4.3.3 Main limitations

In real life, cross-correlators are subject to the performance of the whole receiving system. This comprises the ``analog part'' (the signal path from the receivers to the IF filters at the correlator entry), and the ``digital part'' (everything behind the sampler). Although the analog part is out of the correlator, its performance requires to change our assumptions concerning the input data. This complicates the analysis of the correlator response. The following discussion refers to instantaneous errors only. However, in interferometric mapping, scan-averaged visibilities are used, and the data may be less affected.

4.3.3.1 Analog part

The shape of the bandpass function (amplitude and phase) at the correlator output is mainly due to the correlator's response to the filters inserted in the IF band at the correlator entry. So far, for the sake of simplicity, rectangular passbands, centered at the intermediate frequency $\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}$, have been assumed. A more complex (and more realistic) case may be an amplitude slope where the logarithm of the amplitude varies linearly with frequency. Although the bandpass function will be calibrated (see Eq.4.17, and R.Lucas chapter 5), the effect of such a slope on sensitivity remains. A derivation of the signal-to-noise ratio for that case is beyond the scope of this lecture. To give an impression of the order of magnitude: a slope of 3.5dB (edge-to-edge) leads to a 2.5% degradation of the sensitivity calculated for a rectangular passband. A center frequency displacement of 5% of the bandwidth leads to the same degradation.

As already demonstrated, delay-setting errors linearly increase with the intermediate frequency (Eq.4.6). Table4.3 gives an impression of the decrease of sensitivity due to a delay error. For a range of delay errors $\Delta\tau$, the effect is also shown in Fig.4.3. For example, a delay error of $0.12/\Delta\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}$ accounts for a 2.5% degradation. Delay errors are mainly due to inaccurately known antenna positions (asking for better baseline calibration), or due to errors in the transmission cables.

 

Table 4.3: Effects of delay pattern on the sensitivity

Intermediate frequency bandwidth	$\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}} = 160$MHz
Baseline	b = 100m
Zenith distance of source in direction $\ensuremath{\mathbf s}$	$\Theta = 30^\circ$
Results in geometric delay:	$\ensuremath{\tau_\mathrm{\scriptscriptstyle \rm G}} = \ensuremath{\mathbf b}\cdot \ensuremath{\mathbf s} /c = 0.17\,\mu$s
Attenuation according to Eq.4.4	$1\,\%$

 

Phase errors across the bandpass may also be of random nature. A phase fluctuation of $12.8^\circ$ (rms) per scan leads to a degradation of $(1-\exp{(-\sigma_\Phi^2/2)}) \times 100\,\% = 2.5\,\%$.

  

Figure 4.7: The Gibbs phenomenon. The convolution of the bandpass with the (unapodized) spectral window (sinc function) is shown for the real and imaginary parts. Note that for the real part, the phenomenon is strongest at the band edges, whereas for the imaginary part, it contaminates the whole bandpass.

Fluctuations across the bandpass also appear as ripples. They may have several reasons, and are mainly due to the Gibbs phenomenon, and due to reflections in the transmission cables. A sinusoidal bandpass ripple of 2.9dB (peak-to-peak) yields a 2.5% degradation in the signal-to-noise ratio. The Gibbs phenomenon also occurs in single-dish autocorrelation spectrometers. For the sake of illustration, let us again assume a perfectly flat response of receivers and filters. However, the filter response function is only be flat across the IF passband. Towards its boundary, steep edges occur. We already learned that strong spectral lines may show ripples, if no special data windowing in time domain is applied. The Gibbs phenomen is due to a similar problem (but now the spectral line is replaced by the edge of a flat rectangular band extending in frequency from zero to $\Delta\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}$). The output of the cosine correlator is symmetric, but the sine output (imaginary part) is antisymmetric, thus including an even steeper edge. Convolving this edge with the $\mbox{sinc}$ function (i.e. the spectral window) results in strong oscillations. Let us call this function $f(\nu)$. For calibration purposes, the Gibbs phenomenon has to be avoided: the problem is that calibration uses the system response to a flat-spectrum continuum source. A source whose visibility is $V(\nu)$ is seen as $f(\nu) \ast (G_{\rm ij}(\nu)V(\nu))$ (where $G_{\rm ij}$ is now a frequency-dependent complex gain function). After calibration it becomes

$$\hat{V}(\nu) = \frac{f(\nu)\ast [G_{\rm ij}(\nu)V(\nu)]}{f(\nu) \ast G_{\rm ij}(\nu)}$$ (4.16)

