6.2 Basic Theory

The ``heart'' of a correlator consists of the sampler and the cross-correlator. Eq.6.2 represents an over-simplified case, because the bandwidth of the signals is neglected. The correlator output is rather modified by the Fourier transform of the bandpass function. For the sake of simplicity, let us assume an idealized rectangular passband of width $\Delta\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}$ for both antennas, centered at the intermediate frequency $\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}$, i.e.

$$\vert H_{\rm i}(\ensuremath{\nu_\mathrm{\scriptscriptstyle }})\ve... ...rt > \Delta\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}/2 \end{array}\right.$$

(this assumption will be relaxed later). The correlator response to this bandpass is the Fourier transform of the cross power spectrum $H_{\rm i}(\nu)H_{\rm j}(\nu)^*$, which is shown in Fig.6.3:

$$\int_0^\infty{H_{\rm i}(\ensuremath{\nu_\mathrm{\scriptscriptstyl... ...tyle IF}}\tau} \exp{(i2\pi\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}\tau)}$$ (6.3)

The correlator output consists of an oscillating part, and a $\sin{(x)}/x$ envelope (a sinc function). If the delay $\tau$ becomes too large, the sensitivity will be significantly decreased due to the sinc function (see Fig.6.3). Strictly speaking, this is the response to the real part of the bandpass, which is symmetric with respect to negative frequencies. The imaginary part of the bandpass is antisymmetric with respect to negative frequencies, thus the correlator response is different. The separation of real and imaginary parts in continuum and spectroscopic correlators will be discussed below.

This example shows that accurate delay tracking (fringe stopping) is needed, if the bandwidth is not anymore negligible with respect to the intermediate frequency. In other words, the compensating delay $\tau_{\rm I}$ needs to keep the delay tracking error $\Delta \tau = \ensuremath{\tau_\mathrm{\scriptscriptstyle \rm G}}-\tau_{\rm I}$ at a minimum. The offset $k\Delta t$ introduced in correlator channel $k$ needs to be applied with respect to a fixed delay. In the following, the correlator response to a rectangular bandpass will be expressed by the more general instrumental gain function $G_{\rm ij}(\tau)$, defined by

$$A_0\int_0^\infty{H_{\rm i}(\nu)H_{\rm j}^*(\nu)\exp{(2\pi i\nu\ta... ...{\rm ij}(\tau)\exp{(2\pi i\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}\tau)}$$ (6.4)

$G_{\rm ij}(\tau) = \vert G_{\rm ij}(\tau)\vert\exp{(i\Phi_{\rm G})}$ is a complex quantity, including phase shifts due to the analog part of the receiving system (amplificators, filters)^6.2. After fringe stopping, the single-sideband response of correlator channel $k$ becomes (for details, see R.Lucas, Chapter 7)

$$R_{\rm ij}({k\Delta t}) = \vert V\vert\vert G_{\rm ij}\vert{\rm Re}\left\{\exp{(\pm 2\pi i\... ...tyle SKY}}\pm i\ensuremath{\varphi_\mathrm{\scriptscriptstyle \rm G}})}\right\}$$

$$= \vert V\vert\vert G_{\rm ij}\vert\cos{(\pm 2\pi\ensuremath{\nu_\m... ...ptscriptstyle SKY}}\pm \ensuremath{\varphi_\mathrm{\scriptscriptstyle \rm G}})}$$ (6.5)

where the plus sign refers to upper sideband reception, and the minus sign refers to lower sideband reception. From Eq.6.6, we immediately see that the residual delay error (due to a non-perfect delay tracking) enters as a constant phase slope across the bandpass (with opposed signs in the upper and lower sidebands). The effect of such a phase slope on sensitivity will be discussed later. In order to determine the phase of the signal, the imaginary part of $R_{\rm ij}({\tau})$ has to be simultaneously measured. In a continuum correlator (Fig.6.1), a $\pi/2$ phase shift applied to the analog signal yields the imaginary part. The signals are then separately processed by a cosine and a sine correlator ^6.3. In other words: the pattern shown in Fig.6.3 is measured in the close vicinity of two points, namely at the origin, and at a quarter wave later, i.e. at $\tau = 1/(4\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}})$. Note, however, that due to the sinc-envelope, the decreasing response function cannot be neglected if the bandwidth is comparable to the intermediate frequency.

In a spectroscopic correlator (Fig.6.2), the imaginary part can be entirely deduced from the digitized signal: if $N_{\rm ch}$ is the number of complex spectral channels, $2N_{\rm ch}$ time lags are used, covering delays from $-N_{\rm ch}\Delta t$ to $(N_{\rm ch}-1)\Delta t$. The correlator output is a real signal with even and odd components (with respect to time lags of opposed signs). The $N$ complex channels of the Fourier transform at positive frequencies yields the cross-power spectrum:

$$r_{\rm ij,k}({\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}}) = r_{\rm ij}({k\delta\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}... ...{\rm ij}({t}) \exp{(2\pi i\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}t)}dt}$$ (6.6)

$$= \sum_{l=-N_{\rm ch}}^{N_{\rm ch}-1} \langle {v_{\rm i}(t)v_{\rm j}(t+\tau+l\Delta t)}\rangle \exp{(2\pi i lk/2N_{\rm ch})}$$ (6.7)

(for channel $k$ of a total of $N_{\rm ch}$ complex channels, spaced by $\delta\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}$). The last expression represents the discrete Fourier transform. According to the symmetry properties of Fourier transforms, the even component of the correlator output becomes the real part of the complex spectrum, and the odd component becomes the imaginary part. The Fourier transform is efficiently evaluated using the Fast-Fourier algorithm. In practice, it is rather the digital measurement of the cross-correlation function that is non-trivial. It will be discussed in detail in Section 6.3.3.

Figure: Left: Correlator output (single-sideband reception) for a rectangular passband with $\Delta\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}/\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}} = 0.2$. Due to the signal phase $\ensuremath{\varphi_\mathrm{\scriptscriptstyle SKY}}$, the oscillations move through the sinc envelope by $\ensuremath{\varphi_\mathrm{\scriptscriptstyle SKY}}/2\pi\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}$. The shift may also be due to the phase of the complex gain (in this case, the shift would be in opposed sense for USB and LSB reception). Right: Sensitivity degradation due to a delay error $\Delta \tau$ (with respect to the inverse IF bandwidth). The effect is due to the fall-off of the sinc envelope.

The ensemble of cross-power spectra $r_{ij}({\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}}})$, after tracking the source for some time, becomes (after calibration and several imaging processes) a channel map.