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[
[
"Discrete-Fresnel Domain Channel Estimation in OCDM-based Radar Systems"
],
[
"Abstract In recent years, orthogonal chirp-division multiplexing (OCDM) has been increasingly considered as an alternative multicarrier scheme, e.g., to orthogonal frequency-division multiplexing, in digital communication applications.",
"Among reasons for thar are its demonstrated superior performance resulting from its robustness to impairments such as frequency selectivity of channels and intersymbol interference.",
"Furthermore, the so-called unbiased channel estimation in the discrete-Fresnel domain has also been investigated for both communication and sensing systems, however without considering the effects of frequency shifts.",
"This article investigates the suitability of the aforementioned discrete-Fresnel domain channel estimation in OCDM-based radar systems as an alternative to the correlation-based processing previously adopted, e.g., in the radar-communication (RadCom) literature, which yields high sidelobe level depending on the symbols modulated onto the orthogonal subchirps.",
"In this context, a mathematical formulation for the aforementioned channel estimation approach is introduced.",
"Additionally, extensions to multi-user/multiple-input multiple-output and RadCom operations are proposed.",
"Finally, the performance of the proposed schemes is analyzed, and the presented discussion is supported by simulation and measurement results.",
"In summary, all proposed OCDM-based schemes yield comparable radar sensing performance to their orthogonal frequency-division multiplexing counterpart, while achieving improved peak-to-average power ratio and, in the RadCom case, communication performance."
],
[
"Introduction",
"All-digital radar systems have been increasingly gaining attention in recent years.",
"While they impose the challenge of handling high data rates resulting from the use of DAC and ADC with high resolutions and sampling rates [1], their use enables a wide range of possibilities.",
"Among these are enabling efficient MU or MIMO operation, e.g., for distributed radar sensing or DoA estimation [2], while also yielding high unambiguous velocity and fine range resolution.",
"Further aspects of all-digital radar systems also include higher signal processing flexibility and improved performance for joint RadCom operation [3].",
"The possibilities enabled by all-digital radar systems rely on the use of efficient modulation schemes, among which are the widely-known OFDM, PMCW [4], [5], [6], [7], [8], and OCDM [9].",
"The latter has been recently investigated in an ISAC context [10], [11] to enable joint RadCom operation [12], [13] or applications such as joint sonar-communication [14].",
"On the one hand, OCDM is known to outperform its aforementioned counterparts for communication purposes [3], [15], e.g., due to its multicarrier and spread spectrum characteristics [16], [17] that yield robustness to multipath propagation, Doppler shifts, ISI [18], and NBI [19].",
"On the other hand, the radar sensing performance of conventional OCDM-based systems is limited.",
"Examples include the relatively high range sidelobe level that is influenced by modulated symbols onto the orthogonal subchirps in the OCDM-based RadCom systems with correlation-based radar signal processing [12], [13], [3], [20].",
"In the face of limitations of typical radar processing schemes for OCDM-based systems, a potential alternative is the use of discrete-Fresnel domain channel estimation, which has been introduced in [21] for coherent optical OFDM systems and further investigated in [22] and [23].",
"The aforementioned channel estimation strategy has also been used in the context of sensing, namely for enabling reflectometric sensing of power lines [24] with a baseband OCDM system [25].",
"The simplest discrete-Fresnel domain channel estimation strategy consists of transmitting a pilot OCDM symbol where only the first subchirp is active.",
"Based on the convolution theorem of the DFnT [16], the receive OCDM symbol will ultimately contain a CIR estimate.",
"The aforementioned studies have, however, considered that the estimated channel only introduces delays in the transmit signal.",
"Frequency shifts were either not considered [21], [22] or assumed to be compensated [23].",
"While the assumption that frequency shifts are compensated may be reasonable for communication purposes, where, e.g., Schmidl & Cox's algorithm [26], [27] can be used, it does not necessarily hold for radar systems.",
"Although transmit and receive channels usually share the same oscillator and are fully synchronized, Doppler shifts introduced by the motion of targets are not compensated during radar channel estimation.",
"In fact, the goal of radar sensing is jointly estimating the range and Doppler shift, and consequently relative radial velocity, of targets.",
"The lack of compensation for Doppler shifts, which are essentially frequency shifts, results in the need for a more thorough investigation of the discrete-Fresnel domain channel estimation for OCDM-based radar systems, which is the subject of this article.",
"The main contributions of this article are as follows: A thorough mathematical formulation of the effects of time and frequency shifts in the discrete-Fresnel domain, both applied individually or jointly.",
"It is shown that both aforementioned effects cause shifts and phase rotations of subchirps in the discrete-Fresnel domain, leading to IChI and resembling the ICI in OFDM-based systems [28].",
"These findings extend the state-of-the-art knowledge on propagation effects on OCDM signals, which previously only consider the effects of time shifts described by the convolution theorem of the DFnT, and can be used in the contexts of radar, communication, and RadCom systems.",
"A detailed description of discrete-Fresnel domain radar channel estimation, with closed-form expressions to analyze results for multiple point targets with distinct ranges and relative radial velocities and radar performance parameters including processing gain, range resolution and ambiguity, and velocity resolution and ambiguity.",
"An extension of the investigated discrete-Fresnel domain radar channel estimation to MU/MIMO and RadCom operation based on a strategic subchirp allocation known as FrDM, with remarks on eventually changed radar performance parameters.",
"A performance analysis based on simulation and measurement results.",
"The presented analysis covers an investigation of biases from ideally estimated CIR under different Doppler shifts, which is supported by an analysis of SNR degradation and sidelobe level changes.",
"Additionally, comparisons with the well-known OFDM RadCom are performed.",
"The achieved results show that the proposed OCDM-based schemes present comparable radar sensing performance to OFDM, while presenting lower PAPR due to a similar effect to the known tone reservation in OFDM systems.",
"Furthermore, the proposed extension of the proposed OCDM-based radar system to RadCom operation, which is named sector-modulated OCDM-based RadCom, yields improved communication robustness w.r.t.",
"OFDM, as it retains the spread-spectrum characteristics of conventional OCDM schemes.",
"The remainder of this article is organized as follows.",
"Section presents a thorough mathematical description of the individual and joint effects of time and frequency shifts in the discrete-Fresnel domain.",
"Based on these results, Section mathematically formulates the discrete-Fresnel domain channel estimation for OCDM-based radar systems in detail and presents extensions of the introduced concept to MU/MIMO and RadCom operations.",
"Next, Section presents a performance analysis of the aforementioned schemes based on simulation and measurement results, and, finally, concluding remarks are given in Section ."
