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ALMA Memo 583
Algorithms and formulas for hybrid correlator data correction Giovanni Comoretto INAF - Osservatorio di Arcetri
Last revised: 2008-11-05

Abstract: This report describes the algorithms and formulas used to reconstruct the hybrid auto or cross spectra from the ALMA correlator output. This process includes the quantization correction ("Van Vleck correction") for the 2 bit quantization, Fourier Transform, calibration for tunable filter passband response and for different amplitude among sub-channels, stitching of the sub-channel spectra in a single composite spectrum, and final correction for the sampler 3-bit quantization. The final product consists in normalized ACF and CCF spectra, processed and presented in a consistent and uniform way for all correlation modes.

1

Intro duction

Al l acronyms and mathematical symbols are defined in Appendix C The ALMA correlator produces a series of correlation functions (auto correlations, ACF, or cross correlations, CCF), of a digital representation of the radio signal, quantized either at 2, 3 or 4 bit. Each CF may represent a segment of delay lags (e.g. in CCF positive and negative lags are computed separately), and/or a different segment of the data, either spectral (in frequency division mode) or temporal (in time division mode). For sample representations greater than 2 bits/sample, different parts of a single correlation function are also computed separately, but the relevant re-assembly is done in hardware by the LTA. Having so many different configuration modes result in a quite complex data processing structure, with several alternate branches. In this document we will deal with the data processing required to produce a normalized ACF or CCF spectrum. The data processing for ALMA is described in document "Specifications and Clarifications in ALMA correlator", by S. Scott [11]. The processing described in this report replaces the procedure described there up to point 6. Subsequent processing remains unchanged. Normalization of the spectra is performed as described in Scott [11]: for autocorrelation spectra, the average spectral density across the spectrum is unitary, and the quantity given in the cross spectra is the spectral cross correlation coefficient. The digital total power is given in units of the sampler quantization step. The processing is kept as consistent as possible for all the correlator modes, i.e. time and frequency division, 2, 3 and 4 bit quantizations.


Algorithms for hybrid correlator data correction

2

1.1

Scop e

This document is relevant to the implementation of the correlator quasi real time software for processing the observed spectra. It provides general suggestions on software implementation topics, but do not suggest specific software implementations. Whenever possible different alternative approaches are suggested. It does not address astronomical calibration issues. The input of the process is a set of raw correlation coefficients for each sub-channel, correlator plane and antenna (for ACF spectrum) or baseline (for cross spectrum). The output is a set of normalized auto and cross spectra, together with digital total power data.

1.2

Description of the problem

The usual processing of correlation data, described in [11], consists in lag normalization, correction for quantization, and subsequent calibration using a reference spectrum (usually the ACF signal) to correct for bandpass effects. This approach is not appropriate for the ALMA correlator in frequency division mode, for the following reasons: · In frequency division mode, the signal is quantized twice, i.e. at the analog 3-bit sampler and after spectral processing in the Tunable Filterbank. Both these quantizations introduce a nonlinear distortion in the correlation function, that must be corrected. · The bandpass response due to the digital filter is fully deterministic, both in amplitude and in phase, and therefore can be corrected using the filter design data. · Having several subchannels, it is easier to realign and combine them in a single spectrum at an eaerlier processing stage than to deal with separate spectra that must be individually calibrated.

1.3

Pro cessing steps

The general steps required for the processing of a set of raw correlation lags, to produce a set of autocorrelation and crosscorrelation coefficients is performed in the following steps: 1. Lag averaging: this includes assembling a complete CF from correlator products computed in different correlator planes 2. Normalization of raw lags, to obtain the quantized correlation product (chapter 2) 3. 2-bit (or 4-bit) quantization correction, using the appropriate correction function. The correction is dependent from the total power levels derived by zero-lag ACF (chapter 3) 4. FFT with appropriate taper, and with 1/2 channel frequency shift (chapter 4) 5. Bandshape correction, using a precomputed shape and with data-dependent correction (chapter 5) 6. Re-gridding of the corrected spectra inside the final composite spectrum (chapter 6) 7. Normalization of the output spectrum, and computation of the associate power level 8. Correction for the 3 bit quantization in the sampler (chapter 7) 9. Final rescaling of the output spectra, to comply with the conventions of [11] (chapter 8) The processing corresponds in reverse to the signal processing in the station hardware, as shown in fig. 1. For TDM only steps 1­4 and 9 are required. In frequency division mode, steps 1­5 must be separately performed on all sub-channels, that are combined together in step 6 and further processed in steps 7­9 as a single entity. In FDM, the resulting spectrum does not usually span the full 2 GHz band, and may contain holes in the frequency coverage. In the following chapters, each of the above operations is described in more detail.


