NOISE IN THE CROSS-POWER SPECTRUM OF THE VELA PULSAR

Because the noise in the spectrum is drawn from a nearly Gaussian distribution with zero mean, variances and covariances characterize it. In Section 2, we introduce the theoretical basis for noise in interferometric visibility and present the mathematical descriptions used in the paper. In Section 3, we discuss our observations, correlation, and initial data processing. The remainder of the paper analyzes the noise in these observations. In Section 4, we quantify the distribution of noise and the influence of a signal. In Section 5, we discuss quantization and correlator effects. In Section 6, we summarize our results and discuss implications for future observations and instruments.

Observations of electromagnetic radiation from astronomical sources measure and compare finite samples of electromagnetic fields. We suppose that these are drawn from ensembles of statistically identical measurements. Noise is the difference between the result of a measurement and the average of an infinite ensemble of such measurements. The measurements, as well as their statistical averages, can be expressed either as spectra, varying with frequency, or as correlation functions, varying with lag in the Fourier-conjugate domain. In practice, the samples are digitized, quantized, and time sampled. We introduce conventions for notation that help distinguish among these various domains. The tilde denotes entities in the spectral domain, indexed by the spectral channel as a subscript, for example, $\tilde{r}_k$. Unaccented symbols designate the Fourier-conjugate domain of the correlation function, indexed by lag: r_τ. Angular brackets 〈...〉_n denote a statistical average over many realizations of the noiselike electric field, with the scintillation spectrum held fixed. The subscripted brackets 〈...〉_S denote an average over many samples of the scintillation spectrum. These conventions are consistent with earlier work (Gwinn 2004; Gwinn & Johnson 2011).

As in earlier studies, we suppose that two antennas record time series of zero-mean, complex Gaussian random variables (Gwinn 2004, 2006; Gwinn & Johnson 2011; Gwinn et al. 2011). These time series can be regarded as the amplitude and phase of one polarization of electric field. The antennas are X and Y, and the time series x_ℓ and y_ℓ, where ℓ indices time. The ensemble-averaged cross-power spectrum $\tilde{\rho }_k$ and the ensemble-averaged autocorrelation spectrum $\tilde{\alpha }_k$ fully describe the spectral properties of these series. At shorter wavelengths quantum-mechanical effects introduce shot noise, but the ensemble-averaged spectrum remains the same (Zmuidzinas 2003). For convenience, we assume unit variances for the real and imaginary parts.

An observer forms the cross-correlation function, r_τ, of the time series x_ℓ and y_ℓ. The Fourier transform of r_τ gives the observed cross-power spectrum, $\tilde{r}_k$. This observed spectrum is an estimate of $\tilde{\rho }_k$, multiplied by a correlator-dependent gain factor (see Section 5). Similarly, autocorrelation of data at a single antenna yields an autocorrelation spectrum $\tilde{a}_k$, an estimate of $\tilde{\alpha }_k$. The autocorrelation spectrum may include an offset, from the correlation of spectrally flat background noise, as well as a gain factor. In practice, individual measurements $\tilde{r}_k$ differ from the "true" spectrum $\tilde{\rho }_k$ by a random amount: the noise. If many samples of $\tilde{r}_k$ are averaged together, the noise for the average approaches a Gaussian distribution.

At lag τ, the cross-correlation function r_τ is

$$\begin{equation} r_{\tau } = {\frac{1}{N_{\rm obs}-|\tau |}}\sum _{\ell =1}^{N_{\rm obs}-|\tau |} x_{\ell } y_{\ell +\tau }^*. \end{equation} \tag{ 1 }$$

This equation parallels Equation (6) of Gwinn (2004); in particular, N_obs is the number of samples. However, we have reduced the definition of r_τ by a factor of two, by changing the normalization of x_ℓ and y_ℓ so that 〈|x_ℓ|²〉 = 〈|y_ℓ|²〉 = 1. Note that the sum here runs from 1 only to N_obs − |τ|, so that large lags are averaged over fewer samples. If the lag τ < 0, then the sum runs from |τ| + 1 to N_obs. The finite spans of the time series prevent the correlation of all samples: for τ > 0, no y_{ℓ + τ} exists for the last τ samples of x_ℓ. For τ < 0, no x_ℓ exists for the first |τ| samples of y_{ℓ + τ}. A pulsar gate, for example, can truncate the time series in this way. Equation (1) describes the calculation of r_τ in many correlators of the "XF" design, where correlation "X" precedes Fourier transform "F." The S2 correlator at Penticton, used for the work described here, is of this design. In contrast, our previous theoretical work assumed averaging of all lags τ over N_obs elements, because of the identification $x_{\ell - N_{\rm obs}} \equiv x_{\ell }$ for any ℓ (Gwinn 2006; Gwinn & Johnson 2011). We call this the "wrap" assumption. Most correlators of the alternative "FX" design, where Fourier transform precedes correlation, obey the "wrap" assumption. However, by zero padding the time series they can emulate the sum in Equation (1). We discuss the consequences of these two formulations for the cross-power spectrum in Section 5.2 below.