],
[
"Notation",
"Throughout this article, $t$ denotes time in seconds and $f$ denotes frequency in Hertz.",
"Additionally, $\\delta [\\chi ]$ , $\\chi \\in \\mathbb {Z}$, is the Kronecker delta function, $\\left<\\cdot \\right>_\\upsilon $ is the modulo $\\upsilon $ operator, $\\upsilon \\in \\mathbb {N}_+$ , and $\\mathbb {E}\\lbrace \\cdot \\rbrace $ is the expectation operator."
],
[
"Effects of Time and Frequency Shifts in the Discrete-Fresnel Domain",
"Let the matrix $\\dot{\\mathbf {X}}\\in \\mathbb {C}^{N\\times M}$ be the discrete-Fresnel domain representation of the transmit frame in an OCDM-based system, with $N\\in \\mathbb {N}_+$ denoting both the OCDM symbol length and the number of subchirps in the OCDM system and $M\\in \\mathbb {N}_+$ denoting the total number of OCDM symbols in the frame.",
"Before transmission over a channel, $\\dot{\\mathbf {X}}$ undergoes IDFnT along its columns to produce the discrete-time domain representation of the OCDM frame $\\mathbf {x}\\in \\mathbb {C}^{N\\times M}$ [9], [12], [3].",
"Consequently, the relationship between the element $x_{n,m}\\in \\mathbb {C}$ at the $n\\mathrm {th}$ row, $n\\in \\lbrace 0,1,\\dots ,N-1\\rbrace $, and $m\\mathrm {th}$ column, $m\\in \\lbrace 0,1,\\dots ,M-1\\rbrace $, of $\\mathbf {x}$ and the element $\\dot{X}_{n,m}\\in \\mathbb {C}$ at the same position of $\\dot{\\mathbf {X}}$ is expressed as $x_{n,m} = \\frac{1}{N}^{\\frac{\\pi }{4}}\\sum \\limits _{k=0}^{N-1} \\dot{X}_{k,m}~^{-\\frac{\\pi }{N}(n-k)^2},$ where $k\\in \\lbrace 0,1,\\dots ,N-1\\rbrace $ is the subchirp index for all of the $M$ discrete-Fresnel domain representations of OCDM symbols contained in $\\dot{\\mathbf {X}}$ .",
"To prevent ISI, CP cyclic prefixes (CPs) of length $N_\\mathrm {CP}$ are prepended to each of the $M$ columns of $\\mathbf {x}$ , resulting in $\\mathbf {x}_\\mathrm {CP}\\in \\mathbb {C}^{(N+N_\\mathrm {CP})\\times M}$.",
"For successful ISI avoidance, it must hold that $N_\\mathrm {CP}\\ge N_\\mathrm {h}-1$, where $N_\\mathrm {h}$ is the expected length of the discrete-time domain representation of the CIR.",
"Next, $\\mathbf {x}_\\mathrm {CP}$ undergoes S/P conversion, and the real and imaginary parts of the resulting vector individually undergo D/A conversion with sampling rate $F_\\mathrm {s}\\ge B$, where $B$ is the bandwidth occupied by the OCDM-based system in Hertz.",
"The output signals by the two aforementioned DAC then undergo analog conditioning and I/Q modulation to a carrier with frequency $f_\\mathrm {c}\\gg B$ .",
"Finally, the resulting continuous-time domain transmit signal $x(t)\\in \\mathbb {C}$, which occupies the RF band $f\\in [f_\\mathrm {c}-B/2,f_\\mathrm {c}+B/2]$, is sent out by the transmit antenna of the OCDM-based system and propagates through a channel that has CIR $h(t)\\in \\mathbb {C}$ and may also introduce frequency shifts in its output signal.",
"At the receiver side of the OCDM-based system, the continuous-time domain output signal from the channel $y(t)\\in \\mathbb {C}$ is received by the receive antenna, undergoing down-conversion and analog conditioning in an I/Q receiver and finally A/D conversion with sampling rate $F_\\mathrm {s}$ .",
"The samples from I and Q channels are then combined into the real and imaginary parts of a serial vector, which after S/P conversion becomes the matrix $\\mathbf {y}_\\mathrm {CP}\\in \\mathbb {C}^{(N+N_\\mathrm {CP})\\times M}$ that represents the discrete-time domain receive OCDM frame.",
"In case $F_\\mathrm {s}>B$, it is also assumed that a prior downsampling to $B$ is performed.",
"An inverse processing to the one performed at the transmitter side is then performed on $\\mathbf {y}_\\mathrm {CP}$ , namely, CP removal to produce $\\mathbf {y}\\in \\mathbb {C}^{N\\times M}$, and column-wise DFnT, which ultimately produces the matrix $\\dot{\\mathbf {Y}}\\in \\mathbb {C}^{N\\times M}$ that represents the discrete-Fresnel domain receive OCDM frame.",
"The relationship between the element $\\dot{Y}_{k,m}\\in \\mathbb {C}$ at the $k\\mathrm {th}$ row and $m\\mathrm {th}$ column of $\\dot{\\mathbf {Y}}$ and the element $y_{n,m}\\in \\mathbb {C}$ at the same position of $\\mathbf {y}$ is expressed as $\\dot{Y}_{k,m} = ^{-\\frac{\\pi }{4}}\\sum \\limits _{n=0}^{N-1} y_{n,m}~^{\\frac{\\pi }{N}(n-k)^2}.$ For the sake of simplicity, AWGN is neither considered in (REF ) nor in the following expressions throughout this study since the DFnT operation does not change its statistics.",
"This is supported by the claims in [9] and further discussion on this topic can be found in [15].",
"In systems such as wireless communication, radar, and RadCom systems, additional processing steps to the aforementioned ones are usually performed.",
"In this section, however, the focus is on analyzing the effects of the propagation through a channel on the originally transmitted OCDM symbol representations contained in $\\dot{\\mathbf {X}}$ , i.e., defining the relationship between $\\dot{\\mathbf {Y}}$ and $\\dot{\\mathbf {X}}$ .",
"The two main effects experienced due to the propagation through a channel are frequency and time shifts.",
"The first are caused by the experienced circular convolution of each of the $M$ columns of $\\mathbf {x}$ , which represent the discrete-time domain transmit OCDM symbols, with the discrete-time domain CIR representation $\\mathbf {h}\\in \\mathbb {C}^{N_\\mathrm {h}\\times 1}|\\mathbf {h}=[h_0, h_1, \\dots , h_{N_\\mathrm {h}-1}]^T$.",
"In their turn, the latter effects encompass Doppler shifts caused by reflections off moving scatterers.",
"In this context, Subsections REF and REF analyze the sole effects of frequency and time shifts in the discrete-Fresnel domain frame $\\dot{\\mathbf {Y}}$ , respectively, while Subsection REF addresses the case where time and frequency shifts are jointly experienced."
