2.1 Direct imaging approaches

With the advent of large-N telescopes, there is increased interest in ‘direct imaging’ instruments which follow the pre-correlation gridded path of Fig. 1. In one approach, antennas are physically arranged on a grid, and then a spatial 2D FFT is applied to form images (Tegmark & Zaldarriaga Reference Tegmark and Zaldarriaga2009). This approach is extended by the Modular Optimal Frequency Fourier (MOFF) imaging technique. In the MOFF approach, antennas do not need to be physically placed on a grid; rather, voltages are gridded electronically based on the aperture illumation of consituent antennas (Morales Reference Morales2011; Thyagarajan et al. Reference Thyagarajan, Beardsley, Bowman and Morales2017; Kent et al. Reference Kent2019). Note that a visibility matrix can be retrieved from a grid of beams via an inverse 2D FFT (Tegmark & Zaldarriaga Reference Tegmark and Zaldarriaga2009; Foster et al. Reference Foster, Hickish, Magro, Price and Zarb Adami2014), which is shown in Fig. 1 as a dashed line between pathways.

Here, we focus on non-gridded methods, but note that equivalent trade-offs between resolution and sensitivity could be leveraged with these alternative architectures.

2.2 Tensor formalism

Equation (5) can be written in an equivalent compact form using summation notation:

(6) \begin{equation} B = \textbf{w}\,\textbf{V}\,\textbf{w}^H \equiv\ \textbf{w}^{p} \, \textbf{V}_{pq} \, (\textbf{w}^*)^{q} = \textbf{W}^{pq} \textbf{V}_{pq} \end{equation}

In this notation, summation is implied over all indices that appear in an upper and lower index (^* represents complex conjugation). So, we sum across indexes p and q (summation indices representing antennas) which returns a single value B. The quantity $\textbf{W}_{pq}=\textbf{w}_p \textbf{w}^*_q$ is equivalent to the ( $P \times P$ ) matrix formed from the outer product $\textbf{w} \textbf{w}^H$ .

The visibility matrix can itself be written in summation notation, instead of using $\langle \rangle$ brackets, as:

(7) \begin{equation} \textbf{V}_{p q} = \textbf{v}^{t}_{p} (\textbf{v}^*)_{q t}\end{equation}

where t represents time.

These bold-face quantities are often referred to as data tensors. Tensors can simply be considered as multi-dimensional arrays, or generalisations of a matrix into higher dimensions. Here, we will follow the nomenclature used in the TensorFlow software package.Footnote e As we are using data tensors as a computational tool, we do not need to worry about other uses of tensors found in physics and mathematics, such as their interpretation as mappings between vector spaces, so we do not define a tensor metric here. We note that tensors have indeed been used previously in calibration and imaging (e.g. Smirnov Reference Smirnov2011; Price & Smirnov Reference Price and Smirnov2015; Thekkeppattu et al. Reference Thekkeppattu, Wayth and Sokolowski2024).

Whenever a pair of tensors have matching upper and lower indices, entries are summed across all matching indices; this is known as a pairwise contraction. The order of pairwise contractions for a chain of tensors is known as the contraction path. The computational cost to evaluate a tensor chain depends on the contraction path; finding the optimal path is an NP-hard problem that can quickly become intractable as the number of tensors increases.

2.3 Computing tensor contractions

Numerous software packages for performing tensor contractions are available. We highlight that the Python Numpy package (Harris et al. Reference Harris2020) includes a flexible ‘one-liner’ einsum method for computing contractions on Numpy arrays. An equivalent einsum method exists in the Cupy software package (Okuta et al. Reference Okuta, Unno, Nishino, Hido and Loomis2017), which offloads computations to a graphics processor unit (GPU). If compiled against the NVIDIA cuTENSOR library,Footnote f highly optimised tensor cores on the GPU may be targeted; these offer orders-of-magnitude greater energy efficiency than regular GPU cores.

A highly optimised correlation code that uses tensor cores is detailed in Romein (Reference Romein2021), and a complex general matrix multiply (cGEMM) code has been developed for use for beamforming with a phased array feed.Footnote g Together, these two codes could be used to compute Equation (6), and the einsum method provides a straightforward path for prototyping and developing post-correlation beamforming techniques.