The statistical average of the cross-correlation function r_τ is 〈r_τ〉_n = ρ_τ. This average runs over many realizations of the electric field, with uniform statistics. Our observations are not stationary and do not approximate an ensemble average because both the scintillation spectrum and the intrinsic flux density of the pulsar change with time. However, we can regard each observation as being drawn from an ensemble of observations of statistically identical pulses at the same pulse phase and in the same scintillation state (see Gwinn et al. 2011).

The observed cross-power spectrum is the Fourier transform of the cross-correlation function:

$$\begin{equation} \tilde{r}_k=\sum _{\tau =-N}^{N-1} e^{i {\frac{2\pi }{2 N}} k \tau } r_\tau, \end{equation} \tag{ 2 }$$

where the number of spectral channels is 2N. The statistical average of $\tilde{r}_k$ is $\langle \tilde{r}_k \rangle _{\rm n} = \tilde{\rho }_k$. For autocorrelation, the average over x_ℓx*_{ℓ + τ} analogous to Equation (1) forms the autocorrelation function a_τ. A statistical average of a_τ yields 〈a_τ〉_n = α_τ. The Fourier transform analogous to Equation (2) yields the autocorrelation spectrum $\tilde{a}_k$, and an average over a statistical ensemble yields $\langle \tilde{a}_k \rangle _{\rm n} = \tilde{\alpha }_k$.

Noise in the cross-power spectrum is the difference of an observation and the ensemble average: $\tilde{r}_k - \langle \tilde{r}_k \rangle _{\rm n}$. Typically, the observed spectrum is an average over a number of individually formed spectra, as described by Equations (1) and (2). Consequently, the Central Limit theorem suggests that the noise follows a Gaussian distribution. Because the cross-correlation function is complex, this distribution is an elliptical Gaussian distribution in the complex plane.

When the signal is completely absent, as when the pulsar is off, one expects that $\tilde{\rho }_k=0$ and that $\tilde{r}_k$ consists of noise drawn from a zero-mean, circular complex Gaussian distribution. Our observations match this expectation closely, as we discuss in Section 5.1.

If the signal is present, then the Dicke equation describes the contribution of self-noise. This equation states that the error δT in measurements of antenna temperature varies with total system temperature T, including the contribution of the source (Dicke 1946):

$$\begin{equation} (\delta T)^2 = {\frac{T^2}{ N_{\rm obs} }}. \end{equation} \tag{ 3 }$$

Here, N_obs = Δν × Δt is the number of samples, for an observed bandwidth Δν and integration time Δt. The analogous expression holds for interferometric visibility (Thompson et al. 1986). More generally, this equation describes the noise in the sample variance for draws from a Gaussian distribution.

The real and imaginary parts of statistical averages of $\tilde{r}_k$ suffice to estimate both the signal and the noise (Gwinn 2006, Equations (11), (18), and (19)):

$$\begin{eqnarray} \langle \tilde{r}_k\rangle _{\rm n} &=& \tilde{\rho }_k \nonumber\\ \delta \tilde{r}_k \, \delta \tilde{r}_k^* \equiv \langle \tilde{r}_k \tilde{r}_k^* \rangle _{\rm n} - \langle \tilde{r}_k \rangle _{\rm n}\langle \tilde{r}_k^* \rangle _{\rm n} &=& {\frac{1}{N_{\rm obs}}} \tilde{\alpha }_{Xk} \tilde{\alpha }_{Yk} \nonumber \\ \delta \tilde{r}_k \, \delta \tilde{r}_k \equiv \langle \tilde{r}_k \tilde{r}_k\rangle _{\rm n} - \langle \tilde{r}_k \rangle _{\rm n}\langle \tilde{r}_k\rangle _{\rm n} &=& {\frac{1}{N_{\rm obs}}} \tilde{\rho }_k \tilde{\rho }_k. \end{eqnarray} \tag{ 4 }$$

Again, N_obs is the number of samples gathered, per spectral channel. The subscripted angular brackets 〈...〉_n indicate an average over many realizations of noise. As the first expression indicates, $\tilde{\rho }_k$ is the cross-power spectrum, averaged over an ensemble of statistically identical realizations of noise. Analogously, $\tilde{\alpha }_{{X}k}$ is the autocorrelation function at station X, and $\tilde{\alpha }_{Yk}$ is the autocorrelation function at Y. We assume single-sideband operation. Here we also apply the "wrap" assumption, discussed in Sections 2.1 and 5.2.3. The autocorrelation spectra are always real and are often identical between stations after calibration, with offsets for background noise. For a scintillating source observed on a long baseline, the spectra need not be identical, because the stations may lie in different scintillation elements in the observer plane.

To help visualize the distribution of noise in Equation (4), we divide the noise into components parallel with and perpendicular to the phase of the average visibility. The expression for noise then takes the form

$$\begin{eqnarray} \tilde{\rho }&= |\tilde{\rho }| e^{i\phi } \nonumber\\ \sigma _{||}^2 &= {\frac{1}{2 N_{\rm obs}}}(|\tilde{\alpha }_{X}| |\tilde{\alpha }_{Y}| + |\tilde{\rho }|^2)\nonumber \\ \sigma _{\perp }^2 &= {\frac{1}{2 N_{\rm obs}}}(|\tilde{\alpha }_{X}| |\tilde{\alpha }_{Y}| - |\tilde{\rho }|^2). \end{eqnarray} \tag{ 5 }$$

Here, ϕ is the phase of the average visibility. We have omitted the subscript k for clarity. For most interferometric observations, the intensity of the source is constant over the observer plane: $\tilde{\alpha }_X = \tilde{\alpha }_Y$. The difference between σ_|| and σ_⊥ then produces an elliptical distribution of noise, with major axis aligned with the average phase ϕ. Usually, background noise contributes a constant offset to α, and the source contributes the rest. This form is quite general: it holds for any interferometric observations of a noiselike source, not just observations of a scintillating pulsar.

