Design of Multichannel Spectrum Intelligence Systems using Approximate DFT Algorithm for Antenna Array based Spectrum Perception Applications

1. Introduction

In Tolkien’s classic The Lord of the Rings, an omnipresent “All Seeing Eye” senses, and hitherto inspires inhabitants of Middle Earth to act based on their proximity to objects under the influence of Lord Sauron [1]. In our world, we will enable persistent and omnipresent knowledge of the electromagnetic spectrum via sensing across space, time, and waveform, through radio frequency (RF) perception, and thus enable intelligent action for AI based DSA. We need an RF All-Seeing Eye for the spectrum. The fusion of RF spectrum sensing with AI and machine learning (ML) enables the full perception of the spectrum. Omnipresence, which implies everywhere, and persistence which implies all the time for information on what might be happening in the electromagnetic environment will lead to unimaginably powerful capabilities for tomorrow’s intelligent wireless systems. The spectrum is an immensely valuable and scarce natural resource.; however the exponential growth in wireless connectivity inevitably leads to running out of spectrum. The problem of spectrum shortage can in theory be mitigated by moving communication networks to the mm-wave band in the 30–300 GHz range. Nevertheless, industry has recently learnt a bitter lesson with mm-wave. Indeed, mm-wave technology works in a laboratory environment; yet, a number of issues are present: (i) it is difficult to get to work across real-world environments as the sharp RF beams necessary for maintaining connectivity are prone to blockages, (ii) beam-search is a tough problem to solve specially considering mobility aspects, (iii) power amplifiers (PA) are inefficient and waste a lot of energy, and (iv) channel coherence times are very low requiring faster updates of channel state information (CSI) leading to high computational load.

1.1. FR3 Band; The Goldilocks Spectrum

On the one-hand, FR1/sub-6 GHz frequency bands—the legacy spectrum—is simply not available for expanded growth. Indeed, the 100 MHz sub-band at 3.5 GHz was sold at an eye-popping, $29B clearly showing the demand from the wireless industry. However, all the money in the world cannot get you resources that do not exist. Given that mm-wave is unlikely to lead to practical systems in the near future, industry is turning to a middle ground. This spectral middle ground lies in the FR3 band (6–24 GHz), affectionately known as the upper-mid band by industry players. In our recent work, we show that the upper-mid band is not too low that it is overcrowded as the FR1 (legacy band) and is not too high that building practically scalable infrastructure is not currently viable. The FR3 band sits “just right” as it lies within a range that is favorable for scalable systems. Additionally, bandwidth is available in FR3 for the next generation wireless, provided that a tangle-web of spectrum policy can be navigated [2].

1.2. A Shared Spectrum Economy

There is a pressing problem with using the upper-mid band for wireless connectivity. The 6–24 GHz has already been assigned for mission critical applications and its licenses are owned by the Department of Defense (DoD) and other government entities. Most of the DoD mission critical sensors, including X-band radar, communication networks, science experimental bands (e.g., radio astronomy), and weather radar sensors also exist within the FR3 band. Primary users (PUs) will likely never give up their “precious” spectrum permanently; however, the future will likely necessitate DSA where secondary users (SUs) such as commercial wireless networks can make use of the FR3 bands in a way that poses little or no threat to critical applications pursued by PUs. Although DSA can bring massive benefits and economic growth to society, such great power comes with great responsibility as compliance to sharing agreements

MUST be ensured with non-repudiation and zero trust.

1.3. The Need for Real-Time AI-Spectral Perception

“DoD determined that sharing is feasible if certain advanced interference mitigation features & a coordination framework to facilitate spectrum sharing are put in place” proclaims the 2023 National Spectrum Strategy from the White House [3]. The early detection of acts-of-war sits higher on the list of priorities than commercial wireless network quality of service. Although the chance of a cataclysmic event is small, it is not zero. The only way to ensure SUs do not encroach on PU spectrum allocations across space, time, and waveform domain is through independent measurements assuming zero trust to fully guarantee the compliance to spectrum sharing agreements. In addition to genuine mistakes and unintentional harmful RF interference, there are impending cyber-threats from bad actors hitherto unknown. The possibility of jamming of critical wireless and defense infrastructure should not be taken lightly. Reliable, high-sensitivity measurements, with RF sensing everywhere all the time allows better resilience to such future threats from a spectrum warfare standpoint.

