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The institutional observing guide explains interferometric Fourier sampling, the primary beam and synthesized resolution. Its “Imaging of extended structures” section gives the approximate $0.6\lambda /b_{\min }$ scale and explains the need for short-spacing measurements. This is an operational rule of thumb, not an exact source-independent cutoff.

A complex visibility is a calibrated complex correlation of the signals received at a particular interferometric baseline. In a narrow field, with scalar radiation, the far-field approximation for an outgoing source, a narrow frequency channel, negligible noncoplanar phase curvature and a steady sky of mutually incoherent emitters,Here $l,m$ are small dimensionless sky-direction offsets, $I$ is brightness in the chosen channel, and $A$ is the primary beam. This is the Fourier transform of beam-weighted brightness. The two-dimensional formula requires those approximations: a wide field generally needs the third baseline coordinate and curved sky geometry.

For an explicit one-dimensional derivation, set the primary beam to one over the small field, and place independent emitters at angular offsets ${\theta }_j$. Choose a wave phase convention in which, after removing the phase centre's delay, their received complex amplitudes areHere ${\delta }_{jk}$ is the Kronecker delta and angle brackets denote an expected value estimated by time averaging a steady signal. The last equality states mutual incoherence: different emitters have zero averaged cross term, although each emitter's two received copies remain correlated. Multiplying and averaging givesFor continuous brightness this becomes $V(u)=\int I(\theta )e^{-2\pi iu\theta }d\theta$. Real brightness implies the Hermitian symmetry $V(-u)=V(u{)}^{\ast }$; it does not require brightness to be even. In particular, an asymmetric real source usually has a nonzero imaginary complex visibility. At zero baseline, $V(0)={\sum }_j{I}_j$ is the total beam-weighted source strength. Ordinary separated-element correlations need not measure that zero-baseline value.

The dirty image is the direct weighted inverse Fourier transform of the measured complex visibilities:The final equality follows by substituting the point-source complex visibility and exchanging the finite sums. For continuous brightness it is a convolution with the dirty beam, with the primary beam included in the brightness when necessary. Negative values and subsidiary peaks can arise from that response even when every true source has positive strength.

The center-to-center extent is $18\mathrm{m}$ on each side and $18\sqrt{2}=25.46\mathrm{m}$ diagonally, while the shortest nonzero baseline is $6\mathrm{m}$. At $\lambda =0.02\mathrm{m}$, characteristic scales areThese describe the finest synthesized scale, a largest recoverable scale estimate, and the individual-dish field of view, respectively. Along an array side the longest baseline is only $18\mathrm{m}$, giving $3.82^{\prime }$ rather than the diagonal estimate. Projection, weighting and sidelobes determine an actual synthesized beam.

There is no sharp universal largest-scale cutoff: the commonly used $0.6\lambda /b_{\min }$ convention would give $6.88^{\prime }$. For an ideal uniformly illuminated circular dish, the primary-beam full width at half maximum is approximately $1.02\lambda /D=17.5^{\prime }$, while the first-null angular radius is $1.22\lambda /D=21.0^{\prime }$. These definitions should not be mixed. Uniform lattice spacing also produces grating sidelobes; the primary beam restricts the useful field but does not supply the missing zero-baseline flux.

Delay-compensated geometry. Choose an east-west baseline vector $\mathbf{b}=b\hat{\mathbf{x}}$ and let the upward unit vector be $\hat{\mathbf{z}}$. A ray direction at zenith angle $\beta$ in their plane has $\mathbf{s}(\beta )=(\sin \beta ,0,\cos \beta )$. Its signed geometric path difference is $\mathbf{b}\cdot \mathbf{s}=b\sin \beta$. Removing the phase-centre delay $b\sin \theta /c$ leaves the residual pathWith the receiver ordering chosen to give positive phase for this path, a complex correlator measuresThe reverse ordering complex-conjugates the expression without changing its information. A single real multiplier supplies its cosine component; the stated exponential represents an in-phase/quadrature correlator. Equal cable lengths to the central correlator remove a fixed instrumental difference but do not remove the changing geometric delay. Pointing the antennas and electronically compensating the path are separate operations.

Resolved uniform source. Contributions from mutually incoherent sky elements add as brightness-weighted complex visibilities. Put $I(\alpha )=I_0$ over $-\Delta \alpha /2<\alpha <\Delta \alpha /2$ and zero elsewhere, and define $q=kb\cos \theta$. Integrating givesThereforeThe zero-argument value is defined by continuity as one. Normalization by $V(0)=I_0\Delta \alpha$ isolates the visibility shape from total flux. The first null occurs when the source spans one fringe period, $\Delta \alpha =\lambda /(|b\cos \theta |)$. A negative real visibility beyond a null is a phase reversal, not negative surface brightness.