],
[
"Effects of frequency shifts in the discrete-Fresnel domain",
"If a frequency shift $f_{\\Delta }$ is the only experienced effect during propagation of the transmit OCDM signal through a channel, then the element at the $n\\mathrm {th}$ row and $m\\mathrm {th}$ column of the discrete-time domain frame $\\mathbf {y}$ obtained after CP removal at the receiver side can be expressed as $y_{n,m} &=& x_{n,m}~^{2\\pi f_{\\Delta }\\left[m\\left(N+N_\\text{CP}\\right)+N_\\text{CP}+n\\right]/B}\\nonumber \\\\&=& x_{n,m}~^{2\\pi f_{\\Delta }n/B}^{2\\pi f_{\\Delta }\\left[m\\left(N+N_\\text{CP}\\right)+N_\\text{CP}\\right]/B}.$ For the sake of simplicity, the normalized frequency shift $k_{\\Delta }\\in \\mathbb {R}$ and the phase $\\phi _m\\in \\mathbb {R}$ are defined as $k_{\\Delta } \\triangleq f_{\\Delta }/(B/N)$ and $\\phi _m \\triangleq 2\\pi k_{\\Delta }\\left[m\\left(N+N_\\text{CP}\\right)+N_\\text{CP}\\right]/N,$ respectively.",
"Next, performing column-wise DFnT on $\\mathbf {y}$ yields the matrix $\\dot{\\mathbf {Y}}$ , whose element at the $k\\mathrm {th}$ row and $m\\mathrm {th}$ column is expressed as $\\dot{Y}_{k,m} &=& ^{-\\frac{\\pi }{4}}\\sum \\limits _{n=0}^{N-1} \\left(x_{n,m}~^{2\\pi k_{\\Delta }n/N}^{\\phi _m}\\right)~^{\\frac{\\pi }{N}(n-k)^2}\\nonumber \\\\&=& ^{\\phi _m}^{-\\frac{\\pi }{4}}\\sum \\limits _{n=0}^{N-1} x_{n,m}~^{\\frac{\\pi }{N}(n^2-2n(k-k_{\\Delta })+k^2)}.$ Knowing that $n^2 -2nk + k^2 +2nk_{\\Delta } = [n-(k-k_{\\Delta })]^2 + 2kk_{\\Delta } - k_{\\Delta }^2,$ it is possible to rewrite (REF ) as $\\dot{Y}_{k,m} = ^{\\phi _m}^{\\frac{\\pi }{N}\\left(2kk_{\\Delta } - k_{\\Delta }^2\\right)}^{-\\frac{\\pi }{4}}\\sum \\limits _{n=0}^{N-1} x_{n,m}~^{\\frac{\\pi }{N}[n-(k-k_{\\Delta })]^2}.$ Expressing $x_{n,m}$ as the IDFnT of $\\dot{X}_{k,m}$ in (REF ) and rearranging the resulting expression yields $\\dot{Y}_{k,m} = \\frac{^{\\phi _m}}{N}\\sum \\limits _{\\kappa =0}^{N-1}\\dot{X}_{\\kappa ,m}~^{\\frac{\\pi }{N}\\left(k^2-\\kappa ^2\\right)}\\sum \\limits _{n=0}^{N-1}^{\\frac{\\pi }{N}n\\left(-2k+2k_{\\Delta }+2\\kappa \\right)},$ for $\\kappa \\in \\lbrace 0,1,\\dots ,N-1\\rbrace $.",
"The rightmost sum in (REF ) is a finite geometric series, which, according to the result from the Appendix, can be evaluated to yield $\\dot{Y}_{k,m} = \\frac{^{\\phi _m}}{N}\\sum \\limits _{\\kappa =0}^{N-1}\\dot{X}_{\\kappa ,m}~^{\\frac{\\pi }{N}\\left(k^2-\\kappa ^2\\right)}\\left(\\frac{^{2\\pi \\left(k_{\\Delta }-k+\\kappa \\right)}-1}{^{\\frac{2\\pi }{N}\\left(k_{\\Delta }-k+\\kappa \\right)}-1}\\right).$ Since $k\\in \\mathbb {N}$ and $\\kappa \\in \\mathbb {N}$ , it holds for the specific case where $k_{\\Delta }\\in \\mathbb {Z}$ , which is when the frequency shift $f_{\\Delta }$ is an integer multiple of $B/N$ , that $\\left.\\frac{^{2\\pi \\left(k_{\\Delta }-k+\\kappa \\right)}-1}{^{\\frac{2\\pi }{N}\\left(k_{\\Delta }-k+\\kappa \\right)}-1}\\right|_{k_{\\Delta }\\in \\mathbb {Z}} = \\delta \\left[\\left<\\kappa -(k-k_{\\Delta })\\right>_N\\right].$ The result from (REF ) finally allows rewriting (REF ) as $\\dot{Y}_{k,m} = \\frac{^{\\phi _m}}{N}\\sum \\limits _{\\kappa =0}^{N-1}\\dot{X}_{\\kappa ,m}~^{\\frac{\\pi }{N}\\left(k^2-\\kappa ^2\\right)}~\\delta \\left[\\left<\\kappa -(k-k_{\\Delta })\\right>_N\\right],$ which after further manipulation becomes $\\dot{Y}_{k,m} = ^{\\phi _m}\\dot{X}_{\\left<k-k_{\\Delta }\\right>_N,m}~^{\\frac{\\pi }{N}\\left(2kk_{\\Delta } - k_{\\Delta }^2\\right)}.$ In summary, the results from (REF ) and (REF ) reveal that a frequency shift $f_{\\Delta }$ during the propagation of the transmit OCDM signal through a channel results in a circular shift by $k_{\\Delta }=f_{\\Delta }/B$ samples and a multiplication by the complex exponential $^{\\frac{\\pi }{N}\\left(2kk_{\\Delta } - k_{\\Delta }^2\\right)}$ that rotates the phases of the elements within the columns of $\\dot{\\mathbf {Y}}$ , which represent discrete-Fresnel domain receive OCDM symbols.",
"Additionally, the introduced frequency shift rotates the phase of each $m\\text{th}$ OCDM symbol with respect to the previous one, which is expressed as the phase $\\phi _m$ of the complex exponential $^{\\phi _m}$ that multiplies all elements of the $m\\mathrm {th}$ column of $\\dot{\\mathbf {Y}}$ ."
