PAPR Reduction Algorithm Based on Improved μ-law Companding

Sensors & Transducers, Vol. 171, Issue 5, May 2014, pp. 43-50

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

PAPR Reduction Algorithm Based on Improved μ-law Companding Transform and PTS Technology *

Gewei TAN, Xian Huang

School of information science and engineering, Huaqiao University, Xiamen 361021, China * Tel.: (+86) 13799763568, fax: (+86) 0592-6162380 * E-mail:[email protected]

Received: 11 March 2014 /Accepted: 30 April 2014 /Published: 31 May 2014 Abstract: As a new multicarrier modulation system, orthogonal wavelet packet multiplexing (OWPM) system also has the problem of high peak to average power ratio (PAPR). In order to resolve the problem that of high computational complexity and inadequate effect of PAPR suppression for Partial Transmit Sequence (PTS) and high BER for the method of μ-law companding transform, an algorithm of PAPR reduction based on improved μ-law companding transform and PTS technology is put forward in the paper. Simulation results show that the proposed algorithm compared with the original algorithm has corresponding improvement of performance, and the combined processing with μ-law companding transformation and PTS technology, on the condition that not to affect the original bit error rate(BER) of system, can better reduce PAPR, while decrease the computational complexity. Copyright © 2014 IFSA Publishing, S. L. Keywords: Multicarrier modulation, orthogonal wavelet packet multiplexing, peak to average power ratio, partial transmit sequence, μ-law companding transform.

1. Introduction With the growing of requirement for wireless communication broadband network, the problems of traditional transmission technology have become increasingly prominent. Compared to orthogonal frequency division multiplexing (OFDM) system, OWPM system has more advantages in these aspects such as utilization rate of spectrum, anti-interference and allocation of sub-channel, thus attracting more and more attention. However, as a new technology of multicarrier communication, OWPM system also has the problem of high PAPR which is generated by phase accumulation of multiple subcarrier at some moment, which requires the linear range of front-end power amplifier very large. If the requirement cannot be met, once the peak of signal reaches the non-linear range of power amplifier, signal will distort and

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intermodulation interference between subcarrier will occur, resulting in the destruction of the orthogonality among subcarrier, ultimately reducing the transmitting performance of system, therefore, it is necessary to find a method to reduce PAPR effectively [1]. There have been literatures studying the problem of PAPR, literatures [2-5] attempt to solve the problem from the optimization of wavelet packet modulation. Literature [6] and [7] are method of probability to reduce PAPR at the same time not to affect the BER performance of system, but have high complexity, such as PTS technology. Literature [8] and [9] are method of pre distortion, which implement nonlinear transform to the demodulated signals to decrease PAPR such as the method of μ-law companding transform. Aiming at the shortcomings of PTS technology and μ-law companding transform, the paper presents

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Sensors & Transducers, Vol. 171, Issue 5, May 2014, pp. 43-50 corresponding improved algorithms and their combined algorithm. The proposed method can better reduce PAPR of OWPM system and decrease computational complexity, while as far as possible without affecting the original BER performance.

2. OWPM System and Peak to Average Power Ratio 2.1. Orthogonal Wavelet Packet Multiplexing System The technology of orthogonal wavelet packet multiplexing turns high rate broadband signals into parallel low rate narrowband signal, so effectively overcomes the inter symbol interference, and because protect interval and cyclic prefix are not required, OWPM system has higher spectrum efficiency and anti-jamming performance than OFDM system. In the multicarrier modulation system based on wavelet packet transform, the transmitting data respectively modulates wavelet packet functions of different nodes after series-to-parallel conversion, and the modulated signal x ( t ) can be expressed as:

x (t ) =

å

(l ,m )ÎΓ

å Slm (n )flm (t - nTl ) , n

k

where s01 (k) =

å å f (k - 2 n )S l

lm

(l ,m )ÎΓ

n

lm

(n )

(1)

(2) and flm as

the equivalent filter coefficient from the nodes (l, m ) to the root node ( 0,1) , f01 represents the scale function of scale space V0 . By the above definition, a basic structure of wavelet packet multicarrier modulation system is shown in Fig. 1. Where IDWPT represents inverse discrete wavelet packet transformation, namely wavelet packet reconstruction; DWPT is short for discrete wavelet packet transformation, namely wavelet packet decomposition.