Algorithms for hybrid correlator data correction
TFB SUB-CHAN MIXER SPLITTING FILTER GAIN ADJ 2-BIT SAMPLER CORREL

3

IF

3-BIT SAMPLER

3 BIT QUANTIZ. SPECTRUM CORRECTION Sh ( ) Sh ( ) 8

SUB-CHAN STITCHING S i( )

BANDPASS CORRECTION

x

1 S1*S2 S( )

FFT

CORR PROD 2 BIT QUANTIZ. CORRECTION R ( ) R 4( )

Figure 1: Signal processing in the station hardware (top row), and corresponding data processing in the software (bottom row)

2

Normalization of raw lags

The Short Term Accumulator (STA) in the correlator chip accumulates the correlation product R 4 ( ) over the chosen dump time. For hardware considerations, the quantity accumulated is R 4 + 9, where the offset of 9 has been chosen to avoid negative numbers. The total number of samples in a millisecond is (125000 - 270) where 125000 is the number of clock cycles, 270 is the number of clock cycles lost in the correlator chip dump. Dump time may be either 1 or 16 ms, after which the STA results are accumulated in the LTA. The lower 10 bits of the register are not read (6 bits for 1 ms dump time), and thus the raw lag is divided by 64 or 10241. Therefore the offset in a dump period is Vs16ms = 9(125000 - 270)/64 - 0.5 = 17539.65625. Subtracting 0.5 from Vs compensates the error of truncation vs. rounding during STA readout. The total bias in a complete integration is then Vs = Vs16ms Np Nd (1) where Np is the number of correlator planes co-added in the LTA, and Nd = tint /tdump is the number of correlator chip dumps. 3 and 4 bit modes use 4 planes of the correlator, to synthesize a 4 bit multiplier from four 2 bit multipliers. For 3 bit mode only the lower 3 bits are used, i.e. the sample values have a restrict range, but the same hardware multiplier is used. The 4 planes are co-added with relative weights of 1, 4, 4 and 16, so the total offset implied in this scheme is increased by a factor of 25 (225 instead of 9). The simplest way to consider this is to include an extra factor of 25 in Np . The values of Np for the available correlator modes are listed in tab. 1. This is the product of a factor of 32 for TDM, a factor of 25 for 3x3 and 4x4 quantization, and a factor of 2 for the oversampled modes at 62.5 MHz bandwidth (at 31.25 MHz oversampling is performed in the correlator hardware, without involving the LTA). Mode TDM TDM FDM FDM FDM FDM FDM FDM Quant. 2x2 3x3 2x2 2x2 2x2 4x4 4x4 4x4 Oversample Band 2GHz 2GHz any 62.5 31.25 any 62.5 31.25 Np 32 800 1 2 1 25 50 25

no yes yes no yes yes

Table 1: Np for the available correlator modes. Bandwidth refers to the single sub-channel
1 Actually, one bit is discarded in the multiplication table, as the offset multiplication result is always an even numb er, and 5 or 9 bits are discarded in the STA readout


Algorithms for hybrid correlator data correction

4

The normalization process is then expressed by the formula R4 ( ) = 9K L( ) - V Vs
s

(2)

where L( ) is the LTA raw lag output for delay . The factor K is 1 for 2 bit modes, and 25 for 3 and 4 bit modes. It is important to note that R4 is a correlation product, and includes total power information for the 2 bit quantized signal. In particular R4 is generally not equal to 1.0 for the zero-lag autocorrelation, but is typically around 3.5, 11 and 35 for respectively an optimally quantized 2, 3 and 4 bit signal. 2

