International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 1 ISSN 2229-5518 IJSER © 2012 http://www.ijser.org WiMax PLL's FIR Filter Design Using LMIs Hatem Elaydi, Ayman Alquqa, H Khozondar Abstract—WiMax technology evolved greatly over the last decade. Using optimization techniques to improve the behavior of filters in phase-locked loop (PLL) in terms of overshoot and setlling time is a challenging task. This paper introduces a new PLL's loop filter design methodology that meets mobile WiMax design objectives such as small settling time, minimum overshoot and working mobile WiMax frequency range. LMI optimization based on semidefinite programmimg is used for FIR filter design to optimize conflicting objectives. The obtained results are compared with linear programming results. The LMI results outperformed results of linear programming and other comparable designs. Index Terms— LMI, PLL, FIR, Mobile WiMax, Frequency Synthesizer ——————————  —————————— 1. INTRODUCTION he fourth generation (4G) of mobile broadband networks is based on the foundation of worldwide Interoperability for Microwave Access (WiMax)[1]. Quality of Service (QoS) for different traffic classes, robust security, and mobility are guaranteed by WiMax. The phase-locked loop (PLL) is basically an electronic circuit that maintains a constant phase angle relative to a reference signal by controlling an oscillator [2]; moreover, it plays a significant part in WiMax system. PLLs are most commonly used in frequency synthesizers of wireless systems. A frequency synthesizer generates a range of output frequencies from a single stable reference frequency of a crystal oscillator [3]. In communication systems, many appli- cations require a range of frequencies or a multiplication of a periodic signal. For example, in most FM radios, a PLL fre- quency synthesizer is used to generate 101 different frequen- cies. In order to generate highly accurate frequencies with varying precisely steps, such as from 600 MHz to 800 MHz in steps of 200 KHz, most wireless transceiver designs employ a frequency synthesizer. Frequency Synthesizers are also widely used in signal generators and in instrumentation systems, such as spectrum analyzers and modulation analyzers. A basic configuration of a frequency synthesizer is shown in Figure 1. Fig. 1. Basic Frequency Synthesizer The basic frequency synthesizer includes a very stable crystal oscillator and N- programmable divider in the feedback loop in addition to the PLL. The programmable divider divides the output of the Voltage Controlled Oscillator (VCO) by N and locks to the reference frequency generated by a crystal oscillator. The output frequency of VCO is a function of the control voltage generated by the phase detector/comparator (PD). The output of the phase comparator, is proportional to the phase difference between the signals applied at its two inputs, controls the frequency of the VCO. Thus, the phase comparator input from the VCO through the programmable divider remains in phase with the reference input of crystal oscillator. Therefore, the VCO frequency is maintained at . This relation can be expressed as (1) where fr is fractional frequency and N is an integer number. This implies that the output frequency, f0 is equal to (2) Using this technique, a number of frequencies separated by and a multiple of N can be produced. For example, if the input frequency is 24 KHz and the N is 32, then the output frequency is 0.768MHz. In the same manner, if N is an array of numbers, then the output frequencies will be a proportional array. This basic technique form using phase locked loop technique is used to develop a frequency synthesizer using a single reference frequency. As for the phase detector in Figure 1, represents the phase input, the phase error, and output phase. Phase error (phase detector output) can be calculated such as (3) and the VCO output can be calculated such as (4) where is the feedforward transfer functions, and is the feedback transfer functions [3]. T ————————————————  Hatem Elaydi is currently working as a professor in electrical engineering Department at the Islamic University of Gaza, Palestine. E-mail: helaydi@iugaza.edu.ps  Ayman Alquqa is holding a masters degree in electrical engineering from the Islamic University of Gaza, Palestine. E-mail: aalqouqa@gmail.com  Hala Khozodar is currently working as a professor in electrical engineering Department at the Islamic University of Gaza, Palestine. E-mail: hkhozon- dar@iugaza.edu.ps Phase Detector Loop Filter VCO fr fo Programmable Counter ( )