An Improved Discrete Cosine Transform (DCT) for Peak-to-Average Power Ratio (PAPR) Reduction in OFDM System
Yong Liang Lim, A.A.A. Wahab, S. S. N. Alhady, W. A. F. W. Othman, H. Husin
School of Electrical and Electronic Engineering, Universiti Sains Malaysia, 14300 NibongTebal, Pulau Pinang, MALAYSIA
Submitted: 21 June 2019 / Accepted: 27 September 2019 / Published online: 31 October 2019
ABSTRACT
Orthogonal Frequency Division Multiplexing (OFDM) which is the recognized signal modulator parameter has been widely applied at wideband communication and Digital Signal
Processing. 4G mobile communications and Wireless networks have been accounted for the nobility of OFDM. This due to some advantages of OFDM for instance, high spectral efficiency, high transmission bit rate and immunity against multipath fading which leads to the heavy application of OFDM in our appliances nowadays. However, high Peak-to-Average Power
Ratio (PAPR) value is still the agenda which being investigated by a lot of researchers who are specialize in OFDM. This is because high value of PAPR restrict the achievement of signal transmission. This paper discusses the improved method of Principle Component Analysis
(PCA) to bring down PAPR values of OFDM system. The proposed ideas involve Discrete
Cosine Transform replaces the Fast Fourier Transform in the OFDM signal modulation technique by utilizing the modulation scheme of Quadrature Phase Shift Keying (QPSK). This proposed methods able to provide better performance in PAPR reduction. The proposed method of PCA with DCT with the 1024 numbers of subcarriers improves the greatest performance by
5.72%.
Keywords: DCT; IDCT; OFDM; PAPR; PCA.
Author Correspondence, e-mail: [email protected], [email protected], [email protected]
ROBOTIKA is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. Libraries Resource Directory. We are listed under Research Associations category.
ROBOTIKA, 2019, 1(2), 9-23 8
1. INTRODUCTION
Orthogonal Frequency Division Multiplexing (OFDM) was expected to excel among other techniques for high data rates transmission and multipath transmission over time dispersive channels. A wide variety of wireless digital communication has been relying on OFDM over this past several decades. Most of the wireless LAN standards like IEEE 802.11a/g and Wi-Max are utilizing OFDM to enhance the world of wireless communication in 21st century.
Under the concept of innovation study, the series of mobile technology from 2G to 3G and from
3G to 4G, can be regarded as a sequence of incremental innovation, since each emerges gradually with continuous improvement from the preceding technology [1]. OFDM able to provide substantially high spectral and high-power efficiency, immune to the multipath delay and frequency selective fading [2]. Furthermore, Gu et al. suggested that OFDM can occupy huge space of capacity system than conventional single carriers’ system [3]. It is believed that the transmitted signal has high peak values in the time domain occasionally and unexpectedly.
This can be explained as subcarrier components are included throughout inverse Fast Fourier
Transformation (IFFT) [18,21].
To reduce PAPR, several designated schemes and techniques have been proposed and tested.
Throughout the techniques or schemes proposed so far, they can be distinctively categorized into two types. Distortion based signal helps in differentiating the amplified signal with the directly addition of peak values reduction. Non-distortion Techniques induce the redundancy limitation and help in manipulating the OFDM signal with the objectives of achieving maximum rection of peak value in terms of PAPR.
2. METHODOLOGY
In this paper, one of the major drawbacks of the OFDM system which is high PAPR (Peak-to-
Average Power ratio) is discussed. PAPR is defined as randomly sinusoidal leads occurred during transmission of the OFDM signal. Most of those mentioned above type of communication systems required high data rate. However, problem occurs like PAPR in OFDM ROBOTIKA, 2019, 1(2), 9-23 9 system indeliberately creates resistance and causes reduction of efficiency to these types of facilities in the real life.
The fluctuation value in the time interval of amplitude due to the high number of subcarriers present in OFDM leads to overwhelming exceeded limit of allowed values for Peak Power to
Average Power Ratio (PAPR). A multicarrier signal is the addition of whole set value consists of narrow band signals in a time domain [19]. The value set of multicarrier signals vary from time to time. Thus, the peak value of the modulated data might be larger than the remaining value of the data. This is where PAPR being introduced and it will lead to high power amplifier
(HPA) to work in the nonlinear region [20]. Nonlinear operation acts as an agent to degrade the signal at the output of power amplifier.
