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the signals received by the antennas obey the stationary stochastic process and then ergodic process . the ergodic theory can be applied to the auto - correlation function for a spectrometer and the cross - correlation function for radio interferometer . under such conditions , weinreb ( 1963 ) developed the first digital spectrometer . this digital spectrometer is called the xf correlator in which the correlation is calculated before fourier transform . meanwhile , chikada et al . ( 1987 ) developed the first the fx correlator of an another design , in which fourier transform is performed before cross multiplication . although there is a difference of property between two basic designs , the obtained astronomical spectra of them were confirmed to be identical . determining the number of correlation lags in the xf scheme or of fourier transform points in the fx scheme is essential for the realization of high - dispersion and wideband observation , because the frequency resolution is derived as @xmath0 where @xmath1 is the sampling period , @xmath2 is the number of correlation lags or fourier transform points , and the bandwidth of b is equal to @xmath3 . the material size and cost of the correlator strongly depend on the sampling period , @xmath1 , and the number of correlation lags or fourier transform points , @xmath2 .    the new xf architecture with the digital tunable filter bank that is designed with the finite impulse response ( fir ) has been proposed and developed for the next generation radio interferometers , the expanded very large array ( evla ) and the atacama large millimeter / submillimeter array ( alma ) ( @xcite , @xcite ) . this is called the `` fxf correlator '' . the architecture of the fxf scheme can make the material size smaller in comparison with that of the conventional xf scheme . since the digital filter allows a variety of observation modes [ scientific and observational availability were shown in iguchi et al . ( 2004 ) ] , the fxf scheme will provide us with the most appropriate specifications which meet the scientific requirements . this will lower the risk of over - engineering of the correlator . the improved fx architecture with dft filterbank was developed by bunton ( 2000 ) . the use of polyphase filter banks allows arbitrary filter responses to be implemented in the fx scheme ( bunton 2003 ) . this is called the `` polyphase fx correlator '' . this scheme has a possibility to achieve the spectral leakage of about -120 db . in particular , this performance is significant to suppress the leakage from the spurious lines mixed in receiving , down - converting or digitizing .    the ffx correlator is a new algorithm for correlation process in radio astronomy . the ffx scheme consists of 2-stage fourier transform blocks , which perform the 1st - stage fourier transform as a digital filter , and the 2nd - stage fourier transform to achieve higher dispersion . the first f of the ffx is the initial letter of the word `` filter '' . in this paper , we present a new ffx architecture . the principle of the ffx scheme in section 2 , the properties of the ffx scheme in section 3 , the algorithm verification and performance evaluation with the developed ffx correlator in sections 4 and 5 , and the summary of this paper in section 6 are presented . this section shows the algorithm and the data flow diagram of the signal processing in the fourier transform of the ffx scheme ( see figure [ fig : ffx ] ) . suppose that @xmath4 are the digital waveforms at the correlator input from the astronomical radio signals that are received by the telescope . the inputs , @xmath4 , are real digital signals at sampling period of @xmath1 , and obey the zero - mean gaussian random variable . the suffix @xmath5 is an integer for time . fig1    ( 160mm,200mm)fig1.eps    [ step 1 ] the correlator receives the time - domain digital sampling signals from the analog - to - digital converter ( adc ) , and accumulate them up to @xmath6 points . [ step 2 ] the