You immediately see that the complex gain $G_{\rm ij}(\nu)$ does not cancel out, as desired, and $\hat{V}(\nu) \ne V(\nu)$. Automatic calibration procedures have to flag the channels concerned. As shown in Fig.4.7, for the cosine correlator, the effect is strongest at the band edges, but the output of the sine correlator also shows ripples in the middle of the band (thus, the problem is of greater importance for interferometers than for single-dish telescopes using auto-correlators). If the bandwidth to be observed is synthesized by two adjacent frequency windows, the phenomenon is strongest at the band center anyhow. You should avoid to place your line there, if it is on top of an important continuum. To what extent the Plateau de Bure system is concerned, will be discussed in Section 4.4.1.

The above summary of the system-dependent performance of a correlator is not exhaustive. For example, the phase stability of tunable filters, which depends on their physical temperature, is not discussed. Alternatives to such filters are image rejection mixers (as applied in the Plateau de Bure correlator, see last section).

4.3.3.2 Digital part

Errors induced by the digital part are generally negligible with respect to the analog part. In digital delays, a basic limitation is given by the discrete nature of the delay compensation, which cannot be more accurate than given by the clock period of the sampler. However, digital techniques allow for high clock rates, keeping this error at a minimum.

Evidently, a basic limitation is given by the memory of the counters, setting the maximum time lag (which in turn defines the spectral resolution, as already discussed): with 2K bits, we can exactly represent N^2Knumbers. However, the information contained in the bits is not equivalent. For the 3-level correlator, the output of each channel i=1,...,N is

$$R(i) = \frac{1}{2}\left(N \pm \sqrt{N}\sqrt{1-{\rm erf} {\left(v_0/\sqrt{2}\right)}}\right)$$ (4.17)

(assuming white, Gaussian noise of zero mean and of unit variance, and neglecting the weak contribution of the astrophysical signal). The $1\sigma$-precision of the output is $\approx \sqrt{N}/2$, contained in the last K-1 bits, which thus do not need to be transmitted. The maximum integration time before overflow occurs is set by the number of bits of the counter, and the clock frequency. Table4.4 shows an example.

 

Table 4.4: Maximum integration time of a 16-bit counter

clock frequency:	80MHz
weight for intermediate-level products:	n = 3
positive offset:	n2 = 9
weight for autocorrelation product:	18 (using offset multiplication table)
carry out rate of a 4-bit adder	18/24 = 1.125
maximum integration time:	$2^{16}/(80\,\mbox{MHz} \times 1.125) = 0.73$ms
same with a 4-bit prescaler:	$2^{16}\times 2^4/(80\,\mbox{MHz} \times 1.125) = 11.7$ms

 

The only error cause due to the correlator that is worth to be mentioned is the sampler, i.e. the analog-to-digital conversion. As already shown, the threshold levels are adjusted with respect to the noise in the unquantized signal. However, the noise power may change during the integration. In that case, the correlator does not operate anymore at its optimum level (see Fig.4.4). This error cause can be eliminated with an automatic level control circuit. However, slight deviations from the optimal level adjustment may remain. Without going too far into detail, the deviations can be decomposed in an even and an odd part: in one case, the positive and negative threshold voltages move into opposed directions (even part of the threshold error). The resulting error can be equivalently interpreted as a change of the signal level with respect to the threshold v₀, and leads to a gain error. In the other case, the positive and negative threshold voltages move into the same direction (odd part of the threshold error). This error, however, can be reduced by periodic sign reversal of the digitized samples (if the local oscillator phase is simultaneously shifted by $\pi$, the correlator output remains unaffected). Combining the original and phase-shifted outputs, the error cancels out with high precision. Such a phase shift is implemented in the first local oscillators of the Plateau de Bure system (for details see the lecture by R.Lucas). Note also that threshold errors of up to 10% can be tolerated without degrading the correlator sensitivity too much: the examination of Fig.4.4 shows that such an error results in a signal-to-noise degradation of less than 0.2% for a 3-level system, and of less than 0.5% for a 4-level system (the maxima of the efficiency curves are rather broad).


Another problem is that the nominal and actual threshold values may differ. The error can be described by ``indecision regions''. By calculating the probability that one or both signals of the cross-correlation product fall into such an indecision region, the error can be estimated. With an indecision region of 10% of the nominal threshold value, the error is negligibly small.


Finally, it should be noted that strict synchronisation between the cosine and the sine correlators is needed. Any deviation will introduce a phase error.