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Abstract: Orthogonal frequency division multiplexing (OFDM) is the latest trend in this era. OFDM is a multicarrier modulation technique that has recently found wide adoption in a widespread variety of high-data-rate communication systems, including digital subscriber lines, wireless LANs (802.11a/g/n), digital video broadcasting. Original OFDM signals have high PAPR (Peak to Average Power Ratio) which require expensive radio transmitters and receivers having high power amplifiers operating in linear range. There are numerous methods to reduce the PAPR of the OFDM signal. In this paper, the COFDM system is analyzed using two different Reed Muller codes and the simulation results are compared with the convolution codes. Also, Hanning windowing and peak clipping techniques were investigated for reducing the PAPR in OFDM systems.
Key words : PAPR,COFDM, Reed-Muller Codes, Hanning Window, Peak Clipping.
OFDM has its roots in the military communication systems from the late1950s. OFDM is based on the FFT, which is a mathematical concept; FFT allows individual channels to maintain their Orthogonality for distance to adjacent channels. These techniques allow data symbols to be reliably extracted and multiple sub channels to overlap in the frequency domain for increased spectral efficiency. The basic principle of OFDM is to split a high-rate data stream into a number of lower rate streams that are transmitted simultaneously over a number of sub-carriers. The relative amount of dispersion in time caused by multipath delay spread is decreased because the symbol duration increases for lower rate parallel sub-carriers. The other problem to solve is the inter-symbol interference, which is eliminated almost completely by introducing a guard time in every OFDM symbol. This means that in the guard time, the OFDM symbol is cyclically extended to avoid inter carrier interference. Figure 1 shows the spectrum of OFDM signal.An OFDM signal is a sum of subcarriers that are individually modulated by using phase shift keying (PSK) or quadrature amplitude modulation (QAM).
Figure 1 Spectra of(a)an OFDM subchannel and (b) an OFDM signal
The original OFDM signal has high PAPR due to instantaneous addition of subcarrires at a particular time. The distribution of the data over many carriers means that selective fading will cause some bits to be received in error while others are received correctly. By using an error-correcting code, which adds extra bits at the transmitter, it is possible to correct many or all of the bits that were incorrectly received. The information carried by one of the degraded carriers is corrected, because other information, which is related to it by the error-correcting code, is transmitted in a different part of the multiplex (and, it is hoped, will not suffer from the same deep fade). This accounts for the "coded" part of the name COFDM which is also helpful to reduce PAPR.
II. Coded OFDM
The basic principle of COFDM is to divide a high-rate data stream into N lower rate streams and to transmit them at the same time over a number of subcarriers. Since the symbol duration is increased, the relative amount of dispersion in time caused by multipath delay spread is decreased. Intersymbol interference (ISI) is another problem, which can almost be eliminated by introducing a guard time in every COFDM symbol. In order to avoid the ICI, a COFDM symbol is cyclically extended by adding a guard time.
A general block diagram of the transmitter and the receiver for the COFDM scheme is shown in Figure 2.
Figure 2 Block Diagram of COFDM Transmitter and Receiver
The COFDM is used in 802.11a standard. In the following discussion as we describe the COFDM system, frequent references will be made to this standard
1. Channel Coding
In the IEEE 802.11a standard, data is encoded with a convolutional encoder . The code rate for the convolutional code can be changed by using a puncturing process. In this report, the data are also encoded with RM codes. The details of RM codes are explained later.
2. Block Interleaver
If decoding errors occur in a codeword and these are passed to the next block, they may affect the performance of the entire system. The performance of the system can be improved if these errors are distributed over the other code words. This can be achieved by interleaver/ deinterleaver.
A block interleaver consists of a two-dimensional array, into which the data are read along its rows. When the array is full, the data are read out by the columns, thus the order of the data is permuted. The original order can be received by the corresponding de-interleaver in which the data are read in by columns and read out by rows.
3. Symbol Mapping
The interleaved and rearranged data are mapped onto constellation points in ac-accordance with the modulation type. Figure 3 shows QPSK constellation points.
Figure 3. QPSK Constellation points
4. Discrete Fourier Transform (DFT)
There are 52 subcarriers per channel in a IEEE 802.11a standard WLAN system, where 48 of these subcarriers carry data and the remaining four subcarriers are used as pilot tones. After serial-to-parallel conversion, each COFDM symbol is modulated over 52 subcarriers by applying an inverse fast Fourier transform (IFFT).
