MEP 2006, 7-11 November 2006, Guanajuato, Guanajuato, México.
METHOD FOR PHASE SHIFT MEASUREMENT USING FAREY FRACTIONS Daniel Hernández B.1, Moisés Rivas L.1, Larysa Burtseva1, Oleg Sergiyenko1, Vira Tyrsa2. 1
Autonomous University of Baja California, 21280, Mexicali, BC, México, Phone/Fax: 52-686-5664270, e-mail:
[email protected], {lpb,srgnk}@iing.mxl.uabc.mx. 2 Kharkiv Automobile and Highway National University, 61002, Petrovsky Street 25, Ukraine, Phone: 380 57 700 38 65, Fax: 380 57 700 38 66, e-mail:
[email protected]
Abstract-The phase shift of coherent signals and the time intervals can be measured with the method of coincidences of electrical pulses. The results are automatically obtained and represented like simple fractions. These fractions are rational approximations of the unknown parameter, which generally is an irrational number. Such approximations can be represented with continued fractions or Farey fractions. Definitions and properties of Farey fractions are presented. An original method for phase shift measurement based on the pulses coincidence principle, whose result is a Farey fraction, is described. In order to increase the measurement accuracy, the continuous formation of rational means is performed, to represent the measurement result. These simplify the data processing and provide more accurate results than traditional methods. Keywords: Phase shift measurement, Farey fraction, mediant.
INTRODUCTION On this paper, an application of the method of approach by rational approximations in metrology is presented. This method can be used to find the solution of several measurement problems, particularly, to increase the accuracy and speed of automatic digital measurements of physics magnitudes, such as frequency, phase shift and time intervals. The application of this method can be successful where the possibilities of the classic methods already were exhausted. On this article, an original algorithm based on the Farey succession is presented. The algorithm allows a quick and high accuracy way of measuring the phase shift between two coherent sine signals.
FREQUENCY MEASUREMENT USING THE COINCIDENCE PULSE METHOD Consider an electrical sine signal with unknown frequency fx, in order to measure it, its frequency is converted to an electrical pulse train signal. In the classical frequency measurement method the result is a digital code Nx, which is the output of a digital counter, and represent the amount of periods Tx=1/ fx, in 1 s. The measure frequency result is Nx and has an uncertainty Δf x ≤ 1 Hz. The fractional measurement uncertainty is β=Δfx/ fx, which is inversely proportional to the measure frequency. For high frequencies such as fx=1 GHz, the fractional measurement uncertainty is β≤10-9. For accurate measurements fractional measurement uncertainties in the order of β≤10-12 are necessaries, comparables with the reproducibility of time and frequency standards [1]. In order to increase the measurement accuracy, the frequency measurement method by rational approximations is proposed [2]. In this method the measurement system is a digital system, which counts the amount pulses of a couple of pulse trains. The first one is associated 1-4244-0628-5/06/$20.00 ©2006 IEEE
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with the standard and has a repetition rate f0. The other one is associated with the sine signal of unknown frequency and has an unknown repetition rate fx. Periodic coincidences of these pulse trains are observed in time. Its mean frequency is [2]: f 0 x = 1 / T0 x = (τ 0 + τ x ) f 0 f x , (1) where T0x is the mean coincidence period; τ0, τx are the pulse duration of the pulse trains being compared in time, whose frequencies are f0 and fx respectively. The measurement time is the time interval between a couple of those coincidence pulses. The measurement result is represented like a simple fraction and is a rational approximation of the unknown frequency, which generally is an irrational number. A computation of the number of pulses N0= T0x f0 and N0= T0x fx between a pair of coincidences allows the determination of the unknown frequency fx for a know frequency f0: f x = N x f 0 N 0 .
MEASUREMENT RESULT REPRESENTATION IN FAREY FRACTIONS FORM Let ϕ be the phase shift between two electrical sine signals of frequency f1. And consider two pulse trains of periods T1 associated with the electrical sine signals. The phase shift is representing by the time interval tϕ as shown in Figure 1. Also consider a third pulse train signal of period T0 associated with an electrical sine signal of frequency f0, with T1> T0. In order to explain the method, let us call to the first signal the start pulse train (Figure 1a), and to the second one stop pulse train (Figure 1b) and to the third signal quantizing pulse train (Figure 1c). The start and stop pulses delimit the time interval tϕ, see Figure 1.
a
b
c
Start pulses 0 1 2 3 ...
N
3
t
T1= 1 / f 1 Stop pulses 0 1 2 3 ...
N1
tϕ
t
T1
Quantizing pulses 0 1 2 3 ...
N2
tϕ + N T = N T 1 1
2 0
N4
t
T0 = 1 / f 0
NT =NT 3 1
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Figure 1. Phase shift measurement by pulse coincidences. The start measurement criterion is the first coincidence between the start pulses and the quantizing pulses. This is the starting signal to count of the pulse in the three signals. However two stop measurement criterions exist. The first stop criterion is the second coincidence between the start pulses and the quantizing pulses, and they define the digital codes N1 and N2. The second stop criterion is the second coincidence between the stop pulses and the quantizing pulses, and they define the digital codes N3 and N4 as shown in Figure 1. From Figure 1, is easy to see that N3T1= N4T0 and tϕ+N1T1= N2T0, consequently
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T1 = T0 N 4 N 3 ,
and Since ϕ = tϕ / T1 then
tϕ = T0 (N 2 N 3 − N 1 N 4 ) N 3 .
(2)
ϕ = (N 2 N 3 − N 1 N 4 ) N 4
(3)
whit 0 ≤ϕ
