Digital Noncoherent Demodulation of. the Frequency-Modulated Signals

Applied Mathematical Sciences, Vol. 9, 05, no. 39, 695-6934 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ams.05.50664 Digital Noncoherent Demodulation of the Frequency-Modulated Signals A.N. Glushkov, V.P. Lintvinenko, B.V. Matveev Department of Radio Engineering Voroneh State Technical University, Voroneh, Russia O.V. Chernoyarov Department of Radio Engineering Devices and Antenna Systems National Research University MPEI, Moscow, Russia K.S. Kalashnikov Department of Radio Physics Voroneh State University, Voroneh, Russia Copyright 05 A.N. Glushkov et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We consider the fast algorithm for digital noncoherent processing of frequency-modulated and frequency-shift keyed radio signals. This algorithm requires carrying out the minimum simple arithmetic operations over the input signal period. We are also illustrating its possibilities concerning detecting the analog frequency-modulated signals and demodulating the binary frequency-shift keyed signals. We show that the presented demodulator provides the optimal signal processing and possesses the potential noise immunity. It can be implemented by means of digital signal processors, or with the use of modern programmable logic devices. Keywords: frequency modulation, detection, demodulation, digital signal processing, noncoherent processing

696 A.N. Glushkov et al. Introduction Frequency-modulated (FM) and frequency-shift keyed (FSK) signals [, ] have found wide application in analog and digital signal transfer systems [3-5], with analog and digital methods applied for their processing, accordingly [5-8]. FM signal detectors are implemented by means of primary signals transformation into the amplitude-modulated or phase-modulated signals, or into the pulse signals by means of carrier tracking methods. For generation of such signals, the analog and digital frequency detection methods based on the detuned resonators [5, 8] can be also used. Digital frequency detectors include the narrow-band digital frequency filters requiring really great amounts of addition and multiplication. It does sufficiently sophisticate equipment needed and implies rather high performance of hardware components. In this context, the problem of designing highly efficient digital frequency demodulator (detector) is pressing. Below we are to present a frequency demodulator based on the fast algorithm for digital signal processing. This algorithm is demonstrated in [9], and its hardware implementation is described in the patent [0]. Signal Processing Algorithm The input FM signal can be mathematically described as s t S cos f t t, 0 t is the slowly varying initial phase determined by where S is the amplitude, the modulating signal, and f 0 is the carrier frequency. The fast digital algorithm and FM signal demodulator corresponding to it [9, 0] are designed on the basis of the two digital amplitude demodulators (DADs), tuned to the frequencies f and f : f f 0 f, f f 0 f, where f is the DAD adjustment frequency shift relative to the FM signal carrier frequency. The DADs are described in [], and its hardware implementation is considered in the patent []. Block diagram of the digital FM signal demodulator is shown in Fig.. In the first DAD (DAD) the input signal is quantied by the clock signal t c analog-to-digital converter (ADC) with four samples s i, 0, s i,, s i, and s i,3 within each i-th period T f and with sample rate f Q 4 f. Similarly, the input signal is processed in the second DAD (DAD) by the clock signal c t ADC with sample rate f Q 4 f. In each DAD four samples received within the current (i-th) period are memoried in the multibit shifter (MS4), and two quadrature samples are formed in the subtractors SUB 0 and SUB, as follows xi, 0 si,0 si,, xi, si, si, 3.

Digital noncoherent demodulation of the frequency-modulated signals 697 Then, the sums over N last signal periods are calculated in summators SUM 0k, SUM k, k, n, n log N and in multibit shifters MR 0k, MR k reserved for storing intermediate results: Fig.. Digital FM signal demodulator N N y i, 0 xi j,0 si j,0 si j,, y i, xi j, si j, si j,3. j0 j0 N j0 N j0

698 A.N. Glushkov et al. In the quadratic transformer (QT), the value i i, 0 y i, y is calculated. This value is the DAD response and depends on the input signal amplitude S and frequency f 0. In the fast algorithm (Fig. ), the values x i, 0 and x i, are summaried with the contents of multibit shifters MR 0 and MR. Here each shifter contains one value x i, 0 or x i,, accordingly, obtained earlier, while processing the previous period. At the outputs of summators SUM 0 and SUM, the sums of two samples differences xi,0 xi, 0 and xi, xi, are formed. Thereafter, the new values x i, 0 and x i, are rewritten into the shifters MR 0 and MR, substituting their previous contents. Then in the summators SUM 0 and SUM the sums of four neighboring samples differences are calculated, and so on. DADs responses and move to the substracter SUB, at the output of which the output FM demodulator signal is formed: () Frequency responses of the first and second DADs are determined as H f S sin Nf f cos f f, H f S sinnf f cosf f, correspondingly, and their graphs are shown in Fig. a for N 56, f0 0 MH, and f N. () f 0 a) b) Fig.. DAD and DAD frequency responses (a) and FM signal demodulator frequency response (b) The demodulator frequency response f S H f H f H