time - domain @xmath6-point data are transferred to the frequency - domain by using the @xmath6-point discrete complex fourier transform as follows : @xmath7 where @xmath8 is the spectrum after the 1st fourier transform , the suffix @xmath9 is an integer for frequency , and @xmath1 is equal to @xmath10 at the bandwidth of @xmath11 . the @xmath12 is the minimum frequency resolution of the 1st fourier transform , which is equal to @xmath13 . [ step 3 ] the extraction of the @xmath14 points from the frequency domain @xmath15-point data after the 1st fourier transform is conducted as if filter and frequency conversion are performed simultaneously : @xmath16 where @xmath17 is the minimum frequency channel in the extraction , and the suffix @xmath18 is an integer for frequency . [ step 4 ] the @xmath14-point data after inverse fourier transform is written by @xmath19,\ ] ] where @xmath20 is the time - domain signal after inverse fourier transform , the suffix @xmath21 is an integer for time , and @xmath22 is the sampling period after filtering at the bandwidth of @xmath23 @xmath24 . [ step 5 ] by repeating the procedure from step 1 to step 4 , the data are gathered up to @xmath25 points as follows ; @xmath26 where @xmath27 is @xmath28 , and @xmath29 is the number of repeating times of the procedure from step 1 to step 4 . [ step 6 ] the time - domain @xmath25-point data after gathering are transferred to the frequency - domain by using the @xmath25-point discrete complex fourier transform as follows : @xmath30 where @xmath31 is the spectrum after the 2nd fourier transform , and the suffix @xmath32 is an integer for frequency . the @xmath33 is the minimum frequency resolution after the 2nd fourier transform , which is equal to @xmath34 ( = @xmath35 ) . .definition of functions . [ cols="<,<",options="header " , ]     [ table : fxopmode ]    llllll stage & bandwidth & spectral & spectral & velocity & correlation + & & points & resolution & resolution & + & & & & at 1 mm & + 1@xmath36 & 4096 mhz & 512 & 8 mhz & 8.0 km s@xmath37 & 2ac1cc + 2@xmath38 & 64 mhz & 2048 & 31.25 khz & 0.031 km s@xmath37 & 2ac1cc +   + 1@xmath36 & 4096 mhz & 512 & 8 mhz & 8.0 km s@xmath37 & 2ac1cc + 2@xmath38 & 128 mhz & 4096 & 31.25 khz & 0.031 km s@xmath37 & 1ac +   + 1@xmath36 & 2048 mhz & 512 & 4 mhz & 4.0 km s@xmath37 & 4ac2cc + 2@xmath38 & 32 mhz & 2048 & 15.625 khz & 0.016 km s@xmath37 & 4ac2cc +   + 1@xmath36 & 2048 mhz & 512 & 4 mhz & 4.0 km s@xmath37 & 2ac1cc + 2@xmath38 & 64 mhz & 4096 & 15.625 khz & 0.016 km s@xmath37 & 2ac1cc +   + 1@xmath36 & 2048 mhz & 512 & 4 mhz & 4.0 km s@xmath37 & 2ac1cc + 2@xmath38 & 128 mhz & 8192 & 15.625 khz & 0.016 km s@xmath37 & 1ac +   + 1@xmath36 & 2048 mhz & 512 & 4 mhz & 4.0 km s@xmath37 & 2ac1cc + 2@xmath38 & 96 mhz & 6144 & 15.625 khz & 0.016 km s@xmath37 & 1ac + 2@xmath38 & 32 mhz & 2048 & 15.625 khz & 0.016 km s@xmath37 & 1ac +   +   +   +    [ table : ffxopmode ]    @xmath39 correction is performed based on the baseline . @xmath39 correction coefficient is generated within fpga . firstly , @xmath40 , the gradient of @xmath39 to time change , is evaluated and then the value of each frequency channel is calculated ( see figure [ fig : deltaw ] ) . circuit diagram of the line graphs above is shown in figure [ fig : deltawcir ] . for every 1 millisecond , the initial value ( init ) is provided by the monitor @xmath41 control computer , and read as a initial gradient [ @xmath42 . in the border between segments , the previous value is multiplied by the gradient ( grad ) data to generate a gradient of a new segment [ @xmath40 ] . in the arbitrary time @xmath44 , initial value ( the value of dc ) is set to 0 . @xmath39 is set for every 128 channel in the full bandwidth of 8@xmath45 channel , which means the full bandwidth ( 8@xmath45 channel ) is corrected with 64 steps . the initial value of @xmath40 can be set for every 1 millisecond , however , a given value is set within 128 channel . the gradient of @xmath40 is the gradient variation of @xmath39 per one segment length for fft . the monitor @xmath41 control computer specifies a set of the initial value and gradient approximately every 1 second . ( 80mm,120mm)fig17.eps    ( 130mm,120mm)fig18.eps    final correlation value is calculated by adding the output of 16 correlators . the composition of the long - term accumulation / output board ( ltaob ) is shown in figure [ fig : lta ] . when the correlation result is output , the data is converted into the format of ieee single - precision floating point .    