Direct Antenna Frequency-Hopped M-FSK Modulation With Time-Modulated Arrays

We present an innovative approach that simultaneously enables direct antenna frequency-hopped M-ary frequency shift keying (DAFH-MFSK) modulation and beamsteering through the use of time-modulated arrays (TMAs). The distinctive feature of our approach lies in the modulation of the TMA excitations with binary periodic sequences, which can be easily frequency-adjusted and time-delayed to simultaneously allow for DAFH-MFSK direct antenna modulation and beamsteering. Notably, our TMA proposal offers a distinct advantage over conventional architectures in terms of performance metrics, including reduced insertion losses and enhanced phase resolution for beamsteering, while also simplifying hardware complexity.


I. INTRODUCTION
L OW-power and low-data-rate internet of things (IoT)   wireless devices widely use frequency-shift keying (FSK) modulation due to their simplicity and resilience to noise and attenuation [1].In standard M-ary FSK (MFSK) [2, Chapter 5], a constant-amplitude sine carrier with frequency f c +f m FSK = f c +m∆f FSK , m∈M={1, 2, . . ., M } is selected every symbol period T s , depending on which symbol is to be transmitted, being f c the reference (or base) carrier frequency, and ∆f FSK the separation between two adjacent f m FSK values.Accordingly, B FSK =M ∆f FSK corresponds to the total FSK bandwidth.
Despite their advantages, MFSK signals can be easily intercepted and, moreover, can be seriously distorted by frequency selective channels.These drawbacks can be overcome by means of frequency-hopping (FH) techniques [2, Chapter 13], which randomly change the carrier frequency in every hop period T h .We assume hopping frequencies that are randomly selected from a set of equally spaced frequencies The authors are with the Universidade da Coruña (University of A Coruña), CITIC Research Center, 15071 A Coruña, Spain (email: roberto.maneiro@udc.es;julio.bregains@udc.es;jagarcia@udc.es;luis.castedo@udc.es).
Digital Object Identifier 10.1109/LAWP.2023.3330435Fig. 2. Proposed TMA architecture to jointly perform DAFH-MFSK modulation and beamsteering.The schematic of a TM-FN module is detailed in Fig. 3b).For each modulating waveform h mk n (t), n∈N denotes the corresponding antenna element, m∈M is the FSK symbol transmitted, and k∈K is the hop frequency slot.In the text, the values of m and k will be single digits so that the notation with the superscript mk is not misleading.k∈K={1, 2, . . ., K} with ∆f FH being the separation between any two adjacent f k FH .We consider ∆f FH =M ∆f FSK , hence the transmit frequency for every is the total transmission bandwidth.We focus on slow FH where T h =LT s (L∈N), as shown in Fig. 1, which contains the time-frequency plot of an FH-FSK transmission.Since demodulation requires knowledge of the pseudo-random FH pattern, FH-FSK prevents eavesdropping while increases robustness to frequency-selective channels.
Another concept, in line with IoT, is direct antenna modulation (DAM), which consists in modulating the carrier in the arXiv:2402.03194v1[eess.SP] 5 Feb 2024  antenna itself [3], [4].DAM replaces baseband modulation and significantly reduces transmission hardware (HW) complexity, cost, and power consumption [5]- [7].Furthermore, since modulation occurs after amplification, power amplifiers (PAs) only need to amplify a single carrier, hence avoiding wideband PAs.This letter combines FH-FSK and DAM into a transmission method termed DAFH-MFSK and proposes an innovative approach for its implementation using a highly efficient and versatile single sideband (SSB) 1 TMA [8]- [13] which, in addition, is capable of performing beamsteering.