2.3.1. Distribution of Noise

A strong, scintillating source provides a good laboratory for the study of self-noise because it provides many independent observations of visibility, with different flux densities and (if the baseline is long) phases, under identical conditions. For a scintillating, pointlike source, with flux densities I_X and I_Y at stations X and Y, and background noise equivalent to flux densities n_X and n_Y, the signal and noise are (Gwinn et al. 2011, Equations (5)–(7))

$$\begin{eqnarray} \tilde{\rho }&= \sqrt{I_X I_Y } e^{i \phi _{\rm s}} \nonumber\\ \sigma _{||}^2 &= {\frac{1}{N_{\rm obs}}} \left\lbrace {\frac{n_X n_Y}{2}} + { {\frac{1}{2}}} (n_Y I_X + n_X I_Y) + I_X I_Y\right\rbrace \nonumber \\ \sigma _{\perp }^2 &= {\frac{1}{N_{\rm obs}}} \left\lbrace {\frac{n_X n_Y}{2}} + { {\frac{1}{2}}} (n_Y I_X + n_X I_Y)\right\rbrace. \end{eqnarray} \tag{ 6 }$$

The visibility phase, ϕ_s, arises from phase differences between the pair of scintillation elements in the observer plane. The variance of measurements of $\tilde{r}$ at that phase is σ²_||; the variance at quadrature is σ²_⊥. The subscript for channel k is omitted; all quantities are for one spectral channel. Note that in this expression, the autocorrelation spectra at the two antennas $\tilde{\alpha }_{Xk}, \tilde{\alpha }_{Yk}$ differ both because the intensity of the source at the two antennas may differ (I_X ≠ I_Y) and because of different noise at the two antennas (n_X ≠ n_Y).

2.3.2. Short-baseline Limit

For a short baseline, both antennas will lie within the same scintillation element. In this case, ϕ_s → 0, and I_A = I_B ≡ I. However, the variance of the noise at the two antennas may still be different. We can then express the distribution of noise in the form (Gwinn et al. 2011, Equation (8))

$$\begin{eqnarray} \sigma _{||}^2 &= b_0 + b_1 I + b_2 I^2 \nonumber\\ \sigma _{\perp }^2 &= b_0 + b_1 I \nonumber \\ I &\equiv |\tilde{\rho }|. \end{eqnarray} \tag{ 7 }$$

In this expression, b₂ = 1/N_obs. This expression holds even if the source is mildly resolved by the scattering disk, because the normalized, ensemble-averaged visibility remains nearly 1 (Gwinn 2001).

2.3.3. Long-baseline Effects

2.3.4. Effects of Variability, Quantization, and Correlation

The flux densities of pulsars in general, and the Vela pulsar in particular, vary intrinsically from one pulse to the next and within pulses (Krishnamohan & Downs 1983; Johnston et al. 2001; Kramer et al. 2002). Variations on timescales shorter than the time to accumulate one sample of the spectrum, the "accumulation time," lead to correlations of noise between spectral channels and increase the source noise contribution to σ²_|| (Gwinn & Johnson 2011). If we parameterize these variations by δI/I, with 〈δI〉 = 0, then the quadratic coefficient, b₂, in Equation (7) becomes

$$\begin{equation} b_2 = \left({\frac{\delta I}{I}}\right)^2 + \frac{1}{N_{\rm obs}}. \end{equation} \tag{ 8 }$$

The other coefficients, b₀ and b₁, are unchanged. This calculation follows Section 3.3.2 of Gwinn & Johnson (2011), but with β − 1 = δI/I averaging to zero over the suite of observations rather than over the accumulation time for a single spectrum.

Digitization, or more precisely quantization during digitization, also affects the noise. If the correlation is not extremely strong, ρ < 0.5, and if the data are viewed in the spectral domain, then the effects of quantization can be represented as a change in gain and a spectrally constant offset. This offset is often termed "digitization noise" and contributes to the values of b₀ and b₁, as one would expect. The expressions for noise take the same forms as in Equations (4)–(6) above, but with corrections to station gains and noise levels (see Equations (56) and (57) of Gwinn 2006).

Correlations among spectral channels also characterize noise. For sources of constant intensity, noise is uncorrelated between spectral channels, under the "wrap" approximation (Gwinn 2006). Relaxation of that assumption can lead to an observable correlation of noise between channels, as we discuss further in Section 5.2.3. Variation of flux density within the accumulation time for a single spectrum can also lead to significant correlations of noise (Gwinn & Johnson 2011).