Digital noncoherent demodulation of the frequency-modulated signals 699 is drawn in Fig. b. As it can be seen, if relation () holds, then the sensibly linear discrimination characteristic of the digital FM signal demodulator is realied. While demodulating the binary FSK signals, the decision concerning the received symbol is made, in accordance with the response sign at the SUB output. 3 Noise Features of the FM signal demodulator Demodulator response to the noise-free FM signal is equal to f S H f. where S is the input FM signal amplitude, and f is the instantaneous frequency. In the presence of Gaussian noise with the dispersion n, the responses and of the DAD and the DAD amplitude demodulators have Rician probability distributions w s exp I 0, w s exp I 0 (3) with mean values H f H f, and with dispersion Nn, S S. The joint probability density of the difference () of the random variables and has the form [3] ws ws, if 0, f, (4) 0, if 0. Then, for one-dimensional probability density of the random variable, we can write down 0 f w, d (5) According to [4], the mean value of the difference () is equal to the difference of mean values of terms. Therefore, finding accuracy quite sufficient practically, it is possible to assume The dispersion is equal to f S H. (6) of the difference of the two random variables R, (7) where R is the correlation coefficient of responses and, while, are the dispersions of,, and according to Eq. (3). Considered amplitude demodulators have half-overlapping frequency characteristics, and thus their responses and are correlated. By means of

6930 A.N. Glushkov et al. the statistical simulation, we have shown for various parameters that the value R 0.5 can be used as a good estimator for the correlation coefficient between and, and it agrees well with the common results [5]. Then, from Eq. (7) we get Nn. (8) Numerical calculations conducted according to Eqs. (4), (5) suggest that the probability density w is well approximated by the Gaussian curve of the form exp w, G where the mean value is determined from Eq. (6), and the dispersion from Eq. (8). It is easy to see that for frequency deviation f () the maximum value of the response of the FM signal detector is equal to Z m NS. Then, output voltage signal-to-noise ratio g can be written down in the form of g Zm n N NS n N S N n. Apparently, it is much greater than the input signal-to-noise ratio S N, due to the inherent filtering capacity of the demodulator. Results of the presented calculations are corroborated by statistical simulation. As an example, in Fig. 3 the time diagrams of the FM demodulator response are shown for N 5, f0 0 MH, f 9. 77 kh (), and modulation frequency F kh. a) b) Fig. 3. Time diagrams of the FM demodulator response without noise (a) and for output signal-to-noise ratio g 8 (b) 4 Binary FM signal demodulator The demodulator presented in Fig. allows us to process the binary FSK signals which imply the transfer of binary symbols by duration T at frequencies f and f ( f f ), accordingly. The continuous phase FSK (CPFSK) signal with carrier f 0 (period T 0 ) is formed so that the integer of the periods T and T fits into the duration of the information symbol T. And for the minimum value

Digital noncoherent demodulation of the frequency-modulated signals 693 of T it can be written down T N T N. Then f f N, f N 0 f. 0 Shift of frequencies f and f relative to f 0 is equal to f f0 f f f0 f N f0 N, (9) and it corresponds to Eq. (). In this case the demodulator frequency response has the form presented in Fig. b. The data transfer rate is determined as V T. Then, if condition (9) holds, we have V baud f H. (0) This is the maximum data transfer rate for CPFSK signals. For FSK signals the frequency modulation index b is equal to b f V, and for CPFSK signals b, as follows from Eq. (0). In Fig. 4 the normalied results are presented of the statistical simulation of the binary CPFSK signal demodulator for f 0 MH, N 56, T NT 0 5.6 μs ( V 39. kilobauds), f 9. 5 kh. The modulating signal is drawn by the dashed line, and the decision making moments concerning clock synchroniation pulses are shown by points. 0 Fig. 4. Time diagram of the CPFSK signal demodulator response As it can be seen, the offered CPFSK signal demodulator extracts the received information symbols with the maximum efficiency. Smooth changes of the response are caused by the inherent frequency discrimination of the demodulator. In the presence of noise, the time diagram of the demodulator response is deformed, and there can be erroneous decisions made about the received symbol. 5 Noise immunity of the CPFSK signal demodulator For CPFSK signal the amplitude of the demodulator response is equal to. The probability density NS, and its dispersion is Nn w s of the demodulator response at synchroniation moments and for the case of the signal reception against white noise is described by Rician distribution in the form of