PAPR was generally expressed as the quotient of maximum power in the time domain (highest power) to the average power (mean power) of the OFDM transmission. The mathematical formula is shown as:
Ppeak 푃퐴푃푅 = (1) Paverage where Ppeak = Peak power of the OFDM system, Paverage = Average power of the OFDM system.
OFDM system consists of a set of input data symbol values which are coordinated at channel encoder in which the data are decrypted onto QPSK constellation. The encoded data values will be converted from serial to parallel format.This porocess is needed for Inverse Fast Fourier
Transform (IFFT) to synthesis the time domain identity for the designated arrangement [7].
OFDM system with involvement of subcarriers, OFDM signal can be expressed as::
X n = IFFT {X k} (2)
푁−1 1 j2πkΔft 푋(푡) = ∑ 푋푘e √푁 (3) 푘=1 where subcarrier N is the subcarriers quantity and Xk is transmitted symbol on the Kth . Inverse ROBOTIKA, 2019, 1(2), 9-23 10
Fast Fourrier Transform is also used in Gait Recognition with the aid of Enhancement
Histogram Oriented of Gradient. [8]
The unparalleled bandwidth savings was displayed thrughout OFDM, which leads to high spectral efficiency [9].
2.1 The Proposed Improved Method
Ahmed et al. was the first at 1974 to propose the Discrete Cosine Transform [10] into the world of Mathematics. In the progress for alleviating the PAPR values of an OFDM signal, DCT is also being proposed by researcher to minimize the autocorrelation of the input signal before the implementation of IFFT operation. For instance, the Discrete Cosine Transformin 1-D with length N is expressed as:
푁−1 (2푛 + 1)휋푘 푋[푘] =∝ (푡) ∑ 푥(푛) cos { } 0 ≤ 푘 ≤ 푁 − 1 (4) 2푁 푛=0 At the receiver, inverse discrete cosine transform is calculated. The IDCT is expressed as
(Sulaiman & Darwish, 2009):
푅 = 퐸(푥∗푥퐻) (5)
푁−1 (2푛 + 1)휋푘 푥[푛] =∝ (푡) ∑ 푋(푘) cos { } 0 ≤ 푛 ≤ 푁 − 1 (6) 2푁 푘=0 Devika Rajeswaran and Aswathy K. Nair proposed Principle Component Analysis with the combined method of Clipping and Filtering for PAPR reduction [11]. To decrease peak reduction, the clipped signal will be further filtered by using FFT/IFFT to calculate the minimum values of PAPR. The obtained data from PCA algorithm acts as the stabilizing threshold for the clipping and filtering blocks. In the occasion of PCA is selected compared than ICA (Independent Component Analysis) is that the division of multivariate signals into individual source signal of ICA is not required in this analysis [12]. So far, there is still no research of PCA with DCT on PAPR reduction even though there are a lot of method combined with DCT such as PTS with DCT and SLM with DCT. ROBOTIKA, 2019, 1(2), 9-23 11
2.2 Methodology
The total amount of subcarriers that are incalculated with Phase Shift Keying modulation scheme is owned by OFDM signal. By applying the selection signal mapping. the serial data will be incalculated into the full set of data [13].
The modulation of OFDM value set is utilised by Quadrature Phase Shift Keying (QPSK). The serial data value is transformed to parallel set. In the process of modulation, modulation schemes such as Phase Shift Keying (PSK) and Quadrature Amplitude Modulation (QAM) is applied for parallel data modulation.
Then, serial–parallel converter is being utilised for implementing QPSK into the parallel data sets. The number of subcarriers is important in this process because it is needed for data transmission. The most important process here is the frequency domain samples converted to time domain domain samples by IFFT operation. After the coversion and interpretation of a set of data from QPSK or QAM, DCT is required for undergoing data transformation.
2.3 DCT Technique
After the process of QPSK Mapping, the data is being channeled to Discrete Cosine Transform.
In DCT as an orthogonal basis, every single set of cosines constructed function is being confined into the modulation scheme. This is due to the proposed scheme synchronization, we found that discrete cosine transforms (DCT) fits the best in this situation.