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fc

fc

φ01

Fig. 1. Block diagram of wavelet packet multicarrier modulation system.

2.2. Peak to Average Power Ratio Since the probability of occurrence of signal peak is relatively small, so the ratio of signal peak power to average power is often used to represent signal distribution of multicarrier system. PAPR is defined as follows [10, 11]: 2

PAPR = 10 lg

where

2 E éê s (t ) ùú ë û

is

max s (t ) (dB ) , 2 E éê s (t ) ùú ë û

the

average

(3) power

of

multicarrier signal. Under normal circumstances, the complementary cumulative distribution function (CCDF) is used to measure the distribution of PAPR within the OWPM system, which is defined as CCDF ( PAPR0 ) = P { PAPR ( dB ) > PAPR0 } ,

where flm represents the basis function of the m branch of the l layer of wavelet packet decomposition, namely the scale function of scale space Vl . Slm (n ) represents the digital signal who modulating the scale function flm at node (l, m ) ; Γ represents the collection of wavelet packet functions. Equation 1 may also be equivalent to the following expressions: x (t ) = å s01 (k )f01 (t - nT0 ) ,

φ01

(4)

where PAPR0 is the threshold of PAPR, P { } is probability distribution of signal.

3. PAPR Reduction Algorithm Based on Improved μ-law Companding Transform and PTS Technology 3.1. Improved μ-law Companding Transform μ-law companding transform is the method to reduce PAPR using nonlinear operation. Fig. 2 is the system principle diagram of μ-law companding transform. The basic principle of the conventional μ-law companding transform is to amplify power of small signal, at the same time to reduce the power of large signal. Through adjusting the companding factor μ, to remain average power basically unchanged, not becoming a burden to the power amplifier. The function expression of sending end is: u x) v y= , ln(1 + u ) x vx ln(1 +

(5)

where u is scaling factor, usually less than or equal to 5; v is the average amplitude of signals, namely the turning point of μ-law companding; x is signal after wavelet packet modulation.

Sensors & Transducers, Vol. 171, Issue 5, May 2014, pp. 43-50

Fig. 2. System principle diagram of μ-law companding transform.

ìïx ; ïï y =í ïïv + ln((x + 0.001)/ v ) ; k ïîï

function y=ln((x+0.001)/v)/(1+lnu)+v

In order to reduce the difficulty of hardware implementation, this paper presents a piecewise function algorithm, x £0 x >0

,

(6)

where k = 1 + ln u . This algorithm compresses signal larger than that at the turning point, and keeps the signal less than or equal to the turning point. In this way, the number that the signal generates distortion is not only reduced, but the complexity of hardware implementation is also cut down. Fig. 3 is curve of the new companding function. As the companding function is normalized relative to the turning point, so the turning point can be selected dynamically according to preset value, to ensure that the average power of signals remains basically unchanged after companding transform.

3 2.5 2 1.5 1

function y=x v=1.5 v=2

0.5 0

0

0.5

1

1.5

2

2.5

3

x Fig. 3. Curve of the new μ-law companding function.

xl = IDWPT { X l },

l = 1, 2, ⋅⋅⋅, V

which are weighted by the weighting coefficients bl , finally getting the transmitted signal

3.2. The Technology of PTS Based on Improved Threshold Search Algorithm

V

x =  bl ⋅ xl ,

(7)

l =1

3.2.1. Basic Principle of PTS Technology PTS is the method which reduces the probability of occurrence of transmission signal peak by using linear transformation. Fig. 3 is principle diagram of PTS technology. In Fig. 3, input signal X is divided into several non overlapping sub vectors, V

namely X =  Xl , where V

is the number of

l =1

packets, X l = { X l1 , X l2 ,..., X lN } is the sub vector, N is the length of each sub vector. In order to keep each sub vector the same length, it is need to add zero for some sub vector. Then performing inverse discrete wavelet packet transform to each sub vector, and getting the signals

where

{b = (b , b ,..., b

the

rotation

0 l

l

b = exp { jφ } , i l

i l

),

l = 1, 2,..., V} referred to

factor

vector, mutual

N −1 l

1 l

of sub φ ∈ [ 0, 2π ] , which are i l

statistically independent. The selection for bl is needed to make the value of PAPR to a minimum, so the optimal weighting coefficients must meeting following condition, 