1.4. Spectrum Sensors with Omnipresent Perception

RF autonomy due to AI systems created using deep learning (DL) and reinforcement learning (RL) are expected to generate a deep knowledge—that is “perception”—about the sensed electromagnetic environment [4]. National Spectrum Strategy (The White House, 11/2023) calls for initial studies in the 3.1–3.45 GHz, 5.030–5.091 GHz, 7.125–8.400 GHz, and 18.1–18.6 GHz bands towards DSA between commercial and DoD entities. AI-enabled perception calls for RF spectrum sensing and the detection of waveform parameters (e.g., modulation and perhaps even coding parameters [5]), carrier-aggregation and bandwidth parts or similar information, propagation directions, power levels and channel behaviors such as multipath, Doppler, and inter-symbol interference (ISI) metrics [6]. Typical spectrum sensors likely do not provide actionable intelligence on the spectrum except for usage statistics [7,8,9,10], where the detection of “spectrum holes” determines sub-band(s) possibly available for SUs [11,12,13,14,15,16,17]. Spectrum fraud is a major issue [18]. The presence of harmful RF interference (RFI) can only be proven by independent third-party sensing and perception. To internalize a model of the spectral landscape, we will need a good understanding of the measured RF environment; simple energy binning using an FFT is too limiting for future use cases. Mitigating RFI events everywhere all the time is challenging and calls for detection of jamming, pulse-weapons, and spectrum fraud. We need real-time perception using machine intelligence as events happen everywhere all the time at rates of thousands of events per second thus ruling out any human-in-the-loop approaches. Fully AI-enabled perception of RF environment is of critical importance and serves as an intelligent information source to drive automated DSA [19].

2. Review

2.1. Dynamic Spectrum Access: Signal Processing and ML/DL Approaches

DSA was first considered two decades ago based on the measurements of actual spectrum usage performed by the Spectrum Policy Task Force of Federal Communications Commission (FCC) [20]. On the contrary to the majorities belief on a well-crowded spectrum, actual measurements indicated that many portions of the spectrum are idle at a given time and location, leading to spectral holes. This results from the fixed spectrum access policy. Under DSA, secondary users (i.e, unlicensed users) were allowed to utilize spectral holes of primary users (i.e., licensed users) without interfering to the operation of primary users through a cognitive radio [21,22,23,24], which is an intelligent system built on a software-defined radio. Here, we briefly review spectrum sensing techniques associated with DSA whereas the reader is referred to [25,26,27] for other aspects of DSA, such as spectrum management and spectrum sharing policies.

Several techniques have been proposed for both narrowband and wideband spectrum sensing. For narrowband sensing, the main non-ML techniques are: energy detection [28,29,30,31], matched filter based detection [32,33,34], cyclostationary features detection [35,36], and covariance based detection [37,38,39]. The energy detection and covariance based detection approaches do not require prior knowledge on the primary user’s signal; however, the former is not reliable at low signal-to-noise ratio (SNR) values. Matched filter based detection provides optimal sensing and better detection at low SNR values, but this technique requires prior knowledge on the primary user’s signal, which may not always be available. The cyclostationary features detection is generally robust against noise uncertainly and works well at low SNR values, however, it suffer from large sensing time. In wideband sensing, as an straight forward approach, a wideband signal may be divided into a set of narrowband signals via temporal DFT or a filter bank [40,41,42,43] and sense sequentially or parallelly using narrowband sensing techniques. However, sequential sensing requires longer time whereas parallel sensing requires more resources. Because a wideband signal with spectral can be considered as a sparse signal in the frequency domain, several compressive sensing based techniques have been proposed for wideband sensing [44,45,46,47]. These approaches require lower sampling rate and have lower power consumption in addition to the fast sensing. Thus, compressive sensing based approaches eliminate most of the drawbacks of sequential or parallel narrowband sensing of a wideband signal.