Path-compensation precision. A finite observing band sets a coherence envelope. For a rectangular band of width $\Delta \nu$, an uncorrected residual delay $\delta \tau =\delta \ell /c$ multiplies the correlation byCoarse delay steps must leave $\Delta \nu |\delta \tau |\ll 1$ if decorrelation is to be small. Thus a coarse path-step scale must be well below the coherence length $c/\Delta \nu$, with the precise bound determined by the tolerated amplitude loss and maximum quantization residual. A digital fine-delay or phase-rotation system can separately correct the central-frequency phase.

If there is no such carrier-phase correction, the steps must also be a small fraction of $\lambda$ to avoid large untracked phase jumps or time-averaging loss. For a step $d_{\ell }$ rounded to the nearest setting, $|\delta \ell |\le d_{\ell }/2$ and the maximum carrier-phase error is $\pi d_{\ell }/\lambda$. Bandwidth controls coarse group-delay decorrelation; wavelength controls the precision of untracked carrier phase. No numerical delay step can be inferred from the central frequency alone without a bandwidth, loss tolerance and compensation architecture. Earth rotation additionally sets how rapidly the delay setting must be updated.

Largest sampled angular scale. For a near-zenith view take the shortest projected baseline as $b_{\min }\simeq 0.5\mathrm{m}$. At $\nu =30\mathrm{GHz}$, $\lambda =c/\nu =9.9931\times 10^{-3}\mathrm{m}$. The inverse shortest spatial frequency givesThis is an order-of-magnitude largest fringe scale. A convention defining maximum recoverable source size as $0.6\lambda /b_{\min }$ would instead give about $41$ arcminutes; source profile, weighting and projected baseline coverage determine the operational value. The antenna diameter and primary beam can further restrict the usable field but were explicitly neglected in the source's initial idealization. Ten antennas supply $10\times 9/2=45$ pairwise baselines; their number improves sampling and sensitivity, not the inverse-shortest-baseline scale by a factor of ten. The longest baseline is not supplied, so the finest synthesized resolution is not numerically fixed.

A suitable observing programme. A compact centimetre-wave array is well suited to CMB temperature anisotropy on degree to sub-degree scales sampled by its baselines, and, with sufficiently long baselines and suitable foreground control, thermal Sunyaev-Zeldovich structure in galaxy clusters. At $30\mathrm{GHz}$ the thermal Sunyaev-Zeldovich signal is a decrement relative to the CMB. The compact baselines retain sensitivity to extended brightness features; the spatial frequencies measured by each baseline have a direct statistical interpretation in CMB analysis.

Compared with lower radio frequencies, synchrotron foregrounds are generally reduced, while thermal dust rises toward higher frequencies; observing near this centimetre-wave atmospheric window is therefore useful, although foregrounds are not absent. Correlation and phase switching reject mean uncorrelated receiver offsets and some instrumental drifts. Receiver thermal noise still contributes variance, and atmosphere, ground pickup or cross-talk can be correlated and require control. Additional long baselines can identify compact radio-source contaminants for subtraction, but the question does not specify enough array geometry to guarantee that capability. There is no zero baseline in ordinary pairwise cross-correlation, so the array does not directly measure the sky monopole or arbitrary larger-scale smooth emission.

Choose the baseline vector from the first antenna to the second. Apart from a sign convention for receiver delay, the geometric delay is ${\tau }_g=\mathbf{b}\cdot \mathbf{s}/c$, and the phase difference is $\varphi =2\pi \nu {\tau }_g$. For equal received source powers $P_a$, adding the voltages and averaging over many wave periods givesUncorrelated receiver noise adds a mean level but not the source's coherent cross term. For an east-west baseline and one choice of hour-angle sign, ${\tau }_g=(b/c)\cos \delta \sin H(t)$. Earth rotation therefore sweeps the phase; locally the fringe frequency is $(b/\lambda )\cos \delta \cos H\dot{H}$. Tracking the antenna pointing keeps the source in the primary beam. Electronic fringe stopping is a separate operation: it can remove this phase evolution, making a source at the phase centre constant in the corrected complex visibility. The oscillating power above is the geometrical fringe before that correction.

