In this letter we show first experimental data on the dependence of the lineshape of 1-3 /mi InGaAsP DFB lasers on the measurement time. The lineshape is measured by using delayed self-heterodyne set-ups5 with fibre delay lines having different lengths (1-5 km and 27 km). The result shows that the linewidth with the longer delay line is larger than that measured with the shorter delay line. Furthermore, an analysis of the data suggests that the \/f noise in the FM noise spectrum is the principal cause of the power-independent line broadening. The possibility of linewidth reduction using a narrowband automatic frequency control scheme is also discussed. dominant cause of the spectral broadening in such high-power operation of a DFB laser. (Note that the dependence of the linewidth on the delay-line length is even stronger in the experiment than in the theory.) Lineshape determined by delayed self-heterodyne method: We assume first that the FM noise spectrum S{f) consists of the power-dependent white noise and power-independent \/f noise: S(f) = C + K/f (1) where K and C are frequency-independent, whereas C is inversely proportional to the output power. The lineshape determined by a scanning Fabry-Perot interferometer with a finite measurement time xm is not affected by the FM noise component having frequencies below l/rm, because the finite measurement time is equivalent to a highpass filter which eliminates frequency fluctuations below l/xm. Consequently, as shown theoretically by O'Mahony and Henning,4 the measured lineshape depends strongly on the measurement time only when \/f noise exists in the FM noise. Next, we consider the delayed self-heterodyne method, whose principle is described in Reference 5. The delay time xd in the delayed self-heterodyne method corresponds to the measurement time xm of a scanning Fabry-Perot interferometer. The reason is given in the following. We consider a frequency fluctuation component whose frequency is much lower than l/xd. For such a component the phase difference between the signal beam and the reference beam after travelling through the delay line is small; hence the frequency component of the FM noise below l/xd is eliminated automatically in the delayed self-heterodyne method. Therefore, similarly to the case of a scanning Fabry-Perot interferometer, the measured IF linewidth will depend strongly on the delay-line length only when the 1//noise exists. When it does not exist, the linewidth increase is practically negligible in most cases. The lineshape /(/) of the IF signal obtained by the delayed self-heterodyne method is related to the FM noise spectrum S(/)as 10MHz l^U Fig. 1 IF spectra measured by delayed self-heterodyne method a Length of delay line = 27 km b Length of delay line = 1-5 km The above conclusion is particularly important in the application of such lasers to coherent communications, because it means that the lineshape could be improved effectively by an automatic frequency control (AFC) having a relatively narrow bandwidth. It has generally been believed for the AFCoriented lineshape improvement techniques at microwave frequencies that an AFC circuit with a bandwidth fm reduces the spectral density of noise sideband within ±fm of the carrier. This statement is valid when the origin of the spectral spread is white noise, but not valid when it is 1// noise. When the 1// noise is predominant, an AFC in narrow bandwidth may reduce the linewidth drastically. In heterodyne optical communications, such a reduction can also be achieved in the receiver by an AFC scheme in the local oscillator. Experiments are now being performed and will be reported later. Acknowledgments: We thank Dr. K. Kobayashi of OptoElectronics Research Laboratories, NEC, for supplying DFB lasers. This work is supported by a Scientific Research grantin-aid from the Japanese Ministry of Education, Science & Culture. K. KIKUCHI T. OKOSHI 3rd September 1985 Department of Electronic Engineering University of Tokyo Bunkyo-ku, Tokyo 113, Japan x (1 - cos 2nf'xd) df 'X] (2) References 1 5 where J " denotes a Fourier transform. Derivation of this equation will be given elsewhere. In eqn. 2 the fact that lowerfrequency components are cut off is expressed by the term (1 — cos 2nf'xd). 