3

Quantization correction

The correlation functions must be corrected for the effect of the quantization. This applies both at the quantization of the analog signal in the analog sampler, and at the requantization of the multi-bit digital signal after the Tunable Filterbank. The mathematical treatment is the same, irrespective of the analog or digital nature of the quantized signal. Quantization correction depends on the exact ratio of the quantization step to the RMS amplitude for the two signals correlated (one signal in the case of ACF). In the ALMA quantization process, thresholds (or signal level) are set initially to the optimum value with respect to the signal level, within a relatively coarse accuracy (±0.25 dB for the analog 2 or 3-bit quantization, 1% for the 2-bit quantization inside the TFB), and then kept constant. During the observation, level may change, and thus accurate measurement of this ratio must be performed in order to reconstruct the signal. This aspect is particularly important for a hybrid correlator, as the accurate relative calibration of the individual sub-channels is essential for accurate realignment of the reconstructed spectrum. A total power error of the order of the dynamic range (10-4 to 10-5 ) is thus required to correctly align each sub-channel in the composite spectrum and avoid platforming. The RMS signal amplitude before the quantization can be measured either directly, or inferred from zero-lag ACF. A direct measure of the digital total power in each sub-channel is in principle possible, but would saturate the the control bus bandwidth in most realistic scenarios. The latter method is usually much simpler, as the zero-lag ACF is directly available from the ACF data. Some bookkeeping is required to associate the zero-lag ACF of the two signals composing a CCF.

3.1

Zero delay auto correlations

For Gaussian noise, the zero delay autocorrelation of a 2 bit sampled signal of amplitude expressed in units of the quantization step l is given by the formula: 1 (3) 2 The relation can be directly inverted, using the inverse erf routine derived by Blair et al., and implemented as part of the GBT software by F. Schwab [12]. For multi-bit quantization, a similar relation can be derived. If N is the total number of levels, and assuming multiplicative weight 2k + 1 for level k (the multiplication scheme adopted in the ALMA correlator), the zero-lag ACF is given by: R(0) = 9 - 8 erf
N/2-1

R(0) = (N - 1)2 -

8k erf
k=1

k 2

(4)

It is possible to recover for all the antennas, BBCs and sub-channels by numerically inverting this relation (that is strictly monotonically crescent). A computer routine to invert this relation with standard numeric algorithm is also given by Schwab.
2 The signal level must b e adjusted at the sampler input for the quantization scheme adopted. The signal RMS amplitude at the analog sampler input should b e set to 1.70 times the (3 bit) sampler quantization step in 3 bit TDM and all FDM mo des, and to 2.01 times the quantization step in 2 bit TDM mo de, ±0.5 dB. This implies that in 2 bit TDM mo de the RF level must b e raised by 1.4 dB.


Algorithms for hybrid correlator data correction

5

3.2

Quantization correction

The digital cross correlation R4 of the 2-bit quantized signal is a function of the cross correlation of the two continuous3 signals x1 , x2 before the digitization stage, of the two normalized quantization steps l1 and l2 (or the normalized signal amplitudes 1 , 2 ), and of the multiplication scheme adopted. The 2 bit 4 level multiplication scheme adopted in the ALMA correlator is implicitly assumed here. It is important to note that the function (R4 ) depends strongly on l1 , l2 , i.e. on the amplitude of the input signals. Correcting R4 assuming a constant input level is absolutely not acceptable. Several numeric approaches to compute R4 () are available. The process is however quite computing intensive. It is possible to speed up this process by tabulating as a function of R4 , 1 , 2 , but the number of tabulated points needed for a meaningful interpolation with the required accuracy is quite high. Several efficient algorithms to implement this quantization correction have been developed. For example, a set of routines developed for GBT are publicly available [12], and have been adapted to the ALMA multiplication table, for the 2, 3 and 4 bit cases. These routines compute a spline interpolation table for given values of 1 , 2 , and apply it to the correlation function. Calculating the table is still quite computer intensive, and for short integration times it is beyond the available computing power. If, as it is usually the case, the RMS of the signal stays constant for a significant amount of time, one can compute the table for each sub-channel and baseline only once, and apply it to all the correlation functions for the whole time interval. An efficient relation for R4 (l1 , l2 , ) have been found by Schwab: R4 (l1 , l2 , ) = -8 + 2 arcsin() l2 l1 + erf + + 3erf 2 2 +8 (L(0, l1 , ) + L(0, l2 , ) + L(l1 , -l2 , ) + L(l1 , l2 , ))
1,2