For a small angular displacement $d\theta$ along the projected baseline, $d\varphi =2\pi b_{p}d\theta /\lambda$. One full fringe therefore spans $\Delta \theta \sim \lambda /b_p$. Here $b_p=|\mathbf{b}-(\mathbf{b}\cdot \mathbf{s})\mathbf{s}|$. This is the angular period of the measured Fourier component and the characteristic resolution scale when such long baselines are synthesized; precise beam width and image ambiguity also depend on baseline coverage and weighting. A single pair by itself does not give an unambiguous complete image.

Writing $x=h\nu /(k_{B}T)$, the expansion $e^x-1=x+O(x^2)$ immediately gives the Rayleigh-Jeans law,The factor two counts the two independent modes of polarization of waves. A small brightness-temperature perturbation thus has $\Delta I_{\nu }=2k_B{\nu }^2\Delta T_{\mathrm{RJ}}/c^2$ in this limit.

Hot intracluster electrons inverse-Compton scatter the microwave-background photons. The scattering optical depth is $d\tau =n_e{\sigma }_{T}dl$ and the characteristic thermal fractional energy transfer is of order $k_B{T}_e/(m_e{c}^2)$. The integrated distortion is therefore controlled byScattering conserves photon number while shifting photons toward higher frequencies, giving a low-frequency decrement and a high-frequency increment. Thus the printed estimate correctly gives a negative change of order $y{T}_{\mathrm{CMB}}$. Its numerical coefficient, however, is not the standard Rayleigh-Jeans coefficient. For nonrelativistic electrons and small $y$, the Kompaneets equation gives $\Delta n=y{x}^{-2}{\partial }_x(x^4{\partial }_x{n}_0)$. At small $x$, $n_0\simeq 1/x$, so $\Delta n\simeq -2y/x$ andEquivalently the thermodynamic-temperature spectral factor is $x\coth (x/2)-4\to -2$. The original PDF really prints the estimate with $-y$, not $-2y$; both numerical conventions are retained below. Relativistic corrections at the supplied $T_e$ are small compared with this factor-of-two difference.^[ref]

The uniform cube has a constant line-of-sight depth $L=6\times 10^{22}\mathrm{m}$. Using the supplied cross-section rather than replacing it with a more precise value givesAdopt $T_{\mathrm{CMB}}=2.72548\mathrm{K}$, explicitly an external measured constant, not a temperature supplied in the question. The values $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$ and $m_e\simeq 9.10938\times 10^{-31}\mathrm{kg}$ set the physical units. Hence the printed relation gives $\Delta T\simeq -22\mu \mathrm{K}$; the standard Rayleigh-Jeans result gives $\Delta T\simeq -44\mu \mathrm{K}$. Keeping the answer as $-8.14\times 10^{-6}{T}_{\mathrm{CMB}}$, or twice that, avoids needing a numerical background temperature.^[ref]

The face is a square, so its solid angle is $\Omega =(6\pi /10800{)}^2=3.046\times 10^{-6}\mathrm{sr}$, not the area of a circular six-arcminute aperture. At $15\mathrm{GHz}$,Thus $\Delta S_{\nu }\simeq -4.67\times 10^{-4}\mathrm{Jy}=-0.47\mathrm{mJy}$ using the printed estimate, or $-0.93\mathrm{mJy}$ using the standard factor two. The sign denotes missing brightness relative to the surrounding microwave background. At this frequency $h\nu /(k_B{T}_{\mathrm{CMB}})\simeq 0.264$; the exact nonrelativistic spectral intensity differs from the strict Rayleigh-Jeans approximation by about one percent. This precision does not affect the estimates.

For the array noise, the system temperature sets the receiver noise scale; let $A_e={\eta }_A\pi d^2/4$ be the effective area of each dish, and let ${\eta }_c$ describe correlation losses. No efficiencies or polarization of waves count are supplied, so first take ${\eta }_A={\eta }_c=1$ and a single independent polarization of waves. An unpolarized point source produces antenna temperature $T_a=S_{\nu }{A}_e/(2k_B)$, and the system-equivalent flux density is $\mathrm{SEFD}=2k_B{T}_{\mathrm{sys}}/A_e$. The interferometric radiometer equation gives one baseline an rms flux error $\mathrm{SEFD}/\sqrt{2\Delta \nu t}$. There are $N(N-1)/2=45$ cross-correlation baselines; averaging their independent thermal noise for a point source givesThe ideal single-dish effective area is $4\pi {\mathrm{m}}^2$, giving $\mathrm{SEFD}=6.59\times 10^3\mathrm{Jy}$ andCombining two independent parallel-hand polarization of waves measurements gives $6.3\mu \mathrm{Jy}$. Nonunit efficiencies increase these estimates by $1/({\eta }_A{\eta }_c)$. Here signal equal to noise means a weak-source signal-to-noise ratio of one after integration; it does not mean equating the source to the entire unaveraged receiver-noise power.^[ref]