3 Experimental results: The laser under test was a 1-3 fim InGaAsP DFB laser. Lineshapes were measured by using the delayed self-heterodyne set-ups with 1-5 km and 27 km singlemode fibre delay lines. Fig. 1 shows lineshapes of the IF signal measured when the bias current level was six times the threshold. The FM noise spectrum of the laser was also measured, giving K = 1-44 x 1012 Hz2. Discussion: In Fig. 1 the linewidth (FWHM) measured with the 27 km fibre is 21 MHz, and that with the 1-5 km fibre is 14 MHz; a remarkable change is observed in the linewidth. We calculate here the linewidth of the IF signal using eqns. 1 and 2. We assume temporarily that the white noise component in the FM noise spectrum is negligible (C = 0 in eqn. 1) because the bias current level is fairly high. Under such an assumption, the linewidth calculated for the 27 kmfibreis 21 MHz, and that for the 1-5 km fibre is 17 MHz. These results support the assumption that the 1//" noise is the pre1012 2 4 5 YARIV, A.: 'Quantum electronics' (J. Wiley & Sons, New York, 1975), eqn. 13.2-18 WELFORD, D., and MOORADIAN, A.: 'Output power and temperature dependence of the linewidth of single-frequency CW (GaAl)As diode lasers', Appl. Phys. Lett., 1982,40, pp. 865-867 KIKUCHI, K., and OKOSHI, T.: 'Measurement of spectra of and cor- relation between FM and AM noises in GaAlAs lasers', Electron. Lett., 1983,19, pp. 812-813 O'MAHONY, M. J., and HENNING, I. D. : 'Semiconductor laser linewidth broadening due to 1//carrier noise', ibid., 1983,19, pp. 1000-1001 OKOSHI, T., KIKUCHI, K., and NAKAYAMA, A. : 'Novel method for high resolution measurement of laser output spectrum', ibid., 1980, 16, pp. 630-631 TRANSFER CHARACTERISTIC OF A LINEAR PHASE DETECTOR FOR VARIABLE PHASE Indexing terms: Circuit theory and design, Detector circuits The performance of a linear phase detector for noisy signals and variable phase is investigated. It is shown both theoretically and experimentally that in this case the range of phase differences, which can be measured without systematic errors, decreases. ELECTRONICS LETTERS 24th October 1985 Vol. 21 No. 22 Introduction: It is well known that the performance of the so-called linear (sawtooth) phase detector degrades when the input SNR is small. 12 These detectors are essential parts of many types of phase meters, so a systematic error of phase measurement arises when the input SNR is poor. However, to the author's knowledge, nothing is known about what happens if the phase varies during the measurements. This is of importance, for example, in phase ranging of mobile vehicles. Operating principle: A model of the sawtooth phase detector and its characteristic A($) for noiseless signals is shown in Fig. 1. The positive edge of the first signal sets the RS flip-flop and the positive edge of the second signal resets it. Then the output of the flip-flop is averaged. In this way the output of the detector for noiseless signals is directly proportional to the measured phase shift. limiter inputs limiter _d_ dt — ! RS flip-flop d dt We study in detail the case when the phase varies linearly during the measurements according to the rule tel T T -—,— (5) This corresponds to the uniform motion of the positioned object. Using on eqns. 3 and 5, we can compute the values of <$*> for various <f>0 and A. The results for SNR = 10 are presented in Fig. 2. As is readily seen, when the value of A increases it makes the useful (linear) range of the phase detector narrower. 360* integrator 2 7 7 <(> ex [77O/T1 30* 300* 60° Fig. 1 Linear phase detector and its characteristic 330° 360* [7707F| Characteristic for variable phase: When the phase varies during the time T, the result of measurements in the presence of narrow bandwidth noise is given by T/2 <t>* = - (1) Fig. 2 Mean value of(f>* against parameter (f>0 To compute the variance of </>*, 0^., we note that in the linear (useful) part of the characteristic of the detector this variance is approximately equal2 to o~\. When M independent measurements are taken during time T, the value of <J\* is then given by -772 Here <f>(t) is the true value of the measured phase shift and n(t) is the value of the noise. The function A(<£) is depicted in Fig. 1. The mean of <f>* is Measurements: The measurement system used to verify the theoretical results is depicted in Fig. 3. signal generator T/2 phase meter dt -T/2 filter T/2 ill A[$(f) + n]p{n) dn dt (2) -T/2 - x Here p(n) is the probability density function of n. For additive Gaussian narrow bandwidth noise in one channel and moderate and large SNRs, the function p(n) is also Gaussian1-2 with zero mean and variance a\ ~ 1/(2SNR). In this case we have, from eqn. 2, T/2 T/2 = Y J MWRdt + Y f -T/2 -T/2 where the function G(<f>) is defined as : J e" (f2/2) dt <t> e (0, n) (4) (6) 2M . SNR summer I noise generator / variable phase shifter / impulse generator -output 1 voltagecontrolled phase shifter 1 ^ L r integrator Fig. 3 Block diagram of experimental system To obtain the linearly varying phase, a voltage-controlled phase shifter (VCPS), similar to that described in Reference 3, was used. During the experiment the phase shift of the VCPS had an almost linear dependence on the controlling voltage. The phase meter worked only during the impulse from the generator. The results are marked in Fig. 2 and they agree well with the predictions. Some discrepancies are most probably caused by instabilities of the VCPS. Conclusions: It has been shown that changes of the phase during the measurements decrease the effective range of the phase differences which can be measured without systematic errors. This range becomes smaller if the values of a\ and/or A increase. 