Matlab/Simulink can be a useful tool in analyzing the effects of signal impairments, such as phase noise, on the modulation quality of single-carrier and multicarrier OFDM WiMAX transmitters.

Advanced wireless communications standards rely on complex modulation schemes to achieve high bandwidth efficiency. WiMAX, for example, employs the orthogonal frequency division multiplexing (OFDM) technique.^{1} Due to many challenges with such complex modulation formats, sophisticated simulation and verification tools and approaches are required to achieve optimum system-level performance. For example, the transmit modulation accuracy depends on the transmit filter accuracy, digital-to-analog- converter (DAC) performance, in-phase/ quadrature (I/Q) imbalances, phase noise, and transmitter nonlinearity. To better understand the effects of various performance parameters on WiMAX, a simulation model based on Matlab/Simulink software from The MathWorks was developed to analyze the impact of phase noise, peak-to-average power ratio (PAPR), transmitter (TX) nonlinearity, and I/Q mismatches on the transmitter modulation quality. The WiMAX 64-state quadratureamplitude- modulation (64QAM) OFDM (IEEE 802.16 OFDM) standard was used for this example because it imposes the highest signalto- noise-ratio (SNR) requirements.

The Simulink model consists of the major function blocks shown in **Fig. 1(a)**: the 64QAM modulator, the OFDM modulator, the RF transmitter (Tx), the OFDM receiver (Rx), and the 64QAM demodulator. According to the model, random binary data is first 64QAMmodulated, and then OFDM-modulated to divide the transmission bandwidth into 192 narrow subchannels; the data are then transmitted in parallel over these subchannels at a relatively low rate. Running at this low data rate plus adding the cyclic prefix helps to combat delay spread. The OFDM modulator creates an OFDM symbol through an inverse Fast Fourier Transform (IFFT) by means of these 192 data subcarriers along an additional 28 lower-frequency null guard subcarriers and 27 higher-frequency null guard subcarriers, 8 pilot subcarriers, and one DC null subcarrier. The OFDM signal then is frequency upconverted to 2.35 GHz and further amplified to +23 dBm before it is sent to the receiver (Rx). The PAPR is measured before the signal is OFDMdemodulated. * Figure 1(b)* shows the RF Tx with direct frequency upconversion architecture and three key components: the frequency upconverter (UPC), the pre-power amplifier (PPA), and the power amplifier (PA). The error vector magnitude (EVM) and relative constellation error (RCE) are measured after the signal is OFDM-demodulated. Then the signal is further 64QAM-demodulated before bit-error-rate (BER) monitoring. The Simulink model does not include the channel coding for accurate BER monitoring. The OFDM parameters are listed in

*.*

**Table 1** * Figure 1(b)* addresses an approach to modeling the nonlinearity of the RF Tx, which is a major source of SNR impairment in the system. When multitone signals pass through the RF Tx, intermodulation components are generated, which causes the distortion to the signal. Nonlinearity models for UPC and PPA use the static nonlinearity model by specifying the third-order intercept point (IP3). The nonlinearity model for PA is the well-known Rapp model

^{2}as shown in Eq. 1:

where

V_{sat} = the saturation voltage level and

P = the knee parameter.

The Rapp model addresses amplitude-modulated-related (AM-AM) distortion but no AM-phase-modulation (AM-PM)-related distortion. According to previous work, it has been stated that in the sub-5 GHz band, typical values of the knee parameter are in the range of 2-4.^{3} In this model example, a knee parameter of 2 was used.

The next step in the Simulink model development is to address two key Tx requirements: the RCE and the EVM, according to the WiMAX standard.1 These two parameters are also the metrics for evaluating the modulation quality and the impact of distortion on WiMAX system performance throughout this article. Both EVM and RCE are defined as the vector difference between the actual and ideal symbol position as shown on an I/Q constellation diagram. Their relationship can be expressed by Eq. 2:

The Tx RCE is dictated by the required Rx SNR for a certain modulation and error correction scheme to ensure that the error contribution to system performance is small:

In the case of the WiMAX OFDM Rx, since 64QAM modulation with a channel code rate of 3/4 requires the largest SNR (21 dB) for a BER of 1 10 ^{6} , the 64QAM Tx requires the smallest RCE (-31 dB), corresponding to an EVM of 2.8 percent. This required Tx RCE of -31 dB only causes a SNR degradation of 0.41 dB as calculated by

