II. OFDM TRANSRECEIVER SYSTEMS

OFDM Transciever for Nonlinear Communication Systems 1 Shatrughna Prasad Yadav, 2 Subhash Chandra Bera 1 EEE Department, Indus University, Ahmedabad 2 Satcom & Navigation Systems Eng. Division, SAC, ISRO, Ahmedabad Abstract Orthogonal frequency division multiplexing (OFDM) is a special form of multi-carrier transmission systems where all the subcarriers are orthogonal to each other. It provides high data rate transmission capability with sufficient robustness to radio channel impairments, is spectrally efficient, is ideal for multimedia communications and has been widely accepted for future communication for different services. At the same time, it suffers from sensitivity to timing and frequency synchronization errors, high value of peak-to-average power ratio (PAPR) and co-channel interference (CCI). High value of PAPR drives high power amplifier into its saturation region and causes it to operate in the nonlinear mode. In this work, Matlab simulation of OFDM transceiver has been carried out to provide description of each of the steps involved in the generation and reception of an OFDM signal. The carrier frequency of 2 GHZ has been used for 124 subcarriers. Theoretical and simulated results of PAPR have been compared for different values of subcarriers. There is perfect match between the simulated and the theoretical results when the number of subcarrier (N) is sufficiently large. Index Terms Orthogonal Frequency Division Multiplexing, Peak-to-Average Power Ratio, High Power Amplifiers, Nonlinearity. I. INTRODUCTION The future wireless communication requires high data rates. Dealing of the high data rate in an unpredictable wireless channel is a difficult task [1]. To combat the unpredictability of wireless channel and provide high data rate communications, multi-carrier transmission system has gained popularity [2]. The channel distortion to the data at high data rate is quite significant which demands a complex receiver structure [3]. One such multi-carrier transmission system uses orthogonal frequency division multiplexing (OFDM) in which all the subcarriers are orthogonal to each other [4]. OFDM system has many advantages over traditional communication systems. (a) It uses simple receiver as it turns the frequency-selective fading channel into a flat fading channel and hence a simple one-tap equalizer is sufficient for channel estimation and recovery of data. (b) OFDM is a spectrally efficient and is ideal for multimedia communications. (c) It has been widely accepted for future communication for different services. OFDM system has following disadvantages: (a) It is highly sensitive to time and frequency synchronization errors, (b) It has high value of peak-to-average power ratio (PAPR) and (c) OFDM in cellular systems gives rise to co-channel interference (CCI) [5]. Various uses of OFDM include: wideband data communications over mobile radio FM channels, high-bit-rate digital subscriber lines (HDSL), asymmetric digital subscriber lines (ADSL), very-high-speed digital subscriber line (VDSL), Digital audio broadcasting (DAB), digital video broadcasting (DVB) along with high-definition television (HDTV) terrestrial broadcasting, wireless local area network (WLAN) standards in their physical layers, HiperLAN in physical layer, IEEE 82.11a in its physical layer [6]. II. OFDM TRANSRECEIVER SYSTEMS The OFDM transceiver system is depicted in figure1. The channel coded bits are mapped to the constellation points using 4-QAM modulation. These modulated data are represented in complex number and are in serial form. Then it is converted into parallel form using serial-to-parallel convertor. Inverse discrete Fourier Transform (IDFT) is applied to each stream. The IDFT transform can be implemented very efficiently by the inverse fast Fourier Transform (IFFT) [7]. This equals transition from frequency-domain to time-domain. After IFFT, cyclic prefix is added in every block of data and data are multiplexed in serial fashion. Cyclic prefix is added at the transmitter to maintain orthogonality among the subcarriers and reduce the effect of inter carrier interference (ICI). At this point of time data are OFDM (baseband) modulated and these are reconverted into serial form using 185