In the last decade, a considerable number of ML and DL techniques have been proposed for spectrum sensing. ML algorithms have been mostly used to determine the channel occupancy patterns of a primary used and to classify the channel as either free or occupied. In [48,49,50,51,52], K-means clustering and Gaussian mixture models have been first used to determine the presence of primary users and a classifier such as support vector machine and K-nearest-neighbor have then been employed to determine a channel is free or occupied. DL techniques such as convolutional neural networks and long short-term memory have been employed for spectrum sensing in [53,54,55,56], which outperformed the conventional techniques. Recently, a transformer-based DL architecture has been proposed in [57]. This DL approach outperform the previously proposed approaches based on convolutional neural networks. Another DL approach employed for spectrum sensing in a cooperative way is deep reinforcement learning [58,59,60,61]. These deep reinforcement learning improves the robustness of the spectrum sensing system and allows to make more accurate decisions in dynamic environments. The reader is referred to [62,63,64] for a comprehensive review of spectrum sensing techniques.

Spatio-temporal modeling of cognitive radio systems using multi-dimensional signal processing concepts has been presented in [65]. Here, directional sensing has been exploited using three-dimensional infinite impulse response (IIR) filters as beamformers. Similar directional sensing approaches using multi-dimensional filters were presented in [66,67,68]. Furthermore, antenna array systems have been employed for directional sensing in [69,70,71]. In [72], DFT-based multibeamformer has been employed for real-time directional sensing. Compared to the multibeamformer employed with ADFT in this paper, exact DFT has been employed in [72] with 16 simultaneous beams at 2.4 GHz.

2.2. Radio Astronomy and RFI

Radio astronomy relies on a minimally contaminated radio spectrum to observe the universe [73]. Detecting and estimating weak celestial signals arriving from far away galaxies are particularly required in achieving the key science goals of the next generation radio telescopes, i.e. The Square Kilometre Array (SKA) [74], The 2030 Atacama Large Millimeter/sub-millimeter Array - Wideband Sensitivity Upgrade (ALMA-WSU) [75] and The next-generation Very Large Array (ngVLA) [76]. Despite being located at remote sparsely populated regions in the world, the increasing intensity, frequency, bandwidth, and occurrences of RFI are threatening the utility of the next generation radio telescopes [77]. RFI from Global navigation satellite system (GNSS) [78], terrestrial and airborne wireless communications systems, satellite mega constellations [79] and other technologies are encroaching on the spectra that have been exclusively used by radio astronomers, making it harder to distinguish faint celestial signals from RFI.

The International Telecommunication Union (ITU) and the FCC have important roles in protecting the spectrum for radio astronomy and other passive users. The ITU, through its Radio communication Sector (ITU-R), is responsible for developing international regulations and recommendations for the use of radio frequencies. It recognizes the significance of protecting the spectrum for passive services, such as radio astronomy, Earth exploration-satellite services, and meteorological satellite services. The ITU-R identifies specific frequency bands for these passive services and establishes regulatory provisions to ensure their protection from harmful interference. These provisions include coordination procedures, power limits, and frequency separation requirements to safeguard the sensitive observations and measurements conducted by radio astronomers and other passive users [80,81]. Similarly, the FCC in the United States recognizes the importance of protecting the spectrum for radio astronomy and other passive services. The FCC’s Office of Engineering and Technology (OET) formulates policies and rules to prevent harmful interference to these services. It establishes technical standards and licensing conditions that take into account the needs of passive users, including radio astronomers. Also, the FCC works closely with the National Science Foundation (NSF) and other relevant agencies to coordinate spectrum usage and protect the integrity of scientific observations. However, both organizations are under relentless pressure from the industry and governments for more commercial utilization of the spectrum that has been previously allocated for the non-commercial users.

