PLL FM Demodulator Performance Under Gaussian Modulation

Transcription

1 PLL FM Demodulator Performance Under Gaussian Modulation Pavel Hasan * Lehrstuhl für Nachrichtentechnik, Universität Erlangen-Nürnberg Cauerstr. 7, D Erlangen, Germany hasan@nt.e-technik.uni-erlangen.de Abstract-A wide-spread test of the PLL FM demodulator under Gaussian modulation is simulated using the Monte Carlo method. Unified noise-free and noise performance analyses of the PLL FM demodulator are presented. Substantial reduction of the modulation limit by the input bandpass filter is reported in the region of the input bandwidth of practical interest. Bessel bandpass filters of order greater than two are shown to compare favorably with Butterworth filters in front of the PLL FM demodulator relative to the IM distortion. A lower bound on the loop noise bandwidth is found by minimizing the output click rate for given IM distortion specifications. FM threshold of 4 db and 7 db for the rms frequency deviation-to-message bandwidth ratio 0.1 and 1.0, respectively, is reported on the worst-case IM distortion of 45 db. I. INTRODUCTION The phase-locked loop (PLL) is attractive as an FM demodulator because of its threshold extension capabilities compared to the conventional FM discriminator. In addition to a low threshold the intermodulation (IM) distortion of the PLL FM demodulator is a performance measure of major importance. The PLL FM demodulator remains in lock with the frequency modulation of the FM signal within the modulation limit. If the frequency deviation of the modulating signal is excessive, the phase-locked loop fails to track the modulation and loses lock. The output click rate and the signal-to-noise ratio (SNR) versus the carrier-to-noise ratio (CNR) curve characterize the PLL FM demodulator performance in the presence of noise. A wide-spread test of FM demodulators consists in demodulating Gaussian random signals. Due to nonlinear processing of the filtered FM signal, a complete performance analysis of the PLL FM demodulator under this situation is missing. The IM distortion and the modulation limit of the PLL FM demodulator under Gaussian modulation were derived in [1] and [2], respectively, disregarding the input bandpass filter which is not justified, since in a well-designed loop the input filter bandwidth is comparable to the loop bandwidth. Lindsey [3] and Gardner [4] evaluated the noise performance of the PLL FM demodulator. Lindsey derived the SNR versus CNR curve for additive white noise filtered by an RC filter using the Fokker-Planck technique and Gardner obtained the output click rate for a particular hardware experimentally. In this paper, the input bandpass filter is considered as a part of the PLL FM demodulator. Both the noise-free and the noise performance measures of the PLL FM demodulator are evaluated using the Monte Carlo simulation. The modulating baseband Gaussian signal is simulated by a sum of N sine waves of equally spaced frequencies and random phases [5], [6]. The FM signal filtered by the bandpass filter is derived using the (within the limits of computational accuracy) exact Fourier method [5]. The colored additive Gaussian noise components are simulated by filtering pseudorandom sequences in the frequency domain by the lowpass equivalent of the bandpass filter. The nonlinear stochastic integrodifferential equation of the PLL FM demodulator operation is solved in consideration of the amplitude and the distorted frequency modulation of the filtered FM signal using a fast recurrent algorithm and multirate sampling. The performance measures of the second-order loop PLL FM demodulator with flat amplitude response input bandpass filters, i.e. Butterworth and Bessel filters are evaluated. For the first time, unified noise-free and noise performance analyses of the PLL FM demodulator under Gaussian modulation are reported. * The author is also with the Institute of Radio Engineering and Electronics, Academy of Sciences of the Czech Republic, Prague, Czech Republic. This work was supported by the Deutsche Forschungsgemeinschaft. 1

