Eq. (19), synthesis filter bank can be expressed as

As prototype filter is symmetrical, we can get the expression f k (n)  hk ( N  1  n) , according to Eq. (8) and (9). Synthesis filter bank can be expressed as f  (h1T h 0T ) . Hence, Eq. (15) can be rewritten as xˆ (1)   h1T h 0

 N 1   ck ,l  2 cos  (2k  1) (l  )  (1) k  , 2 M 2 4 

where

h1 h0

 x(2)    T h 0 h1   x(1)   x(0)   

We define that

(16)

h 0 and h1 are orthogonal to each other. Hence, we can get that xˆ (1) x(1) . The input signal can be a perfect reconstruction and the delay is only a frame. Frequency characteristics of the designed semisinusoidal filter and rectangular filter are shown in Fig. 4. We can see that the amplitude-frequency characteristic of semi-sinusoidal window is improved by nearly 10 dB compared with that of rectangular window. According to CMFB, cosine-modulated transmultiplexer (CMT) can be implemented by exchanging the analysis and synthesis filter bank of CMFB. It has fast algorithm.





 E0  z 2 M  F0  z       z 1 E1  z 2 M  F1  z    Cˆ           2 M 1 F z   z  M 1  E2 M 1  z 2 M 









   ,    

(20)

where Cˆ k ,l  ck ,l , 0  k  M  1 , 0  l  2 M  1 . Reference Cˆ  ck ,l  J  I

[22] has proved that J  I  , where I and J mean identity matrix and reverse identity matrix, respectively. The output signal can be expressed as

   

 E0 z2M   z1E z2M 1 Y z X z F z X z C J I J I      2M1 z E2M1 z2M 

 

   ,    

(21)

where X  z  is the input signal vector and C is the discrete cosine transform (DCT) matrix. According to

195


Eq. (21), Fig. 1 can be decomposed into polyphase form as shown in Fig. 5. In the implementation

process, the type IV butterfly architecture.

DCT

is

realized

by

Fig. 5. Polyphase structure of CMT.

4. Simulations The block diagram of the modulation system based CMT is shown in Fig. 3. First of all, the input signal s (n) will pass through the baseband modulation. Then the baseband modulated signals will be processed by the synthesis filter bank for the multicarrier modulation. The output signals will be added into the cycle prefix before going through the radio channel. In the receiving part, the signal will be processed in an inverse process. Firstly, the cyclic prefix should be removed. In order to eliminate the influence of the channel, the equalization is necessary. In this paper, we choose the frequency domain equalizer. After equalizing, the signal will be demodulated by the analysis filter bank. Finally, the signal will be demodulated in the baseband and reconstructed. Symbol error rate (SER) and peak to average power ratio (PAPR) are two important performance parameters in MCM system. In the communication systems, the primary purpose is to receive the sent signals as accurately as possible. Hence, SER is the first element that we should consider. PAPR is always used to describe the large fluctuation of the signal envelope. High PAPR will lead power amplifier not to maintaining linearity over the whole dynamic range of the signals. Hence, PAPR is the other factor that we have to take into account. For these reasons, in this paper, we choose SER and PAPR as the parameters to compare with conventional OFDM system. In general, complementary cumulative distribution function (CCDF) is chosen to describe PAPR performance. And CCDF means the probability that PAPR exceeds to a fixed threshold value. In the simulations, we will choose Quadrature Phase Shift Keying (QPSK) and 16 Quadrature Amplitude Modulation (16QAM) as baseband

196

modulations, respectively. Fig. 6 is the SER curves of MCM system based CMT and OFDM.

Fig. 6. SER comparison between OFDM system and CMT based system with 16QAM and QPSK.

With the increasing ration of the signal to noise ratio (SNR), the SERs of both systems are improved. However, as the SNR increases to a certain value, the SER performance of CMT system is gradually better than OFDM system. And it can also be seen that QPSK is more suitable for MCM system as the baseband modulation method compared with 16QAM. Fig. 7 are the CCDF curves of MCM system based CMT and conventional OFDM system with 16QAM and QPSK as the baseband modulation, respectively. From the figure, it can be seen that whether 16QAM or QPSK that is used as the baseband modulation, the performance of PAPR for CMT system is obviously better than OFDM system. In general, the simulations show that the proposed system is better than the conventional OFDM system in the performance of SER and PAPR.


[4].

[5].

[6].

[7]. Fig. 7. PAPR comparison between OFDM system and CMT based system with 16QAM and QPSK.

5. Conclusion OFDM is widely applied in MCM system. However, the spectral character and high PAPR are limiting factors in its application. In this paper, we applied CMT into the MCM system. CMT is a special transmultiplexer based on CMFB. In CMFB, all the analysis and synthesis filters are derived by modulation of a low-pass prototype filter. For this reason, design of CMT can be simplified as design a prototype filter. In this paper, according to Parks-McClellan algorithm, we choose the semisinusoidal window as the prototype filter to design and implement a perfect reconstructed filter bank. The synthesis and analysis filter banks are all implemented by the polyphase component and DCT IV is realized by butterfly architecture. Simulations show that the CMT is more suitable for MCM system and it has a better performance than OFDM system.

Acknowledgment

[8].

[9].

[10].

[11].

[12].

[13].

This work is supported by National Natural Science Foundation of China (No. 61071195) and Sino-Finland Cooperation Project (No. 1018). [14].

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