parallel to serial convertor [8]. These serial data are converted into analog form using digital to analog converter. Passband (RF) modulation is performed and the signal is up converted to transmission frequency of 2 GHz for transmission. At the receiver, the signal is down-converted and reconverted into digital domain using analog-to-digital converter (ADC). At this point of time carrier frequency synchronization is done. These serial data are converted into parallel form using serial to parallel converter. Then cyclic prefix is removed before using FFT block to demodulate the OFDM signal and subsequently parallel to serial conversion is done. The received data are demapped according to the transmission constellation diagram [9]. III. OFDM TRANSMITTER MODEL The OFDM transmitter diagram is shown in figure 1. The OFDM signal is generated using IDFT. Continuous transmitter output signals are constructed using a Fourier series representation within each OFDM symbol interval. m(t) S/P IDFT Add Cyclic Prefix P/S Transmit Filter PT (ω) OFDM TRANNMITTER CHANNEL WITH ADDED NOISE Channel H (ω) n(t) P/S DFT Remove Cyclic Prefix S/P Receive Filter PR (ω) OFDM RECEIVER Fig. 1 OFDM transceiver model Each OFDM symbol contains N subcarriers, where N is an even number and a power of 2. The OFDM symbol duration is T u seconds.the subcarrier spacing is maintained as Δω, which is the shortest and defined in equation (1). (1) The Fourier series spectrum for the duration of the n th OFDM symbol is given as in equation (2). The spectrum in equation (2) is inverse Fourier transformed and limited to a time interval of T u in order to provide the OFDM symbol in the time domain. The time-domain signal is given as in equation (3). After frequency- to time-domain conversion, the signal is summed up, and the cyclic prefix is added as given in equation (4). (4) Where T g is the cyclic prefix duration (guard interval) and T s = T u + T g is the total OFDM symbol duration. The transmitted baseband signal is formed by linking together all OFDM symbols in time domain as in equation (5). 186

This signal is finally up converted to a carrier frequency and transmitted as given in equation (6). Where u(t) is the transmitted RF signal and f c is the RF carrier frequency. Equation (6) can be further expanded as given in equation (7) and (8). and, Here, N is the number of subcarrier; T is symbol duration and equation (7) for the emitted signal is given in equation (9). is the carrier frequency. A particular version of. Where Equation (9) and (1) describes an OFDM transmission system. Here, n is the carrier number, l is the symbol number, m is transmission frame number, N is the number of transmitted carriers; T S is the symbol duration; T U is the inverse of the carrier spacing; T g is the duration of the guard interval; f c is the central frequency of the radio frequency (RF) signal; C m,,n is the complex symbol for carrier n of the data symbol n o.1 in frame number m; C m,1,1 complex symbol for carrier n of the data symbol no.2 in frame number m, etc. Carrier frequency (1) Table 1: Numerical values for the OFDM parameters Parameters Values 2 GHz Number of subcarriers, N 124 Useful OFDM symbol period, T U Guard Interval (Cyclic Prefix), (T g = T u /32) Total OFDM symbol period (T S = T g + T U) 5.12μs.16μs 5.28 μs 2L- IFFT/ IFFT length, (L= 124) 248 Considering equation (9) for the period from t= to t=t S, we get Equation (11). With n = n ( N max + N min ) / 2 Equation (11) is similar to the inverse discrete Fourier transform (IDFT) as given below: The transmitted OFDM signal is organized in frames. Each frame has duration of T F seconds, and consists of 68 OFDM symbols. Four frames constitute one super-frame. Each symbol is constituted by a set of N=1,24 carriers 187