Therefore, ideally, the awareness of the key characteristics of RFI (i.e., frequency range, modulation scheme, source locations, intensity, etc.) would help the radio telescopes to plan and schedule the observations. Nevertheless, this is not practical for all cases of RFI, therefore, records of radio spectral strength in terms of strength, direction, duration, and time of incident would also help the radio astronomers to flag and excise the contaminated observations [82]. In this case, the omnipresent perception of the EM environment would be a huge boon in the post processing stage for radio astronomical observations [83].

2.3. Approximate DFT

The discrete Fourier transform is a linear operator that relates an input N-point signal $\mathbf{x}={\left[\begin{array}{cccc}x\left[0\right]& x\left[1\right]& \dots & x[N-1]\end{array}\right]}^{\top }$ to the output signal $\mathbf{X}={\left[\begin{array}{cccc}X\left[0\right]& X\left[1\right]& \dots & X[N-1]\end{array}\right]}^{\top }$ according to the following expression:

$$\begin{array}{c}\hfill X\left[k\right]=\sum _{n=0}^{N-1}x\left[n\right]·{\omega }_N^{-nk},k=0,1,\dots ,N-1,\end{array}$$

(1)

where ${\omega }_N=exp(2\pi j/N)$ is the Nth root of the unity and $j\triangleq \sqrt{-1}$. Computed by definition, the DFT presents a computational complexity in $O\left(N^2\right)$ which is prohibitively high for real-time applications.

In practice, the DFT is computed by means of one of many efficient numerical routines—fast algorithms, which are collectively referred to as “FFTs”. Particular popular choices of FFTs are the radix-2-based algorithms, such the Cooley-Tukey FFT, which can recursively decompose the N-point DFT block in two $N/2$-point DFT sub-blocks [84].

In general, the FFTs are capable of reducing the computational complexity of the DFT computation from $O\left(N^2\right)$ to $O(NlogN)$ [85]. In many cases [86,87,88,89,90,91], such computational cost reduction achieves the theoretical minimum multiplicative complexity of the DFT [92]. As a consequence, it is impossible to further reduce the multiplicative complexity. Being a quite mature area of research, the proposition of new FFTs capable of offering significant reductions in computation cost is an unlikely event.

However, noticeable reductions can be accomplished under a fault-tolerant paradigm. If the DFT computation is permitted a given error level, then the computational effort can be re-adjusted to meet the lower, but acceptable, precision level. As a consequence, fewer arithmetical operations are needed for the DFT estimation [93].

Systematically, a way to obtain such low-complexity DFT estimators is by means of matrix approximation theory. In matrix formalism, the DFT is described by the following operation

$$\begin{array}{c}\hfill \mathbf{X}={\mathbf{F}}_N·\mathbf{x},\end{array}$$

(2)

where the N-point DFT matrix, denoted by ${\mathbf{F}}_N$, is the $N\times N$ matrix whose elements are given by $f_{k,n}={\omega }_N^{-nk}$, $k,n=0,1,\dots ,N-1$.

In the context of approximations for discrete transforms, qualitatively, an approximate discrete transform, ${\widehat{\mathbf{F}}}_N$, is a transform matrix such that $\widehat{\mathbf{X}}={\widehat{\mathbf{F}}}_N·\mathbf{x}\approx \mathbf{X}={\mathbf{F}}_N·\mathbf{x}$. In other words, the exact and the approximate transform-domain signals are “close” in some relevant sense to be quantitatively defined. Generally, an approximation transform must satisfy the following conditions: (i) it can largely preserve meaningful properties of the exact transform; (ii) mathematical proximity between the exact and approximate transform matrix according to a contextually relevant metric function—usually a performance figure of merit; and (iii) it possesses a significantly lower computational cost, compared to the exact computation by FFTs. In [94] an overview of the theory is made available.