V



l =1



{b2 , bl , ⋅ ⋅ ⋅, bV } = arg min  1max  bl ⋅ xl  , ≤n≤ N {b2 , bl ,⋅⋅⋅, bV } 

(8)

where arg min ( ) representing the decision condition when the function takes the minimum.

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Sensors & Transducers, Vol. 171, Issue 5, May 2014, pp. 43-50 X1

x1

X2

x2

b1

b2

xv bv

Xv

Fig. 4. Block diagram of PTS technology for sending end.

The technology of PTS can effectively reduce PAPR and does not produce distortion because it is a method of linear processing, but it has high computational complexity. Therefore, the research for PTS technology with low complexity is very necessary. The paper is mainly focus on how to search the optimal rotation factor to reduce computational complexity.

3.2.2. Improved Threshold Search Method Traditional searching method is an optimal method, but the high computational complexity makes it difficult in application. Then, Cimini proposed a suboptimal algorithm that of Cimini searching method, which is a good solution to the high computational complexity, but whose performance of PAPR suppression decreases. In order to solve the problem, literature [12] proposed a threshold searching method which can avoid the searching of phase factor being trapped in local optima when the threshold gets smaller, thus increasing the probability of searching to the optimal phase factor sequence, eventually greatly enhancing the performance of PAPR suppression, but the system complexity increases. Pointing to the deficiency of literature [12], the paper puts forward a method of improved threshold searching which increases the module block of acceptable probability to reduce the cycle operation of system in the premise of guaranteeing the performance of PAPR suppression, so as to achieve fast convergence of the optimal value. Finally reducing the complexity of system. Supposing the peak to average power ratio of the current phase factor sequence is PAPR , the acceptable probability is [13], 1, P =  −λ / T , e

λ≤0 λ >0

,

(9)

where λ = PAPR − threshold , T is the variable decreasing with the increase of the number of loop iterations. In the paper, processing for T by

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geometry cooling of the simulated annealing [14], that is Ti +1 = nTi , where i is the number of loop iterations, n = (T f / Te )

1/ I

as the declining rate of T ,

I as the total number of loop iterations, T f and Te as

the initial and final value of T respectively. The principle of the algorithm is: if λ ≤ 0 , then P = 1 , which indicates accepting the current phase factor sequence and stopping search; if λ > 0 , calculating the value of acceptable probability P = e − λ / T , at the same time randomly generating a number r which is evenly distributed in (0,1). It is known from formula 9 that the greater the value of λ , the smaller the value of P . So, as long as the value of λ in an acceptable range, namely as long as P > 0.5 , the probability of r < P will be greater than that of r > P , then accepting the phase factor sequence and stopping the search, otherwise continuing to search. Processing flow of the method of improved threshold searching is shown in Fig. 5. Specific steps of the algorithm are as follows: 1. At the start, all phase factor bl = 1 { l = 1,2,...,V} , and calculating the peak to average ratio PAPR0 . 2. Supposing threshold = 4dB . If PAPR0 ≤ threshold , stopping search; otherwise cycle search and the process is: let i = 1, l = 1 , where i as the cycle number, turning bl = −bl , and recalculating the peak to average ratio PAPRil . If PAPRil ≤ threshold , stopping search; otherwise randomly generating a number r which is evenly distributed in (0,1), and computing the acceptable probability P = e − λ / T , if r < P , stopping search; otherwise recovering bl = −bl at the same time turning bl +1 = −bl +1 , computing the current peak to average ration PAPRil +1 , until finishing optimization of all phase factor bl , l = 2,3,..., V . The remaining steps are same with the original method of threshold search, finally getting the optimized phase factor sequence {b1 , b2 , ⋅ ⋅ ⋅, bV } = min ( PAPR1 , PAPR2 , ⋅ ⋅ ⋅, PAPRV ) . The improved method of threshold search adds a module block of acceptable probability, which decides whether to accept the current phase factor

Sensors & Transducers, Vol. 171, Issue 5, May 2014, pp. 43-50 sequence by comparing the value of a random number r and acceptable probability P , so as to prevent cycle operating when the selection of threshold is too low, finally to achieve fast convergence to the optimal value.