5.Guard Interval and Cyclic Extension
A guard time is added to each COFDM symbol to eliminate the ISI and ICI, and it is removed before the FFT operation at the receiver. Since the other parameters are chosen according to the guard interval time, it is an important parameter for the COFDM system. As long as the guard time is larger than the expected delay spread, multipath components from one symbol do not cause interference with the other symbol.
The COFDM signal then is up converted to the 5-GHz band and transmitted over the channel.
The presence of noise in the channel affects the ability to make correct decisions about the received symbols at the receiver part of the communication system, thereby limiting the data transmission rate.
On the receiver side of the COFDM system, the reverse operations are performed. At the front end, a low-noise amplifier (LNA) that reduces the effective noise temperature of the receiver and an automatic gain control (AGC) that estimates the power of the pilot tone and controls the power at the demodulator output are used. The guard interval is removed once the symbols are detected. The symbol constellations are recovered by passing the signal through FFT. The resulting data are deinterleaved and channel decoded.
This section describes the encoding/decoding algorithm of the Reed-Muller (RM) coding  and how it is used for the reduction of PAPR in COFDM systems. Since the encoding and decoding algorithms are complicated and different from the other schemes, some binary operations used with RM codes  are first defined and then the encoding and decoding algorithms are presented.
An r th order Reed Muller code (r, m) is the set of all binary strings (vectors) of length n = 2m associated with the Boolean polynomials p (x1 , x2 , . . . , xm) of degree at most r . The 0th order Reed Muller code (0, m) consists of the binary strings associated with the constant polynomials 0 and 1; that is,
Thus, (0, m) is just a repetition of either zeros or ones of length 2m . At the other extreme, the mth order Reed Muller code (m, m) consists of all binary strings of length 2m , see.
, when m = 3 we have
The rows x1 x2 = 11000000, x1 x3= 10100000, and x2 x3 = 10001000 are added to form
Finally, the row x1 x2 x3 = 10000000 is added to form
Another example of a Reed Muller encoding matrix is
Encoding a message using Reed Muller code (r, m) is straightforward. Take the code we are using to be (r, m). Its dimension is
In other words, the encoding matrix has k rows, see . We send messages in blocks of length k. Let m = (m1, m2 , . . . mk) be a block, the encoded message Mc is,
where Ri is a row of the encoding matrix of (r, m). For example, using (1, 3) to encode m = (0110) gives0 âˆ- (11111111) + 1 âˆ- (11110000) + 1 âˆ- (11001100) + 0 âˆ- (10101010) = (00111100). Similarly, using (2, 4) to encode m = (10101110010) gives (0011100100000101).
Decoding Reed Muller encoded messages is more complex than encoding them. The theory behind encoding and decoding is based on the distance between vectors. The distance between any two vectors is the number of places in the two vectors that have different values. The distance between any two code words in (r, m) code is 2mâˆ’r. The basis for Reed Muller encoding is the assumption that the closest codeword in (r, m) to the received message is the original encoded message. Thus for e errors to be corrected in the received message, the distance between any two of the code words in (r, m) must be greater than 2e.
The decoding method used is not very efficient, but is straightforward to implement. It checks each row of the encoding matrix and uses majority logic to determine whether that row was used in forming the encoding message. Thus, it is possible to determine what the error-less encoded message was and what the original message was. It must be noted that the error-correcting capability of the RM codes can be increased by increasing the minimum Hamming distance.
IV. SIMULATION PARAMETERS
During the simulations, in order to compare the results, the same random message was used. Although the code makes it possible to select a different number of symbols and interleaver pairs, all simulation runs were performed with 1,000 symbols and a (20, 50) interleaver pair.
After construction of the subblocks and modification,four different channels were formed. Channel 0 is a noise-free channel with no AWGN and multipath effects. Channels 1, 2 and 3 incorporate AWGN, multipath, and AWGN and multi-path, respectively which are tabulated in table I.
In Channel 1, the standard deviation Ïƒ of white Gaussian noise is varied from 0 to 0.06 for different coding options.