Figure 1 shows some sample data, along with the scheme of pulse gates. To produce this figure, we averaged the real part of a segment of data for IF1 on the short Mopra–Tidbinbilla baseline, from 14:15:05 UT to 14:20:00 UT. This averaging reduced the depth of scintillation. We display the resulting average in the five gates as a function of pulse phase, including pulse gate and spectral dispersion. Each gate sampled the IF bandwidth over a short range of pulse phase; however, because of pulse dispersion, the low-frequency end of the gate sampled earlier parts of the pulse than the high-frequency end. The plot gives a rough idea of the pulse profile, although the effects of scintillation are still quite large and each sample is averaged over the 1 ms gate. As the figure shows, each phase in the pulse profile is represented twice, at two different frequencies. Each frequency is represented five times, in each of the gates (as well as in the empty sixth gate, outside the plot).

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Figure 2 shows cross-power scintillation spectra. It compares the same 1 MHz spectral range for a single 2 s integration, in Gates 1, 2, and 6 (off-pulse). The scintillation appears as dramatic variations in amplitude, with large amplitudes concentrated in a few spectral regions. As measured from all of the Mopra–Tidbinbilla interferometric data, the typical scintillation bandwidth was 15 kHz (half-width at half-maximum of the autocorrelation function, in frequency; Gwinn et al. 2012). This frequency scale is apparent in the spectra. Note, however, that noise modulates the scintillation peaks on finer scales and introduces differences in detailed shapes of the peaks between gates. The typical scintillation timescale was 9 s (1/e point of autocorrelation function, in time). This timescale is longer than the integration time of the spectra. These scintillation scales are in good agreement with results extrapolated from single-dish measurements by other observers at other frequencies (see, for example, Roberts & Ables 1982; Cordes et al. 1985). The spectrum in Gate 6 appears completely noiselike: indeed, the samples are drawn from a circular Gaussian distribution in the complex plane, as discussed in Section 5.1 below.

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4.1.1. Noise Estimates

Differences between samples separated by much less than the scales of scintillation reflect noise and intrinsic variability; these differences can therefore be used to estimate properties of the noise. For example, consecutive samples in time can estimate both the signal (as their average) and the noise (as $\sqrt{2}$ times their difference). More generally, we can estimate the signal and noise as averages over, and differences among, groups of nearby samples. These estimates are unaffected by the scintillation if all of the samples are contained well within the characteristic frequency and timescales of the scintillation.

Specifically, if we assume that the statistically averaged signal s_k is identical in N samples r_k(t_ℓ) and that the noise n_{k, ℓ} is uncorrelated, then we can estimate both the signal and the noise as

$$\begin{eqnarray} s_{k} &= \frac{1}{N} \sum _{\ell } r_{k}(t_{\ell }) \nonumber\\ n_{k,\ell } &= \sqrt{\frac{N}{N-1}} (r_{k}(t_{\ell }) - s_{k}), \end{eqnarray} \tag{ 9 }$$

where the index ℓ runs over the N samples compared. By binning the estimated noise, n_{k, ℓ}, according to the estimated signal, s_k, and then estimating the variance in each bin, we can identify the changes of the distribution of noise as a function of signal and so identify the three coefficients {b₀, b₁, b₂} in Equation (7).

4.1.2. Effects of Scintillation, Variability, Correlation, and Binning on Noise Estimates

The interferometric visibility is complex, and the distribution of noise varies with it over the complex plane. Figure 3 shows estimated noise as a function of average visibility in the complex plane, for the Hartebeesthoek–Tidbinbilla baseline. This baseline was long enough to span much of one scintillation element, so that the interferometric phase varies over 2π. Moreover, the antennas are large, so the resulting moderate S/N allows the complex plane to be divided into many bins. Note that although the distribution extends over 2π in phase, it is concentrated toward the right: the average visibility lies on the positive real axis.

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We used data from IF2, Gate 2, channels 2048–3072, for the entire time span of the observation on that baseline. We formed spectra, and fringed the data, as described in Section 3 above, to align the average visibility with the positive real axis. We then differenced pairs of consecutive time samples to find the estimated noise n_k and signal s_k. The data were binned by real and imaginary parts of the average signal s_k, with bin increments of 0.015 correlator units. For bins containing more than 100 samples, we found error ellipses from the standard deviation of our estimates of the complex noise n_k in that bin. The figure shows these ellipses. The displayed ellipses extend to one-half standard deviation of the noise in each bin, to reduce confusing overlap.

The error ellipses have the form that Equation (7) suggests, with size increasing with distance from the origin, indicating increasing noise with increasing signal amplitude. Ellipses close to the origin are nearly circular, as they must be: the noise is independent of signal phase, when signal amplitude is near zero. Both dimensions of the error ellipses grow with increasing signal; however, the noise in phase with the signal grows faster, so that the ellipses become elongated farther from the origin. Of course, amplitude variations in time, between pulses, would produce the same effect. However, the contribution of such amplitude variations is small for our 2 s averages.

Figure 3 and the form of Equation (7) suggest that we find noise parallel with and perpendicular to the signal. Thus, as long as we consider perpendicular and parallel components separately, we can group together noise estimates with the same magnitude of visibility. Figure 4 shows this analysis for IF2, Gate 1 channels 5120–6144 on the Mopra–Tidbinbilla baseline. This short baseline provided a long observation, yielding many individual measurements of noise. The chosen gate and channel range are near the pulse peak, where the source was strong and S/N is high, providing for many, relatively narrow bins. As in Equation (9), we averaged the signal over four samples, or 8 s. We then found noise by differencing individual samples from the average. For each bin in signal amplitude, we present histograms of noise in phase with the signal and at quadrature.