II. JOINT DAFH-MFSK MODULATION AND BEAMSTEERING: A TMA APPROACH
A. SSB Time Modulating Feeding Network Fig. 2 plots the proposed SSB TMA architecture for DAFH-MFSK, equipped with periodic time-modulating feeding networks (TM-FNs) to jointly perform DAFH-MFSK modulation and beam steering.We consider a linear array of N isotropic elements with unitary static excitations, I n =1, n∈N , and N ={1, . . ., N }.The n-th element excitation is modulated during a symbol period T s by the periodic pulsed signal h mk n (t), which is a time-shifted version of the stairstep approximation to h mk (t) (see Fig. 3a), an LP-TM waveform [12] with unit amplitude and phase varying from 0 to 5π/3 (six steps).Therefore, h mk n (t)=h mk (t−D mk n ), being D mk n a variable time-delay.Notice that h mk n (t) has a fundamental period T mk TMA =1/f mk TMA ≪T s , where m∈M refers to the transmitted m-th FSK-modulated level, and k∈K accounts for the hop frequency slot selected during T s .
The synthesis of h mk n (t) is described using a switched TM-FN (see Fig. 3b).Considering the rectangular pulse signal p mk (t)=1 when 0≤t<T mk TMA /6, and p mk (t)=0 otherwise, we can express a single period of h mk (t) as .Hence, the exponential Fourier series coefficients of the periodic signal h mk (t) are: The Fourier coefficients H mk q are the same for all values of m and k because h mk (t) is always a time-scaled version of 1 SSB TMAs remove unwanted frequency-mirrored beam patterns produced by conventional TMAs to achieve high efficiency levels.
the waveform shown in Fig. 3a.In view of (1), the Fourier coefficients of h mk n (t)=h mk (t−D mk n ) are given by and the exponential Fourier series expansion of h mk n (t) is According to (2), Fig. 4 shows the normalized Fourier series power spectrum of h mk n (t) in dB, namely 20 log 10 H mk nq /H mk n1 .We can see, apart from the SSB property of the waveform, that the most meaningful unwanted harmonic is the one with order q= − 5, whose relative level is at −13.97 dB with respect to the useful harmonic q=1.
TMA periodic modulating signals, h mk n (t), have the following features: (1) they have no frequency-mirrored harmonics (hence the term SSB) and their first positive harmonic concentrates almost all the transmitted energy; (2) this harmonic is located at f mk TMA =f k FH +f m FSK , hence the TMA transmits the m-th level within the k-th FH slot; and (3) the phase term of the first positive Fourier coefficient of h mk n (t) is proportional to the time delay D mk n (see (2)), which is instrumental to determine the steering direction of the TMA beampattern and, unlike digital variable phase shifters (VPSs) in standard phased arrays, D mk n can be adjusted almost continuously [14], [15].In addition to the rejection threshold of unwanted harmonics shown in Fig. 4, the time-modulation efficiency of the TMA is η TM =P TM U /P TM R , where P TM U and P TM R are the respective useful and total average power radiated by the TMA.According to [16,Eq. 16], η TM is given by Since h mk n (t) has unit amplitude, then thus obtaining (see ( 2) and ( 4)) This means that the proposed SSB TMA architecture ensures that more than 91 % of the total energy is transmitted over the first positive harmonic.On the basis of this result, the following approximation is applicable in (3)

B. Signal Radiated During a Symbol Period
As shown in Fig. 2, the input to the proposed SSB TMA is the single-frequency carrier signal s(t)=e j2πfct .During a symbol period T s , the TMA excitations are time modulated by h mk n (t) and the signal radiated by the TMA in the spatial direction θ is given by where z n is the position of the n-th array element on the z axis and β c =2π/λ c is the wavenumber for a carrier wavelength λ c =c/f c .Normalizing (7) with respect to H mk 1 , which is the same for all values of m and k (see (1)), Eq. ( 8) can be rewritten as where the term AF mk (θ)= N n=1 e j2π( zn λc sin θ−f mk TMA D mk n ) is the spatial array factor during T s and provides the beamsteering ability to the TMA.Indeed, the maximum of the radiation pattern can be pointed to the direction θ 0 by adjusting the delays D mk n , n∈N , so that the following equation is satisfied e j2πf mk TMA D mk n = e j2π zn λc sin θ0 (10) in which case the array factor is e j2π zn λc (sin θ−sin θ0) (11) On the other hand, the term e j2πf mk FH-FSK t in (9) allows the TMA to transmit the m-th FSK level over the k-th FH slot, and thus perform DAFH-MFSK modulation.

III. CASE STUDY AND COMPARATIVE ANALYSIS
This section has a twofold purpose: (1) Demonstrate the feasibility of the proposed technique by means of numerical simulations, and (2) compare it with conventional architectures performing FH-MFSK and beamsteering simultaneously.

A. Numerical Example
Let us consider a DAFH-MFSK modulator architecture based on TM-FNs (refer to Fig. 2).We assume the following parameters: carrier frequency: f c =2.5 GHz; number of antenna elements: N =4, spaced λ c /2 apart; modulation scheme: 4-FSK (M =4); hopping frequencies: 6 possible values (K=6); symbol period: T s =10 ms; hop duration: L=4, thus T h =4T s =4 ms; frequency separation for FSK: ∆f FSK =50 kHz, with f 1 FSK =50 kHz; and frequency spacing for FH: ∆f FH =200 kHz, with f 1 FH =0 Hz.The minimum frequency of the TMA modulating waveforms is given by f 11  TMA =f 1 FH +f 1 FSK =50 kHz, corresponding to their maximum possible period T max TMA =T 11 TMA =1/f 11 TMA =20 µs ≪ T s =10 ms.Additionally, the TMA offers a minimum period of T min TMA =T 46 TMA =1/f 46 TMA =833 ns.As an example, let us consider that the TMA radiates sequentially the four 4-FSK levels m={1, 2, 3, 4} over the second (k=2) FH slot of duration T h and centered at frequency The frequencies to be transmitted and the antenna elements time delays D m2 n , n∈N , (determined according to (10)) to point the maximum of the radiation pattern towards θ 0 =30 • are shown in Fig. 5 and summarized in Table I.
Fig. 6.Block diagram of a conventional FH-MFSK transmitter followed by a standard phased array with digitally tuned passive VPS to perform beamsteering.