The result signal of DCT based technique in OFDM can be expressed as formula below:
2 푁푠−1 푛휋푡 푋(푡) = √ ∑ 푑푛퐵푛cos ( ) (7) 푁푠 푛−0 푇푠
2.4 PCA Implementation
Principal component analysis (PCA) helps in alleviating the problem of processing datasets, improve interpretation and at the same time reduce to the minimum data loss. The new uncorrelated variables is also essential for obtainable variance in maximum values. ROBOTIKA, 2019, 1(2), 9-23 12
If we mention in a simpler way, the technique is used to learn the minimum values of latent variables, called “principal components,” which indicates the largest size of calculated variability.
For instance, PCA known as conservative data analysis method because the implementation method are expected to have and can be enhanced conditionally in terms of data and structures.
[14]
PCA can produce the maximum variability in the data happened along the axis by correlating the structure of multivariate observations throughout the whole analysis. [15]
Principle Component Analysis can be applied to exhibit the relative “locations” of the samples.
The method is expected to simplify the large quantity of the synthesized data variable in different condition. We are still in the progress in determining that is it essential to evaluate every single original variable inversely of the significantly differential data. Based on the PCA results acquired, the number of variables that are inspected should not be reduced.
Covariance Matrix method and Singular Value Decomposition Method (SVD) are the two most frequent used PCA technique by most of the researcher [16]. In this study, Covariance Matrix method is chosen rather than Singular Value Decomposition Method (SVD).
Eigen vector acts as the guide to lead the direction of principle component. For Eigen value, the maximum variance value set is needed for data set correlation.
The proposed algorithm of PCA is shown below.
1. Obtain the mean of all samples.
2. Subtract the mean from all samples
3. Obtain the covariance matrix
4. Obtain the eigenvectors V and eigenvalues of the covariance matrix ROBOTIKA, 2019, 1(2), 9-23 13
1 ⅀ = 퐷푥퐷푇 (7) 푁 − 1
Where N number of iterations, D is the mean samples and T is the dimension of matrix
5. Calculate the eigenvectors in conjunction of their corresponding eigenvalues
6. The largest eigenvalues is chosen among the eigenvectors.
푤 = {푣1, … , 푣푘} (8) Where v=eigenvectors of the value set
7. All selected data will be projected into dimensional space of PCA(W) at the lower part
of the dimension.
푌 = 푊푇퐷 (9) Where D is the mean of samples, W is the selected eigenvectors and T is the dimension of matrix set.
Table 1 shows parameters for QPSK modulations for the improved PCA technique.
Table 1. Parameter for QPSK modulations No. Parameter Value 1 Total number of subcarriers 64, 512, 1024 2 Modulation Scheme QPSK 3 Number of symbols 10000 4 Number of iterations 1000
Figure 1 shows the block diagram of PCA DCT implementation
Fig. 1. the block diagram of PCA DCT implementation ROBOTIKA, 2019, 1(2), 9-23 14
Figure 2 shows the progress diagram of design implementation of improved PCA method with
DCT.
Fig. 2. Progress diagram of design implementation
3. RESULTS AND DISCUSSION
3.1 PAPR performance
Complementary cumulative distribution function (CCDF) [17] is usually implemented in ROBOTIKA, 2019, 1(2), 9-23 15 studies of the PAPR value. The parameters in terms of number of subcarriers is investigated in this paper.
The improvement comparison of PAPR value reduction in term of the numbers of subcarriers for QPSK modulation scheme is shown in the next three graphs with N=64, N=512 and N=1024 respectively.
Table 2 shows PAPR value of Original, PCA and Improved technique with DCT for N = 64,
N=512 and 1024.
Table 2. PAPR value of Original, PCA and Improved technique with DCT for N = 64, N=512 and 1024 Number of subcarriers PCA PCA-DCT Improvement (PCA with PCA-DCT) 64 9.79 9.27 0.52 (5.31%) 512 10.94 10.36 0.58 (5.60%) 1024 11.37 10.70 0.67 (5.72%)
(a) ROBOTIKA, 2019, 1(2), 9-23 16
(b)
(c)
Fig. 3. (a) Graph of CCDF with QPSK N=64; (b) Graph of CCDF with QPSK N=512; (c) Graph of CCDF with QPSK N=1024
ROBOTIKA, 2019, 1(2), 9-23 17
From Figure 3, MATLAB coding is utilized in implementing DCT-PCA technique to monitor the improvement of PAPR reduction. QPSK modulation scheme is being implemented in this paper with 3 comparison of N=64, 512 and 1024. Figure 3 (a), (b) and (c) depicts that the CCDF graph which shows the PAPR performance of the proposed OFDM systems. From Table 2, the huge PAPR reduction improvement by 5.72% is achieved by number of subcarriers with N=
1024 compared to N= 64 which gives 5.31% improvement and N= 512 which gives 5.60% improvement. This is due to the larger the number of subcarriers, the larger the data transmitted simultaneously, the lower the probability of the distorted signal produced, the lower the amplitude of the signal, the better the improvement of PAPR reduction.