693 A.N. Glushkov et al. w s exp I 0. () If the signal is absent, then noise response has the Raileigh probability distribution: exp w n. () The true decision is made, if the DAD channel response in the presence of the signal is greater than the DAD channel response in the absence of the signal. Then the probability of correct reception Q is determined as v Q ws v wn d dv. (3) 0 0 Substituting Eqs. (), () in Eq. (3), we get Q exp 0 v v exp I Let us introduce the power signal-to-noise ratio 0 v v exp dv. (4) h 4N S 4Nn NS n and designate v. Then, from Eq. (4) we have h exp I g Q exp 0 exp d. (5) 0 While calculating integral (5), we find the value for the probability of correct reception Q exp h and for the error probability p err h Q exp. (6) The expression (6) is referred to in the specialied literature [] as error probability for optimal noncoherent reception of orthogonal signals in the channel with an uncertain phase. So, this FSK signal demodulator does provide potential noise immunity. In Fig. 5 the dependence of error probability p err (6) upon the signal-to-noise ratio h (in decibels) is shown by continuous line, and the results of statistical simulation by points. As it can be seen, the satisfactory correlation between the theoretical and the experimental data is observed, indeed.

Digital noncoherent demodulation of the frequency-modulated signals 6933 Fig. 5. Dependence of the error probability upon the signal-to-noise ratio 6 Conclusion The suggested fast digital algorithm for demodulation of frequency-modulated and frequency-shift keyed signals requires carrying out the minimum simple arithmetic operations over the input signal period and provides potential immunity to white noise. The algorithm can be implemented by means of digital signal processors, or with the use of modern programmable logic devices [6]. Acknowledgements. The reported study was supported by Russian Science Foundation (research project No. 5--00). References [] G. Kennedy, B. Davis, Electronic Communication Systems, McGraw-Hill, Blacklick, USA, 99. [] L.M. Fink, Discrete-message Communication Theory, Sovetskoe Radio, Moscow, 970. (in Russian) [3] R.K. Rao Yarlagadda, Analog and Digital Signals and Systems, Springer, New York, 00. http://dx.doi.org/0.007/978--449-0034-0 [4] R. Rudersdorfer, Ulrich Graf, Hans Zahnd, Radio Receiver Technology: Principles, Architectures and Applications, Wiley, New York, 04. http://dx.doi.org/0.00/978864788 [5] N.N. Fomin, Radio Receivers, Radio i Svya, Moscow, 003. (in Russian) [6] J. Proakis, M. Salehi, Digital Communications, McGraw-Hill, New York, 007.

6934 A.N. Glushkov et al. [7] B. Sklar, Digital Communications: Fundamentals and Applications, Prentice Hall, New Jersey, 00. [8] R.G. Lyons, Understanding Digital Signal Processing, Prentice Hall, New Jersey, 00. [9] A.N. Glushkov, V.P. Litvinenko, B.V. Matveev, O.V. Chernoyarov, Basic Algorithm for the Noncoherent Digital Processing of the Narrowband Radio Signals, Applied Mathematical Sciences, 9 (05), no.95, 477-4735. http://dx.doi.org/0.988/ams.05.5435 [0] A.V. Glushkov, V.P. Litvinenko, Digital demodulator for frequency-modulated signals, Certificate of Authorship 5039, Russia, IPC H04B /0, H03D 3/00 (patent application 054699/08, appl. 7..0, publ. by Federal Service for Intellectual Property Patents and Trademarks of the Russian Federation 0.07.04, Bull. 9). [] A.N. Glushkov, V.P. Litvinenko, P.A. Popov, Fast digital detection algorithm of a narrowband signal, Herald of the Voroneh State Technical Universit,. Series Radio Electronics and Communication Systems, 4. (00), 6-8. (in Russian) [] A.V. Glushkov, V.P. Litvinenko, Ju.D. Proskuryakov, Digital narrowband signals detector, Certificate of Authorship 5767, Russia, IPC H04B /0 (patent application 0033587/09, appl..09.003, publ. by Federal Service for Intellectual Property Patents and Trademarks of the Russian Federation 7.07.005, Bull. ). [3] H. Stark, J.W. Woods, Probability and Random Processes with Applications to Signal Processing, Prentice-Hall, New Jersey, 00. [4] A.I. Solonina, D.A. Ulakhovich, L.A. Yakovlev, Digital Signal Algorithms and Processors, BHV-Peterburg, St.-Petersburg, 00. (in Russian) [5] V.I. Tikhonov, Optimal Signal Reception, Radio i Svya, Moscow, 983. (in Russian) [6] R.E. Haskell, D.M. Hanna, Introduction to Digital Design Using Digilent FPGA Boards - Block Diagram / Verilog Examples, LBE Books Rochester Hills, MI, 009. Received: November 5, 05; Published: November 8, 05