],
[
"Effects of time shifts in the discrete-Fresnel domain",
"Let an OCDM signal be transmitted through a channel with a CIR represented in the discrete-time domain by $\\mathbf {h}$ .",
"In the absence of frequency shifts, the columns of the discrete-time domain receive OCDM frame after CP removal $\\mathbf {y}$ are the result of the circular convolution between the columns of $\\mathbf {x}$ and $\\mathbf {h}$ .",
"In other words, the element at the $n\\mathrm {th}$ row and $m\\mathrm {th}$ column of $\\mathbf {y}$ is expressed as $y_{n,m} = x_{n,m}\\circledast h_n$ , where $\\circledast $ is the circular convolution operator.",
"Alternatively, one can write $y_{n,m} = \\sum \\limits _{\\nu =0}^{N-1} h_\\nu x_{\\left<n-\\nu \\right>_N,m},$ for $\\nu \\in \\lbrace 0,1,\\dots ,N-1\\rbrace $.",
"By performing column-wise DFnT on $\\mathbf {y}$ , the matrix representation $\\dot{\\mathbf {Y}}$ of the discrete-Fresnel domain frame is yielded.",
"Its element at the $k\\mathrm {th}$ row and $m\\mathrm {th}$ column is expressed as $\\dot{Y}_{k,m} = ^{-\\frac{\\pi }{4}}\\sum \\limits _{n=0}^{N-1}\\left(\\sum \\limits _{\\nu =0}^{N-1}h_\\nu x_{\\left<n-\\nu \\right>_N,m}\\right)~^{\\frac{\\pi }{N}(n-k)^2}.$ Substituting $l=\\left<n-\\nu \\right>_N$ in (REF ) and rearranging the resulting expression yields (REF ).",
"Figure: NO_CAPTIONBased on the obtained results in Subsection REF , more specifically in (REF ), (REF ) can be rewritten as in (REF ).",
"Figure: NO_CAPTIONSince $k\\in \\mathbb {N}$ , $\\kappa \\in \\mathbb {N}$ , and $\\nu \\in \\mathbb {N}$ , the same principle from (REF ) can be applied to the rightmost term in (REF ), which then becomes (REF ).",
"Figure: NO_CAPTIONFigure: NO_CAPTIONAfter further manipulation, (REF ) can be finally rewritten as $\\dot{Y}_{k,m} = \\sum \\limits _{\\nu =0}^{N-1}h_\\nu \\dot{X}_{\\left<k-\\nu \\right>_N,m}$ The relationship between $\\dot{Y}_{k,m}$ and $\\dot{X}_{k,m}$ as in (REF ) reveals that the same circular shifts and amplitude weightings suffered by the columns of $\\mathbf {x}$ are observed in the columns of both $\\mathbf {y}$ and its discrete-Fresnel domain representation $\\dot{\\mathbf {Y}}$ .",
"In other words, $y_{n,m} = x_{n,m}\\circledast h_n$ results in $\\dot{Y}_{k,m} = \\dot{X}_{k,m}\\circledast h_k$, which is also known as the convolution theorem of the DFnT [16], [9]."
],
[
"Joint effects of time and frequency shifts in the discrete-Fresnel domain",
"Should the propagation of the OCDM transmit signal through a channel results not only in time shifts represented by the CIR $\\mathbf {h}$ , but also in a normalized frequency shift $k_{\\Delta }$ , the element at the $n\\mathrm {th}$ row and $m\\mathrm {th}$ column of the discrete-time domain receive frame $\\mathbf {y}$ obtained after CP removal at the receiver side can be expressed as $y_{n,m} = \\left(x_{n,m}\\circledast h_n\\right)~^{2\\pi k_{\\Delta }n/N}^{\\phi _m}.$ For the sake of simplicity, the auxiliary matrix $\\mathbf {r}\\in \\mathbb {C}^{N\\times M}$ is defined, being the element $r_{n,m}\\in \\mathbb {C}$ located at its $n\\mathrm {th}$ row and $m\\mathrm {th}$ column given by $r_{n,m} \\triangleq x_{n,m}\\circledast h_n.$ Consequently, (REF ) can be alternatively expressed as $y_{n,m} = r_{n,m}~^{2\\pi k_{\\Delta }n/N}^{\\phi _m}.$ Defining $\\dot{\\mathbf {R}}\\in \\mathbb {C}^{N\\times M}$ as the output of column-wise DFnT on $\\mathbf {r}$ , the relationship between the element $\\dot{R}_{k,m}\\in \\mathbb {C}$ at the $k\\mathrm {th}$ row and $m\\mathrm {th}$ column of $\\dot{\\mathbf {R}}$ and the element $\\dot{X}_{k,m}$ at the same position of $\\dot{\\mathbf {X}}$ can be defined based on the result from (REF ).",
"Additionally, the element $\\dot{Y}_{k,m}$ at the $k\\mathrm {th}$ row and $m\\mathrm {th}$ column of $\\dot{\\mathbf {Y}}$ can be derived from $\\dot{R}_{k,m}$ based on the result from (REF ).",
"Consequently, $\\dot{Y}_{k,m}$ can be expressed as a function of $\\dot{X}_{k,m}$ as in (REF ).",
"In the specific case where $k_{\\Delta }\\in \\mathbb {Z}$ , the property used in (REF ) can be finally applied to (REF ), yielding $\\dot{Y}_{k,m} = ^{\\phi _m}\\left(\\sum \\limits _{\\nu =0}^{N-1}h_\\nu \\dot{X}_{\\left<(k-k_{\\Delta })-\\nu \\right>_N,m}\\right)~^{\\frac{\\pi }{N}\\left(2kk_{\\Delta }-k_{\\Delta }^2\\right)},$ i.e., $\\dot{Y}_{k,m} = ^{\\phi _m}(\\dot{X}_{\\left<k-k_{\\Delta }\\right>_N,m}\\circledast h_k)~^{\\frac{\\pi }{N}\\left(2kk_{\\Delta }-k_{\\Delta }^2\\right)}$ .",
"The results from (REF ) and (REF ) indicate that the convolution with the CIR followed by a frequency shift $f_{\\Delta }$ during the propagation of the transmit OCDM signal through a channel results in a circular shift by $k_{\\Delta }=f_{\\Delta }/B$ samples on top of the circular shifts caused by the convolution with the multiple taps of the discrete-time domain CIR representation $\\mathbf {h}$ .",
"This effect is similar to known range- or delay-Doppler coupling in chirp-based radar systems [29].",