where L(l1 , l2 , ), is the cumulative bivariate probability that x1 > l1 and x2 > l2 , and l 3.2.1 Polynomial approximation

= 1/

1,2

.

An alternative approach has been described in [3]. The relation R4 (l1 , l2 , ) has been expanded in powers of up to 5 , and inverted up to the same accuracy. The resulting relation is quite accurate up to = 0.2, i.e. for almost all the correlation points. The approximate relation has the form R4 = a + b3 + c5 , where a, b, c are given by: 2 2 2 1 + 2 exp(-l1 /2) 1 + 2 exp(-l2 /2) (5) 1 2 2 2 2 b= 1 + 2(1 - l1 ) exp(-l1 /2) 1 + 2(1 - l2 ) exp(-l2 /2) (6) 3 1 2 4 2 2 4 2 c= 3 + 2(3 - 6l1 + l1 )) exp(-l1 /2) 3 + 2(3 - 6l2 + l2 )) exp(-l2 /2) (7) 60 and the inverse relation is 1 b2 3b2 - ac 5 = R4 - 4 R4 + R4 (8) a a a7 These relations are symmetric, i.e. (-R4 ) = -(R4 ), if no DC bias is present in the sampled data. The correlation coefficient ( ) must then be multiplied by 1 2 , to obtain the cross product between the input continuous signals, R( ). This approach requires less computing time than the spline interpolation technique, as the computing intensive expression R4 (l1 , l2 , ) needs not to be computed to build the spline interpolation table. If it is not known in advance whether the interpolation will be required (i.e. if there are correlation products with || > 0.2) the spline interpolation table must be computed anyway, and the polynomial relation can be used to speed up the construction of the spline table for the inner part of the correction curve. The speed-up is modest, around a factor of 2, but worth the extra effort. a=
3

Here with continuous we may intend either an analog signal or a digital signal represented with many bit accuracy


Algorithms for hybrid correlator data correction

6

3.2.2

3 and 4 bit quantization correction

The 3 and 4 bit cases can be treated in a similar way. From [3] for the generic N level case the N bit correlation coefficient RN can be expressed as a polynomial in odd powers of , truncated to 5 . The particular case N = 4 gives the coefficients in equations 5-7. 3 bit quantization requires N = 8 even if the samples are actually quantized with 4 bits, as only 8 separate values are used. 2
5

RN () =

k=1

k Ck (l1 )Ck (l2 ) k! exp(-(il)2 /2) (1 - (il)2 ) exp(-(il)2 /2) (3 - 6(il)2 + (il)4 ) exp(-(il)2 /2)

(9)

N/2-1

C

1

= 1+2
i=1 N/2-1

(10)

C

3

= 1+2
i=1

(11)

N/2-1

C where il = i/ is the i RMS amplitude.
th

5

= 3+2
i=1

(12)

threshold, and l = 1/ is the threshold spacing normalized to the signal

4

Fourier Transform

For detailed discussion on the Fourier Transform, we refer to chapter 2.1 in [2]. There it is shown that, if the sub-channels are overlapped by a even number of spectral points, it is convenient to compute the Fourier transform in a set of frequencies i given by j = (j + 1/2), j = 0 . . . (N - 1) (13)