2.4 Multiple beams and frequency channels

We can extend Equation (6) with extra indices (subscripts l and m) to represent a grid of $(L\times M)$ beams on the sky and across F frequency channels (subscript $\nu$ ):

(8) \begin{equation} \textbf{B}_{l m \nu} = \left(\textbf{w}_{l m p \nu} \textbf{v}^{p t}_{\nu} \right)\left( \textbf{w}^*_{l m q \nu} (\textbf{v}^*)^{q}_{\nu t}\right) \end{equation}

That is, we sum across indexes p and q (summation indices) and output an N-dimensional array with indices $(l, m, \nu)$ . We can also write Equation (6) as:

(9) \begin{equation} \textbf{B}_{l m \nu} = \textbf{W}^{p q}_{l m \nu} \, \textbf{V}_{p q \nu}. \end{equation}

If power beams are evaluated at a series of timesteps, we may write $\textbf{B}_{l m \nu}=\textbf{B}(t)_{l m \nu}$ as a function of time:

(10) \begin{equation} \textbf{B}(t)_{l m \nu} = \textbf{W}^{p q}_{l m \nu} \, \textbf{V}(t)_{p q \nu}. \end{equation}

There are two key differences between Equations (8) and (11). The first is that the tensor $\textbf{W}_{p q l m \nu}$ has $(P^2 \times L \times M \times F)$ entries, but $\textbf{w}_{p l m \nu}$ has only $(P \times L \times M \times F)$ entries. The second is that Equation (8) requires three pairwise contractions to compute, while Equation (11) is only one pairwise contraction – two contractions have already been performed to produce $\textbf{W}_{lmpq\nu}$ and $\textbf{V}_{pq\nu}$ .

To extend to a polarisation-aware version, we may simply add a subscript x:

(11) \begin{equation} \textbf{B}(t)_{l m \nu x} = \textbf{W}^{p q }_{l m \nu x} \, \textbf{V}(t)_{p q \nu x}. \end{equation}

where x represents a set of four polarisation coherency measurements (e.g. XX^*, XY^*, YX^*, and YY^* for a linearly polarised dual-polarisation antenna).

2.5 Comparison to interferometric imaging

In synthesis imaging, an image I(l,m) is created by evaluating

(12) \begin{equation} I(l, m) = \frac{1}{M} \sum_{m=1}^{M} V (u_k, v_k) e^{2\pi (u_k l + v_k m)},\end{equation}

where M is the number of baselines and $(u_k,v_k)$ are coordinates relating to the baseline k between a (p, q) antenna pair (Equations 7–3, Briggs et al. Reference Briggs, Schwab, Sramek, Taylor, Carilli and Perley1999). This equation can written as a tensor contraction:

(13) \begin{align} \textbf{I}_{lm} &= \textbf{W}^k_{lm} \textbf{V}_k \end{align}

(14) \begin{align} \textbf{W}_{klm} &= \frac{1}{M} exp\left(-i 2 \pi (u_k l + v_k m)\right),\end{align}

which by splitting $k=pq$ can be rewritten as:

(15) \begin{align} \textbf{I}_{lm} &= \textbf{W}^{pq}_{lm} \,\textbf{V}_{pq}, \end{align}

(16) \begin{align} \textbf{W}_{pqlm} &= \frac{1}{M} exp\left(2 \pi i (u_{pq} l + v_{pq} m)\right).\end{align}

Comparison of Equations (5) and (16) reveals that $\textbf{I}_{pq}\equiv \textbf{B}_{pq}$ . That is, each pixel in an image is equivalent to a power beam, or alternatively, an image can be created out of a grid of power beams. Note that in synthesis imaging, autocorrelations are generally not included; that is, the weights tensor $\textbf{W}_{pq}=0$ if $p=q$ . From a power beam interpretation, this is equivalent to subtracting the incoherently summed beam from the tied-array power beam (Roy et al. Reference Roy, Chengalur and Pen2018).

2.6 Historical perspective

While the interpretation of image pixels as power beams is not commonly discussed, the link between imaging and beamforming is fundamental to radio astronomy and was leveraged by early interferometers, including the Mills Cross (Mills et al. Reference Mills, Little, Sheridan and Slee1958). However, for the most part, the historical development of pulsar search pipelines has run parallel to high-fidelity synthesis imaging systems, leading to disparate science communities and techniques.