"Besides the aforementioned effect, a multiplication by the complex exponential $^{\\frac{\\pi }{N}\\left(2kk_{\\Delta } - k_{\\Delta }^2\\right)}$ of the shifted versions of $\\dot{\\mathbf {X}}$ takes place as observed in Subsection REF , increasingly rotating the phases of the $N$ elements of each column of $\\dot{\\mathbf {Y}}$ .",
"As in the case of Subsection REF , every $m\\mathrm {th}$ column of $\\dot{\\mathbf {Y}}$ is additionally multiplied by the complex exponential $^{\\phi _m}$ , which rotates the phase of all elements of the $m\\mathrm {th}$ discrete-Fresnel domain OCDM symbol.",
"The aforementioned effects ultimately distort the transmit subchirps and cause IChI in the discrete-Fresnel domain, which is somewhat similar to the ICI in the discrete-frequency domain in OFDM-based systems.",
"The discussion in Section described the effects of time and frequency shifts on the originally transmitted discrete-Fresnel domain OCDM symbols represented by the columns of $\\dot{\\mathbf {X}}$ , which becomes $\\dot{\\mathbf {Y}}$ after considering the effects of the propagation through a channel.",
"Based on the achieved results, it is possible to predict the relationship between the channel estimates in OCDM-based systems and the actual channel CIR.",
"For this purpose, the discrete-Fresnel domain pilot design strategy introduced in [21] for coherent optical OFDM systems, which was also investigated for reflectometric sensing of power lines in baseband OCDM systems as reported in [24] and further improved in [22], is considered.",
"The original pilot design from [21] performs unbiased channel estimation under the assumption that no frequency shifts take place.",
"This is done by exploiting the convolution theorem of the DFnT [16] and having a Kronecker comb with $P\\in \\mathbb {N}_+$ active subchirps in the discrete-Fresnel domain at the transmitter side, and averaging the resulting $P$ sections of length $N/P$ in the discrete-Fresnel domain at the receiver side to estimate the CIR.",
"Conversely, the improvement proposed in [22] achieves optimal CIR estimates in the MMSE sense by activating only the first subchirp in the discrete-Fresnel domain at the transmitter side, and discarding noisy samples beyond the maximum expected CIR length $N_\\mathrm {h}$ in the obtained discrete-Fresnel domain receive OCDM symbol to obtain a CIR estimate.",
"Since dividing the transmission power among $P$ active subchirps and averaging their corresponding CIR estimates yields the same SNR as in the case where the full transmission power is allocated to a single subchirp and only one CIR is obtained, the superiority of the improved pilot design approach is solely due to the noise-rejection windowing performed in the discrete-Fresnel domain at the receiver side.",
"Considering the two aforementioned characteristics of the pilot design strategies, namely the one from [21] and the one from [22], the latter one is adopted in this study.",
"Consequently, the element at the $k\\mathrm {th}$ row and $m\\mathrm {th}$ column of the matrix $\\mathbf {\\dot{X}}$ that represents a frame containing $M$ discrete-Fresnel domain pilot OCDM symbols in its columns is given by $\\dot{X}_{k,m} = \\delta [k].$ As all $M$ OCDM symbols within the frame are equal, no CP is required.",
"The implications of the pilot OCDM symbol design on the performance of radar channel estimation is analyzed in Subsection REF .",
"Furthermore, extensions of the discussed concept to MU or MIMO scenarios as well as to RadCom applications are discussed in Subsections REF and REF , respectively."
],
[
"Radar Channel Estimation",
"For the sake of simplicity, it is assumed in this subsection that the OCDM-based system acts as a monostatic radar, in which transmit and receive antennas are virtually collocated.",
"In practice, however, only a quasi-monostatic radar is possible, in which the distance between transmit and receive antennas is negligible in comparison to the range of expected target distances w.r.t.",
"the radar.",
"To estimate the range and relative radial velocity of multiple point targets w.r.t.",
"the radar, consecutive radar CIR are estimated.",
"Each CIR is composed of multiple taps at delays corresponding to the ToF taken by the OCDM signal to be sent out by the transmit antenna, reflected off targets, and captured by the receive antenna.",
"If no effects such as range migration occur within the measurement time, all of the estimated radar CIR allow estimating the range of resolved point targets present in the monitored scenario.",
"The observed difference among consecutive CIR, which is the phase progression of its taps, ultimately allows obtaining a radar image that has not only range information, but also estimates of the experienced Doppler shifts related to the relative radial velocities of the point targets.",
