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Subsections

5 Correlators

The three operating bands of the telescope, the 1420-MHz continuum band, the Hi spectroscopy band, and the 408-MHz continuum band, use digital correlators of different designs, which we label the C21, S21, and C74 systems respectively. The three correlators were built at different times; their design differences are partly dictated by that history but are mostly driven by the significantly different requirements. Table 4 lists the specifications of the three correlators.


   
Table 4: Correlator characteristics
Correlator C21 S21 C74
Bandwidth (MHz) 7.5 ($\times 4$) 2n, $-3 \leq n \leq 2$ 4.0*
Sampling rate (MHz) 20.0 2 $f_{\rm N}^\dagger$ 16.0
Correlator efficiency 0.985 0.88 0.88
Number of quantizer levels 14 3 3
* Bandwidth actually used is 3.5 MHz, defined by RF filters
$^\dagger f_{\rm N}=$ Nyquist frequency.

Digital representation of signals introduces quantization noise, some of which can be recovered by increasing the sampling rate (Bowers & Klingler 1974; Klingler 1974). Figure 3 shows correlator efficiency for the two quantization schemes used in this telescope.


  \begin{figure}
\includegraphics[width=8.8cm,clip]{H2136F3.PS}\end{figure} Figure 3: Correlator efficiency. Curves (a) and (b) show the efficiency of a 3-level correlator at Nyquist and double-Nyquist sampling rates respectively, and (c) and (d) show the same for a 14-level correlator. The horizontal axis is labelled with the voltage setting of the first quantizer threshold in units of the rms value of the input signal. In each correlator on the telescope, the input signal level is chosen to optimize correlator efficiency

Sections 5.1, 5.2 and 5.3 describe the three correlators in turn. Section 5.4 describes routines developed to correct the inherent correlator non-linearity. These corrections are currently applied to C21 and S21 data only.

5.1 The 1420-MHz continuum correlator

The C21 correlator (Karpa 1989) uses fourteen-level representation of the input signals; with moderate over-sampling, it achieves an unusually high correlator efficiency ( $\eta_{\rm c}=0.985$; see Fig. 3). The correlator was designed to provide products from all antenna pairs, and to form all four polarization products from the LHCP and RHCP inputs. However, the total continuum bandwidth is too wide to permit mapping of the entire field of view of the telescope without bandwidth smearing (decorrelation arising from differential delay of off-axis, wide-bandwidth signals; see Bridle & Schwab 1989). The 30 MHz band is therefore divided into four 7.5 MHz sub-bands as illustrated in Fig. 2. Each sub-band input is digitized to 14 levels and the products are formed in one of four identical correlators.

Input signals are digitized using four-bit monolithic flash converters (Analog Devices AD9688) with the sampling clock running at 20 MHz, slightly faster than the Nyquist rate for a 7.5-MHz bandwidth. Of the 16 possible A/D output levels, only 14 are used. The two extreme levels (outputs 0000 and 1111) are reserved for use as control signals embedded in the data stream.

Each correlator module uses a ROM look-up table where an 8-bit product is stored at the address specified by the two 4-bit inputs. The product is accumulated for 5.6 s. The occurrence of the code 0000 at either input of the multiplier causes accumulation to stop. The accumulator (a combination of synchronous and ripple counters) is allowed to settle, and is then read by the system microprocessor.

The occurrence of code 1111 on one of the multiplier inputs causes the multiplier to produce an 8-bit output corresponding to the square of the 4-bit number appearing on the other input. Extra correlator cards are used in this way as auto-correlators, and they generate outputs corresponding to the power in each input signal. Small noise signals are injected into each antenna (see Sect. 4.1), and the autocorrelation outputs are used to measure both system gain and noise in each channel. Additional correlator channels are used to compute the mean of each input signal (by multiplying the input signal by a fixed number), a measure of the error in each A/D converter. Similarly, "self-correlations'' are formed between in-phase and quadrature signals from a given antenna to measure any deviation from orthogonality (which is corrected for in subsequent processing).