Since their discovery in 1967 by Jocelyn Bell Burnell, a majority of pulsars have been discovered using single-dish instruments, such as the Parkes Murriyang, Arecibo, Lovell, and Green Bank telescopes (Manchester et al. Reference Manchester, Hobbs, Teoh and Hobbs2005). As such, most observing techniques and knowledge within the pulsar community is derived from single-dish approaches. Pulsar search codes such as PRESTO Footnote h (Ransom Reference Ransom2011) were also designed for single-dish telescopes and are, in general, incompatible with interferometric data products.

In contrast, radio interferometers have predominantly been used for long-exposure synthesis imaging. Earth rotation synthesis, popularised by the Cambridge One-Mile Radio Telescope (Ryle Reference Ryle1962), combines data across many hours of observation to improve imaging fidelity. Instruments were not designed to output high time resolution data products; rather, they were designed to integrate data for as long as possible to minimise output data rates. Despite their lack of time resolution, pulsars have been detected in synthesis images via other characteristics; exceptionally, the first millisecond pulsar was detected in synthesis images as a source with an anomalous spectrum (Backer et al. Reference Backer, Kulkarni, Heiles, Davis and Goss1982).

The quest to discover new pulsars has nonetheless led pulsar astronomers towards radio arrays. Modern interferometers have good sensitivity, a large field of view, and better localisation capability; on a technical level, tied-array beamforming is increasingly feasible. Fast imaging techniques – where each pixel is treated as a power beam – have been used for searches of millisecond transient pulses (Law et al. Reference Law2015, Reference Law2018), as have FFT-based beamforming approaches (Ng et al. Reference Ng2017), but are yet to be eagerly adopted in pulsar periodicity searches.

Techosignature searches have followed a similar trajectory. Frank Drake performed the first SETI search using the Tatel single-dish telescope in 1961 (Drake Reference Drake1961); single-dish telescopes remained the primary technology used until the construction of the Allen Telescope Array (Welch et al. Reference Welch2009). Many technosignature experiments have targeted nearby stars and have sought to maximise frequency coverage (e.g. Isaacson et al. Reference Isaacson2017); hence, field of view has not historically been a science driver. But more ambitious surveys of nearby star targets (Czech et al. Reference Czech2021) motivate faster survey speed, and there is a growing movement towards non-targeted widefield technosignature searches (Houston et al. Reference Houston, Siemion and Croft2021).

3.1 Sensitivity

From the radiometer equation, the expected thermal noise (in Jy units) of a beam formed via post-correlation beamforming can be written as:

(17) \begin{equation} \Delta S_{\mathrm{post-x}} = \frac{k_B T_{\mathrm{sys}}}{A_e \sqrt{\Delta \nu \tau N_{\mathrm{pol}} (2 N_{\mathrm{baselines}} + N_{\mathrm{ant}})}} \end{equation}

where we introduce the following quantities:

Equation (17) differs slightly from the standard interferometer radiometer equation (Equation 9.26, Wilson et al. Reference Wilson, Rohlfs and Hüttemeister2013) as it includes a $N_{\mathrm{ant}}$ term to account for autocorrelations. Noting that

(18) \begin{align} N_{\mathrm{baselines}} &= N_{\mathrm{ant}} (N_{\mathrm{ant}} - 1) / 2 \end{align}

(19) \begin{align} N_{\mathrm{ant}}^2 &= 2N_{\mathrm{baselines}} + N_{\mathrm{ant}}\end{align}

one finds that Equation (17) reduces to the tied-array radiometer equation if all baselines are included:

(20) \begin{equation} \Delta S_{\mathrm{tied}} = \frac{2 k_B T_{\mathrm{sys}}}{A_{\mathrm{ant}} N_{\mathrm{ant}} \sqrt{\Delta \nu \tau N_{\mathrm{pol}} }}. \end{equation}

Additionally, if cross-correlation terms are not included (i.e. $N_{\mathrm{baselines}} = 0$ ), then we retrieve the radiometer equation for an incoherent beam:

(21) \begin{equation} \Delta S_{\mathrm{incoherent}} = \frac{2 k_B T_{\mathrm{sys}}}{A_{\mathrm{ant}} \sqrt{\Delta \nu \tau N_{\mathrm{ant}} N_{\mathrm{pol}} }}. \end{equation}

If we choose a subset of baselines, based on maximum length, the resulting noise will lie between the ideal $\Delta S_{\mathrm{tied}}$ thermal noise (all baselines included) and the incoherent $\Delta S_{\mathrm{incoherent}}$ noise (only autocorrelations included). Put another way, reduced-resolution beamforming provides access to resolution and sensitivity regimes that lie between that of incoherent and coherent beamforming techniques.