2 II. MONTE CARLO SIMULATION Consider an FM signal v i ( t) = 2S cos t + t c [ ( ) ] (1) with power S and carrier frequency c. The signal is frequency modulated by Gaussian random process Ω( t) = ( t) to be demodulated by the PLL FM demodulator and accompanied by additive white Gaussian noise (AWGN) with one-sided spectral density N 0 W/Hz. Using the Monte Carlo method [5], [6], the Gaussian message of bandwidth W rad/s and rms frequency deviation rad/s has been simulated by a sum of N sine waves where ( ) = Ω n sin( n a t + n ) Ω t N (2) n=1 Ω n = 2 N, n = 1,2,...N (3) is the peak frequency deviation of the nth tone, a is the fundamental modulating frequency so that N a = W corresponds to the message bandwidth, and n is a random phase distributed uniformly over (, ). If the number N of tones simulating the random modulation is large enough, the statistical properties of (2) approach those of Gaussian noise [5]. The FM signal v i ( t) is filtered by the bandpass filter with center frequency and c bandwidth B W rad/s in front of the PLL FM demodulator. Omitting the details, the signal v o ( t) at the filter output has been derived using the Fourier method [5] by processing the spectral components of v i ( t) at the frequencies ± n c a by the transfer function of the bandpass filter. Accordingly, the filtered FM signal is given by where the amplitude modulation and the distorted phase modulation v o ( t) = 2Sa t [ ( ) ] (4) ( )cos c t + t a( t) = c 2 ( t) + s 2 ( t) (5) ( t) = arctan s t c t ( ) ( ) + 2 k( t) (6) have been calculated with arbitrary accuracy depending on the number of spectral components considered. Here, ( ) = 1 2 c t s( t) = 1 2 ( F c n cosn a t + G c n sin n a t) (7) n =0 ( F s n cosn a t + G s n sin n a t) (8) n= 0 2

3 ( ) changes by +(-)1 when the and the coefficients F c n,g c n, F s s n,g n are given in [5]. In (6), k t phasor 2Sa( t)exp[ i ( t) ] passes the negative real axis in the positive (negative) direction. Finally, the distorted frequency modulation ( t) of the filtered FM signal has been obtained analytically by differentiating (6). The AWGN filtered by the bandpass filter in front of the PLL FM demodulator has been considered as narrowband Gaussian process with sample function In (9), n c t spectral densities n( t) = n c ( t)cos c t n s ( t)sin c t. (9) ( ) and n s ( t) are quadrature and in-phase noise components, respectively, with Φ nc ( ) = Φ n s ( ) = 2N 0 H LP ( i ) 2 (10) ( ) is the transfer function of the lowpass equivalent of the bandpass filter. The where H LP i colored Gaussian noise processes n c ( t) and n s () t have been simulated by filtering pseudorandom sequences generated on a computer as follows. Uniformly distributed pseudorandom sequences (uniform deviates) have been generated by a routine based on three linear congruential generators [7]. Normal deviates have been obtained from the uniform deviates using the Box-Muller method [7]. The discrete Gaussian processes have been filtered by the transfer function H LP ( i ) in the frequency domain using the FFT and the complex Fourier series have been inverted to the time domain. The PLL FM demodulator with a multiplier phase detector has been simulated by solving the nonlinear stochastic integrodifferential equation t [ ]d (11) ( t) = Ω 0 + ( t) K f ( t ) a( )sin ( ) + N( ) 0 where Ω 0 = c 0 is the carrier frequency offset relative to the loop quiescent frequency, 0 ( t) = Ω 0 t + ( t) ˆ ( t) is the phase error with ˆ ( t) as the loop estimate of the phase Ω 0 t + ( t), ( t) is the frequency error, K is the dc gain of the linearized loop, f () is the impulse response of the loop filter, and N( ) = n c( ) 2S sin [ ( ) ( ) ] + n s( ) 2S cos [ ( ) ( ) ] (12) is the loop noise. A fast recurrent algorithm based on [8] has been used to solve (11). Computational efficiency of the simulation has been enhanced by multirate sampling with amplitude and frequency modulation of the filtered FM signal cubic spline interpolated [7]. The noise-free performance measures have been evaluated by solving (11) with N( ) = 0 over the fundamental period T a = 2 / a. The modulation limit has been obtained as the maximum rms frequency deviation such that no cycle slips have been observed for an ensemble of M sample functions of (2). The IM distortion has been evaluated at the frequency gaps L a of the modulation spectrum with L = 8, 36,and100. The linear distortion has been avoided by setting the peak frequency deviation at the Lth gap to Ω L and 0, respectively, and calculating the signal-to-distortion ratio (SDR) as the average ratio of the respective powers in the gap. In the presence of noise, the performance measures have been evaluated as averages of M simulations over a period of length T k = kt a, k = 16. The CNR is the carrier-to-noise ratio at the output of the bandpass filter and the SNR is the signal-to-noise ratio at the output of the PLL 3

