Carrier recovery for QPSK signals in the presence of AWGN

INTRODUCTION

A carrier recovery system is a circuit used to estimate and compensate for frequency and phase differences between a received signal's carrier wave and the receiver's local oscillator for the purpose of coherent demodulation. In the transmitter of a communications carrier system, a carrier wave is modulated by a baseband signal. At the receiver the baseband information is extracted from the incoming modulated waveform. In an ideal communications system, the carrier frequency oscillators of the transmitter and receiver would be perfectly matched in frequency and phase, thereby permitting perfect coherent demodulation of the modulated baseband signal. However, transmitters and receivers rarely share the same carrier frequency oscillator. Communications receiver systems are usually independent of transmitting systems and contain their own oscillators with frequency and phase offsets and instabilities. Doppler shift may also contribute to frequency differences in mobile radio frequency communications systems. All these frequency and phase variations must be estimated using information in the received signal to reproduce or recover the carrier signal at the receiver and permits coherent demodulation.

 1. QUADRATURE PHASE SHIFT KEYING (QPSK)

     
Imagine a ship. Its captain thinks up a different signaling arrangement. Here he has marked out four spots on the deck, to the East, West, North and South. He assigns four different combinations to each of the spots as shown below. He can send two bits, with each flash of the light. 

                

Figure 1.1:  Two dimensional signaling system

     By creating four signaling spots he has added another dimension. This gives two basis functions, the East-West and the North-South movements. Now there are four different symbol positions possible and we can assign 2 bits to each unique symbol.

     QPSK uses two basis functions, a sine and a cosine whereas BPSK uses just one. By varying the phase of each of these carriers (in the ship example, the position) we can send two bits per each signal. The dimensionality of a modulation is defined by the number of basis functions used. That makes QPSK a two dimensional signal. Not because it sends two bits per symbol, but because it uses two independent signals (a sine and a cosine) to create the symbols. Here are the four symbol mapping definitions for QPSK. Each packet is defined in terms of a sine or a cosine but with a different phase. (The phase is the angle at which the signal starts).

Table 1.1: Mapping rules for QPSK

     In QPSK the four symbol definitions, written in sine or a cosine, can be decomposed further so we can compute the I and Q channel amplitudes. Let’s take the symbol s1 as example.

Using the trigonometric identity,

cos(x+y) = cosx cosy – sinx siny                                                                                (1.1)

We can write this equivalent expression

        √((2  E_s)/T) cos(ωt+π⁄4) = √((2  E_s)/T) (cosωtcos π⁄4-sinωtsin π⁄4)            (1.2)

                                                   =√(E_s/T)(cosωt-sinωt)

     

We see that the little packet of carrier signal representing a particular symbol can be created by a free running sine and cosine wave of certain amplitudes. This makes hardware realization possible.

     Though we created the BPSK modulated signal by stringing together the appropriate packets of signals, in real systems, we cannot create a modulated carrier this way. We have to make use of oscillators that produces sines and cosines. We cannot just use a certain part of the signal as if they are sitting on a shelf for us to grab a piece. We need a way to create a signal packet of a particular phase when needed out of a fre-running sine or cosine. This is where Quadrature Modulation and I and Q channels come into play. I and Q channels are not just concepts but also how modulators are designed.

Figure 1.2: QPSK Constellation mapping