3.2 Fractional sensitivity

Consider the case where a maximum baseline length d is imposed on an array with longest baseline D. For a tied array beamformer, the sensitivity can be treated as a function of baseline length, $\Delta S_{\mathrm{coherent}}(d)$ (Equation (20)), with the number of antennas in the sub-array is $N^{\prime}_{\mathrm{ant}}(d)$ . For a post-correlation beamformer (Equation (17)), the thermal noise level reached $\Delta S_{\mathrm{post-x}}(d)$ depends on $N^{\prime}_{\mathrm{baselines}}(d)$ .

It is useful to define a fractional sensitivity factor, $f_{S}$ , that relates the sensitivity of the selected subarray to the full array:

(22) \begin{equation} \Delta S(d) = \Delta S(D) \big/ f_{S}.\end{equation}

For a reduced-resolution beam with all autocorrelations, $N^{\prime}_{\mathrm{ant}} = N_{\mathrm{ant}}$ , and

(23) \begin{equation} f_{S} = \frac{\Delta S_{\mathrm{post-x}}(D)}{\Delta S_{\mathrm{post-x}}(d)} = \frac{\sqrt{2 N^{\prime}_{\mathrm{baselines}}(d) + N_{\mathrm{ant}} } }{N_{\mathrm{ant}}}\end{equation}

whereas for a tied-array beam

(24) \begin{equation} f_{S} = \frac{\Delta S_{\mathrm{tied}}(D)}{\Delta S_{\mathrm{tied}}(d)} = \frac{N^{\prime}_{\mathrm{ant}}(d)}{N_{\mathrm{ant}}}\end{equation}

and $N^{\prime}_{\mathrm{ant}} \leq N_{\mathrm{ant}}$ .

Fig. 3 shows the fractional sensitivity for a reduced-resolution beam and tied-array beam as a function of maximum baseline length, using the EDA2Footnote i array as an example (Wayth et al. Reference Wayth2021). The reduced-resolution beam always has a larger fractional sensitivity than a tied-array beam from a sub-selection of antennas. This fact is a key motivation for reduced-resolution beamforming.

Figure 3. Comparison of EDA2 fractional sensitivity for a tied-array beam (red) and a post-x beamformed beam (black), as a function of maximum baseline length $d_{\mathrm{max}}$ . The post-x approach allows short baselines to be included, boosting sensitivity.

The slope of $\Delta S_{\mathrm{post-x}}(d)$ depends on the antenna configuration. Fig. 4 considers the application of reduced-resolution beamforming to three radio arrays: the 64-antenna MeerKAT telescope (Camilo Reference Camilo2018), the 128-tile MWA (in compact configuration, Wayth et al. Reference Wayth2018), and the EDA2 (Wayth et al. Reference Wayth2021). The top panels show the antenna layout for each array, and the middle panels show baseline distribution histograms as a function of baseline length. The lower panel shows the fractional sensitivity for each array as a function of maximum baseline length (solid black line), and the number of beams required to fill the primary field of view, assuming the beam width scales as $\lambda/d$ (dashed red line).

Figure 4. Sensitivity analysis for reduced resolution beamforming applied to MeerKAT, the MWA (compact configuration), and the EDA2. The top panels show antenna location, and the middle panels show histograms of baseline lengths for each telescope. The bottom panel shows the fractional sensitivity for reduced resolution beamforming, as a function of maximum baseline length (black lines). The minimum sensitivity is equivalent to incoherent beamforming, and maximum sensitivity is equivalent to standard beamforming. The number of beams required to cover each telescope’s field of view scales with $d_{\mathrm{max}}^2$ (dashed red lines).

A key takeaway from Fig. 4 is that all telescopes reach a reasonable fractional sensitivity ( $\sim$ $0.5$ ) with an order of magnitude fewer beams to fill their field of view than the full array. In the coming sections, we will explore how this finding could be applied on these telescopes.

3.3 Survey speed

Survey speed is a useful metric for optimising the performance of an array. The point-source survey speed figure of merit (PFoM) relates to the time taken to survey a field with solid angle $\Omega_{\mathrm{survey}}$ to a required thermal noise level in Jy, $\Delta S_{\mathrm{survey}}$ . Following Equation 3 of Cordes (2009),Footnote j we may define a PFoM modified in a similar fashion to Chen et al. (Reference Chen, Barr, Karuppusamy, Kramer and Stappers2021) to account for number of beams and maximum baseline length.