3.3. Method of PAPR Reduction Based on Improved μ-law Companding Transform and PTS Technology Because of various method of PAPR reduction for OWPM system has its advantages and disadvantages, the ideal effect of PAPR reducing can not be obtained if only taking one algorithm. In view of this situation, the paper puts forward a combination algorithm based on the improved μ-law companding and PTS technology in order to achieve the complementary of performance. Fig. 6 is the block diagram of sending end. But it should be noted that PTS technology is a kind of linear operation and the μ-law companding transform as nonlinear operation, in order to prevent nonlinear operation increases the distortion of signal, so execution of the joint algorithm is taking the first on the signal by PTS technique, then by the method of μ-law companding transform. As long as the two methods are chosen properly, not only the computational complexity of the system can be reduced, noise and radiation out of band but also can be cut down, at the same time, to ensure the system performance of PAPR reduction.

4. Simulations and Analysis In order to prove the effectiveness and feasibility for the proposed algorithms, simulations for the performance of OWPM system are performed. Simulation parameters are shown in Table 1.

4.1. Simulation for the Method of Improved μ-law Companding Transform Make the companding factor μ = 5 , respectively simulating the performance of PAPR suppression and the BER for OWPM system taking the conventional or the improved μ-law companding transform, then analyzing the simulation results.

Fig. 5. PTS algorithm based on improved threshold search method.

X1

x1

X2

x2

b1 b2

xv Xv

bv

Fig. 6. The transmitter’s functional block diagram of improved joint algorithm based on the μ-law companding and PTS technology.

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Sensors & Transducers, Vol. 171, Issue 5, May 2014, pp. 43-50 Table 1. Simulation parameters. Parameter The length of original signal sequence The number of total symbols Wavelet packet function Mapping mode Number of sub channel companding factor

Value N=2048 10000 db4 4QAM 8 μ =5

Fig. 7 is comparison of the distribution curves of PAPR between taking the original and the improved μ-law companding transform and taking no companding transform. As can be seen, the performance of PAPR suppression of system has improved greatly when the system signals are processed by companding transform, and the improved method to the performance improvement of PAPR reduction is the best.

CCDF (Pr[PAPR>PAPR0])

10

10

10

-1

based on the improved threshold search algorithm selects signal variance as the initial value Ts , and making T f / Ts = 0.2, I = 9 .

-2

Improved μ-law companding No companding transform

-3

2

4.2. Simulation for PTS Technology Based on the Improved Threshold Search Simulations for the performance of PAPR suppression of PTS technology and for the BER performance of the OWPM system based on PTS technology are as follow. In experiment, signal packet using random segmentation, and the segmentation number V = 4 , phase rotation factor bi ∈ {+1, −1} . PTS technology

0

Original μ-law companding

10

Fig. 8 is the statistical results of simulating 1000 times in Gauss white noise channel. As can be seen, the BER performance of improved μ-law companding transform is slightly worse than that of conventional μ-law companding transform when SNR is less than 15 dB; but when SNR is high, the curve shows better BER performance of the improved μ-law companding transform, which has proved the validity and correctness of the improved algorithm. Comprehensive analysis shows the improved method of μ-law companding transform not only reduces the complexity of system implementation, but also as far as possible without affecting BER greatly enhances the system performance of PAPR suppression.

3

4

5

6

7

8

9

10

11

12

PAPR0 [dB]

Fig. 7. Comparison of the distribution curves of PAPR between taking the original and the improved companding transform and no companding transform.

Fig. 9 shows the performance of PAPR suppression has a certain degree of improvement after the signals are processed by PTS technology, and Cimini search for the performance improvement of PAPR reduction is the worst, while the improving performance of the other three kinds of search algorithm is almost the same.