B. Comparison with Conventional Techniques
Fig. 6 shows the block diagram of a conventional FH-MFSK transmitter followed by a standard phased array equipped with digitally tuned passive VPSs to perform beamsteering.Every T s , the incoming binary data bits are employed, via a multiplexer (MUX), to select the transmitting carrier frequency from a pool of M possibilities.The FH-MFSK signal is generated by mixing the resulting MFSK modulated signal with a carrier obtained from a digital frequency synthesizer under the control of a code generator.
Given that the mixer produces both sum and difference frequency components, but only the sum frequency is intended for radiation, a bandpass filter (BPF) is placed after the mixer.Following amplification through the PA, the signal is radiated to a given direction by adjusting the VPSs within the standard phased array.
Compared with the conventional scheme in Fig. 6, our proposed TMA approach in Fig. 2 offers several key advantages: 1) Reduced hardware complexity: In our approach, there is no need for a MUX, mixer, frequency synthesizer, or BPF, and the PA only needs to amplify a single carrier.2) Minimal oscillator requirements: Unlike the conventional scheme illustrated in Fig. 6, which necessitates multiple oscillators corresponding to the number of MFSK levels, M , our approach requires just one oscillator.Refer to Table II where we employ big O notation [17] for a comprehensive hardware comparison.

FH-MFSK & BS Scheme
# Oscillators # SPDT switches η (dB) Conventional (Fig. 6) This work: TMA approach (Fig. 2) 3) Reduced number of SPDT switches: The number of SPDT swtches in our approach is only half that of the conventional scheme employing 6-bit VPSs (minimum resolution comparable to that of the TMA [14]).This results in a linear complexity, as detailed in Table II.4) Improved insertion losses: For both the conventional architecture and the TMA approach, we consider offthe-shelf devices with the lowest insertion loss, denoted as η, within the specified frequency band.Specifically, η MUX =0.7 dB (single-pole four-throw (SP4T) switch) [18], η mixer =4.5 dB [19], η BPF =2 dB [20], η VPS =4 dB [18], η SPDT =0.5 dB [18], and ideal 1:N power splitters are taken into account.Under these circumstances, the conventional architecture exhibits an insertion loss of η conv = η MUX + η mixer + η BPF + η VPS = 11.2 dB.In contrast, the TMA approach [21] demonstrates significantly lower insertion losses, with η TMA = −10 log 10 η TM + 6η SPDT = 3.4 dB.Consequently, assuming equal power levels in the respective carrier signals, the same PA gain, and that the performance of all components remains consistent across the entire bandwidth, the proposed TMA approach achieves a substantial insertion loss reduction of ∆η=η TMA −η conv = − 7.8 dB.This insertion loss reduction leads to a significant performance improvement of MFSK demodulation in terms of bit error ratio (BER) versus received signal-to-noise ratio (SNR) per bit, as illustrated in Fig. 7. Fig. 7. Performance of MFSK demodulation for the proposed TMA approach (see Fig. 2) and the conventional one (see Fig. 6) in terms of BER versus received SNR per bit in the conventional scheme.

IV. CONCLUSIONS
We have introduced an innovative TMA approach that seamlessly combines DAFH-MFSK modulation and beamsteering (BS), making it particularly well suited for low-power and low-data-rate applications.This approach offers several key advantages over comparable existing architectures, including better energy efficiency, simplified design, and the ability to achieve continuous phase-sensitivity beamsteering.

Fig. 3 .
Fig. 3. (a) Periodical (T mk TMA ) six-level LP-TM signal which allows for the direct transmission of the m-th FSK symbol over the k-th FH slot using a TMA.Notice that only ∡h mk (t) is sketched since |h mk (t)|=1 ∀m, k.To additionally perform beamsteering, in the n-th antenna element, h mk (t) is subject to a time-delay D mk n , i.e., h mk n (t)=h mk (t−D mk n ).(b) Switched TM-FN of the n-th TMA element, which consists of two SP3T switches and two SPDT switches or, equivalently, six SPDT switches, to time modulate an input signal s(t) with h mk n (t); and (c) simplified block diagram of (b).

Fig. 4 .
Fig.4.Normalized Fourier series power spectrum of h mk n (t).All unwanted harmonics have a minimum rejection level of 13.97 dB with respect to the exploited harmonic at q=1.Notice that H mk nq is the same for all values of m, k, and n for a given q because h mk n (t) is either a time-scaled version (when m and/or k changes) or a time-shifted version (when n changes) of the same waveform.

Fig. 5 .
Fig. 5. (a) Time delays D m2 n specified in Table I as a function of n∈N and time.During T h , the sequence of symbols sm, corresponding to the frequencies f m FSK , m={1, 2, 3, 4}, are transmitted.(b) Relative power radiated pattern of the proposed TMA.When its modulating waveforms are subject to D m2 n during the considered T h , the beampattern points towards θ 0 =30 • .
th FH slot using a TMA.Notice that only ∡h mk (t) is sketched since |h mk (t)|=1 ∀m, k.To additionally perform beamsteering, in the n-th antenna element, h mk (t)

TABLE II COMPARATIVE
ADVANTAGES OF THE PROPOSED TMA APPROACH.