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LETTERS International Journal of Recent Trends in Engineering, Vol 2, No. 6, November 2009

Performance of Uncoded and LDPC Coded MSK and GMSK in Nakagami Fading

Simranjit Singh1, Student Member, IEEE, Surbhi Sharma2, Member, IEEE, Rajesh Khanna3, Fellow, IETE

Electronics and Communication Engineering Department, Thapar University, Patiala, India Email: [email protected] Email: {surbhi.sharma, [email protected]

Abstract--Low density parity check (LDPC) codes are one of the best error correcting codes in today's coding world and are known to approach the Shannon limit. As with all other channel coding schemes, LDPC codes add redundancy to the uncoded input data to make it more immune to channel impairments. In addition to coding, the best modulation techniques available today are MSK (Minimum Shift Keying) and GMSK (Gaussian Minimum Shift Keying). By using these modulation techniques in combination with LDPC codes, the paper compares the performance of coded MSK and coded GMSK with each other and with their uncoded counterparts over Nakagami fading channel for different values of m. The results show that a significant coding gain is obtained by using LDPC codes. It is also seen that although the error performance of LDPC coded MSK is slightly better than LDPC coded GMSK, the former is spectrally inefficient. Index Terms--LDPC, MSK, GMSK, iterative decoding, fading, Nakagami

I. INTRODUCTION The fundamental problem of communication systems is how to reproduce the originally transmitted signal at the receiver. Till date, many different techniques have been proposed to overcome this problem. Channel coding is one such technique of detecting and correcting transmission errors. It gives an improvement of about 10dB without much change in the hardware or increase in the cost of the overall system. In channel coding, redundant bits are added to the uncoded sequence, which help in error detection and correction. Low density parity check (LDPC) codes were developed by Robert Gallager in his PhD thesis at MIT in 1962 [1]. These codes were ignored for about 30 years and rediscovered in the late 1990s by D. J. C. MacKay and R. M. Neal [2]. In 2001, T.J Richardson, A. Shokrollahi, and R. Urbanke [3] proved that the performance of LDPC codes is close to the Shannon limit (the limit of reliable communication over unreliable channels). It has been further demonstrated by simulations that LDPC codes of block length 107 approach the Shannon limit within 0.0045dB [4]. Because of their excellent forward error correction properties, LDPC codes are set to be used as a standard in Digital Video Broadcasting (DVB-S2) and 4G mobile communication. Another advantage of LDPC codes is that they are highly parallelizable in hardware. Also, their minimum distance (dmin) increases proportionally with an increase in the block length.

In addition to using channel coding for better error performance, the technique used for modulating the coded signal is also very important as it transforms the signal waveforms and enables them to better withstand channel distortions. A number of modulation techniques are currently in use out of which MSK and GMSK are known to be the best. MSK was proposed in the late 1960's and developed later by Subbarayan Pasupathy [5]. As stated above, channel coding adds redundancy to the uncoded signal and thus increases the bandwidth in the process. So, a modulation technique is needed, which is spectrally efficient and also has good error performance. Much research has been done on the concatenation of LDPC codes with different modulation techniques [6-7]. Also, the earlier studies reveal that uncoded MSK has a better error performance than uncoded GMSK but the latter is spectrally more efficient [8-9]. What needs to be established is whether to use MSK or GMSK with channel coding, specifically LDPC codes. When the modulated signal travels through the channel, it gets distorted by noise and fading. The noise is generally modeled as AWGN (Additive White Gaussian Noise) as it is easier to treat noise as additive rather than multiplicative. A variety of models for fading have been proposed by researchers over the years, out of which Rayleigh, Rician and Nakagami have become very popular [10-12]. Rayleigh fading is used for modeling severe fading conditions and Rician fading is used for modeling fading conditions where a LOS (line-of-sight) exists between transmitter and receiver, that is, where fading conditions are less severe than Rayleigh fading. The Nakagami fading provides a very good fit for all fading conditions ranging from very severe to no fading because of the presence of an adaptive fading parameter m called shape factor [13-14]. Applications of Nakagami fading include modeling the wireless channel in microcellular systems, indoor wireless systems and mobile satellite systems. This paper is organized as follows. Section II gives an overview of the system model. In section III, LDPC codes are presented. GMSK and MSK modulation is discussed in Section IV. The fading model used is described in Section V. The simulation details are given in section VI. Finally, in section VII, the simulation results and conclusion are presented. II. SYSTEM MODEL The system model used is shown in Fig. 1. The data to be transmitted over the channel was randomly generated