The multipath fading parameters used in Channels 2 and 3 are tabulated in table II. The multipath loss in dB and the delay in ms are listed for indoor channel environment. There are 18 taps and delay coefficients in the channel. Three different Doppler frequencies of 5 Hz, 15 Hz, and 25 Hz were considered; the corresponding velocities of these frequencies were 0.29, 0.87 and 1.45 m/s, respectively. They represent walking speeds in an indoor environment.
Table I: Types of Channels
Noise Free Channel
Multipath effect Channel
Multipath + AWGN Channel
Table II: Simulation parameters
V. SIMULATION RESULTS
Channel 0 was used to test whether the system was configured properly and working correctly. In this case, the received message is the same as the source message. Numerous simulations performed for different types of RM and convolution codes demonstrated that the code ran correctly. Figure 4 shows the transmitted and received QPSK Constellations.
Figure 4: Transmitted and Received QPSK Signal Constellations
The performance of the COFDM system is next tested using Channel 1, which includes AWGN without multipath fading effects. The received QPSK constellations for different levels of noise (ï³) are shown in Figure 5.
Figure 5: The Effects of AWGN over QPSK Signal constellation
The performance of the COFDM system was studied by adding multipath effects to the channel. First, multipath effects without AWGN were simulated using Channel 2. Figure 6 shows the received QPSK constellation. The constellation points are scattered from their original position due to the effects of multipath fading. Figure 6 shows that, with the addition of differential decoding, the constellation points realigned somewhat within their respective spaces.
Figure 6. The Effects of Multipath on QPSK Signal Constellation
Channel 3 takes AWGN, multipath effects, and Doppler shift into account, hence more realistic than Channel 2. Channel 3 may be considered a good representation for indoor environments. Figure 7 shows the magnitude of the QPSK signal at the input of the receiver. Figures 8.5 show the QPSK constellation prior to and after the differential decoding. QPSK constellation points are shifted from their original phase sectors because of the effects of multipath fading. The spreading of the constellation points increases with the noise variance. As Figure 8.5 illustrates, the constellation points can be realigned to their respective phase sectors to some extent by using differential decoding.
Figure 7: The Effects of Channel 3 on QPSK Signal Constellation
2. PAPR Reduction
The RM codes used in this work are R(2,4) and R(2,5).The RM codes in general are not efficient in reducing the PAPR in COFDM. Table 3 shows the effect of different codes on PAPR reduction.
Table 3 Types of code and PAPR reduction
Peak clipping and Hanning windowing were used to reduce the PAPR. It can be seen that peak clipping reduced PAPR by 3 dB. However, the reduced PAPR value introduced a high BER. Table 8.5 shows the results when the COFDM signal was clipped at 18 through 12 it shows performance of the system at CL = 18 is very close to no-clipping performance. However, as the peak clipping level is decreased, the required Eb/No to achieve the same BER performance increased because of the increasing probability of existence of the COFDM signal magnitudes higher than the clipping level.
TABLE 4: Effect of Clipping levels on PAPR reduction
Hanning windowing was also implemented to reduce the PAPR. The values of kc =0.1 and ka = 0.2 were assigned as Hanning windows coefficients during the simulations to achieve the best performance.
When the windows were applied to the COFDM signal, the resulting spectrum was the spectrum of the windowed signal. The windowing process was tested with 3-, 5- and 9-point Hanning windows. The PAPR values are shown in Table 5. It shows that the improvement in PAPR reduction is limited with Hanning windowing
TABLE 5: Reduction in PAPR by windowing and clipping
Type of PAPR reduction
No PAPR reduction technique
Figure 8 shows the received QPSK constellation for channel 1(ï³ = 0.001) and R(1,3)code.
Figure 8: Signal constellation plot with Hanning windowing
Figure 9: Signal constellation plot with Peak Clipping
The constellation points are more scattered from their original position due to the effects of peak clipping (fig.9) as compared to Hanning windowing
In this paper, a COFDM based digital communication system with Reed-Muller error correction coding was successfully simulated. The results showed that the COFDM system is robust in indoor channel environments. Reed-Muller codes are straightforward to implement, and they provide a wide range of coding options. The results showed that they have almost the same performance in PAPR reduction as convolution codes. The addition of Hanning windowing and peak clipping improve PAPR reduction by 1 dB and 3 dB respectively, improvement with Hanning windowing is limited. But Hanning window is introduced less bit error as compare to Clipping. Peak clipping provides considerable reduction in PAPR but at the cost of increased bit error.