The following sections will use the Matlab/Simulink model to address the impact on Tx design caused by Tx nonlinearity, PAPR, local oscillator (LO) phase noise, I/Q gain/phase mismatch, and DAC parameters. The LO phase noise degrades SNR as additive noise effects. The UPC, the phase noise is superimposed on all subcarriers when the OFDM modulation signal is mixed with the LO signals. The required Tx RCE dictates the integrated LO double-sideband (DSB) RMS phasenoise (FM_{rms}) requirement, which is -31 dB for 64QAM OFDM with a 3/4 coding rate mode. Parameter F_{rms} has been widely used to calculate the phase noise degradation to the SNR for a single-carrier (SC) system. Earlier reports claimed that OFDM systems were orders of magnitude more sensitive to phase noise than SC systems.4,5 But it was also claimed oppositely in later articles that SNR degradation caused by phase noise was the same in OFDM and SC systems.^{6,7} The Matlab/Simulink model can be useful in modeling the impact of LO phase noise on OFDM systems. The phase-noise profile in **Fig. 2a**^{8} is used here as the reference phase noise (RPN), and the phase noise with 30 dB better than RPN at each frequency offset as "30-dB better" in * Table 2*. The calculated F

_{rms}with RPN is -45.37 dB using the method in ref. 9. In test case 1, the contribution to the SNR distortion is all from the LO phase noise since the UPC, PPA, and PA are configured to have no nonlinearity by setting output third-order intercept point (OIP3) and saturated output power (P

_{sat}) to +100 dBm. The simulated RCE is -48.35 dB as in

*, which is almost same as the calculated F*

**Table 2**_{rms}value. Therefore, the Matlab/Simulink model shows that the phase noise causes almost the same SNR degradation to both multicarrier and single-carrier systems. It is worthwhile to mention that some algorithms have been used for tracking, estimating, and correcting common phase error (CPE) of the LO phase noise using pilot signals, but this is beyond the scope of this report.

A major problem with the OFDM modulation is its relatively high PAPR, which requires a high IP3 specifications for the RF Tx. When the number of subcarriers is large, the subcarrier signals are uncorrelated and their amplitudes add to produce the peak power. When all individual subcarriers reach their peak at the same time, it could produce a maximum PAPR equal to the number of used data and pilot subcarriers, N_{used} + N_{pilot}, or 10 x log_{10}(200) or about 23 dB in WiMAX OFDM systems. In reality, this maximum PAPR rarely occurs. * Figure 3* shows the simulated complementary cumulative distribution function (CCDF) for 64QAM OFDM envelope signal powers with 18,992,933 samples collected. The probability of occurrence of signal peaks that are 12.4 dB greater than the average power (20.5 dB) is only 2 x 10

^{-6}.To guarantee the system BER to be less than 1 x 10

^{-6}at a certain PAPR, the probability of that PAPR should be less than 2 x 10

^{-6}. Therefore, a 12.4-dB PAPR should be appropriate for 64QAM WiMAX if a system BER of 1 x 10-6 is to be achieved.

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Typically, PAs are main sources for Tx nonlinearity. In WiMAX systems, PAs must deliver high power levels with stringent linearity requirements and handle a high PAPR. Otherwise, the transmitted signals may suffer spectral spreading and in-band distortion. Achieving such PA linearity results in a trade off in efficiency. Therefore, finding the maximum allowable nonlinearity is very important for WiMAX applications.

In test case 2, the LO phase noise is configured as "30-dB better" to reduce any impairments due to phase noise. The IP3s of the UPC and PPA are reduced to +37 and +38 dBm, respectively, and the PA's P_{sat} is reduced to +32.35 dBm. The output 1-dB compression point (OCP1) was measured to be +30.2 dBm. Since the Tx output power is +23 dBm and the PAPR is 11.7 dB, the peak power could be as high as +34.7 dBm, or 4.5 dB above the OCP1. The simulated EVM is 2.489 percent. The SNR distortion at OCP1 could be slightly different for different PA designs. Therefore, other than knowing the PAPR value, building a PA model that matches the design will be useful for deciding the power backoff from the OCP1. In test case 3, P_{sat} is further reduced to +29 dBm to increase the nonlinearity. The simulated EVM and RCE are 6.7 percent and -23.5 dB, a severe degradation due to the nonlinearity. The obvious spectral regrowth could also be observed from the plots in * Fig. 4*.