LETTERS International Journal of Recent Trends in Engineering, Vol 2, No. 6, November 2009

and was in the uncoded form. This data is coded by using LDPC codes. After the coded bit sequence has been obtained, it is applied to a GMSK/MSK modulator. This modulated waveform is transmitted over the Nakagami channel in the presence of AWGN (Additive White Gaussian Noise). The received signal is passed through demodulator and decoder where the errors are detected and corrected. The various blocks used in the model have been described in detail below. dominate digital modulation are MSK (Minimum Shift Keying) and GMSK (Gaussian Minimum Shift Keying). MSK is a type of continuous phase frequency shift keying that was developed in the late 1960s. In MSK the modulation index of 0.5 corresponds to the minimum frequency spacing that allows two FSK signals to be coherently orthogonal. MSK is unique due to the relationship between the frequencies of logic 0 and 1. However, the fundamental problem with MSK is that it has side lobes and is spectrally inefficient. To remove this problem, a pre-modulation Gaussian filter is placed before MSK modulator which helps in improving the spectral efficiency and reducing the side lobes. This technique is called GMSK (Gaussian Minimum Shift Keying). The bandwidth reduction comes at a cost of increased ISI (Inter Symbol Interference). The premodulation filter has an impulse response according to the Gaussian distribution and has a bandwidth-time product BT which governs the width of the pulses. The GSM standards use GMSK as their modulation scheme with BT product equal to 0.3. The impulse response of the Gaussian filter is:

Figure 1: System model

III. LDPC CODES LDPC codes are acknowledgedly one of the best forward error correction codes. These codes have been shown to approach the Shannon capacity and have come as close as 0.0045dB. As the name suggests, LDPC codes are characterized by a parity check matrix which is sparse. A sparse matrix is one in which the number of 1's is very less as compared to the number of 0's. Due to the sparse property of the matrix, the size of the matrix can be increased without an increase in the number of 1's, which means that we can achieve better distance properties without increasing the decoding complexity. The LDPC codes are represented graphically by Tanner graphs in which there are two types of nodes, check nodes and bit nodes [15]. This graphical representation helps to easily understand the iterative decoding algorithm of LDPC codes. Unlike other codes which are decoded by ML (Maximum Likelihood) detection, LDPC codes are decoded by iterative decoding algorithm called message passing algorithm. The decoding can be hard decision or soft decision. At the time when these codes were developed, they were ignored partly because of the parallel development of concatenated codes and also because the hardware at that time could not support such a complex decoder design. In today's world with the rapid development of DSP (Digital Signal Processing) and VLSI (Very Large Scale Integration), these codes can be effectively implemented and hence they are set to be the codes of the coming wireless generations. IV. MSK AND GMSK V. NAKAGAMI FADING Modulation is the process by which signal waveforms are transformed and enabled to better withstand the channel impairments. The two modulation techniques which Nakagami fading channel has received considerable attention in the study of various aspects of wireless systems. It is also called m-distribution and fits for 0 BbT

Figure 2: PSD of GMSK for different values of BbT [9]

Where Q (t) is the Q-function

Bb is the bandwidth of the low pass filter having a Gaussian shaped spectrum, T is the bit period and BbT is the normalized bandwidth. The analysis of MSK and GMSK with LDPC codes in Nakagami fading has been left as a topic of future research with a few scattered references on this topic [16-19].



LETTERS International Journal of Recent Trends in Engineering, Vol 2, No. 6, November 2009

experimental data collected in a variety of fading conditions better than Rayleigh, Rician or Lognormal distributions. It can be used to model fading conditions more or less severe than Rayleigh fading. In this type of fading, we have an m parameter by which we can model the severity of fading. The probability density function of Nakagami distribution is given as:

Figure 3: Comparison of uncoded MSK with LDPC coded MSK for m=1

m 1/2; r 0 m is Nakagami parameter or shape factor, describing the fading degree due to scattering and multipath interference is the average power of multipath scatter field. And, (m) is the gamma function. Theoretical and experimental results on Nakagami fading show that it is the best model for data obtained from many urban multipath radio channels. VI. SIMULATION DETAILS In this paper, the performance of coded MSK and coded GMSK is compared with each other and with their uncoded counterparts over Nakagami fading channel. Also, the performance of LDPC coded MSK has been compared with LDPC coded GMSK. A (500,600) irregular LDPC coded bit stream was used with the mean column weight (wc) taken as 3. The length of the coded bit stream was taken to be 600. The bandwidth-time (BT) product for GMSK is 0.25 and the oversampling period is 8. The number of iterations is taken to be 1000. The channel is modeled as Nakagami fading and the complex noise is added randomly. The path amplitudes in Nakagami fading were taken to be Weibull distributed. Unless otherwise stated, it is assumed that: 1. The Nakagami fading model that we use in our simulations is flat. 2. The BbT product of the premodulation Gaussian filter in GMSK is always taken to be 0.25. 3. The receiver has complete knowledge about the channel. VII. SIMULATION RESULTS AND DISCUSSION Many simulation runs have been done for MSK and GMSK in both LDPC coded and uncoded format for different values of m. The obtained results are presented in the following figures.

Fig. 3 shows that LDPC coded MSK modulated signal gives about 3.3 dB coding gain over the use of uncoded MSK. So the performance of coded MSK is much better than the performance of uncoded MSK.

Figure 4: Comparison of uncoded MSK with LDPC coded MSK for m=2

Fig. 4 shows that LDPC coded MSK modulated signal gives about 4.5 dB coding gain over the use of uncoded MSK. So the performance of coded MSK is much better than the performance of uncoded MSK.