where N is the number of (positive) correlation lags and = 1/(2N ) is the spectral point spacing. In this way, it is possible to discard an integer number of spectral points on each side of the band. Incidentally, this is convenient also for time division mode, or for modes with a single sub-channel, as the spectral points are spaced evenly across the observed spectrum, without a "special" first and last point. Computationally, the simplest way to obtain this shift in the frequency channels is by multiplying the correlation function R( ) before Fourier transform by an exponential: S (j ) = F C (k ) exp 2 i k 4N (14)

where F [C ] denotes the Fourier transform, and the index k assumes the values [-N . . . (N - 1)]. An efficient algorithm to perform modified FFT on real data is described in appendix A. If the sub-channels are overlapped by an odd number of channels, the conventional FFT, computed on frequencies j = j with j = 0 . . . N , can be used. Since the first and last channels of conventional FFT of a real data set has half width, it is possible to discard a semi-integer number of channels from each side of the spectrum. To simplify the data reduction pipeline, and to guarantee a coherent interpretation of the output channel frequencies, the 1/2 spectral point offset is always applied. This constrains the number of overlapped points to be even (or zero). In the present correlator firmware, the subchannel overlap is fixed to 1/16 of the subchannel width, and the number of overlapped channels is always a power of 2. In any case, discarding of unwanted spectral points is performed after the calibration described in the next section.


Algorithms for hybrid correlator data correction

7

4.1

Tap ering functions

Appropriate tapering T ( ) must be used before Fourier transform for apodization of the spectrum. ALMA specifications require that at least the following tapering functions w(i) must be provided. The index i runs from 0 (zero lag ACF) to N/2, with N the two sided size of the correlation. The function is even, w(-i) = -w(i). · Uniform: constant weight (no tapering) · Bartlett (triangular): w(i) = 1 - |2i/N i| · Welch (parabolic): w(i) = 1 - (2i/N )
2

· Hanning: w(i) = 0.5 + 0.5 cos(2 i/N ) · Hamming: w(i) = 0.54 + 0.46 cos(2 i/N ) · Blackmann: w(i) = 0.42 + 0.5 cos(2 i/N ) + 0.08 cos(4 i/N ) · Blackmann-Harris: w(i) = 0.35875 + 0.48829 cos(2 i/N ) + 0.14128 cos(4 i/N ) + 0.01168 cos(6 i/N ) According to [11], in the ALMA processing software the FFT routine is normalized so that the average spectral density over the whole spectrum is equal to unity for the autocorrelation spectra, and scaled accordingly for cross spectra. The quantity recorded for cross spectra is thus the correlation coefficient as a function of frequency. In this way, absolute amplitude is lost, and is recovered in later processing by appropriate astronomical calibration. In a hybrid correlator however it is important to keep track of the actual signal amplitude, and therefore the correlation is normalized to the amplitude 1 2 of the two input data streams. In this way the average (real) spectral density for ACF is equal to the input total power, in units of the quantization step. The final normalization to conform to the ALMA convention is done only at the end of the processing, as described in chapter 8, after the sub-channels are stitched together and the 3 bit quantization correction has been applied.

5

Bandshap e correction

The resulting spectrum must be corrected for the filter shape and slope, as described in [2]. The combined response for the two digital filters and the tapering can be precomputed for all the observation modes. For each spectral channel j in the sub-channel, the integral response a(j ) and its first momentum m(j ) are tabulated in a response file. For symmetric filter taps and tapering function, these values are real. a(j ) is symmetric with respect to the sub-channel center, while m(j ) is antisymmetric. A separate response file is available for each spectral resolution, sub-channel width and tapering function. A procedure to compute these values from the filter taps and the tapering function w(j ) is reported in appendix B. With the 7 tapering functions above, 2 sub-channel widths (62.5 and 31.25 MHz) and 8 resolutions (64-8192 spectral points), a total of 102 calibration files are required. The calibration algorithm is then: 1. Each (complex) spectral point is divided by a(j ), obtaining a first corrected spectrum, S (j ). 2. S is differentiated, obtaining d(j ) = (S (j + 1) - S (j - 1))/2. The first and last points are linearly extrapolated from the two next ones, as in d(0) = 2d(1) - d(2). 3. The spectrum is corrected for the small offset in channel barycenter induced by the filter slope: S (j ) = S (j ) + m(j )d(j ). 4. A new estimate of d(j ) is obtained, and used to estimate a better correction for S (j )