4 FM demodulator after a lowpass post filter. The rectangular passband post filter with cutoff frequency equal to the message bandwidth W has been considered. The SNR has been evaluated by the spectral analysis of the recovered message as the ratio of the power at the spectral lines a = k k, =1,2,...N to the power at the spectral lines k, =1,2,...Nk 1, k with k = 2 / T k = a / k. III. SIMULATION RESULTS The simulation results reported have been obtained from M = 100 Monte Carlo trials with N =100 tones. This guarantees both statistical properties of (2) close to those of Gaussian noise [5] and good convergence of the Monte Carlo method [6]. The simulated conditions are as follows: The carrier is tuned to the loop quiescent frequency, c /W = 70, and the control characteristic of the voltage-controlled oscillator is linear. In accordance with [4], the input bandpass filters with flat amplitude response, i.e. Butterworth and Bessel filters of the same 3 db bandwidth B W have been considered. The results reported apply to the second-order loop PLL FM demodulator with an active loop filter and the damping factor =1. A. Modulation Limit Fig. 1 shows the noise-free modulation limit normalized by the message bandwidth versus the relative bandwidth B W W of the input bandpass filter for the one-sided loop noise bandwidth B L = W. In practice, the input bandwidth should be as narrow as possible in order to filter the RF noise. A lower bound on the input bandwidth [6] B W = 2( W + 4 ) (13) given by the IM distortion specifications yields B W <10W for <W. In this region of practical interest, the modulation limit is substantially reduced by the bandpass filter. For B W >10W, the modulation limit approaches that derived from experimental data [2] disregarding the input filter. The input bandwidth given by (13) is considered in what follows. B. Intermodulation Distortion The IM distortion of the PLL FM demodulator with the loop noise bandwidths B L = W and B L = 4W has been evaluated. For B L = W, the IM distortion of the PLL FM demodulator is primarily determined by the loop bandwidth. The SDR increases with increasing loop bandwidth and for B L = 4W approaches that of the bandpass filter within a fraction of db. In this case, the SDR is by 6 9 db and 8 14 db higher at low- and high-frequency gaps, respectively, for the Bessel filter compared to the Butterworth filter, depending on the W ratio. The SDR increases with the order of the Bessel filter as opposed to the Butterworth filter. Fig. 2 shows the IM distortion of the PLL FM demodulator with the sixth-order Bessel filter. C. Output Click Rate Slight dependence of the output click rate of the PLL FM demodulator on the input bandpass filter has been observed. Fig. 3 compares the mean output click rate of the PLL FM demodulator with the sixth-order Bessel bandpass filter normalized by the message bandwidth versus the relative loop noise bandwidth B L W to that of the conventional FM discriminator [9]. For a given message bandwidth, the mean click rate is minimum at the optimum loop noise bandwidth. The optimum is a weak function of the CNR and a strong function of the W ratio. A mean click rate by several orders of magnitude lower compared to the conventional FM discriminator has been obtained by optimizing the loop bandwidth of the PLL FM demodulator. The output click rate of the broadband loop PLL FM demodulator ( B L >> W ) approaches that of the conventional FM discriminator. The optimum loop noise bandwidth of the PLL FM demodulator for minimum output click rate is less than the message bandwidth at W < 1. 4