Figure 5: Comparison of uncoded and LDPC coded GMSK for m=1 and m=2

Fig. 5 shows that LDPC coded GSK modulated signal gives about 3.9 dB coding gain over the use of uncoded MSK for m=1 and a coding gain of 5 dB for m=2. So the performance of coded GMSK is much better than the performance of uncoded GMSK.



LETTERS International Journal of Recent Trends in Engineering, Vol 2, No. 6, November 2009

Electronics Letters, vol. 32, no. 18, pp. 1645-1646, Aug. 1996. T. J. Richardson, A. Shokrollahi, and R. Urbanke, "Design of capacity approaching irregular low-density parity-check codes," IEEE Trans. Inform. Theory, vol. 47, no. 2, pp. 619­637, Feb. 2001. S.Y. Chung, G. David Forney, T J. Richardson, and R. Urbanke "On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit" IEEE Communications Letters, vol. 47, no. 2, pp. 58-60, February 2001. Subbarayan Pasupathy, "Minimum shift keying: a spectrally efficient modulation", Communications Magazine IEEE, vol. 17, no. 4, pp. 14-22, July 1979 K.R. Narayanan, I. Altunbas, R. Narayanaswami, "On the design of LDPC codes for MSK", Global Telecommunications Conference, 2001 (GLOBECOM 2001), vol: 2, pp.1011-1015, Nov 2001 D. Sridhara, T.E. Fuja, "LDPC Codes Over Rings for PSK Modulation", IEEE Transactions on Information Theory, vol 51, no. 9, pp.3209-3220, Sep 2005 S. Elnoubi, S. Abou Chahine, H. Abdallah, "BER performance of GMSK in Nakagami fading channels", 2lst National Radio Science Conference (NRSC2004), pp. 1-8, March 2004. K. Murota, K. Hirade, "GMSK modulation for digital mobile radio telephony," IEEE Trans. Commun., vol. 29, no.7, pp.1044-1050, July 1981. R.H Clarke, "A statistical theory of mobile radio reception," Bell Labs System Technical Journal, Vol. 47, pp.957-1000, July-August 1968 W.C. Jakes, "Microwave mobile communications", Wiley, 1974, New York M. Nakagami, ``The m-distribution. A general formula of intensity distribution of rapid fading," in Statistical Methods in Radio Wave Propagation, W.C. Hoffman, Ed. Elmsford, NY, Pergamon Press, 1960. Li Tang, Zhu Hongbo, "Analysis and simulation of Nakagami fading channel with matlab" Asia-Pacific Conference on Environmental Electromagnetics CEEM 2003, pp. 490-494, Nov. 2003 Gayarti S. Prabhu, P.Mohana Shankar "Simulation of flat fading using MATLAB for classroom instruction", IEEE Transactions on Education Vo1.45. No.1, pp. 1925, Feb. 2002. R. M. Tanner, "A recursive approach to low complexity codes", IEEE Trans. Inform. Theory, vol. 27, no. 5, pp. 533-547, September 1981. LiDuan MA, David ASANO, "Performance of GMSK and Reed-Solomon code combinations" IEICE Trans. Fundamentals, vol. 88, No. 10, pp. 2863-2868, October 2005. M.K. Caldera and K.S. Chung, "Trellis coded GMSK in frequency-selective fading channels", Electronics Letters, Vol. 36, No. 25, pp. 2082-2084, December 2000. Taki M., Nezafati M.B., "A new method for detection of LDPC coded GMSK modulated signals", Wireless Communications, Networking and Mobile Computing, WiCOM 2006. International Conference, pp. 1 ­ 5, Sept. 2006 G.L Lui, Tsai Kuang, Ye Zhong, S. Dolinar, K. Andrews, "Coded performance of a quaternary GMSK communication system", Military Communications Conference, 2003. MILCOM 2003, Vol. 1, pp. 36- 40, Oct 2003




Figure 6: Comparison of LDPC coded MSK with GMSK for m=1


Fig. 6 shows that the error performance of both MSK and GMSK is excellent even in very severe fading conditions. Out of the two, MSK has a slightly better error performance.





11. 12.

Figure 7: Comparison of LDPC coded MSK with GMSK for m=2


Fig. 7 shows that the error performance of LDPC coded MSK is slightly better than LDPC coded GMSK for m=2.


CONCLUSION It is concluded that the overall performance of LDPC coded GMSK is better than LDPC coded MSK. Even though a slightly better BER was obtained by using MSK, its spectrum utilization is poor. GMSK is spectrally efficient and ideal to use with LDPC codes as the latter introduces redundancy that leads to spectral widening. The use of GMSK with LDPC codes ensures the efficient use of the spectrum. ACKNOWLEDGEMENT

18. 15.



This work was supported by UGC and AICTE India vide letter no. 31-1 (PUN) (SR) 2008 TU and letter no 8023/BOR/RID/RPS-105/2008-09. REFERENCES

1. 2. R.G. Gallager, "Low density parity-check codes." M.I.T. Press, Cambridge, MA, 1963. D. J. C. MacKay and R. M. Neal, "Near Shannon limit performance of low density parity check codes," 19.




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