Conversion from Linear to Circular Polarization and Stokes Parameters in FPGA. Koyel Das, Alan Roy, Gino Tuccari, Reinhard Keller

Transcription

1 Conversion from Linear to Circular Polarization and Stokes Parameters in FPGA Koyel Das, Alan Roy, Gino Tuccari, Reinhard Keller

2 Purpose 1. Conventionally, for the formation of circular polarization, analogue circular polarizers are used 2. Circular polarizers need flat phase response - existing circular polarizers are narrow band due to imperfect 90 degrees phase shift over broad bands 3. VLBI is simplest with circular polarization but hard with new broad-band receivers 4. Better bandwidth and polarization purity is possible with formation of circular polarization in the digital domain. 5. Field Programmable Gate Arrays are used because of high processing power, low cost, portability and reconfigurability. 6. The instrument under development will cover 500 MHz bandwidth

3 Processing Steps 1) Receive two linear polarizations X and Y time domain samples at a rate of 1 GS/s. 2) Reduce the rate to 128 MHz by parallel processing techniques in FPGA. 3) Equalize the instrumental phase and gain between X and Y. 4) Form circular polarization by adding 90º phase shift to one of the polarizations. 5) Obtain the Stokes parameters in terms of LHC and RHC polarizations. Implementation 1) The implementation is aimed to be done in the Core2 board of the DBBC that has a Virtex 5 FPGA chip (xc5vlx220 or xc5vlx330) on board.

4 Theory Introduction 1. The field components x(t) and y(t) from a crossed dipole undergo unequal phase and amplitude distortions due to dissimilar transfer characteristics of the two receiving systems. 2. Phase and gain equalization is performed to make the phase and amplitude difference between x(t) and y(t) zero. 3. Calibration is already performed in software on stored data by astronomers. Our effort is to develop an instrument to form circular polarization in real time using digital techniques to improve on the analogue counterparts. 4. We consider here a linear time invariant system (ie no channel non-linearities or multipath effects)

5 Phase, Gain Calibration And Conversion to Circular Polarization Let us consider a broadband source s(t) sending signals continuously in time and is located equidistant from the two dipoles X and Y. We will consider a finite number of time domain spectra N s each consisting of N samples spaced continuously in time. N s depends on the number of samples N, the sampling rate f s and the total integration time t integ N s = f s. t integ / N For a simple analysis, we consider just one frequency component in a spectrum and the results will hold good for all other spectral components. (1) The time domain signals x i t and y i t where i = 1,2,3,..., N s are represented by the following equations. x i t =h 1 t s i t n 1i t y i t =h 2 t s i t t d xy n 2i t (2) (3) where h 1 t and h 2 t are the transfer functions of channel x and channel y; t d xy is the initial time delay that the input signal s i t undergoes before entering channel y w.r.t channel x.

6 The two interfering sources n 1i t and n 2i t include the thermal fluctuations and external spurious sources. Going directly to the discrete form frequency domain representation where the samples of x i and y i are located at uniform intervals of 0 and considering only the rth samples of x i and y i for clarity. x i r. 0 = x i r. 0.e j. x r. 0 = h 1 r. 0. e j. 1 r. 0. s i r. 0. e j.r. 0.0 n 1i r. 0 y i r. 0 = y i r. 0. e j. y r. 0 = h 2 r. 0.e j. 2 r. 0. s i r. 0. e j.r. 0. k 1.T s n2i r. 0 (4) (5) Phase Equalization The phase difference x r. 0 y r. 0 cross power spectrum, is obtained from the accumulated N s z r. 0 = i=1 N s z i r. 0 = x i r. 0. y i r. 0. e j x r. 0 y r. 0 i=1 (6)

7 The accumulated rth spectral component consists of the summation of the following product terms. 1. h r. 0. e j. r. 0. s i r. 0 2.e j.r. 0. k 1.T s (7) where, h r. 0. e j. r. 0 = h 1 r. 0. h 2 r. 0. e j. 1 r. 0 2 r h 1 r. 0.e j. i r. 0. s i r. 0. e j.r. 0.0 '. n 2i r. 0 (8) 3. h 2 r. 0. e j. 2 r. 0. s i r. 0.e j.r. 0. k 1 T s.n1i r. 0 (9) 4. h r. 0. e j. r. 0 '. n 2i r. 0. n 1i r. 0 (10) Term 4 gets canceled in a subtraction operation performed between the on source and off source accumulation results. 2 and 3 will average down to 0 unless an external time invariant source is present, which needs separate treatment and is not a part of this project.

8 The first term contains the phase difference due to different transfer functions of the two channels and due to the initial phase difference. instrument = r. 0 and initial =r. 0. k 1.T s Hence, the measured phase difference = instrument initial initial = instrument if is zero then,,which is required for instrumental phase calibration. can be set to zero by appropriate measurement setup. initial (11) (12) (13) Hence, if one of the polarization vectors is rotated with respect to the other by the measured phase difference then the instrumental phase errors are calibrated. Rotation is done by determining the rotation parameters of the rotation matrix [ R y' r. 0 I y' r. 0 where ]=[ cos sin sin cos =R z r. 0 / z r. 0 cos ][ R y r. 0 I y r. 0 ] and sin =I z r. 0 / z r. 0 (14)

9 Hence, the phase difference between x r. 0 and y ' r. 0 reduces to zero after rotation of the vector y r. 0 by the phase difference r. 0 or, x r. 0 y ' r. 0 =0 (15) Gain Equalization In order to compensate for the attenuations caused by the channel properties, all the spectral components of x i and y i are scaled to one same level determined by the maximum signal level A max in the passbands of the two spectra. The accumulated power spectra are expressed as follows N s N s x r. 0 2 = x i r. 0 2 = x i r. 0. x i r. 0. e j x r. 0 x r. 0 i=1 i=1 N s N s y r. 0 2 = y i r. 0 2 = y i r. 0. y i r. 0. e j y r. 0 y r. 0 i=1 i =1 (16) (17)