The flow of input signals from the A/D converters to the correlators, and the flow of output data from the correlators, is handled by the Control/Interface (CI) modules. These modules include a $4096 \times
4$-bit RAM, located at the boundary between data acquisition and data processing, which serves two purposes. The RAM can store A/D converter outputs for later analysis (used in testing A/D converters). Alternatively, it can be loaded from the microprocessor with data which can be passed to the correlators in place of the normal input signals. In this way all correlator channels can be subjected to a test which uses realistic input signals while giving exactly known outputs. The entire correlator system undergoes self-test in this fashion at the start of every observation.

The correlator modules are arranged in a matrix. Signals move one step along the rows and diagonals of the matrix at each clock cycle and are re-timed at each correlation site. Since control signals are embedded in the data stream, timing is very simple, and the correlation matrix is (in principle) extendible to handle any number of antennas.

5.2 The Hi correlation spectrometer

The S21 correlator forms only those baseline products needed for complete sampling of the u-v plane, the 12 combinations of fixed and movable antennas (Fig. 1). For maximum sensitivity, two identical correlators form products from LHCP and RHCP signals. Cross-correlation between signals from the two hands of polarization is not currently available. This correlator design, in auto-correlator mode, is also used on the DRAO 26-m Telescope for observation of large-scale Hi structure for combination with Synthesis Telescope images.

The S21 correlator uses many modules from the C21 system, including the A/D converter, the CI modules, the backplanes, and the microprocessors. Many functions from the C21 system are reproduced, including the autocorrelation circuits and the circuits used to compute means of input signals (see Sect. 5.1). The significant difference between the systems is the correlation module itself.

Data streams, which leave the A/D converters encoded to 14 levels, are used with 14-level precision for auto-correlation and other functions, but are converted to 3-level coding for processing by the correlation modules. The 14-level auto-correlators provide accurate measurement of input power levels, and this information is used to linearize the cross-correlation functions formed from 3-level data (see Sect. 5.4).

The correlator was designed around a CMOS application-specific integrated circuit (ASIC) developed for this project (Hovey 1998). The ASIC computes a four-point correlation function from input signals quantized to 3 levels, and 128 ASICs implement a 512-lag correlator (256 complex channels) for each baseline. The ASIC architecture is flexible, allowing both cross- and auto-correlation of data sampled at the Nyquist rate, $f_{\rm N}$, or at $2f_{\rm N}$. Sampling at $2f_{\rm N}$ is used because it increases correlator efficiency from 0.81 to 0.88.


  \begin{figure}
\includegraphics[width=8.8cm,clip]{H2136F4.PS}\end{figure} Figure 4: Functional diagram of the ASIC developed for the spectral-line correlator (the S21 system)

Figure 4 shows the organization of the ASIC. The x and y inputs are 3-level quantities, taking values (1, 0, -1). The shift registers are arranged so that x and y streams flow in opposite directions. Correlation function values are formed by connecting a multiplier-accumulator across each shift-register output. The multiplier output is biased, so that only positive products (0, 1, 2), are formed, allowing accumulation in a simple up-counter. Accumulation begins with a 2-bit adder, whose output synchronously clocks a 23-bit ripple counter. Adder outputs are clocked into this counter at a fixed rate so that the mean value of the accumulator is constant even though the sample rate is adjusted when the overall bandwidth is changed. The system microprocessor detects overflows of the most significant accumulator bit, and extends the accumulation interval to 5.6 s.

The cross-correlation function is under-sampled in the ASIC because the delay interval is $2\tau$, where $\tau$ is the period of the sampling clock. By adding a second ASIC with one input delayed by $\tau$, the correlation function can be fully sampled. Alternatively, if the input signals are sampled at $2f_{\rm N}$, a fully sampled correlation function is obtained (this is the normal mode of operation). For auto-correlation, a correlation function with one point at zero lag can be created by delaying one input by one half-lag.