10

10

BER

10

10

10

-1

10

CCDF=Pr[PAPR>PAPR0]

10

-2

-3

-4

10

10

10

10

-1

-2

-3

-5

-6

No companding transform Original μ-law companding Improved μ-law companding

-20

-15

-10

-5

0

5

10

15

SNR [dB]

Fig. 8. Comparison of BER between taking various companding transform and no companding transform.

48

0

0

20

10

-4

5

Original PTS based on traversal searching PTS based on Cimini searching PTS based on threshold searching PTS based on improved threshold searching 5.5

6

6.5

7

7.5

8

8.5

PAPR0[dB]

Fig. 9. Comparison of the distribution curves of PAPR for PTS based different search method.

9

Sensors & Transducers, Vol. 171, Issue 5, May 2014, pp. 43-50 Fig. 10 is the statistical results of simulating 1000 times in Gauss white noise channel. As can be seen, the BER curve of OWPM system based on PTS technology is almost the same as that of the OWPM system without PTS algorithm when SNR is less than 12 dB; but when SNR is more than 12 dB, the curve shows better BER performance of the improved threshold search algorithm. Simulation results have proved the PTS algorithm can gain accurate demodulation signals on the condition that not to affect the BER performance of system.

transformation and PTS technology based on improved threshold search algorithm, and the cascade algorithm requires the signal should be processed firstly by PTS algorithm, then by the pre distortion operation. Fig. 11 shows the OWPM system based on the combined algorithm has better effect of PAPR suppression than that of the system only based on the improved μ-law companding transform or only based on the improved PTS algorithm. 10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

CCDF =Pr[PAPR>PAPR0]

10

BER

10

0

10

Original PTS based on traversal searching PTS based on Cimini searching PTS based on threshold searching PTS based on improved threshold searching 10

-7

10 -20

-15

-10

-5

0 SNR [dB]

5

10

15

The measure of complexity only considers the number of calculation times of PAPR in the paper. Table 2 shows the complexity of the three kinds method that of Cimini search, threshold searching and the improved threshold search can be greatly reduced than that of the original search method.

-2

-3

2

3

8

4

8

24 5

28 9

24 5

28 9

5

9

10

36

2

6

10

34

4.3. Simulation for the Combined Algorithm of PAPR Reduction The combined algorithm is to process signals using the improved method of μ-law companding

5

6

7

8

9

10

11

Fig. 12 is the statistical results of simulating 1000 times in Gauss white noise channel. As can be seen, the BER curve of OWPM system based on the combined algorithm is slight lower than that of the system only based on the improved PTS algorithm, but better than that of the system only based on improved method of μ-law companding transform.

The maximum number of iterations

4

4

Fig. 11. Comparison of the distribution curves of PAPR between the combined algorithm and other methods.

10

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

BER

Number of segmentation module Original search Cimini search Threshold search (threshold=4 dB) Improved threshold search (threshold=4 dB)

-1

PAPR0 (dB)

Table 2. The number of loop iterations of various search algorithms. The minimum number of iterations

Original Improved μ-law companding PTS based on improved threshold searching The new algorithm

20

Fig. 10. Comparison of BER for PTS based different search method.

No.

0

-7

10 -20

Original Improved μ-law companding PTS based on improved threshold searching The new algorithm -15 -10 -5 0 5 SNR [dB]

10

15

20

Fig. 12. Comparison of BER between the combined algorithm and other methods.

In summary, compared to the improved μ-law companding transform or the PTS based on improved

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Sensors & Transducers, Vol. 171, Issue 5, May 2014, pp. 43-50 threshold search method, the proposed combined algorithm can obtain higher PAPR reduction properties at the expense of loss small part of BER performance, which can seek a balance point between the performance of PAPR suppression and the BER performance.

5. Conclusions On the basis of studying for the improved μ-law companding transform and the improved PTS technology, and for the cascade processing on the two kinds of improved algorithms, this paper presents an improved method of PAPR reduction for OWPM system. Theoretical analysis and simulation results show that, the improved μ-law companding transform or PTS technology compared with the former algorithm has corresponding performance improvement, while the combined method can better reduce PAPR and the complexity of processing in premise of littler affecting the original BER performance.

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