PSD (db/hz) and transmitted with a duration T S. A useful part with duration T U and a guard interval with a duration T g compose T S. The specific numerical values for the OFDM parameters are given in Table 1. The guard interval is added by taking copies of the last NT g / T U of these samples and appending them in front. A subsequent up-conversion then gives the real signal u(t) centered on the frequency f c. Matlab simulation of OFDM transmitter as given in figure 2 has been performed using equation (9) for the parameters depicted in table 1 at the carrier frequency 2 GHz. 4-QAM modulation has been used and 2L - IFFT of length 248 (L= 124) has been considered. To achieve over sampling zeroes are padded to the signal. Bandpass output BASEBAND QAM MOD 1 2 3 4 5 OFDM SIGNAL (IFFT) PULSE SHAPING LOW PASS FILTER X Carrier Frequency, fc Fig. 2 OFDM signal generation at transmitter The base band modulation is done using 4-QAM modulation. After serial to parallel conversion OFDM signal is obtained using IFFT. The frequency domain response of the signal at point 2 of figure 3 is given in figure 3. 1.5 Frequncy response of Carrier FFT at point 1 1.5.5 1 1.5 2 2.5 3 3.5 4 x 1 8 Carrier Welch PSD estimate at point 1-4 -6-8 -1-12 5 1 15 2 25 3 35 Frequency (MHz) Fig. 3 Frequency response of signal at point 2 The OFDM modulated signal is passed through a pulse shaping filter. Its impulse response is depicted in fig. 4. 1 Pulse shape g(t).9.8.7.6.5.4.3.2.1.5 1 1.5 2 2.5 Time (second) x 1-9 Fig. 4 Impulse response of the D/A reconstruction filter The output of the transmit filter is shown in Figure 5. 188

(db) PSD (db/hz) 4 3 2 1 Frequency Response of signal (FFT) at point 3 1 2 3 4 5 6 7 8 x 1 9 Welch PSD estimate at point 3-5 -1-15 1 2 3 4 5 6 7 Frequency (GHz) Fig. 5 Frequency response of filter at point 3 The D/A reconstruction filter is a Butterworth filter of order 13 and cut-off frequency of approximately 1/T. Its response is shown in Figure 6. 1 Frequency Response of D/A Reconstruction Filter -1-2 -3-4 -5-6 -7.5 1 1.5 2 2.5 3 3.5 4 x 1 9 Fig. 6 Frequency response of D/A reconstruction filter The filter introduces a delay of approximately 5 μs. The RF signal is generated using bandpass modulation for the in-phase signal m l (t) and quadrature signal m q (t) as per the equation (13). The frequency responses for signal, u(t), are shown in figure 7 and 8 respectively. Here, it can be seen that the modulated signal contains large value of PAPR in the time response of signal in figure 7. 8 Time Response of Passband Signal at point 5 6 4 2-2 -4-6 -8 1 2 3 4 5 6 7 Time (second) x 1-8 Fig. 7 Time response of passband signal at point 5 189

PSD(dB/Hz) 2 15 1 5 Frequency Response of passband Signal at point 5 1 2 3 4 5 6 7 8 x 1 9 Welch PSD estimate at point 5-5 -1-15.5 1 1.5 2 2.5 3 3.5 4 Frequency (GHz) Fig. 8 Frequency response of passband signal at point 5 IV. OFDM RECEIVER The transmitted signal pass through the channel which can be as modeled as a time-domain complex-baseband transfer function, and convolved with the transmitted signal to estimate the signal at the receiver side. The channel baseband equivalent impulse response function for nth user, h n (t) is given in equation (14). Where is the complex gain of the mth multipath component for the nth user at time t. The channel is assumed to be static for the duration of one OFDM symbol. For static channel equation (15) is redefined and is given in equation (15). Where The corresponding frequency-domain channel transfer function, H n,s, is obtained using Fourier transformation as given in equation (16). 6 7 8 9 1 RF Txr signal X LPF SAMPL ER FFT QAM DEMOD QAM CONST Carrier, fc Fig. 9 Receiver model of OFDM systems The signal at the receiver consists of multiple echoes of the transmitted signal, as well as white Gaussian noise and interference. The RF signal received by the nth user is written as in equation (17). Where, is a real-valued, passband signal combining additive noise and interference. The receiver now has to recreate the transmitted signal. Apart from noise and multipath effects, timing and frequency error may also occur 19