The fraction of the maximum PFoM (all antennas, beams covering instrumental FoV) reached with a reduced-resolution beamforming system is

(25) \begin{equation} f_{\mathrm{PFoM}} = \frac{{\mathrm{PFoM}}(d)}{\mathrm{PFoM_{full}}} = f^2_S \times f_{\mathrm{FoV}}\end{equation}

where the fractional coverage of the instrument’s field of view ( $\Omega_{\mathrm{full}}$ ), depends on the number of beams formed:

(26) \begin{equation} f_{\mathrm{FoV}} = \frac{\Omega_{\mathrm{FoV}}(d)}{\Omega_{\mathrm{full}}} \approx \frac{N_{\mathrm{beam}}}{\Omega_{\mathrm{full}}} \left(\frac{\lambda}{d} \right)^2,\end{equation}

assuming resolution follows the $\lambda/d$ rule of thumb. However, the actual beam width depends on both the array configuration and the beam’s declination in a complex fashion. Many radio arrays have decreasing antenna density for longer baselines, for which $\lambda/d$ underestimates the beam width. Nevertheless, if the number of beams is fixed, we expect the fractional field of view to decrease rapidly as d increases.

3.4 Computational cost

The approximate computational cost of a correlator and multi-pixel beamformer can be written in terms of complex-multiply accumulate operations per second (CMACs/s) as:

(27) \begin{align} {\mathrm{C_X}} &= \left(N_{\mathrm{ant}} N_{\mathrm{pol}}(N_{\mathrm{ant}} N_{\mathrm{pol}} + 1)\right) \times \Delta \nu \end{align}

(28) \begin{align} {\mathrm{C_B}} &= \left(N_{\mathrm{beam}} \, N_{\mathrm{ant}} \, N_{\mathrm{pol}} \right) \times \Delta \nu\end{align}

where $\Delta\nu$ is the bandwidth of data processed. To form post-correlation beams from the correlator output requires an additional step, with computational cost

(29) \begin{equation} {\mathrm{C_{pxb}}} = f_{\mathrm{baselines}} \times \left( N_{\mathrm{pxb}} N^2_{\mathrm{ant}} N^2_{\mathrm{pol}} \right) \times \frac{\Delta \nu}{N_{\mathrm{int}}} \end{equation}

where $N_{\mathrm{int}}$ is the number of time samples summed during integration, $f_{\mathrm{baselines}}$ is the fraction of baselines included, and $N_{\mathrm{pxb}}$ is the number of post-correlation beams. Note that the cost of FFT-based imaging scales proportionally to $N_{\mathrm{pxb}} {\mathrm{log}}_2 N_{\mathrm{pxb}}$ , and for most arraysFootnote k

(30) \begin{equation} {\mathrm{log}}_2 N_{\mathrm{pxb}} \ll N_{\mathrm{ant}}^2,\end{equation}

so we stress that the PXB approach is generally far more computationally expensive than the 2D FFT approach employed in imaging pipelines. That said, for fair comparison with computationally expensive searches – where only a few beams can be processed on any one compute server – we do not compare against FFT-based imaging any further in this article.

Based on Equations (28)–(29), tied-array beamforming is more computationally expensive if $N_{\mathrm{beams}} \lessapprox N_{\mathrm{pxb}}$ and/or if the correlator does not integrate many time samples, that is, $N_{\mathrm{int}}$ is large.

This conclusion is consistent with Roy et al. (Reference Roy, Chengalur and Pen2018), which considers post-correlation beamforming techniques for pulsar studies with the Giant Metrewave Radio Telescope. Roy et al. (Reference Roy, Chengalur and Pen2018) also concludes that post-correlation beamforming is computationally cheaper for a large number of beams at low time resolution, and regular beamforming is cheaper for a small number of high time-resolution beams.

As with imaging, the cost of post-correlation beamforming (Equation (29)) can be reduced significantly if visibilities are gridded and a 2D FFT is used (Briggs et al. Reference Briggs, Schwab, Sramek, Taylor, Carilli and Perley1999). However, the FFT approach is not be suitable for a small number of beams and/or widefield imaging; further, when distributing processing tasks across a supercomputer, it may be advantageous to have only a small number of beams per node. As such, we present values for the non-gridded approach.