"To enable the aforementioned estimation of range and Doppler shifts of $H\\in \\mathbb {N}$ point targets, the matrix $\\tilde{\\mathbf {y}}^\\mathrm {rad}\\in \\mathbb {C}^{N\\times M}$ that would ideally represent the discrete-time domain receive OCDM frame has elements at its $n\\mathrm {th}$ row and $m\\mathrm {th}$ column expressed as in (REF ).",
"In this equation, $n_{\\Delta ,\\eta }\\in \\mathbb {R}|n_{\\Delta ,\\eta }=2R_\\eta B/c_0$ is a normalized range term, being $R_\\eta \\in \\mathbb {R}_+$ the range in meters of the $\\eta \\mathrm {th}$ target and $c_0$ the speed of light in vacuum in meters per second.",
"Additionally, $k_{\\Delta ,\\eta }\\in \\mathbb {R}$ and $\\phi _{m,\\eta }\\in \\mathbb {R}$ are the normalized Doppler shift and the Doppler phase associated to the $\\eta \\mathrm {th}$ point target.",
"In the specific case where $n_{\\Delta ,\\eta }\\in \\mathbb {Z}$, (REF ) can be rewritten as $\\tilde{y}^\\mathrm {rad}_{n,m} = \\sum \\limits _{\\eta =0}^{H-1}x_{n,m}~\\delta [n-n_{\\Delta ,\\eta }]~^{2\\pi k_{\\Delta ,\\eta }n/N}^{\\phi _{m,\\eta }}.$ With the OCDM pilot symbols contained in the columns of $\\dot{\\mathbf {X}}$ having their $k\\mathrm {th}$ elements defined as in (REF ), the resulting ideal discrete-Fresnel domain receive OCDM frame matrix representation $\\tilde{\\dot{\\mathbf {Y}}}^\\mathrm {rad}$ can be alternatively defined as the matrix $\\tilde{\\mathbf {h}}^\\mathrm {rad}\\in \\mathbb {C}^{N\\times M}$ that contains $M$ consecutive radar CIR estimates of length $N$ , which can be expressed as $\\tilde{h}^\\mathrm {rad}_{n,m} = \\sum \\limits _{\\eta =0}^{H-1}\\left(\\frac{^{2\\pi \\left(n_{\\Delta ,\\eta }-n\\right)}-1}{^{\\frac{2\\pi }{N}\\left(n_{\\Delta ,\\eta }-n\\right)}-1}\\right)~^{2\\pi k_{\\Delta ,\\eta }n/N}^{\\phi _{m,\\eta }}$ or simply $\\tilde{h}^\\mathrm {rad}_{n,m} = \\sum \\limits _{\\eta =0}^{H-1}\\delta [n-n_{\\Delta ,\\eta }]~^{2\\pi k_{\\Delta ,\\eta }n/N}^{\\phi _{m,\\eta }}$ for $n_{\\Delta ,\\eta }\\in \\mathbb {Z}$.",
"In reality, however, a matrix $\\dot{\\mathbf {Y}}^\\mathrm {rad}\\in \\mathbb {C}^{N\\times M}|\\dot{\\mathbf {Y}}^\\mathrm {rad}\\ne \\tilde{\\dot{\\mathbf {Y}}}^\\mathrm {rad}$ is obtained at the receiver side of the OCDM-based system.",
"Since pilot OCDM symbols are transmitted, $\\dot{\\mathbf {Y}}^\\mathrm {rad}$ can be alternatively defined as the matrix $\\hat{\\mathbf {h}}^\\mathrm {rad}\\in \\mathbb {C}^{N\\times M}|\\hat{\\mathbf {h}}^\\mathrm {rad}\\ne \\tilde{\\mathbf {h}}^\\mathrm {rad}$ that contains the actual consecutive radar CIR estimates in its columns.",
"Assuming a single target scenario and considering only the $m\\mathrm {th}$ of the $M$ radar channel estimates, a corresponding continuous-frequency domain CFR $\\hat{H}^\\mathrm {rad}_m(f)\\in \\mathbb {C}$ of the aforementioned channel presents linear phase progression along with the frequency in the RF band $f\\in [f_\\mathrm {c}-B/2,f_\\mathrm {c}+B/2]$ as represented at the top of Fig.",
"REF .",
"In the discrete-frequency domain, the CIR contained in the $m\\mathrm {th}$ column of $\\hat{\\mathbf {h}}^\\mathrm {rad}$ is represented by the vector $\\hat{\\mathbf {H}}^\\mathrm {rad}_m\\in \\mathbb {C}^{N\\times 1}$.",
"Since the samples $k=0$ to $k=N/2-1$ of $\\hat{\\mathbf {H}}^\\mathrm {rad}_m$ can be used to reconstruct spectral information on the RF frequencies $f_\\mathrm {c}$ to $f_\\mathrm {c}+(N/2-1)B/N = f_\\mathrm {c}+B/2-B/N$, while the samples $k=N/2+1$ to $k=N$ allow reconstructing spectral information on the frequencies $f_\\mathrm {c}+(-N/2)B/N = f_\\mathrm {c}-B/2$ to $f_\\mathrm {c}-B/N$, the phase of $\\hat{\\mathbf {H}}^\\mathrm {rad}_m$ is circularly shifted by $N/2$ samples w.r.t.",
"an ideally linearly increasing phase [30].",
"Considering a single-target scenario, an exemplary phase progression in the BB of $\\hat{\\mathbf {H}}^\\mathrm {rad}_m$ can be seen in Fig.",
"REF .",
"Figure: Channel frequency response phase progression due to experienced propagation time delay in RF and BB spectra .In order for the ToF, and consequently target ranges, to be appropriately estimated, the effectively obtained matrix $\\mathbf {y}^\\mathrm {rad}\\in \\mathbb {C}^{N\\times M}|\\mathbf {y}^\\mathrm {rad}\\ne \\tilde{\\mathbf {y}}^\\mathrm {rad}$ that represents the discrete-time domain receive OCDM frame must consist of circularly-shifted versions of $\\mathbf {x}$ .",
"If the experienced phase folding is not compensated, the columns of the estimated radar CIR matrix $\\hat{\\mathbf {h}}^\\mathrm {rad}$ are equal to the corresponding columns of the ideally obtained matrix $\\tilde{\\mathbf {h}}^\\mathrm {rad}$ with the addition of discrete-frequency domain shift by $N/2$ samples.",
"In other words $\\hat{h}^\\mathrm {rad}_{n,m} = \\tilde{h}^\\mathrm {rad}_{n,m}^{-2\\pi (N/2)n/N} = \\tilde{h}^\\mathrm {rad}_{n,m}^{-\\pi n},$ where $\\hat{h}^\\mathrm {rad}_{n,m}$ denotes the element of $\\hat{\\mathbf {h}}^\\mathrm {rad}$ at its $n\\mathrm {th}$ row and $m\\mathrm {th}$ column.",