PSD(dB/Hz PSD(dB/Hz in the receiver.a basic receiver model which is the inverse of the transmission process is shown in figure 9. Mainly it consists of down conversion of RF passband signal to recover in-phase signal and quadrature component of the signal. It is passed through low pass filter and sampler. Sampler is used to convert the analog signal in discrete sequence form. OFDM demodulation is done by FFT operation. Finally, baseband 4-QAM demodulation is done to recover the signal constellation. 2 15 1 5 Frequency Response (FFT) of received signal at point 6.5 1 1.5 2 2.5 3 3.5 4 x 1 9 Frequency Response Welch PSD estimate of received signal at point 6-5 -1-15.5 1 1.5 2 2.5 3 3.5 Frequency (GHz) Fig. 1 Frequency response of bandpass signal at point 6 Matlab simulation is performed for the receiver model shown in figure 11. To recreate the transmitted subcarriers, N correlators are used, each one correlating the incoming signal with the nth subcarrier frequency over an OFDM symbol period as per equation (18). The received signal subjected to multipath effects and AWGN is down converted to baseband signal as shown in figure 1. 4 Frequency Response (FFT) of received signal at point 7 3 2 1.5 1 1.5 2 2.5 3 3.5 4 x 1 9 Frequency Response (PSD estimate) of received signal at point 7-5 -1-15.5 1 1.5 2 2.5 3 3.5 Frequency (GHz) Fig. 11 Frequency response of low pass filter at point 7 The Cyclic Prefix (CP) is removed from each block, and the signal is then correlated with each subcarrier frequency.response of the low pass filter is plotted in figure 11. The OFDM receiver uses discrete signal processing to obtain the estimate of the transmitted subcarriers. The received signal is sampled using Dirac impulse train and is written as given in equation (19). Where, T is the sample duration and is given by equation (2). 191

Imaginary axis PSD(dB/Hz) 1.5 Frequency Response (FFT) of received signal at point 8 1.5.2.4.6.8 1 1.2 1.4 1.6 1.8 2 x 1 8 Welch PSD Estimate of received signal at point 8-4 -6-8 -1-12 2 4 6 8 1 12 14 16 18 Frequency (MHz) Fig. 12 Frequency response of sampler at point 8 The sampler frequency response is plotted in figure 12. Then FFT operation is performed to demodulate the OFDM signal as given in equation (2). 1.5 QAM Constellation at point 1 1.5 -.5-1 -1.5-1.5-1 -.5.5 1 1.5 Real axis Fig. 13 scatterplot of 4-QAM signal constellation at point 1 Finally, the baseband demodulation is done to obtain the 4- QAM constellation which is plotted in figure 13. V. PEAK TO AVERAGE POWER RATIO PAPR is defined as the ratio of the maximum power and the average power of the complex passband signal as given in equation (22). Crest factor (CF) is also used to define the power characteristics in terms of its magnitudes (not power) as given by equation (23) which is square root of PAPR. Maximum peak power occurs when all the N subcarrier components are added with same phases and is a case of constructive addition. When average power, = 1, it results in PAPR = N, which is, the maximum power equals to N times of average power. Value of PAPR is more for QAM with M > 4 than M-ary PSK. The probability of the occurrence of the maximum power signal decreases with increase in the value of N. The amplitude of the OFDM signal u(t) follows a Rayleigh distribution. If {Z n } is magnitudes of complex samples and if average power of u(t) is equal to one, then {Z n } is a Rayleigh random variable and normalized with its average power. It has the following probability density function defined by equation (24). 192