Usually, in order to preserve the physical interpretation of the approximate spectrum, approximate transformation matrices are sought to be close in the Euclidean sense to the associate exact transformation matrix. This can be accomplished by the minimization of the Frobenius norm of the difference between the exact and candidate matrices for approximation, subject to the restriction of low-complexity. Therefore, a low-complexity approximation can be obtained by restricting its elements to numerical sets of trivial multiplicands, such as $\{0,\pm 1\}$ or $\{0,\pm 1,\pm 2\}$.

In symbols, a possible formulation for the problem of deriving a DFT approximation is given by [94]:

$$\begin{array}{cc}\hfill \underset{{\widehat{f}}_{k,n}}{min}& \parallel {\mathbf{F}}_N-\mathbf{S}·{\widehat{\mathbf{F}}}_N{\parallel }_{\mathrm{F}}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \mathrm{subject}\mathrm{to}& {\widehat{f}}_{k,n}\in {\{0,\pm 1\}}^2,\hfill \end{array}$$

(4)

where ${\widehat{f}}_{k,n}$, $k,n=0,1,\dots ,N-1$, are the entries of the approximate matrix ${\widehat{\mathbf{F}}}_N$, ${\parallel ·\parallel }_{\mathrm{F}}$ denotes the matrix Frobenius norm, and $\mathbf{S}$ is a normalization matrix to ensure that the basis vectors of $\mathbf{S}·{\widehat{\mathbf{F}}}_N$ have energy equal or near to one.

In this work, we adopted the ADFT introduced mathematically in [95] and further elaborated in theory and in hardware in [96,97]. Figure 1 shows a comparison of the magnitude responses for the 32-point exact DFT and the 32-point ADFT.

Represented by the low-complexity matrix ${\widehat{\mathbf{F}}}_{32}$, the selected ADFT admits the following factorization:

$$\begin{array}{c}\hfill {\widehat{\mathbf{F}}}_{32}={\mathbf{W}}_8·{\mathbf{W}}_7·{\mathbf{W}}_6·{\mathbf{W}}_5·{\mathbf{W}}_4·{\mathbf{W}}_3·{\mathbf{W}}_2·{\mathbf{W}}_1,\end{array}$$

(5)

where ${\mathbf{W}}_i$, $i=1,2,\dots ,8$, are sparse, low-complexity matrices. In the Appendix A, we provide the explicit definition of matrices ${\mathbf{W}}_i$, $i=1,2,\dots ,8$.

For comparison, Table 1 lists the arithmetic operation count for the exact 32-point DFT computed according to its definition and to the radix-2 Cooley-Tukey FFT. For this operation counting, we assume that (i) the input signal is purely real-valued; (ii) trivial multiplications by 0, $\pm 1$, or $\pm j$ are not counted; and (iii) a multiplication between a real number and a complex number is performed as two real multiplications—not a complex multiplication [98]. Due to (i) and (iii), the 704 non-trivial multiplicands of ${\widehat{\mathbf{F}}}_{32}$ are equivalent to 1408 real multiplications. The evaluation of the number of additions considered the sum of the number of real- and imaginary-part additions. It is also provided the arithmetic complexity for the discussed 32-point ADFT by its definition and by means of its fast algorithm (factorization). The complexity analysis of the radix-2 Cooley-Tukey FFT is detailed in [98].

Table 2 shows the arithmetic complexity for each individual matrix term of the ${\widehat{\mathbf{F}}}_{32}$ factorization. The matrix ${\mathbf{W}}_8$ does not present arithmetic costs because it consists of only sign-changing and data-swapping (real- and imaginary-part interchanging) operations. The data interchange is a consequence of the multiplication by j (a rotation by $\pi /2$).

3. System Overview

3.1. Proposed Architecture

To experimentally validate and measure the beams generated by the proposed ADFT algorithms, a 5.7 GHz RF system with a digital processing back-end was meticulously designed and implemented. Figure 2(a) shows the overall architecture of the system based on a 32-element antenna array, encompassing key subsystems such as the antenna array itself, the RF receiver chain, and the digital hardware unit. Figure 2(b) shows the experimental setup of the proposed architecture. In addition to the digital cores responsible for executing the spatial DFT of the sampled signals, supplementary circuitry was developed within the FPGA. This additional circuitry facilitates real-time data manipulation and beam measurements across different angles, enhancing the flexibility and capability of the system.