"Although the amplitudes of the elements of $\\hat{\\mathbf {h}}^\\mathrm {rad}$ are the same as the corresponding ones of $\\tilde{\\mathbf {h}}^\\mathrm {rad}$ , the introduced phase rotations would ultimately prevent correctly estimating the ranges of targets when reconstructing the analog equivalent of $\\hat{\\mathbf {h}}^\\mathrm {rad}$ , e.g., via ZP.",
"Consequently, the experienced phase folding must be corrected by performing a circular shift in the discrete-frequency domain by $N/2$ on $\\dot{\\mathbf {Y}}^\\mathrm {rad}$ to produce $\\dot{\\mathbf {Y}}^\\mathrm {rad,corr}\\in \\mathbb {C}^{N\\times M}$, which can be alternatively defined as the matrix $\\hat{\\mathbf {h}}^\\mathrm {rad,corr}\\in \\mathbb {C}^{N\\times M}$ that contains an estimate of the desired radar CIR matrix $\\tilde{\\mathbf {h}}^\\mathrm {rad}$ .",
"The aforementioned phase folding compensation can be achieved in two manners.",
"The first one is by multiplying every $k\\mathrm {th}$ element of $\\dot{\\mathbf {Y}}^\\mathrm {rad}$ by $^{\\pi k}$ .",
"A second solution is based on the OCDM receiver #2 structure from [9], where the DFnT is calculated by means of a DFT, followed by an element-wise multiplication by the vector $\\mathbf {\\Gamma }\\in \\mathbb {C}^{N\\times 1}|\\mathbf {\\Gamma }=[\\Gamma _0, \\Gamma _1, \\dots , \\Gamma _{N-1}]^T$, whose $k\\text{th}$ element is expressed for even $N$ as $\\Gamma _k = ^{-\\pi k^2/N},$ and an IDFT.",
"In this approach, a circular shift in the discrete-frequency domain by $N/2$ samples is performed between the element-wise multiplication by $\\mathbf {\\Gamma }$ and the IDFT.",
"Assuming that no Doppler shifts take place, i.e., $k_{\\Delta ,\\eta }=0$ and $\\phi _{m,\\eta }=0$ $\\forall m,\\eta $ , both described phase folding correction approaches yield the CIR matrix estimate $\\hat{\\mathbf {h}}^\\mathrm {rad,corr}$ , whose element at its $n\\mathrm {th}$ row and $m\\mathrm {th}$ column is given by $\\hat{h}^\\mathrm {rad,corr}_{n,m} = ^{\\pi n}\\hat{h}^\\mathrm {rad}_{n,m} = \\tilde{h}^\\mathrm {rad}_{n,m}.$ If, however, a frequency shift also takes place, the discrete-time domain CIR matrix representation without phase folding correction $\\hat{\\mathbf {h}}^\\mathrm {rad}$ has the element at its $n\\mathrm {th}$ row and $m\\mathrm {th}$ column expressed as in (REF ) or, for the specific case where $n_{\\Delta ,\\eta }\\in \\mathbb {Z}$ and $k_{\\Delta ,\\eta }\\in \\mathbb {Z}$ $\\forall \\eta $ , as in (REF ).",
"Figure: NO_CAPTIONFigure: NO_CAPTIONAfter performing phase folding correction, $\\hat{\\mathbf {h}}^\\mathrm {rad,corr}$ is obtained such that the element in its $n\\mathrm {th}$ row and $m\\mathrm {th}$ column is expressed as in (REF ) or, for $n_{\\Delta ,\\eta }\\in \\mathbb {Z}$ and $k_{\\Delta ,\\eta }\\in \\mathbb {Z}$ $\\forall \\eta $ , as in (REF ).",
"Figure: NO_CAPTIONFigure: NO_CAPTIONThe obtained expressions in (REF ) and (REF ) reveal that the delay-Doppler coupling mentioned in Subsection REF still takes place after phase folding correction.",
"Additionally, it is observed for the $\\eta \\mathrm {th}$ target contribution to the aforementioned equations that the columns of $\\hat{\\mathbf {h}}^\\mathrm {rad,corr}$ are circularly-shifted versions of the columns of $\\tilde{\\mathbf {h}}^\\mathrm {rad}$ by $k_{\\Delta ,\\eta }$ samples.",
"Compared to their counterparts in $\\tilde{\\mathbf {h}}^\\mathrm {rad}$ , the $n\\mathrm {th}$ elements of the aforementioned columns are also multiplied by complex exponentials with linearly increasing phase along with the index $n$ .",
"Since each of the $M$ columns of $\\hat{\\mathbf {h}}^\\mathrm {rad,corr}$ contains information on the range of the $H$ targets, performing row-wise DFT on $\\hat{\\mathbf {h}}^\\mathrm {rad,corr}$ converts the phases $\\phi _{m,\\eta }$ into Doppler shift information.",
"The aforementioned processing produces the matrix $\\mathbf {I}^\\mathrm {rad}\\in \\mathbb {C}^{N\\times M}$ , which allows to clearly extract information on the range and relative radial velocity of all $H$ targets if they are resolved in at least one of the radar image dimensions.",
"Based on the analyses in [31], [3], [12], [13], the aforementioned radar image experiences a processing gain $G_\\text{p}$ and is associated with range resolution $\\Delta R$ and maximum unambiguous range $R_\\text{max,ua}$, as well as relative radial velocity resolution $\\Delta v$ and maximum unambiguous relative radial velocity $v_\\text{max,ua}$ values as shown in Table REF .",
"Since all transmit OCDM symbols within the frame are equal and therefore no CP is required, no further restriction on the maximum range value such as in [3], [12], and [13] is considered.",
"Furthermore, since a restriction on the maximum tolerable relative radial velocity for the considered OCDM-based system will be accurately investigated in Section , it is not listed in Table REF .",
"Table: Performance parameters in an OCDM-based radar system with discrete-Fresnel domain channel estimation.",
"If all OCDM symbols within the frame are equal, no CP is needed and therefore N CP N_\\mathrm {CP} can be set to zero."