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As the figure shows, noise was equal in the two directions at zero signal, at lower left, and increased with signal amplitude to the right and upward. Moreover, noise increased more in phase with the signal than in quadrature with it, so that the widths of the two distributions in each panel diverge as signal increases. The number of points decreased with increasing signal as well, representing the effect of the underlying distribution of visibility. The vertical scales are logarithmic, so that a parabolic shape indicates a Gaussian distribution. We do not display fits of Gaussian distributions to the histograms in the figure because these are nearly indistinguishable from the histograms.

Figure 5 shows the variances of the best-fitting Gaussian distributions to noise, for the data shown in Figure 4. The noise in phase with the signal (shown by circles in the figure) increases quadratically with signal amplitude, whereas noise in quadrature with the signal (shown by crosses) increases linearly. This is precisely the behavior expected, as discussed in Section 2.2 above. We fit these two curves with polynomials of the form given by Equation (7). We assume that the coefficients (except the quadratic coefficient b₂) are the same for σ²_|| and σ²_⊥. The fit is to points in the range 0.006 < I < 0.056. Points at larger I are based on a small number of samples. In the lowest bin, noise is slightly higher than expected in both components, compared with extrapolation from larger bins and the fit. The noise increases in all directions away from this bin, so that every point "leaked" from an adjacent bin tends to increase the noise, as discussed in Section 4.1.2 above. The y-intercept of the fit is close to the noise level estimated for the empty spectral range of this gate, discussed in Section 5.1 below. The figure also shows, as a dotted line, results for noise in a fit to the global distribution of visibility discussed in Gwinn et al. (2012). The two techniques usually agree well, for gates and spectral ranges with high amplitude, so that leakage of noise into adjacent bins is small, and the distribution can be characterized well.

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As a measure of the quality of these fits, we note that in the absence of self-noise, all 16 points in Figure 4 would have a single value b₀. The mean square residual about the best-fitting single value is 6.5 × 10⁻¹. Our model included two additional parameters, b₁ and b₂, and reduced the mean square residual to 2.4 × 10⁻³, or by a factor of 268. The F-ratio test gives the likelihood of this improvement occurring by chance of less than 10⁻¹² (Bevington & Robinson 2003). Thus, the three-parameter model is excellent. If we adopt the parameters from Gwinn et al. (2012) shown by the dashed lines in Figure 4, the mean square residual is 3.3 × 10⁻³.

The difference between σ_|| and σ_⊥ reflects self-noise and effects of amplitude variations. The contribution of self-noise was 1/N_obs = 1/33, as discussed in Section 3. As Equation (8) shows, amplitude variations on timescales longer than the accumulation time, approximately 1 ms, and shorter than the 2 s integration time contributed to b₂. For the data in the figure, we found b₂ = 0.12. This value indicates that (δI/I)² = 0.09, as would be expected after integrating over 22 or 23 pulses (Krishnamohan & Downs 1983; Johnston et al. 2001; Kramer et al. 2002).

Figure 6 shows the behavior of noise on the Hartebeesthoek–Tidbinbilla baseline. This baseline had projected length of over 9000 km during the observation; however, it is intermediate in length in the sense that it is comparable to the length scale of the scintillation pattern. Moreover, the baseline was short compared with the baseline to the VSOP spacecraft. We included data from the entire time span recorded, in IF2, Gate 1, channels 5120–6144. This is the same gate and frequency range as shown for the Mopra–Tidbinbilla baseline in Figure 5. This range was chosen for comparison, and because Gate 1 allows comparison with an empty region of the gate. We fit these two curves with polynomials of the form given by Equation (7), to points in the range 0.006 < I < 0.056. Our three-parameter noise model reduced the mean square residual over the 10 points in the fit, relative to a single variance, by a factor of 1112. The F-ratio test gives probability of this occurring by chance of less than 5 × 10⁻¹².

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Comparison of Figures 5 and 6 shows interesting consequences of the longer baseline length. The maximum amplitude was smaller, because as the baseline length approaches the scale of the scintillation, occurrences of high intensity at both stations become less likely. The contribution of background noise b₀ was smaller and the self-noise was greater, perhaps because Hartebeesthoek has larger area than Mopra (26 m rather than 22 m diameter).

The points at high amplitude show scatter about the fitted curves; for this intermediate baseline, the average for b₁ converges over some timescale intermediate between the many samples of electric field for a short baseline and the many scintillations required for a long baseline. The first bin is elevated relative to the extrapolated, fitted curves, and the y-intercept of the fitted curves lies above the noise found in the empty part of the gate. This may result from leakage of noise into adjacent bins and the different convergence statistics.

As an example of noise on a long baseline, we analyze the long baseline from the VSOP spacecraft to Tidbinbilla, using procedures similar to those for the short Mopra–Tidbinbilla baseline. On this long baseline the phase of the cross-power spectrum varied through many turns because of scintillation. Instrumental phase variations were larger and more rapid than on the Mopra–Tidbinbilla baseline discussed above, and of course the 8 m spacecraft antenna is smaller than Mopra. Fringe fitting the data, as described in Section 3, was challenging because of the rapid variation of phase and rate from spacecraft motion, the large variations of phase with scintillation, and the low average visibility.