The signal transmitted through the RF Tx could experience signal distortion due to gain and phase mismatches. The imperfections are typically generated by relative differences between the transceiver I and Q signal branches from the DAC to all analog components in the RF Tx I/Q. In * Fig. 1a*, the I/Q imbalance block following the OFDM modulator adds any of those two imperfections into the 64QAM OFDM modulation signal.

In test cases 4 and 5, the IP3 and P_{sat} are set to +100 dBm to eliminate the nonlinearity and the "30-dB better" phase noise is applied to reduce the effects of LO phase noise on system distortion. The SNR degradation due to the 0.15-dB amplitude mismatch and 0.5-deg. phase mismatch are simulated as shown in * Table 2*. For test case 7, all impairments are applied (OCP1 = +26.8 dBm), including LO phase noise, nonlinearities, and I/Q gain and phase mismatches) and the simulated EVM is 2.72 percent while the simulated RCE is -31.29 dB, which are slightly better levels than the WiMAX specifications. In reality, some tolerance should be allowed for performance variations due to changes in temperature, process, and voltage, as well as impairments from the DAC and the filters, which are not included in the Simulink model. The RCE for test case 6 can be estimated based on the results from test cases 1, 2, 4, and 5, as shown in Eq. 5, where RCE

_{n}is the measured RCE for test case n.

In summary, signal-quality requirements for a WiMAX Tx were analyzed using Matlab/Simulink software. The simulator makes it possible to study the impact of impairments on Tx modulation quality for a 64QAM OFDM WiMAX system. The analysis showed that LO phase noise has almost the same impact on multicarrier or signal carrier systems. The simulation created a CCDF of a WiMAX OFDM PAPR, showing a PAPR of 12.4 dB to be an appropriate power backoff for WiMAX.

REFERENCES

1. IEEE Standard 802.16-2004, "Part 16: Air interface for fixed broadband wireless access systems," October 2004.

2. C. Rapp, "Effects of HPA-Nonlinearity on an 4-DPSK/ OFDM-Signal for a Digital Sound Broadcasting System," Proceedings of the 2nd European Conference On Satellite Communications, Liege, Belgium, Oct. 22-24, 1991, pp. 176-184.

3. Tal Kaitz, BreezeCOM, Performance aspects of OFDM PHY proposal, IEEE 802.16.3c-01/49, 2001-03-14.

4. T. Pollet, M. van Bladel, and M. Moeneclaey, "BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise," *IEEE Transactions on Communications*, Vol. 43, February/March/April 1995, pp. 191-193.

5. A. Garcia Armada and M. Calvo, "Phase Noise and subcarrier spacing effects on the performance of an OFDM communication system," *IEEE Communications Letters*, Vol. 2, No. 1, January 1998.

6. M. Moeneclaey, "The effect of synchronization errors on the performance of orthogonal frequency-division multiplexed (OFDM) systems," in Proceedings of COST 254 (Emergent Techniques for Communication Terminals), Toulouse, France, July 1997.

7. A. Garca Armada, "Understanding the effects of phase noise in orthogonal frequency division multiplexing (OFDM)," IEEE Transactions on Broadcasting, Vol. 47, June 2001, pp. 153-159.

8. A 2.4-GHz Direct Conversion Transmitter for WiMAX Applications, Cecile Masse, 2006.

9. Jonathan Y. C. Cheah, "Analysis of Phase Noise in Oscillators," RF Design, November, 1991. pp. 99-105.

10. Hyunchul Ku, and J. Stevenson Kenney, "Behavioral Modeling of Nonlinear RF Power Amplifiers Considering Memory Effects," *IEEE Transactions on Microwave Theory and Techniques*, Vol. 51, No 12, December 2003.

11. L. Ding, "Digital Predistortion of Power Amplifiers for Wireless Applications," Ph.D thesis, Georgia Institute of Technology, March 2004.