From Equation 1,
2 2 According to graph (a), for PCA application, when 푀푎푥|푥푘| = 200.70 and 퐸[|푥푘| ] = 20.5 value from Matlab,
2 푀푎푥|푥푘| 푃퐴푃푅 = 2 퐸[|푥푘| ]
200.70 푃퐴푃푅 = PCA 20.5
= 9.79dB
2 2 When 푀푎푥|푥푘| = 183.55 and 퐸[|푥푘| ] = 19.8 value from Matlab, for PCA and DCT application,
183.55 푃퐴푃푅 = 푃퐶퐴DCT 19.8
= 9.27dB
The difference between 푃퐴푃푅DCT 푎푛푑 푃퐴푃푅푃퐶퐴퐷퐶푇 is
푃퐴푃푅PCA−푃퐴푃푅푃퐶퐴DCT ROBOTIKA, 2019, 1(2), 9-23 18
=9.79-9.27
=0.52dB
2 2 According to graph (b), for PCA application, when 푀푎푥|푥푘| = 248.34 and 퐸[|푥푘| ] = 22.7 value from Matlab,
2 푀푎푥|푥푘| 푃퐴푃푅 = 2 퐸[|푥푘| ]
248.34 푃퐴푃푅 = DCT 22.7
= 10.94dB
2 2 When 푀푎푥|푥푘| = 211.03 and 퐸[|푥푘| ] = 20.37 value from Matlab, for PCA and DCT application,
211.03 푃퐴푃푅 = 푃퐶퐴DCT 20.37
= 10.36dB
The difference between 푃퐴푃푅DCT 푎푛푑 푃퐴푃푅푃퐶퐴DCT is
푃퐴푃푅DCT−푃퐴푃푅푃퐶퐴DCT
=10.94-10.36
=0.58dB
2 2 According to graph (c), for PCA application, when 푀푎푥|푥푘| = 287.66 and 퐸[|푥푘| ] = 25.3 value from Matlab,
2 푀푎푥|푥푘| 푃퐴푃푅 = 2 퐸[|푥푘| ] ROBOTIKA, 2019, 1(2), 9-23 19
287.66 푃퐴푃푅 = PCA 25.3
= 11.37dB
2 2 When 푀푎푥|푥푘| = 254.66 and 퐸[|푥푘| ] = 23.8 value from Matlab, for PCA and DCT application,
254.66 푃퐴푃푅 = 푃퐶퐴DCT 23.8
= 10.70dB
The difference between 푃퐴푃푅퐷퐶푇 푎푛푑 푃퐴푃푅푃퐶퐴퐷퐶푇 is
푃퐴푃푅퐷퐶푇−푃퐴푃푅푃퐶퐴퐷퐶푇
=11.37-10.70
=0.67dB (the greatest improvement)
3.2 Bit Error Rate Performance
This parameter is usually calculated by researcher to specify the level of system performance when sending data information, it is one of the digital communication parameter where BER is formed by comparing the ratio between the total number of bits received and the numbers of bits received mistakenly.
Table 3 shows the BER comparison between Original OFDM, DCT and PCA DCT
Table 3. the BER comparison between Original OFDM, DCT and PCA DCT Number of subcarriers Bit Error Rate Ratio Original OFDM 9.79 DCT 10.94 PCA-DCT 11.37
ROBOTIKA, 2019, 1(2), 9-23 20
Fig. 4. BER comparison between Original OFDM, DCT and PCA DCT.
From Table 3, the proposed method of PCA DCT shows the lowest bit error rate values which shows the best performance among the 3 proposed techniques
4. CONCLUSION
The proposed PCA-DCT method is implemented to reduce PAPR based on the number of subcarriers being compared shown. This improved technique is used due to DCT helps in reducing the amplitude values of the transmitted signal. Therefore, the more the subcarriers involved, the better the improvement of PAPR decreasing. Number of subcarriers of 1024 shows the greatest reduction improvement of PAPR by 5.72%. Throughout the research, the proposed improved PCA method with DCT can lower the PAPR value compared to PCA only technique.
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