At the band centre, $\lambda =0.01999\mathrm{m}$. Taking a representative maximum projected baseline of $20\mathrm{m}$ gives ${\theta }_{\mathrm{res}}\sim 3.4$ arcminutes. A corner-to-corner baseline of $20\sqrt{2}\mathrm{m}$ would give about $2.4$ arcminutes before projection and weighting. The exact antenna locations and source declination are not supplied, so neither a unique synthesized beam shape nor an exact FWHM is determined. The individual-dish primary-beam scale, roughly $\lambda /4\mathrm{m}\sim 17$ arcminutes, is the field of view, not the synthesized resolution.

To extract the cluster signal, retain frequency-resolved complex visibilities and fit an extended SZ template together with contaminating compact sources. For the ideal uniform square centred at the phase centre and with a nearly constant primary beam across it, the Fourier transform isThis is why the integrated decrement must not be treated as a point-source flux on every baseline. The first axial visibility null occurs at $b=\lambda /\Theta \simeq 11.5\mathrm{m}$. Short baselines carry most of the large-scale decrement; the longest baselines are useful for separating unresolved positive radio sources. natural visibility weighting or a suitable taper gives a clearer extended-source image than emphasizing the longest baselines, while a visibility-domain matched filter uses all samples with weights proportional to template amplitude divided by noise variance. Its amplitude estimate is $\hat{a}=\mathrm{Re}(T^{†}{C}^{-1}d)/(T^{†}{C}^{-1}T)$ for a real template amplitude. A joint fit propagates radio-source uncertainty instead of subtracting exact, noise-free source models.

Earth-rotation and multifrequency synthesis improve Fourier coverage, but each channel must use its own baseline in wavelengths, primary beam and SZ spectrum. Use the longer baselines and spectra to identify variable point sources, flag bad data, and test the decrement with time, antenna and frequency splits. Primary-beam and absolute-flux calibration must be included. Large-scale microwave-background fluctuations and residual ground or atmospheric signals can remain even below the thermal-noise estimate. The narrow low-frequency band alone gives limited separation of SZ and primary microwave-background temperature structure. There is no zero-spacing interferometric datum: integrated flux is model dependent unless sufficiently short baselines or additional total-power information are available. Exact cluster detection significance cannot be obtained from the point-source noise alone without the actual projected baseline distribution and weighting.

The wavelength is $\lambda =c/f\simeq 0.020\mathrm{m}$. The longest interferometric baseline measures the finest angular structure and the shortest-baseline angular scale describes the coarsest:These are order-of-magnitude scales, not sharp cutoffs. A commonly used maximum-recoverable-scale convention is $0.6\lambda /B_{\min }$, giving about $8.2^{\prime }$; reconstruction also depends on baseline coverage and source structure. The individual antenna's primary beam is a separate field-of-view limit.^[ref]

The primary beam is the individual antenna pair's directional sensitivity envelope in the complex visibility equation. For identical, identically pointed elements it is the individual element's power pattern. Its angular width sets the useful field of view; the much longer interferometric baselines set finer measured scales. The primary beam and the dirty beam describe different responses.

The shortest useful projected baseline sets the lowest sampled spatial frequency, of order $b_{\min }/\lambda$, and hence an angular scale $\lambda /b_{\min }$. Operational maximum recoverable source sizes also depend on source shape, coverage, weighting and primary beam. The longest baseline sets a different scale, the synthesized angular resolution. The number of antennas alone does not fix either limiting baseline or supply zero-spacing total power.

The synthesized beam is the angular response produced by the sampled and weighted complex visibilities. Its characteristic width is set by the longest well-weighted projected interferometric baselines, of order $\lambda /b_{\max }$, while sidelobes reflect incomplete Fourier coverage. It differs from the individual antenna primary beam, which limits the field of view.

With constant surface brightness over a centred angular rectangle of widths ${\Theta }_x,{\Theta }_y$, and a nearly uniform primary beam over that rectangle, the Fourier transform separates into two one-dimensional integrals. The result is $V(u,v)=S_{\nu }\mathrm{sinc}(\pi {\Theta }_{x}u)\mathrm{sinc}(\pi {\Theta }_{y}v)$, where $\mathrm{sinc}x=\sin x/x$. The zero-spacing value is the integrated spectral flux density, while the first axial null is at $|u|=1/{\Theta }_x$. Long baselines therefore do not measure the total flux of a resolved rectangle.