Algorithms for hybrid correlator data correction

8

Only 2 iterations of steps 2-3 are sufficient, as more iterations just spread around numeric noise. The correction function contains an appropriate scale factor that includes all the processing in the two stage filters. The spectrum amplitude is affected by four more factors: · A factor of 222 or 218 , to account for 11 or 9 bits discarded before the second quantization unit, respectively for 2 and 4 bit quantization · A multiplicative, programmable, factor, used to rescale the filter output for optimal 2 bit quantization. This factor is determined in the SCC before the correlation, and is typically different for each sub-channel. The cross spectrum must be divided by these factors, one for each of the two baseline antennas · A conversion gain of the digital mixer. This value is dependent on the particular sine table used in the mixer, but is constant and deterministic. For the mixer table adopted in the TFB, it is equal to 5.21 [1] · A factor equal to the frequency division factor used (32 for normal TFB mode, or 64 for half bandwidth mode) 4 At the end of this process, the rescaled spectra from each sub-channel are calibrated and corrected to the same scale, corresponding to the output of the 3-bit sampler. The quantity in the spectra is proportional to a spectral density, i.e. power per unit frequency and the scale is the same irrespective of spectral resolution and observing mode.

6

Sub-channel stitching

Each corrected spectrum must be re-gridded in the final hybrid spectrum, dropping the first and last channels. The digital local oscillator for each sub-channel has a tuning step of (2/2 16 ) GHz. i.e. 1/32 of a channel at the minimum resolution. It is thus possible to tune each sub-channel in order to place all spectral points on a common frequency grid. In this case the re-gridding operation corresponds to computing the correct index in the composite spectrum. The amount of spectrum to be dropped is a fixed quantity in frequency, related to the filter shoulder shape and the tapering function T ( ). This is independent from the spectral resolution, and from the sub-channel width (this latter is either 62.5 MHz or 31.25 MHz). Thus if the sub-channel width is B , the number of spectral points N , the spectral resolution per point = B /N , the number of spectral points to delete at each end is: 62.5MHz 62.5MHz N = (15) 32 B 32 Let Sk (j ) be the spectral point j of sub-channel k . Let assume that sub-channels are numbered from 0 to (M - 1), spectral points from 0 to (N - 1) and the spectra produced with the modified FFT routine that computes spectral points centered at frequency j = (j + 1/2). Let 0k be the local oscillator frequency for sub-channel k . It is always possible to set 0k to an exact multiple of , and this restriction can be enforced without losing significant flexibility. We assume thus that 0k = n0k . For continuous coverage starting at frequency 0 = n0 , we have n0k = n0 + (k + 1/2)(N - 2Nd ). Then the frequency for spectral point j of sub-channel k is given by Nd =
4 This factor is due to the convention used for the FFT normalization. The FFT routine normalizes the average sp ectral density to the total p ower in the time domain. In the frequency division mo de, the average sp ectral density is calculated on a numb er of sp ectral p oints that is 1/32 or 1/64 of the total numb er of p oints in the whole channel. For example, for uniform sp ectral density, the total p ower in each sub-channel is 1/32 the total p ower over the whole IF bandwidth, and thus the average sp ectral density calculated for each sub-channel is 1/32 the total average. The factor is physically linked to the change in sampling rate, or in delay spacing in the correlator, and is not dep endent on the actual bandwidth observed.


