# Performance of Quasi-Constant Envelope Phase Modulation through Nonlinear Radio Channels

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3 s o (t) = C 1 sin(ω c t + φ(t)) + s o (t) = C 1 s I (t) cos ω c t s 2 I (t)+s2 Q (t) C 1 s Q (t) + sin ω c t s 2 I (t)+s2 Q (t) + s o (t). In practice there is a low pass network at the output end of the power amplifier to eliminate harmonics [5]. The last term can be ignored in the transmitted signal. The transmitted signal can be written as s o (t) = C 1s I (t)cosω c t + C 1s Q (t)sinω c t. (2) s 2 I (t)+s2 Q s (t) 2 I (t)+s2 Q (t) For BPSK, the output of the power amplifier can be written as Kcos(ω c t) if s i (t) > 0 s o (t) = 0 if s i (t) =0 Kcos(ω c t) if s i (t) < 0 For OQPSK and π/4 QPSK, we can write the transmitted signal as s o (t) =s oi (t)+s oq (t) (3) where s oi (t) = s oq (t) = K s I (t) s 2 I (t)+s2 Q (t) cos ω c t (4) K s Q (t) s 2 I (t)+s2 Q (t) sin ω c t (5) and K is the gain of the fully saturated power amplifier. The fully saturated power amplifier does not change the phase of the input signal. It completely removes the amplitude information. IV. DEMODULATION The nonlinearity makes it very difficult to analyze the performance of a communication system employing a fully saturated power amplifier. Convolution can not be applied in a nonlinear system. With the nonlinear model, we can ignore the up-converter, treat the fully saturated power amplifier as a baseband power amplifier, and analyze the system in baseband. Let the baseband input signal to the power amplifier be s i (t) =s I (t)+j s Q (t) (6) where s I (t) and s Q (t) are the in-phase and the quadrature baseband signals respectively and j 2 = 1. The output of the power amplifier can be written as s o (t) =s oi (t)+j s oq (t) (7) where s oi (t) = s oq (t) = Ks I (t) s 2 I (t)+s2 Q (t) (8) Ks Q (t) s 2 I (t)+s2 Q (t). (9) Define d(t) =s o (t) Ks i (t) as the distortion. The system is of the following properties: (1) The nonlinearity of the system is time invariant. Both the input signal and the output signal are stationary random processes; (2) The input signal and distortion are dependent. The distortion is the complement of the linear part Ks i (t) to make the envelope constant; (3) The shaping filter plays a key role in the relation among s i (t), s o (t) and d(t). For BPSK, Eq. (7) can be simplified as s o (t) =sign(s i (t)). For M-ary PSK with M 4, the in-phase signal and the quadrature signal cannot be treated independently. Without the knowledge of the shaping filter, it leads to unpredictable envelope variation of the power amplifier output. The in-phase signal and the quadrature signal interfere with each other more for those not symbol-aligned modulations such as OQPSK. We introduce a model to analyze the nonlinear system. The equivalent linear model is derived from the statistic characteristics of the nonlinear power amplifier output. In the receiver, one cares only about the SNR at the sampling point. Since s o (t) is stationary and correlated with s i (t), we can take it as an equivalent output of a linear system. The equality is valid at the sampling points. The equivalent linear model of a communication system employing a fully saturated power amplifier depends on the shaping filter and the modulation scheme. For OQPSK, we assume the source generates a sequence of symmetric binary impulses x(t) = A i δ(t it b ) (10) i= where A i = ±1 and T b is the bit time. The autocorrelation R x (t) =δ(t) and the power spectral density P x (f) =1.The shaping filter is a finite impulse response low-pass filter with the frequency response H s (f) and the time response h(t). Suppose the duration of h(t) is 2n symbols, the baseband signal can be written as n s I (t) = A Ii+j h(t it s jt s ) (11) s Q (t) = j= i= n n j= i= n A Qi+j h(t it s jt s T s /2) (12) where {A Ii } and {A Qi } are sequences of elements in {+1, 1} and T s is the symbol time. By (7), (8) and (9) the output signal y(t) can be written as Ks I (t) Ks Q (t) y(t) = + j. (13) s 2 I (t)+s2 Q s (t) 2 I (t)+s2 Q (t) GLOBECOM /03/\$ IEEE

4 By the statistic characteristics of the binary impulse sequence, we can calculate the autocorrelation R y (t) of the output signal. The power spectral density P y (f) = R y (t)e j2πft dt. (14) Since R x (t) =δ(t) and P x (f) =1, we have the equivalent impulse response H s(f) H s(f) = P y (f)/p x (f) = P y (f). (15) The frequency response of the matched filter is H m (f). Therefore, the output of the matched filter is y (t) = the sample value can be written as y (0) = = H s(f) H m (f)exp(2πjtf)df (16) H s(f) H m (f)exp(2πjft)df t=0 H s(f) H m (f)df. Let the output of a communication system employing the linear power amplifier be So(t) =K 0 H s (f)exp(2πjft)df (17) where K 0 is the gain for the linear system, which keeps the transmitted power being the same as that in the compatible nonlinear system. The output of the matched filter in the receiver can be written as y(t) =K 0 H s (f) H m (f)exp(2π.jft)df. (18) Then, y(t =0) = K 0 H s (f) H m (f)df. Define the degradation caused by the nonlinearity as D = 20 lg y (0) = 20 lg H s(f) H m (f)df y(0) K 0 H s(f) H m (f)df. (19) Table II shows the degradation caused by the nonlinear power amplifier to the modulated waveforms using π/4 QPSK or OQPSK. The shaping pulse is the square root raised cosine function with the roll off factor β. TABLE II THE SNR DEGRADATION FOR π/4 QPSK AND OQPSK. β π/4 QPSK OQPSK β π/4 QPSK OQPSK V. NUMERICAL RESULTS We study the effect of the nonlinear radio distortion to the performance of convolutional code and turbo code. Fig. 4 is the simulated BER for the coded π/4 QPSK. When the rate 1/2 convolutional code with K =7and the Viterbi decoding are employed, the SNR degradation is about 0.4 db at BER=10 6. The system employing π/4 QPSK and the rate 1/2 turbo code with K = 4 is also simulated. The turbo decoding algorithm is the modified BCJR algorithm [7], [8]. When the 3-iteration decoder is implemented, the performance is much better than that of the convolutional code. BER Turbo code (K=4, R=1/2, 3 iterations) Linear =1.0 =0.9 =0.7 Convolutional code (K=7, R=1/2) E b /N 0 Fig. 4. The bit error rate of the communication system using π QPSK. The 4 power amplifier is fully saturated. Fig. 5 shows the simulation results for commuincation systems employing OQPSK. There is almost no difference between the BER when a linear amplifier is employed and the BER when a fully saturated power amplifier is employed. The simulated SNR degradation matches the theoretical results in the previous section. For π/4 QPSK, the input of the shaping filters can be written as x I (t) = A i δ(t it s ) x Q (t) = i= i= B i δ(t it s ) where A i,b i {0, ±1, ± 2}, A 2 i + B2 i =2and T s is the symbol time. During one symbol time, if the in-phase input GLOBECOM /03/\$ IEEE

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