The re-configurable open architecture computing hardware (ROACH-2) based digital signal processing (DSP) platform is shown in Figure 3 which plays a crucial role by serving as an intermediary layer, enabling communication between the FPGA hardware (specifically its memory) and the onboard computer. The ROACH-2 board interfaces with the main host Linux server via a 1 Gbps Ethernet connection. A Python routine running on the host server is programmed to interact with the ROACH-2 platform, performing control functions and retrieving data from the FPGA memory while iteratively scanning through different angles. This unique system architecture not only ensures precise beam measurements but also enhances the overall efficiency and automation of the experimental providing a robust platform for validating advanced beamforming algorithms.

3.2. Analog Front-Ends

The RF front end of the receive-mode beamformer is constructed using a 32-element Uniform Linear Array (ULA) operating at 5.7 GHz, paired with 32 direct conversion RF receiver chains on printed circuit boards (PCBs). The array elements are spaced at $0.6\lambda$, which equates to approximately 32 mm at 5.7 GHz. Each antenna element is designed as a vertical subarray consisting of $4\times 1$ patch antennas, facilitating passive beamforming in the vertical plane. This analog beamforming operates independently of the spatial FFT-based beamforming algorithms discussed in this work.

The outputs from the antenna elements are directly connected to 32 heterodyne receivers. These receivers utilize a centralized local oscillator (LO) system, which includes a low-phase-noise oscillator linked to a 32-output power divider network. Each receiver’s initial stage features a low-noise amplifier (LNA) that delivers 16 dB gain at 5.8 GHz with a noise figure of 2.4 dB. Following amplification, the signals are filtered through a bandpass filter spanning 4.7–6 GHz to eliminate out-of-band noise and interference. The filtered signals are then mixed and passed through low-pass filters to produce downconverted low intermediate frequency (IF) inputs.

Subsequently, these 32 low-IF signals are amplified by approximately 30 dB and digitized using two ADC16x250-8 analog-to-digital converter (ADC) cards. Each ADC card supports 16 single-ended input channels, provides 8-bit resolution, and operates at sampling rates up to 250 $\mathrm{M}$$\mathrm{sample}/$$\mathrm{s}$ per channel. This setup ensures high-fidelity digitization of the received signals for further processing [96].

3.3. ADFT Cores and Digital ADFT Spectral Estimation

The discussed 32-point ADFT was implemented by utilizing Xilinx tools. The sparse matrices from the factorization shown in (5) were structured as parallel input-output blocks to optimize performance [96,97]. Verification of these blocks was conducted through MATLAB Simulink, ensuring the reliability of the design before hardware deployment. Subsequently, the verified ADFT design was mapped onto the Xilinx Virtex-6 sx475T FPGA.

To integrate the ADFT core into the ROACH-2 FPGA platform, we utilized two libraries from the Collaboration for Astronomy Signal Processing and Electronics Research (CASPER) community [100,101]. This integration enabled the system to generate 32 simultaneous RF beams in parallel. Each beam corresponds to a discrete direction, calculated using the function ${\theta }_k=arcsin(k/32)$, k ranges from $-16$ to 15. The successful implementation of the ADFT on the ROACH-2 FPGA not only validated the algorithm effectiveness but also demonstrated the practical application of advanced digital hardware design techniques. This work highlights the potential of ADFT in various applications requiring efficient RF beamforming.

We implemented a spatial windowing technique to address the issue of high side lobes in the ADFT, where the worst-case side lobe was observed to be 1.5 dB higher than that of the (exact) DFT. Four configurations were tested: three windowing functions (Butterworth, Hamming, and Hanning) and a baseline with no windowing [102]. The selection among these windowing functions was managed using a multiplexer equipped with a control selector pin.