For analysis of noise, we used fewer, larger bins in visibility so that the noise in a bin does not greatly exceed bin width. For our tests, we used data from the first orbit, from IF1. We used Gate 2, which is near the peak of the pulse and has high flux density across the observing band, to maximize signal and so ease fringe fitting. To improve the quality of the noise measurement, and to reduce leakage into adjacent bins, we found averages for the real and imaginary parts of the visibility for spans of four channels in frequency and eight samples in time (7.8 kHz × 4 s). The frequency span lies well within the scintillation bandwidth of 15 kHz, and the time span within the scintillation timescale of 9 s and the fringing time of 16 s.

The noise is different in phase and in quadrature with the signal. Figure 7 shows the variances σ_|| and σ_⊥ plotted with the averaged magnitude of the visibility |s_k|. We fit to the range 0.005 < |s_k| < 0.020. Again, quadratic and linear models, with linear terms identical, fit the data well for small signal amplitude. The three-parameter model reduces the mean square residual by a factor of 31, relative to a single-parameter model. The F-test gives probability of chance occurrence of less than 6 × 10⁻⁵.

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The quadratic term is small compared with the linear term, in comparison with the other baselines, as expected for the lower S/N for this less-sensitive baseline. The point at smallest amplitude shows considerably higher noise than that extrapolated from larger amplitudes, again most likely from "leakage" of noise from other bins into this bin. As discussed in Section 2.3.3, the noise converges to the expected form only in an average over many scintillation elements. This slow convergence and the relatively small number of samples at large visibility contribute to the variation of the points about the expected form.

5.1.1. Variance: Short Baseline

In spectral regions empty of signal, the average value of the cross-power spectrum is zero, and the noise closely approximates a Gaussian distribution. Figure 8 shows two examples for the Mopra–Tidbinbilla (MT) baseline: the distribution of the real and imaginary part of $\tilde{r}_k$ for IF1 channels 1024–2048 during the period 19:08:17 UT to 21:12:52 UT, in Gates 1 and 6. Gate 6 was correlated only for IF1. The statistics of the real and imaginary parts are nearly identical, as expected for the noise, as the figure shows. The noise is clearly smaller in Gate 1, when the pulsar is "on" in another part of the gate. The variances are 58.3 × 10⁻⁶ correlator units² for Gate 1 and 96.5 × 10⁻⁶ correlator units² for Gate 6.

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The change in noise is an artifact of digitization (Gwinn 2006). The reduction is the consequence both of the change in autocorrelation spectrum $\tilde{\alpha }$ and of the different levels of the quantizer relative to the standard deviation of the signal (Table 1). Equation (56) of Gwinn (2006) gives the noise in the absence of signal:

$$\begin{eqnarray} b_0 &=\Gamma _C \frac{2 N}{N_{\mathrm{obs}}} \left(A_{X2} + B_X(\tilde{\alpha }_k-1)\right)\left(A_{Y2}+ B_Y(\tilde{\alpha }_k-1)\right). \quad\quad \end{eqnarray} \tag{ 10 }$$

This expression relates the notation of this paper on the left side of the equation with that of Gwinn (2006) on the right. Evaluation requires the autocorrelation spectrum $\tilde{\alpha }_k$, subject to the normalization condition: $\sum \tilde{\alpha} _k = 1$, where the sum runs over the 2N spectral channels (Gwinn 2006, Equation (1)). Effects of the change in the threshold for the quantizer v₀ are contained in the constants A_X2, B_X for station X and A_Y2, B_Y for Y. These quantities can be calculated from the statistics of the reduced-table two-bit correlator with n = 3 and the values for v₀ given in Table 1. The correlator-dependent gain Γ_C parameterizes instrumental effects. The number of spectral channels is 2N, and the number of observations is N_obs. For our observations, on a short baseline, 2N/N_obs ≈ 33 as discussed in Section 3.

For Gate 6, empty of any pulsar flux, the autocorrelation function was flat, and $\tilde{\alpha }_k \equiv 1$. When the pulsar turns on in part of the spectrum, the autocorrelation function in the off-pulse portion must fall, because $\tilde{\alpha }_k$ is normalized. This accounts qualitatively for the reduced noise level in the empty portion of Gate 1 relative to the completely empty Gate 6, although changes in v₀ also play a role.

A quantitative calculation of the autocorrelation spectrum when the pulsar is "on" is necessarily indirect. We estimated the relative background noise of the antennas from tabulated system-equivalent flux densities and used the average cross-power spectrum in Gate 1 (a smoothed, larger average similar to that shown in Figure 1) to estimate the contribution of the source to the spectrum. Using the normalization condition, we estimated $\tilde{\alpha }_k$ in regions of the spectrum without pulsar flux. We used the tabulated average values of v₀ and the correlator parameter n = 3 to determine the constants A_X2, B_X, A_Y2, and B_Y, for Gates 1 and 6. Equation (10) then provided the noise in the two cases; their ratio is independent of Γ_C. We found an expected ratio of 1.23 of the standard deviation of noise in Gate 6 to that in Gate 1. This is in approximate agreement with the measured ratio of 1.29, from fits to the histograms shown in Figure 8. A more sophisticated calculation might include use of the autocorrelation spectra for the two antennas and quantizer populations measured synchronously with them; it might also include a spectral model for noise with frequency at each antenna.