Algorithms for hybrid correlator data correction

9



jk

=

0k

-

B B (j + 1/2) + 2 N 2

(16) (17)

= (n0 + k (N - 2Nd ) + j - Nd ) +

If the resulting spectrum is represented by an array with the same indexing convention, i.e. spectral point l has frequency l = 0 + (l + 1/2), the stitching process is a copy of arrays Sj (k ) to the final h spectrum S8 (l). The suffix 8 here indicates that S8 is computed on a 3 bit, 8 level signal representation, i.e. has not been corrected for 3 bit quantization.
h S8 (l) = Sj (k ) j = Nd . . . (N - Nd - 1), k = 0 . . . (M - 1) l = k (N - 2Nd ) + j - Nd l = 0 . . . (M (N - Nd ) - 1)

(18) (19)

For example, in full bandwidth mode, 32 spectra of 64 points each are computed with an overlap of 4 points. 2 points are dropped from each sub-channel edge, and the spectral points j of spectrum k correspond to points 60k + j - 2 in the final spectrum (k = 0 . . . 31, j = 2 . . . 61). A total of 1920 spectral points are computed. The 4/64 overlap has been chosen considering the phase performance near the band edges of a of a 64-point correlator (see [2]). TFB filter shape has been designed accordingly, so this is the optimum value for the current hardware implementation. It is possible to use different values, with the constrain of a even number of spectral points in the overlap, but this is not currently supported by the firmware correlator. At the end of this process, the composite spectrum corresponds to a portion (likely with holes, at least at the two edges) of the spectrum at the sampler output. It is thus equivalent to a portion of the spectrum observed with a conventional high speed 3-bit correlator on the same digitized signal.

7

Quantization correction for 3 bit sampler

The spectrum S8 ( ) is still affected by quantization losses and distortions due to the 3-bit quantization in the analog 4 GHz sampler. In general for a frequency division scheme it is not possible to completely correct these effects, as the input spectrum is not completely observed. By dropping part of the spectrum, or in narrow-band modes by observing only a small portion of it, we lose information necessary to correct for quantization effects. To reconstruct the original spectrum it would be necessary to have at least a rough estimate of the spectrum, e.g. by a short and coarse observation in time division mode. The relation between the 3-bit, 8 levels correlation function R8 () and the original correlation is much more linear than in the 4-level case, and in most cases the correction could be approximated with a linear function in the spectral domain. The sensitivity of the correction to the signal level is also less severe than for the 4 bit case, so a fixed relation could be used, assuming a nominal signal level at the sampler input. Therefore we have the following possibilities to correct for 3-bit quantization, in order of decreasing complexity (and accuracy): · Perform a short TDM observation, to estimate the spectrum in the unobserved portions of the IF band. The TFB itself can be used as a 32-point filterbank, to obtain a coarse spectrum of the whole IF band. From that, the algorithm described in chapter 7.1 can be used · Estimate the spectrum in the unobserved portions of the IF band using a digital total power measurement to constrain the integrated spectrum · Measure the digital total power at the TFB input ad use a linear correction for the auto and cross spectra, as described in chapter 7.3 · Use the linear correction with the nominal value for the digital total power


Algorithms for hybrid correlator data correction

10

7.1

Direct approach

The usual approach in FFT spectrometers (the hybrid design can be considered equivalent to a N = 32 Fourier instrument) is to derive the correlation function from the uncorrected spectrum, apply the quantization correction in the time (delay) domain, and re-transform the correlation function to the spectral domain. In our case this is not possible, as the spectrum is not known on the whole spectral domain (2 GHz bandwidth). Different methods to estimate the spectrum in the unknown regions are thus necessary. The direct approach is to measure, with a coarse and quick time-division mode observation, the complete IF band. The derived spectrum is then used to fill the unobserved portions of the band. The method works both for ACF and CCF. The coarse spectrum can be observed in time multiplexed mode, with a very short integration time. A 16 ms observation determines the spectrum with an accuracy of a fraction of a percent, much better than the absolute calibration of the individual IF channels. The observation must be performed in the same instrumental configuration, in particular the amplitude of the signal at the sampler input should be the same used for 3 bit quantization, even if 2 bit correlation is performed. Let call this reference spectrum S r , while the spectrum to be corrected is S h . The corresponding r correlation functions are Rr and Rh . The same symbols with a suffix, e.g R8 indicate the corresponding quantities computed with a quantized representation with the indicated number of levels. The coarse spectrum S r ( ) must be