Figure 4 shows the spatial windowing unit coupled with the ADFT sparse matrices. In mathematical terms, the windowing is introduced as diagonal matrix. The ADFT-based spectral estimation can be described by combining the windowing diagonal matrix (${\mathbf{D}}_{\mathrm{windowing}}$), the low-complexity ADFT core (${\widehat{\mathbf{F}}}_{32}$), and a power-normalizing diagonal matrix (${\mathbf{D}}_{\mathrm{normalizer}}$). The mathematical representation of the proposed approximate spectrum estimation is given by:

$$\begin{array}{c}\hfill {\mathbf{ADFT}}_{\mathrm{estimator}}={\mathbf{D}}_{\mathrm{normalizer}}·{\widehat{\mathbf{F}}}_{32}·{\mathbf{D}}_{\mathrm{windowing}}.\end{array}$$

(6)

Figure 5 illustrates the ADFT response and its modifications under the windowing techniques. It is noteworthy that the introduction of windowing functions resulted in wider main lobes, which is a compromise made to suppress the side lobes. This trade-off between main lobe width and side lobe suppression is a consideration in the application of spatial windowing techniques. However, implementing spatial windowing increases the hardware complexity due to the multiplexers introduced by the windowing unit. Therefore, user must select the appropriate configuration based on their specific requirements and constraints.

3.4. Digital FFT Spectrometers

The system described involves the implementation of a spatial FFT that generates 32 parallel outputs, each corresponding to a distinct spatial RF beam. Subsequent to this spatial FFT, a temporal FFT spectrometer is applied to each of these 32 spatial RF beams, achieving channelization in the temporal frequency domain. Each temporal FFT is computed with 1024 points, necessitating the use of 32 parallel FFT cores. This parallel processing is facilitated by leveraging real-sampled biplex FFT blocks available in the Xilinx Virtex-6 SX475T FPGA.

The real-sampled biplex FFT blocks exploit the Hermitian conjugation method, enabling the computation of the real-sampled FFT. This technique utilizes a complex FFT core to simultaneously transform two real-valued data streams. As a result, the FFT operation produces all frequencies, both positive and negative, with negative frequencies being the mirror images of their positive counterparts[100].

The temporal FFT outputs for each beam yield 1024 complex values corresponding to 1024 discrete frequency bins. The power spectral density (PSD) is estimated by squaring the magnitude of each frequency bin. This estimate is derived by averaging across 1024 temporal FFT evaluations, resulting in a new PSD measurement every 1 million clock cycles. A Block random access memory (BRAM) vector accumulator block collects the PSD values and transfers them to the software process supported by the CASPER design flow. Figure 6 shows the overall ADFT and temporal FFT flow, including energy integration and BRAM registers. The casperfpga Python library is used to interact with CASPER hardware, extracting values from the shared BRAM.

The FPGA operates with a clock frequency set to 200 MHz. Given this configuration, the system frequency resolution for each beam is determined to be $\Delta F=100/1024$ MHz, which is approximately 100 kHz. The Nyquist bandwidth of the system is thus 100 MHz. This configuration ensures precise channelization and high-resolution frequency analysis for each of the 32 spatial RF beams.

3.5. Digital High-Speed Connectivity

The ROACH-2 boards are equipped with four SFP+ ports on a single 10 GbE mezzanine card, where the Ethernet interface is driven by an on-board 156.25 MHz crystal oscillator. This clock frequency is internally multiplied by a factor of 66 within the FPGA, resulting in an effective wire speed of 10.3125 Gbps.

The CASPER 10 Simulink core facilitates the transmission and reception of UDP packets over IPv4, encapsulated within Ethernet frames. Each Ethernet frame incurs an overhead of 38 bytes for the frame header, 20 bytes for the IPv4 header, and 16 bytes for the UDP header, totaling at least 74 bytes of overhead per packet. To align with 64-bit boundaries, some headers may also include zero-padding, further impacting the effective throughput. The maximum payload size for the CASPER 10 core is 8192 bytes, implemented in BRAM, with an additional 512 bytes available in distributed RAM for application-specific headers [100].