in the fx processing , the number of the output frequency channels is normally 4 @xmath45 ( = 4096 ) . thus the output data size per one correlation is @xmath46 where @xmath47 is the number of single - precision floating bits , @xmath48 is the complex , and @xmath49 is the number of correlations ; auto - correlations of @xmath50 and @xmath51 , and a cross - correlation between them . assuming the minimum integration time is 0.1 second , the estimated output speed is approximately 7.7 mbps . in the ffx processing , the number of the output frequency channels is 512 + 2 @xmath45 , thus the data size per one correlation is 480 @xmath45 bits . consequently , the estimated output speed with 0.1-second integration time is approximately 4.8 mbps . correlation results are sent to the monitor @xmath41 control computer using tcp / ip protocol of 100 baset - ether . the ffx correlator are divided into four f parts in principle ( not physically ) . each f part in a fx mode and the first fft stage of ffx mode are operated at 2048 mhz , and each f part at the second fft stage of ffx mode are operated at 32 mhz . also , the digital signals input from eib are distributed to arbitrary f parts by using the command ( `` fchsel '' ) from the monitor @xmath41 control computer . in fx mode and the first fft stage of ffx mode , the digital signals of 8192 msps are operated by combining two f parts (= 2@xmath522048 mhz ) , while the signals of 4096 msps are operated by using one parts ( = 1@xmath522048 mhz ) . at the second fft stage of ffx mode , the digital signals of 8192 msps are operated by combining two f parts ( = 2@xmath5232 mhz ) , while the signals of 4096 msps are operated using only one parts ( = 1@xmath5232 mhz ) . by using the command ( `` fchsel '' ) , the effective bandwidth after the second fft stage can be changed . all of the operation modes available in this ffx correlator are listed in table [ table : fxopmode ] and [ table : ffxopmode ] . the hardware of the fx correlator is shown in figure [ fig : farm ] . the dts - r module consists of two electrical input interface boards ( eibs ) , two delay correction and data configuration boards ( dcdcbs ) , and one dts - r monitor @xmath41 control boards ( drmcbs ) . the correlation module consists of eight correlation boards ( corbs ) , long - term accumulation / output board ( ltaob ) , and one correlation monitor @xmath41 control board(cormcb ) . each module is connected to the independent power module . the power consumption of the filter module is 400 w , while that of the output module is 600 w. the total ac power is 750 w at 1-phase 100 - 220 vac 50/60 hz ( 100 v @xmath53 or 220 v @xmath53 ) . the total weight is 71.3 kg . ( 80mm,65mm)fig19.eps    figure [ fig : measure ] shows a block diagram of the measurement setup of the frequency response of the ffx correlator . to investigate the frequency response , it is useful to use the cw signal , which can measure the folding effects by sweeping the frequency range of 0 to 4096 mhz . the white noise is important to measure the frequency response , because the astronomical signals obey the gaussian random variable . to obtain the input signals that are approximated to the zero - mean gaussian probability , mixing of the cw signal with the white noise is necessary . ( 147mm,100mm)fig20.eps    the frequency response of the ffx correlator when cw is included is written as @xmath54 \nonumber \\ & & \cdot h_{1}(f ) h_{2}^{\ast}(f ) \cdot |h_{\mathrm{d}}(f)|^2 ,   \label{eq : oncor}\end{aligned}\ ] ] where @xmath55 is the frequency response of the cw signal , @xmath56 is the frequency response of the white noise from the aste analog backend subsystem , @xmath57 and @xmath58 are the frequency response by different transmission paths , in which @xmath59 and @xmath60 ( see figure [ fig : measure ] ) , and @xmath61 is the frequency response of the ffx correlator , including the effects of requantization and the folding noise after downsampling . the bandpass calibration is essential for estimating the cw power accurately , because the bandpass response becomes a time - variable due to outdoor air temperature . the frequency response without the cw signal is written as @xmath62 the adcs work as 1-bit performance ( @xcite ) . in that case , it is important to adjust the power as precisely as possible so as to avoid the high - order spurious effects . the frequency responses of @xmath63 and @xmath64 depend on the relative power of the cw signal to the white noise , and also the threshold levels in quantization . to correct these effects , we need to calibrate the bandpass by sensitively adjusting the continuum floor level of @xmath65 to that of @xmath66 in the data analysis . these values are @xmath67 and @xmath68 .    from equations ( [ eq : oncor ] ) and ( [ eq : coroff ] ) , the frequency response in a ffx mode is written as @xmath69 \cdot h_{1}(f ) h_{2}^{\ast}(f ) \cdot |h^{\mathrm{f}}_{\mathrm{d}}(f)|^2 ,   \label{eq : ffxon}\end{aligned}\ ] ] while the frequency response without the cw signal can be written as @xmath70 and then the cw frequency response including the response of the measurement system is derived as @xmath71 similarly , the frequency response in a fx mode is written as @xmath72 \cdot h_{1}(f ) h_{2}^{\ast}(f ) \cdot |h^{\mathrm{n}}_{\mathrm{d}}(f)|^2 ,   \label{eq : fxon}\end{aligned}\ ] ] while the frequency response without the cw signal can be written as @xmath73 and then the cw frequency response including the response of the measurement system is derived as @xmath74    from equations ( [ eq : ffxpwr ] ) and ( [ eq : fxpwr ] ) , we can derive the frequency response in a ffx mode from the correlated spectra obtained in ffx and fx modes as @xmath75 since the frequency response of @xmath76 in the fx mode is well known ( @xcite ) , the frequency response in a ffx mode can be derived finally . figure [ fig : response ] shows the frequency response of the ffx correlator at a bandwidth of 64 mhz in the range of 2048 to 2112 mhz , and that the lower limits of about -33 db to -40 db are successfully measured with this method . the measurement frequency resolution was 31.25 khz . the measurement results show that the effective bandwidth is about 59.28 mhz , which was obtained by the passband responses of about @xmath77 db at 2046.40625 and 2106.28125 mhz , @xmath78 db at 2045.3125 and 2107.28125 mhz , and by a stopband response with the first sidelobe of about @xmath79 db . the measurement results are well consistent with the theoretical curve in the passband , both bandedges ( sharpness ) , and first sidelobe levels . in the stopband response except these responses , however , it is shown that there are differences between theoretical curve and measured results . this problem is probably due to the non - linear response of 1-bit adc and the precision of the data reduction process in this measurement method . the cross - modulation distortion is strongly generated in digitizing the cw signals at 1 bit . this character complicates the data reduction method , and will reduce the measurement precision . if the adcs with 3 bits or more are feasible , this problem will be relaxed . finally , the measurement results show that the theory of the ffx scheme can be confirmed , and the development of the ffx correlator was successfully realized . there are two basic designs of a digital correlator : the xf - type in which the cross - correlation is calculated before fourier transformation , and the fx - type in which fourier transformation is performed before cross multiplication . to improve the fx - type correlator , we established a new algorithm for correlation