5.3 408-MHz continuum correlator

The C74 correlator was designed to carry out interferometer signal-processing operations in simple software procedures to the maximum extent feasible (Lo et al. 1984). Figure 5 is a diagram of the system, showing the distribution of signal-processing functions between hardware and software.


  \begin{figure}
\includegraphics[width=8.8cm,clip]{H2136F5.PS}\end{figure} Figure 5: Functional diagram of the correlator used with the 408 MHz continuum channel of the telescope (the C74 system). The division of functions between hardware and software is indicated

Incoming signals are sampled at 16 MHz and coarse delay is inserted in steps of one cycle of this clock (62.5 $\mu$s). A 16-point correlation function is generated. The in-phase visibility is retrieved from this function by interpolating to the exact geometrical delay. The quadrature visibility, which contains independent information, is also derived from this function. It is calculated by convolving the in-phase correlation function with the kernel function of a band-limited Hilbert transform. Since the in-phase and quadrature visibility outputs are required at only a single value of delay, the convolution degenerates to a simple dot product, requiring multiplication of each value of the correlation function by a matching coefficient from the kernel function. This is a simple and quick operation in the microprocessor (see Lo et al. 1984 for details). A small error (of the order of 5%) arises in the derivation of the quadrature visibility because of truncation of the kernel. The error is mostly corrected by a multiplication factor. The residual error is much smaller than the decorrelation arising from bandwidth smearing.

Delay equalization is a continuous process, since the required path compensation is continuously changing as the Earth rotates. Precision of the compensation is high, limited only by the numerical precision of the interpolation arithmetic and the rate at which interpolation is performed. This confers an advantage over systems where delay is inserted in discrete steps. Such steps affect the phase of the correlator output, and must be corrected for, either before or after correlation.

5.4 Correlator linearity

Quantization of the baseband signals introduces non-linearities in the correlator output, which can lead to substantial artefacts in images (both Hi-line and continuum) when bright Galactic objects lie in the field. Because of its sensitivity to extended structure, this telescope is more prone to this problem than other synthesis telescopes which resolve out most of the broad emission.

There are two approaches to the derivation of accurate visibilities using a digital correlator. In one approach, the signal level at the input to the A/D converter is held constant at the optimum level (see Fig. 3). Under these circumstances, a 3-level correlator yields a value of correlation coefficient, $\rho$, linear within 1% up to values of $\rho=0.38$. However, to recover accurate visibility amplitudes requires accurate tracking of variations of system noise temperature. The alternative, which we have adopted, is to stabilize receiver gain through the system but to allow the signal level at the A/D converter input to vary (caused, for example, by variations of atmospheric emission and received ground radiation as a function of antenna elevation angle). This approach requires the development of algorithms to linearize correlator output. The algorithms convert the output of the correlators (separate algorithms have been developed for the 3-level and 14-level correlators) into an accurate measure of cross-power in the face of changes of system temperature of as much as 30%. The algorithms must be applicable to both auto- and cross-correlators, and must therefore be able to handle correlation coefficients as high as $\rho=1.0$. However, $\rho$ can also be high in the cross-correlator: when observing Hi emission with the 12.86-m baseline, $\rho$ can rise to values as high as 0.7.

  \begin{figure}
\includegraphics[width=17cm,clip]{H2136F6.PS}\end{figure} Figure 6: The telescope control software. C21, S21 and C74 are the three correlators described in Sect. 5. F21 denotes the task and microprocessor which control the 1420 MHz receiver (frequency, phase, and delay). SID CLK is the sidereal clock. At the completion of an observation all files indicated are transferred via an ethernet connection to another computer where further processing occurs

The algorithms proceed by first linearizing the autocorrelator response to derive an accurate measure of the variance of the input signal. Using this measurement, the cross-correlator response is linearized to obtain an accurate value of cross-power. The equations describing the correlator output in terms of its inputs cannot be directly inverted. Instead we use series approximations, following the ideas of Kulkarni & Heiles (1980) and D'Addario et al. (1984). However, the correlator becomes increasingly non-linear as the input signal increases, spending a greater fraction of time above the highest quantizer decision level. The coefficients of the series inversion formula therefore must vary with input signal level. In our algorithm, these coefficients are themselves estimated by evaluating a series. Details of the linearization algorithms can be found in Hovey (1998). Their application reduces errors of as much as 30%, which occur in common observing conditions with this telescope, to less than 0.1%.


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