Where, E{Zn} 2 =2 = 1. The maximum value of Z n is equivalent to the crest factor (CF) which is defined in equation (23). If Z max denotes the crest factor then the cumulative distribution function (CDF) of Z max is given as per equation (25). Where, = Complementary cumulative distribution function (CCDF) is used to find out the probability that the crest factor (CF) exceeds a particular value z as given in equation (27). It has been assumed that N samples are independent and sufficiently large while deriving equations (24) and (26) and not applicable for bandlimited or oversampled signals. Derivation of exact CCDF for oversampled signal is more involved and therefore, the following simplified CCDF is used as indicated in equations (28). Fig. 14 CCDF of PAPR Where value of is obtained by fitting the theoretical CCDF into the actual one. For sufficiently large value of N simulation results have shown that 2.8 is appropriate. Figure 16 shows the theoretical and simulated CCDFs of OFDM signals with N = 64, 128, 256, 512, 124 for QPSK/QAM modulation. It can be seen that the simulation results deviate from the theoretical ones for smaller value of N. This implies that the CDF described by equations (28) is accurate for large value of N. VI. CONCLUSION Orthogonal frequency division multiplexing is used as multi-carrier communication systems where all the subcarriers are orthogonal to each other. It offers high data rate transmission capability, shows robustness to channel impairments, is spectrally efficient and ideal for multimedia communications. It has been widely accepted for future communication for different services. But exhibits large peak to average power ratio. Larger value of PAPR drives high power amplifiers into its saturation region and causes it to operate in the nonlinear mode. In this paper, Matlab simulation of OFDM transceiver has been carried out and theoretical and simulated results of PAPR have been compared for different number of subcarriers. It has been observed that the simulation results deviate from the theoretical ones as the number of subcarrier becomes small. 193

REFERENCES [1] R. Prasad, OFDM for Wireless Communications Systems, Artech House Publishers, Norwood, MA, USA, September 24. [2] S. Hara and R. Prasad, Multicarrier Techniques For 4G Mobile Communications, Artech House Publishers, Norwood, MA, USA, 23. [3] Cho, Kim, Yang and Kang. MIMO- OFDM Wireless Communications with Matlab, IEEE Press, John Wiely and Sons (Asia) Pvt. Ltd., 21. [4] Park and Song, A New PAPR Reduction Technique of OFDM System with Nonlinear High Power Amplifier, IEEE Transactions on Consumer Electronics, Vol. 53, No. 2, May 27. [5] Han and Lee, An Overview of Peak-To-Average Power Ratio Reduction Techniques for Multicarrier Transmission, IEEE Wireless Communications, April 25. [6] Bo, Zhi-Xing, Chang-Yong, Tao-Tao and Jian-Hua, Effects of PAPR Reduction on HPA Predistortion IEEE Transactions on Consumer Electronics, Vol. 51, No. 4, November 25. [7] W.Y. Zou and Y. Wu, COFDM: An Overview, IEEE Trans. Broadcast. vol. 41, no. 1, March 1995. [8] R.V. Nee and R. Prasad, OFDM for Wireless Multimedia Communications, Artech House Publishers, Norwood, MA, USA, 2. [9] ETSI TR 11 19 V1.3.1 (28-1), Digital Video Broadcasting (DVB); Implementation guidelines for DVB terrestrial services; Transmission aspects. AUTHOR BIOGRAPHY Shatrughna Prasad Yadav received B. Tech. degree in electronics and communication engineering from the Institution of Engineers (India) in 1992, MBA in marketing management from Indira Gandhi National Open University, New Delhi in 1998, M. Tech. in digital systems from Motilal Nehru National Institute of Technology, Allahabad in 22 and currently pursuing Ph.D. in Electronics and Communication Engineering from Gujarat Technological University, Ahmedabad. From 1986 to 26, he was with Indian Air force. From 26 to 27, he worked as senior lecturer in electronics and communication engineering department at Institute of Technology, Nirma University, Ahmedabad. Since 27, he is with the Institute of Technology and Engineering, Indus University, Ahmedabad as a head of electrical and electronics engineering department. Subhash Chandra Bera received the B.Sc. degree (with honors) in physics from Presidency College, Calcutta, B. Tech. and M. Tech. degrees in radio physics and electronics from the Institute of Radio Physics and Electronics, University of Calcutta, and Ph.D. degree in Microwave Engineering from Gujarat University. Since 1994, he is with the Space Applications Centre, (ISRO), India, where he has been involved in design and development of various microwave active subsystems that are being used in many communication and navigation payload projects such as the INSAT-2, INSAT-3, INSAT-4 and GSAT of spacecraft. Presently, he is serving as head of the Satcom and navigation Systems engineering Division, Space Applications Centre, ISRO. He is Ph.D. research supervisor of Nirma University and Gujarat Technological University (GTU) in the field of Electronics and communication engineering. 194