5.1.2. Noise and Number of Pulses

The observed distributions of noise are superpositions of underlying distributions. For example, not all integration times contained the same number of samples. On the MT baseline, the signal was integrated for 2 s, or 22.4 pulse periods; more precisely, 61% of the integrations contained 22 pulses, and 39% contained 23. Although the data are normalized to the integration time, the variance of the noise, parameterized by b₀, will be 4% greater for the integrations with fewer pulses.

Of course, the number of pulses was the same for all channels within a given sample of the spectrum. A spectral average provides an estimate of variance. We demonstrate the effect of integrating over different numbers of pulses in Figure 9. We plot a histogram of the variances over channels 1024–2048, for samples from the time span of the data in Section 4.3, in IF1. The histogram shows two clear peaks. The centroids of the peaks lie at horizontal positions close to the expected ratio of 22:23. Their populations are close to the expected ratio of 61:39. The vertical lines show these positions, constrained to match the overall variance of the data set. The dotted line shows a simple fit of a model for the sum of two Gaussian distributions to the two peaks, with the normalizations and locations of the peaks set as for the lines, with net normalization equal to the number of samples and the widths of the peaks equal.

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5.1.3. Variance: Long Baseline to Spacecraft

Variances and covariances completely characterize Gaussian noise. Covariance of noise in different spectral channels can arise from quantization (Gwinn 2006), from source variability on short timescales (Gwinn & Johnson 2011; Gwinn et al. 2011), and from pulsar gating, as described in the following section. In this section, we calculate the covariance for noise in an empty gate and compare with observations. We then discuss how, for gates containing signal, the covariance introduced by source variability masks that from pulsar gating.

5.2.1. Pulsar Gate without Wrap

Pulsar gating can introduce spectral correlations. Indeed, any temporal modulation is predicted to introduce spectral correlation (Gwinn & Johnson 2011). For our observations, the maximum lag correlated approaches the width of a pulse gate. Therefore, at large lag, we obtain fewer measurements of the correlation function than at small lag. The covariance of noise between two spectral channels is given by Fourier transform of the product of elements of the correlation function. The Fourier transform of two copies of Equation (1) leads to an expression similar to Equation (22) of Gwinn (2006):

$$\begin{eqnarray} \langle \tilde{r}_k \tilde{r}_{k+\ell }^*\rangle - \langle \tilde{r}_k\rangle \langle \tilde{r}_{k+\ell }^*\rangle &=& \sum _{\upsilon,\mu =-N}^{N-1} {\frac{1}{(N_{\rm obs}-|\upsilon +\mu |)(N_{\rm obs}-|\upsilon |)}} \nonumber\\ && \times \sum _{n,m} e^{\left[ i {\frac{2\pi }{2 N}} \left(k \mu + (k-\ell) \upsilon \right)\right] } \alpha _{n-m} \alpha _{-(n-m)+\mu }. \nonumber\\ \end{eqnarray} \tag{ 11 }$$

Note that, in our case, the limits of the sums over n and m depend on both $\upsilon$ and μ. This is because the correlation functions r_τ, r*_ν are not averaged over all of the samples at both stations. The correlation of noise between spectral channels given by Equation (11) depends on the specific form of the autocorrelation function, α_τ. However, the autocorrelation function is always maximum at α₀ = 1 for lag τ = 0 and usually falls rapidly to zero for larger lags τ.

5.2.2. Correlation of Noise: White Signal and White Noise

We suppose in this section that the original time series is spectrally flat or "white" noise (Papoulis 1991). Consequently, the correlation function is zero except at the central lag: α₀ = 1, and $\alpha _\upsilon =0$ for $\upsilon \ne 1$. We then find

$$\begin{equation} \langle r_{\upsilon } r_{\upsilon }^* \rangle - \langle r_{\upsilon }\rangle \langle r_{\upsilon }^* \rangle ={\frac{1}{N_{\rm obs}-|\upsilon |}}. \end{equation} \tag{ 12 }$$

Because we have normalized the correlation function and the signal is white, the mean square noise is just the reciprocal of the number of samples.

In the spectral domain, this variation of the number of samples leads to correlation. In Equation (11), the product of α values is zero unless m = n and μ = 0. Performing the remaining sums, we find for the correlation between channels

$$\begin{equation} \langle \tilde{r}_k \tilde{r}_{k+\ell }^*\rangle - \langle \tilde{r}_k\rangle \langle \tilde{r}_{k+\ell }^*\rangle = \sum _{\upsilon =-N}^{N-1}e^{\left[ i {\frac{2\pi }{2 N}} (k-\ell) \upsilon \right] } {\frac{1}{N_{\rm obs}-|\upsilon |}}. \end{equation} \tag{ 13 }$$

This correlation is largest for close spectral channels (ℓ ≪ 2N) because the noise in the correlation function is largest for large lag ($\upsilon \rightarrow N$).

5.2.3. Observed Correlation of Noise in an Empty Gate

We compared observations on the Mopra–Tidbinbilla baseline with the theoretical prediction of Equation (12) and find quantitative agreement. Figure 10 shows the results. To make the figure, we re-transformed the spectrum for each time interval and found the mean-square correlation function 〈r_τr*_τ〉. The data used to make Figure 10 are from Gate 6, IF2, of the Tidbinbilla–Mopra baseline, from 14:15 to 15:41 UT. We used the full recorded bandwidth. The downward-pointing spike at τ = 5000 apparently results from interference. The solid curve shows the form predicted by Equation (12), as expected for N_obs = 16,000 complex samples, for the 1 ms length of the pulsar gate and our 16 MHz bandwidth.