5 This results in unacceptable high IM distortion (SDR < 20 db) of the bandwidth-optimized loop. In consequence a lower bound on the loop noise bandwidth B L = W has been obtained by minimizing the output click rate for the worst-case IM distortion SDR = 45 db. D. Signal-to-Noise Ratio Fig. 4 shows the SNR versus CNR curves of the PLL FM demodulator with the sixth-order Bessel bandpass filter for the loop noise bandwidths B L = W and B L = 4W. The corresponding curves for the Butterworth filter differ slightly. Using the formal definition of FM threshold as the CNR for 1 db deterioration from the extended straight line [4], FM threshold of 4 db and 7 db at W = 0.1 and 1.0, respectively, has been obtained for the loop noise bandwidth B L = W compared to the respective FM thresholds of 6 db and 8 db for B L = 4W. IV. CONCLUSIONS A wide-spread test of the PLL FM demodulator under Gaussian modulation has been simulated using the Monte Carlo method in consideration of the input bandpass filter. The relevant performance measures of the second-order loop PLL FM demodulator with Butterworth and Bessel bandpass filters have been evaluated. The main results of the performance analysis reported are as follows. The input filter with bandwidth B W <10W of practical interest substantially reduces the modulation limit of the PLL FM demodulator as reported in [2]. Owing to their better phase linearity, Bessel bandpass filters of order greater than two compare favorably with Butterworth filters in front of the PLL FM demodulator relative to the IM distortion. A minimum output click rate by several orders of magnitude lower compared to the conventional FM discriminator is obtainable by optimizing the loop bandwidth of the PLL FM demodulator. The optimum loop noise bandwidth is less than the message bandwidth for the ratios W < 1 of practical interest which leads to unacceptable high IM distortion of the recovered message. A lower bound on the loop noise bandwidth B L = W has been obtained by minimizing the output click rate for given IM distortion specifications. For B L = W, FM threshold of 4 db and 7 db at W = 0.1 and 1.0, respectively, has been reported on the worst-case IM distortion of 45 db. The loop noise bandwidth B L > 4W yields no improvement of the IM distortion but excessively high output click rate. In conclusion, the recommended loop noise bandwidth of the PLL FM demodulator lies within W B L 4W for the considered range W 1 of the rms frequency deviation-to-message bandwidth ratio. REFERENCES [1] D. L. Schilling and M. Smirlock, Intermodulation distortion of a phase locked loop demodulator, IEEE Trans. Commun. Technol., vol. COM-15, pp , Apr [2] F. M. Gardner and J. F. Heck, Angle modulation limits of a noise-free phase lock loop, IEEE Trans. Commun., vol. COM-26, pp , Aug [3] W. C. Lindsey, Synchronization Systems in Communication and Control. Englewood Cliffs, NJ: Prentice-Hall, 1972, Chap. 15. [4] F. M. Gardner, Phaselock Techniques, 2nd ed. New York: Wiley, 1979, Chap. 9. [5] R. G. Medhurst and J. H. Roberts, Evaluation of distortion in f.m. trunk radio systems by a Monte Carlo method, Proc. IEE, vol. 113, pp , Apr [6] C. L. Ruthroff, Computation of FM distortion in linear networks for bandlimited periodic signals, Bell Syst. Tech. J., vol. 47, pp , July-Aug [7] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes. Cambridge: Cambridge University Press, 1989, Chap. 7. [8] P. Hasan, Algorithm for solving a class of phase-lock-loop equations, Electron. Lett., vol. 19, pp , March [9] S. O. Rice, Noise in FM Receivers in Time Series Analysis. New York: Wiley, 1963, Chap

6 FIGURE CAPTIONS Fig. 1. Modulation limit. Second-order Butterworth (Bessel) (blank marks), sixth-order Butterworth (semifilled marks), sixth-order Bessel (filled marks), and no (dashed line) [2] bandpass filter, B L = W. Fig. 2. Intermodulation distortion. Sixth-order Bessel bandpass filter, B L = W (filled marks) and B L = 4W (blank marks). Fig. 3. Mean click rate. Sixth-order Bessel bandpass filter, PLL FM demodulator: CNR = 3 db (blank marks), CNR = 6 db (filled marks) and conventional FM discriminator: CNR = 3 db (dashed lines), CNR = 6 db (dotted lines) [9]. Fig. 4. Signal-to-noise ratio versus carrier-to-noise ratio. Sixth-order Bessel bandpass filter, B L = W (filled marks) and B L = 4W (blank marks). 6

7 Fig. 1 7

8 Fig. 2 8

9 Fig. 3 9

10 Fig. 4 10