10 The accumulated spectra in equation (17) consists of the following three terms h 2 r s i r. 0 2 h 2 r n 2i r h 2 r s i r. 0. n 2i r. 0 cos r. 0. k 1.T s n2i r. 0 (18) (19) (20) A subtraction operation between the on-source accumulation results and offsource accumulation results will eliminate the second term. If an external time invariant source is present in the background then n 2i r. 0 will remain the third term as a source of interference. If this interfering source is not present then we have only the first term. The equations for gain to compensate the distortion caused by as follows. 2 g y r. 0 = A max / y r. 0 2 Similarly, 2 g x r. 0 = A max / x r. 0 2 h 2 r. 0 2 is (21) (22)

11 Note: Gains are calculated using accumulated power spectra and are applied to the voltage spectra. There will be difference since the square root of the denominator in equation (21) or (22) doesn't match exactly the modulus of a single voltage spectrum. However, if the voltage spectral components are accumulated for sufficient amount of time then g y r. 0. y r. 0 2 ~ g y r. 0. y j r. 0 ~ A max, (23) g x r. 0. x r. 0 2 ~ g x r. 0. x j r. 0 ~ A max where j is not equal to i. (24) Windowing The window function applied to the resulting spectra is a unit delta function at 0 th frequency channel superimposed on a shifted rectangular function in the passband. Formation of RHC and LHC After phase and gain equalization, the two spectra are added with degrees phase shift to obtain the two circular polarizations, which are later used for obtaining the stokes parameters.

12 Design Overview x (t) y (t) ADC 1GS/s ADC 1GS/s Clock Rate Reduction 128MHz Clock Rate Reduction 128 MHz Serial Spectra Generator Serial Spectra Generator FFT1 FFT2.. streaming. 1k FFT... FFT8 decoder1 decoder decoder8 Cross Spectra Z=X.Y* X 2 Y 2 Power Spectra Accumulators Zr Zi X 2 Y Zr On-Off Zi On-Off X 2 On-Off Y 2 On-Off 8 accumulators Equalization Parameters Latch Gx.W Gx = Gy = Amax 2 / X 2 Amax 2 / Y 2 L.H.C R.H.C X' ± j. Y' (X,Y) Phase & Gain Corrections: (X,Y ) Fig. 1 Synchronization Gy.W Latch Cosø.W Sinø.W Window Function W Cosø = Zr/ Z Sinø = Zi/ Z

13 Design Logic Verification and Simulation Results Lab setup for collecting test data: - noise source was split and passed through two filters, - outputs were taken as X and Y. - sampled with two channels of a digital storage oscilloscope - test is to verify the design architecture so very small number of spectra were used Filter 1 Ch1 Noise diode Oscilloscope 2 GHz Filter 2 Fig. 2 Total number of samples = Ch2 Time series is divided into /1024 = 24 number of spectra Simulation carried out for an ensemble of 24 spectra.

14 Simulation results : True Values Matlab processed results Measured Values Results from block in fig.1 phase difference between X & Y after equalization. Phase Difference (Radians) Freq / MHz Fig. 3

15 True & measured phase difference between X & Y zoomed into the passband Phase Difference (Radians) Freq / MHz fig. 4

16 True vs measured phase difference True Phase Difference (Radians) Measured Phase Difference (Radians) fig. 5

17 Phase difference before & after equalization zoomed into the passband Phase Difference (Radians) Freq / MHz fig. 6

18 Comments : Phase Equalization 3. Figure 5. shows an exact match between the measured and the actual value though there is a little truncation error. 4. Much better results are expected for a larger number of spectra. 5. The design logic is verified and is found correct. Additional Comments : 2. Deviation from 0 : Residual thermal fluctuations for such a small number of spectra are causing the small fluctuations in the equalized phase.

19 Gain equalized X for the passband X Freq / MHz fig. 7

20 X before & after equalization X Freq / MHz fig. 8

21 Gain equalized Y for the passband Y Freq / MHz fig. 9

22 Y before & after equalization Y Freq / MHz fig. 10

23 True vs measured X & Y True Magnitudes Measured Magnitudes fig. 11

24 Circular power spectra after equalization Power Freq / MHz Fig. 12

25 Gain Equalization : 3. Figure 11 shows an exact match between the actual and measured values. 4. Design logic is verified and is found correct. Additional Comments : 1. These data are not sufficient to equalize the signal levels to high accuracy since equations (23) and (24) are not completely satisfied for 24 spectra. g y r. 0. y r. 0 2 ~ g y r. 0. y j r. 0 ~ A max g x r. 0. x r. 0 2 ~ g x r. 0. x j r. 0 ~ A max, (23) (24) 2. Thermal fluctuations are reducing but not all the way to zero for 24 spectra.

26 Current Status : Implementation- Placing and Routing Number of FPGAs : Three or more since we are constrained by speed and DBBC layout structure. Next : Test in the anechoic chamber with a noise diode feeding the crossed dipole of the multi-beam receiver to measure the polarization purity. Future possible work: D-term (cross-polarization) correction.

27 Summary The simulation results for the phase and gain equalization shows that the algorithm and the design logic are correct. The same data is used for calibration and for measurement for verification of the technique. Next experiments will have different contiguous data sets for calibration and for result verification. Circular polarization with high purity is expected. If it works as expected then broadband receivers can use native linear polarization.