process , that is called the ffx scheme . the ffx scheme demonstrates that the realization of a stopband response with first and second sidelobes of @xmath80 db and higher - order sidelobes of @xmath81 db is technically feasible . the ffx scheme consists of 2-stage fourier transform blocks , which perform the 1st - stage fourier transform as a digital filter , and the 2nd - stage fourier transform to achieve higher dispersion . the ffx scheme provides flexibility in the setting of bandwidth within the sampling frequency . the input data rate of the developed ffx correlator is about 48 giga bit per second ( gbps ) with 3-bit quantization at the sampling frequency of 8192 or 4096 msps , which is 8192 msps x 3 bits x 2 if or 4096 msps x 3 bits x 4 ifs . we have successfully evaluated the feasibilities of the ffx correlator hardware . also , this hardware will be installed and operated as a new spectrometer for aste . we successfully developed the ffx correlator , measured its performances , and demonstrated the capability of a wide - frequency coverage and high - frequency resolution of the correlation systems . our development and measurement results will also be useful and helpful in designing and developing the next generation correlator . the authors would like to acknowledge yoshihiro chikada for his helpful technical discussions . the author would like to express gratitude to brent carlson who provided constructive comments and suggestions on this paper . this research was partially supported by the ministry of education , culture , sports , science and technology , grant - in - aid for young scientists ( b ) , 17740114 , 2005 . bunton ,  j ,  2000 , alma memo 342 , ( charlottesville : nrao ) bunton ,  j ,  2003 , alma memo 447 , ( charlottesville : nrao ) carlson ,  b.  2001 , nrc - evla memo 014 chikada ,  y. , ishiguro ,  m. , hirabayashi ,  h. , morimoto ,  m. , morita ,  k. , kanzawa ,  t. , iwashita ,  h. , nakazima ,  k. , et al . 1987 , proc . ieee , 75 , 1203 escoffier ,  r.  p. , comoretto ,  g. webber ,  j.  c. , baudry ,  a. , broadwell ,  c.  m. , greenberg ,  j.  h. , r. r. treacy ,  r.  r. , cais ,  p. , et al . 2007 , a@xmath41a 462 , 801 ezawa ,  h. , kawabe ,  r. , kohno ,  k. , yamamoto ,  s.  2004 , spie 5489 , 763 iguchi ,  s. , okuramu ,  s.  k. , okiura ,  m. , momose ,  m. , chikada ,  y. ursi general assembly ( j6:recent scientific developments ) , maastricht , 2002 . iguchi ,  s. , kurayama ,  t. , kawaguchi ,  n. , @xmath41 kawakami ,  k.  2005 , pasj 57 , 259 narayanan , d. , groppi , c.  e. , kulesa , c.  a. , & walker , c.  k.  2005 , , 630 , 269 okuda , t. , iguchi , s.  2008 , pasj 60 , 315 okumura , s. k. , chikada , y. , momose , m. , iguchi , s.  2001 , alma memo no.350 ( charlottesville : nrao ) rabiner ,  l.  r. , schafer ,  r.  w.  1971 , ieee trans . audio electroacoust . , au-19 , 200 thompson ,  a.  r. , moran ,  j.  m. , @xmath41 swenson ,  g.  w.jr . 2001 , interferometry and synthesis in radio astronomy , 2nd ed . , ( new york : john wiley @xmath41 sons ) , 289 weinreb , s.  1963 , digital spectral analysis technique and its application to radio astronomy , ( r. l. e. , mit , cambridge , mass . ) , tech . rep .
we established a new algorithm for correlation process in radio astronomy . 
 this scheme consists of the 1st - stage fourier transform as a filter and the 2nd - stage fourier transform for spectroscopy . 
 the `` ffx '' correlator stands for filter and fx architecture , since the 1st - stage fourier transform is performed as a digital filter , and the 2nd - stage fourier transform is performed as a conventional fx scheme . 
 we developed the ffx correlator hardware not only for the verification of the ffx scheme algorithm but also for the application to the atacama submillimeter telescope experiment ( aste ) telescope toward high - dispersion and wideband radio observation at submillimeter wavelengths . 
 in this paper , we present the principle of the ffx correlator and its properties , as well as the evaluation results with the production version .