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Figure 10 shows less noise at small lag, as Equation (12) suggests. From Fourier transform of the theoretical curve shown in the figure, we find that noise in the spectrum $\tilde{r}_k$ is anticorrelated in adjacent channels by approximately −6%, expressed as normalized correlation. This correlation falls off rapidly, however: it is only +0.4% in the second channel, −0.7% in the third, and so on.

5.2.4. Correlation of Noise: Signal Present

We compare theoretical predictions for the distribution of noise for cross-power spectra with observations of a scintillating pulsar, the Vela pulsar. We describe observations made with Earth-based very long baseline interferometry baselines and with baselines from an orbiting spacecraft to Earth. These observations extend previous studies (Gwinn & Johnson 2011) to the regime of high S/N and large variations in interferometric phase.

In Section 2, we argue that, in the presence of signal, noise on a short baseline should be drawn from an elliptical Gaussian distribution in the complex plane. The theory was previously presented in Gwinn (2006) and Gwinn & Johnson (2011). The major axis of the distribution is aligned with the direction of the signal. The variance along the minor axis depends linearly on signal strength; the variance along the major axis has the same linear dependence, plus a quadratic term. At zero signal, the major and minor axes are equal and the distribution of noise is a circular Gaussian, as for a gate or spectral region empty of signal.

We test this theory with observations on the baselines from Mopra, Hartebeesthoek, and the VSOP spacecraft to Tidbinbilla in Section 4. We estimate noise by comparing samples within the characteristic scales of the scintillation and binning their differences by average interferometric visibility. We find that the distribution of noise closely follows the expected elliptical Gaussian form for each visibility, and the scaling with visibility of the major and minor axes corresponds to the quadratic and linear noise polynomials, respectively. The quadratic coefficient accurately reflects the number of samples and the contribution of intrinsic amplitude variations; the constant coefficient agrees with that estimated from empty portions of the spectrum for the Mopra–Tidbinbilla and Hartebeesthoek–Tidbinbilla baselines.

In Section 5, we demonstrate that quantization, gating, and integration each affect the properties of the noise. One interesting consequence is that the noise in the presence of signal is less than that in a completely empty spectrum—a result of the combination of quantization and spectral variations. Also, pulsar gating leaves fewer samples, thus larger noise, at larger lags; this effect incurs correlations in the spectral noise. In principle, complete knowledge of the quantizer levels for each integration period, and of the autocorrelation functions at the two antennas, allows calibration of these effects. Alternatively, recording the signal with many quantizer levels increases the dynamic range and reduces effects of variation in quantizer levels. Flexible software correlators, such as the DiFX correlator (Deller et al. 2007), can control these artifacts, while Nyquist-sampled spectra of individual pulses obviate the difficulties in characterizing inhomogeneities within the integration (Johnson & Gwinn 2012).

We present observations for a scintillating pulsar, but the effects of self-noise hold for any interferometric observation. A careful evaluation of these effects is essential for a priori estimates of the accuracy of pulse timing and spectroscopy using single-dish observations and for scintillation studies and astrometry using interferometry.

Many telescopes now under construction or being planned, such as LOFAR, ASKAP, and SKA, will operate as interferometers, with many baselines among many antennas of a particular design. Each baseline will have the distribution of noise we describe above in Section 2 and as we observe for the Vela pulsar. Because each telescope in a large array receives precisely the same noiselike signal from the source, increases in the number of antennas and averaging of many baselines do not change self-noise, when expressed in terms of flux density. However, averages over many baselines do decrease the background noise. Stated in terms of the notation introduced in Section 2, increasing the number of identical antennas N_A reduces b₀ as N⁻²_A and b₁ as N⁻¹_A but does not change b₂. For true "tied-array" operation, where electric fields from all antennas are phased and summed before correlation, statistics are those of a single dish (Gwinn & Johnson 2011, Equation (11)). As the number of antennas becomes larger, self-noise becomes more important. When the source dominates the system temperature, further improvements demand more samples N_obs, as produced by wider bandwidth or longer integration time: a greater aperture does not provide more accuracy. Our expression Equation (5) generalizes this result to interferometry.

Astrometry depends on measurements of interferometric phase. Equations (5) and (7) show that the maximum attainable phase accuracy is $\delta \phi \approx \sigma _{\perp }/s \approx \sqrt{n/(N_{\rm obs} s)}$, or the inverse square root of the S/N, divided by $\sqrt{N_{\rm obs}}$. Similarly, the maximum attainable accuracy in measurement of flux density by an interferometer, or a large single dish, is approximately the flux density of the source divided by $\sqrt{N_{\rm obs}}$. Likewise, the maximum attainable accuracy in pulsar timing is approximately the width of the narrowest feature in the profile, divided by the S/N and by $\sqrt{N_{\rm obs}}$. However, when self-noise is the limiting factor, the maximum attainable accuracy is simply the width of that feature, divided by $\sqrt{N_{\rm obs}}$.