2x) [ K. HOLEJKO J It follows from eqn. 3 that errors of measurement arise even in the absence of the noise if the measured phase exceeds the 0-2TC range. To avoid this effect it is necessary to equip the detector with some kind of memory able to record 0—*2n (27i-»0) transitions. The second term in eqn. 3 describes the errors due to noise. ELECTRONICS LETTERS 24th October 1985 27th August 1985 J. SIUZDAK Vol. 21 Instytut Telekomunikacji PW ul. Nowowiejska 15/19 00-665 Warszawa, Poland References 1 HOLEJKO, K.: 'Phase measurements in the presence of narrowband noise', Rozpr. Elektrotech., 1977, 23, pp. 487-506 No. 22 1013 RAAB, F. M., and WAECHTER, J. R.: 'The counting phase detector with VLF atmospheric noise', IEEE Trans., 1977, AES-13, pp. 522-532 TAY, E. w., and MURTI, v. G. K.: 'Unity-gain frequency-independent quadrature phase shifter', Electron. Lett., 1984, 20, pp. 431-432 Let q be the difference between quantiser levels. The probability density function (PDF) of the quantisation noise is assumed to be uniform between — q/2 and q/2, and zero elsewhere. Since we assumed that the receiver's A/D levels are identical to the transmitted levels, the receiver quantiser only affects the WGN. If a 'zero level' is transmitted, the PDF of the WGN will be modified by the quantiser to the function shown in Fig. 2. In Fig. 2, the delta functions have the areas a0, at, a2,... as shown and the quantisers have 2n levels (i.e. log2 2n bits). It can be easily shown that PERFORMANCE OF A FOURIER TRANSFORM DATA TRANSMISSION SYSTEM 9/2 exp (-£)*,-erf (1) -9/2 Indexing terms: Data transmission, Fourier transforms Results on the performance of a Fourier transform data transmission system in white Gaussian noise are presented for various quantisation widths and transform sizes. Introduction: Recently, there has been some interest in the Fourier transform data transmission (FTDT) system,1-2 which was first proposed by Weinstein and Ebert.3 The Fourier transform is used in this line coding/modulation technique to process the incoming data prior to transmission and this allows the spectral properties of the signal to be controlled. This leads to the following advantages: where erf (x) is the error function and a2 is the noise power. Similarly k 4= n at positive end k =(= n — 1 at negative end (2) and at the positive end (a) Injection of pilot tones provides the receiver with timing information, automatic channel equalisation ability and immunity to channel fades and Doppler shifts. (b) A constant tone interference in the channel can be neutralised by avoiding the use of the part of the spectrum affected. (c) Using a variable data rate for different parts of the spectrum for optimum error performance when the channel noise is coloured. In addition to these, the system may be designed to detect and remove isolated impulsive interference.4 This letter examines the error performance of the FTDT system in the presence of white Gaussian noise (WGN). Results comparing the error rate using different quantisation levels and transform sizes are given. data in transmitter modem channel jantise antiser noise receiver modem (3) and at the negative end a=h erfc (2n- (4) where erfc (x) = 1 — erf (x). For this analysis, it is convenient to make the following approximations: (a) Although the quantised noise PDF varies with the transmitted signal level, it is simpler to use only the one presented above in the evaluation instead of averaging over the various PDFs. This should not affect the final result significantly, as most of the transmitted signal levels are near to and centred around zero. (b) The quantised noise PDF is made symmetric by equating an to zero and setting both a n _! by eqn. 3. The effects of this should be negligible. data out O" a- P(s) Fig. 1 System model Theory: Let Fig. 1 represent the system. The following assumptions have been made: cyq (a) There is negligible rounding noise in the transform process, compared to the amount of quantisation noise. (b) The transmitted signal levels are identical to the receiver's A/D levels. (c) There is negligible correlation between noise samples. 5q/2 9q/2 (d) A constant power transform is used and no arithmetic overflow occurs. •"0 a -nq 2» ifl -Aq -2q 0 *a2 On-, Fig. 3 PDF of total received noise p(s) The quantised noise is added to the transmitter quantisation noise that is originally present and the PDF of the sum, p(s) (which is a convolution of their respective PDFs), is as shown in Fig. 3. The N-point DFT can be represented by 2q Aq [gisTi] Fig. 2 PDF of quantised noise 1014 ELECTRONICS LETTERS 24th October 1985 Vol. 21 No. 22

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