4. Experimental Results

4.1. Calibration

Calibration of the RF array system is essential for optimal beamforming performance and was carried out in two distinct phases. Initially, the calibration focused on the ADCs, utilizing open-source routines previously developed by CASPER group members for this hardware. The second phase addressed digital correction of mismatches in the microwave front end. Relative gain and phase mismatches of the IF outputs for each chain were determined in relation to a reference chain using a 5.71 GHz reference carrier input. The relative gain differences were compensated using digital multipliers and gain scale factors were calculated by measuring average power values of each ADC output. Ideally, for phase difference compensation, fractional delay filters are required, as the RF chains could have smaller delays than sampling period value. As an interim solution, integer delay filters were designed and implemented in the hardware to mitigate phase differences.

4.2. Beam Measurements

Power patterns were obtained by transmitting a continuous-wave (CW) signal at ${\mathbf{f}}_{\mathrm{RF}}=5.71$ GHz. The LO signal frequency ${\mathbf{f}}_{\mathrm{LO}}$ determines the IF as ${\mathbf{f}}_{\mathrm{RF}}-{\mathbf{f}}_{\mathrm{LO}}$. For the measurements in this study, ${\mathbf{f}}_{\mathrm{LO}}$ was set to 5.7 GHz, resulting in an IF of 10 MHz, with the downconverted outputs digitized at ${\mathbf{f}}_{\mathrm{CLK}}=200$ MHz.

Digital integrator at each ADFT bin output were used to compute the received energy over a fixed duration. The geometry of the measurement setup significantly influences the measured array factor of the beams. Ideally, the transmitter and receiver should be sufficiently far apart so that the waves incident on the receiver array can be approximated as plane waves. According to standard guidelines [103], the separation between the transmitter and receiver should exceed 20 $\mathrm{m}$ at 5.7 GHz to ensure that the receiver aperture is in the far-field. The beams were measured on an open parking deck with a transmitter-receiver separation of approximately 20 $\mathrm{m}$.

Figure 7 illustrates the measured array factor patterns for beams 0 to 16. Due to the input of real-valued IF signals into the ADFT, the resulting beam patterns exhibit symmetry, producing mirror images for the remaining bins. Additionally, the presence of two mirror peaks is observed within bins 0 to 16. The measured beams conform closely to the expected patterns, maintaining a separation of more than 10 dB from the other beam.

Figure 8 depicts the measured temporal PSD distribution of respective beams at multiple frequencies when the signal source is positioned on broadside angle of 15°, 35°, and 50°. The corresponding experimental setup with relative source locations and RF frequencies being emitted for testing purposes is shown in Figure 9.

5. Discussion

This paper presents a low-complexity digital beamforming architecture designed to generate 32 simultaneous RF beams using a 32-point DFT approximation, effectively eliminating multiplication operations. We validated the proposed method on a fully operational 32-element digital 1-D receive array operating at 5.7 GHz. The 1-D array employs 32 parallel ADCs for sampling the antenna outputs and uses an ADFT implemented on a Xilinx FPGA to compute 32 RF beams in real time. Additionally, the paper discusses the enhancement of our existing 32-channel 5.7 GHz spectrum sensing system through the implementation of 32 parallel 1024-bin temporal FFTs on a ROACH-2 hardware platform. This experimental verification with real-time measurements across a band of frequencies and angles of incidence demonstrates the capability of the architecture to sense information from multiple directions with a frequency resolution of 100 and directional resolution of about 6°.

Appendix A.1. 32-Point ADFT Fast Algorithm

The fast algorithm for the discussed 32-point approximate DFT ${\widehat{\mathbf{F}}}_{32}$ consists of a factorization in low-complexity, sparse matrices, as shown in (5). The approximate matrix and its factorization terms are explicitly given below:

(A1)

where $\gamma =1+j$ and ${}^{*}$ denotes complex conjugation,

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

and

(A9)