4.1 Trading sensitivity against compute requirements

The post-processing requirements for SMART scale proportionally with the number of beams searched. By applying reduced-resolution beamforming, the SMART survey volume can be searched with fewer beams, potentially improving computational requirements by an order of magnitude for a moderate drop in sensitivity. However, any improvement in downstream processing costs must be offset against any increases in beamforming costs. Using Equations (28)–(29) and the SMART survey parameters (Table 1), the computational costs are as follows:

(31) \begin{align} {\mathrm{C_X}} &= 16.2 \ {\mathrm{TCMAC/s}} \end{align}

(32) \begin{align} {\mathrm{C_B}} &= 767.6 \ {\mathrm{TCMAC/s}}.\end{align}

The comparable computational cost for reduced-resolution beamforming ( $C_{\mathrm{PXB}}$ ) as a function of fractional sensitivity $f_s$ is shown in Table 2. The computational cost is considerable, as the integration time $\tau=100\,\mu$ s only allows $N_{\mathrm{int}}=2$ . For such a large number of beams, computational cost could be reduced significantly by using a 2D FFT imaging approach. Regardless, Selecting $f_s=0.5$ yields $C_{\mathrm{PXB}}=1\,146$ TMAC/s, that is, roughly 1.5 $\times$ the cost of tied-array beamforming, and the current SMART tiling approach could be maintained.

Table 1. MWA SMART survey parameters for a single zenith pointing.

Table 2. Computational requirements and corresponding sensitivity limits for MWA reduced-resolution beamforming, for increasing fractional sensitivity.

While the beamforming cost is higher, the number of beams to search drops from 6 100 to 277. It follows that post-processing requirements will drop by a factor of 22 $\times$ , while sensitivity drops only by a factor of 2 $\times$ . We extrapolate that the post-processing requirements on OzSTAR would drop from $\sim$ $14\,000$ to $\sim$ 636 kSU. The SMART team anticipates securing 500–600 kSU per annum (Bhat et al. Reference Bhat2023a), meaning it is plausible that the entire survey could be processed within a year if reduced-resolution beamforming is adopted (instead of an implausible 22 years).

The processing requirements for SMART may also be alleviated by planned improvements to the performance of their pulsar search pipeline. The SMART pipeline currently uses PRESTO, a CPU-only code written primarily in ANSIC (Ransom Reference Ransom2011). GPU search codes, such as astroacclerate (Armour et al. Reference Armour2020) and peasoup (Barr Reference Barr2020), are hoped to offer an order-of-magnitude improvement. For example, the astroacclerate Fourier domain acceleration search is up to $8\times$ faster than its PRESTO equivalent (Adámek et al. Reference Adámek, Dimoudi, Giles, Armour, Ballester, Ibsen, Solar and Shortridge2020); recent work to exploit half-precision data types may offer an additional $1.6\times$ speedup (White et al. Reference White2023). Alleviating I/O bottlenecks, such as disc read speed, may also improve efficiency. Regardless, any speedup from pipeline optimisation is complementary to the reduced-resolution beamforming approach.

4.2 Application to technosignature searches

A narrowband technosignature search of SMART data has been proposed, which would be the first all-sky technosignature search in the Southern Hemisphere. As such, a SMART technosignature search would place some of the most stringent constraints on the prevalence of putative engineered transmitters in the Galaxy. It is argued that all-sky SETI searches at low frequency are one of the most compelling methods to detect evidence of technologically capable life beyond Earth (Garrett et al. Reference Garrett, Siemion and van Cappellen2017; Houston et al. Reference Houston, Siemion and Croft2021). Narrowband technosignature searches do not require high time resolution, which decreases the computational cost of post-correlation beamforming. For example, decreasing the time resolution from $100\,\mu$ s to 1 s decreases $C_{\mathrm{PXB}}$ (Equation (29)) by a factor of 10 000.

However, narrowband technosignature searches require high-frequency resolution, meaning that large data volumes must be stored in memory on which a large FFT can be performed. This will limit the number of beams that can be processed at any time. As with the pulsar search, the post-processing requirements for narrowband technosignatures scale linearly with the number of beams. Similar trade-offs between sensitivity and number of beams are thus well motivated for technosignature surveys.