],
[
"Extension to MU or MIMO Radar Operation",
"If $P\\in \\mathbb {N}_+$ synchronized OCDM radar transmitters labeled as $p\\in \\lbrace 0,1,\\dots ,P-1\\rbrace $ are to operate simultaneously in a MU or MIMO scenario, then the elements of the matrix $\\mathbf {\\dot{X}}^p\\in \\mathbb {C}^{N\\times M}$ that represents the discrete-Fresnel domain transmit frame of the $p\\mathrm {th}$ transmitter are given by $\\dot{X}^p_{k,m} = \\delta [\\left<k - pN/P\\right>_N].$ This equation indicates that the FrDM scheme [12], which was originally proposed in [24] for distributed reflectometric sensing of power lines, is adopted to enable orthogonal transmission of signals associated with each of the $P$ transmitters as depicted in Fig.",
"REF .",
"Figure: Subchirp allocation in the discrete-Fresnel domain among distinct transmit channels for MU or MIMO operation based on FrDM.Assuming that there are $Q\\in \\mathbb {N}_+$ synchronized OCDM radar receivers labeled as $q\\in \\lbrace 0,1,\\dots ,Q-1\\rbrace $ and following the formulation in Section REF , the elements of the matrix $\\mathbf {\\dot{Y}}^q\\in \\mathbb {C}^{N\\times M}$ representing the receive frame of the $q\\mathrm {th}$ receiver are given by $\\dot{Y}^{q}_{k,m} = \\sum \\limits _{p=0}^{P-1} \\hat{h}^{\\mathrm {rad,corr},p,q}_{\\left<k- pN/P\\right>_N,m},$ where $\\hat{h}^{\\mathrm {rad,corr},p,q}_{k,m}\\in \\mathbb {C}$ is the element at the $k\\mathrm {th}$ row and $m\\mathrm {th}$ column of the matrix $\\hat{\\mathbf {h}}^{\\mathrm {rad,corr},p,q}\\in \\mathbb {C}^{N\\times M}$ that represents consectutive CIR estimates associated with the $p\\mathrm {th}$ transmitter and $q\\mathrm {th}$ receiver.",
"If the CIR in $\\hat{\\mathbf {h}}^{\\mathrm {rad,corr},p,q}$ are assumed to have length equal to or smaller than $N/P$ , i.e., $\\hat{\\mathbf {h}}^{\\mathrm {rad,corr},p,q}$ can be redefined as $\\hat{\\mathbf {h}}^{\\mathrm {rad,corr},p,q}\\in \\mathbb {C}^{N/P\\times 1}$, than its elements $\\hat{h}^{\\mathrm {rad,corr},p,q}_{k^{\\prime },m}\\in \\mathbb {C}$, $k^{\\prime }\\in \\lbrace 0,1,\\dots ,N/P-1\\rbrace $, can be extracted from $\\dot{\\mathbf {Y}}^q$ as $\\hat{h}^{\\mathrm {rad,corr},p,q}_{k^{\\prime },m} = \\dot{Y}^{q}_{k^{\\prime }+pN/P,m}.$ With the described FrDM approach for enabling MU or MIMO radar operation, most performance parameters from Table REF are kept as all OCDM symbols in the transmit frame are equal and CP are not needed.",
"The only exception is the maximum unambiguous range, which becomes $R^{\\mathrm {MU/MIMO},P}_\\mathrm {max,ua} = (N/P)~c_0/(2B)$ .",
"It is worth highlighting that, although the ultimatelly obtained CIR estimates have the reduced length of $N/P$ , the processing gain remains proportional to $N$ since it is already experienced after the DFnT at the receiver, which performs a pulse-compression like operation on the $N$ discrete-time domain samples and takes place before the $N/P$ samples associated with each transmitter are selected at each receiver.",
"Figure: NO_CAPTIONFigure: Subchirp allocation in the discrete-Fresnel domain for enabling joint RadCom operation.",
"The first active, unmodulated subchirp and the next N CP -1N_\\mathrm {CP}-1 inactive subchirps are used for radar sensing, while the N-2N CP +1N-2N_\\mathrm {CP}+1 active, modulated subchirps are used for comunication.",
"The last N CP -1N_\\mathrm {CP}-1 inactive subchirps are used as a guard interval."
],
[
"Extension to RadCom Operation",
"To enable joint RadCom operation of an OCDM-based radar system with discrete-Fresnel domain estimation, the FrDM principle explained in Section REF is applied.",
"In this context, a sector-modulated OCDM symbol structure in the discrete-Fresnel domain based on the autocorrelation pattern of ZCZ sequences used in PMCW radars is adopted.",
"Unlike in the radar-only case from Section REF , each of the $M$ OCDM symbols in the transmit frame will be unique as they carry different modulated data.",
"Consequently, a CP of length $N_\\mathrm {CP}\\in \\mathbb {N}_+|\\lbrace 2N_\\mathrm {CP}-1<N\\rbrace $ must be appended to each OCDM symbol in the discrete-time domain to avoid ISI.",
"The resulting maximum tolerable range in the considered OCDM-based RadCom system is $R_\\mathrm {max,CP} = N_\\mathrm {CP}~c_0/(2B)$ , which is assumed to be lower than the maximum unambiguous range $R_\\mathrm {max,ua}$ listed in Table REF .",
"For a given $N_\\mathrm {CP}$ , the FrDM-based design of the OCDM symbol depicted in Fig.",
"REF is adopted and described as follows.",
"First, the active, unmodulated subchirp of index $k=0$ from (REF ) is kept and allocated and energy $\\mathcal {E}^\\mathrm {Rad}\\in \\mathbb {R}_+$ , being followed by $N_\\mathrm {CP}-1$ inactive subchirps.",
"At the receiver side, these first $N_\\mathrm {CP}$ elements of the discrete-Fresnel domain OCDM symbol will ultimately contain the radar CIR estimate.",
"Next, subchirps $k=N_\\mathrm {CP}$ to $k=N-N_\\mathrm {CP}+1$ , which comprise a total of $N-2N_\\mathrm {CP}+1$ subchirps, are modulated with symbols belonging to a digital modulation constellation that are contained in the matrix $\\mathbf {C}\\in \\mathbb {C}^{\\left(N-2N_\\mathrm {CP}+1\\right)\\times M}$ .",
"Finally, the last $N_\\mathrm {CP}-1$ are kept inactive, which, taking into account energy factors unlike in the previous sections, yields a discrete-Fresnel domain transmit frame whose elements are given by $\\dot{X}_{k,m} = &\\sqrt{\\mathcal {E}^\\mathrm {Rad}}\\delta [k]\\nonumber \\\\&+ \\sum \\limits _{k^{\\prime \\prime }=0}^{(N-2N_\\mathrm {CP}+1)-1} C_{k^{\\prime \\prime },m}\\delta [\\left<k-(N_\\mathrm {CP}+k^{\\prime \\prime })\\right>_N].$ In this equation, $k^{\\prime \\prime }\\in \\lbrace 0,1,\\dots ,(N-2N_\\mathrm {CP}+1)-1\\rbrace $ and $C_{k^{\\prime \\prime },m}$ denotes an element of $\\mathbf {C}\\in \\mathbb {C}^{\\left(N-2N_\\mathrm {CP}+1\\right)\\times M}$ , which belongs to a digital modulation constellation with mean constellation energy that can be expressed as $\\mathcal {E}^\\mathrm {Com} = \\mathbb {E}\\lbrace |C_{k^{\\prime \\prime },m}|^2\\rbrace $ if $C_{k^{\\prime \\prime },m}$ is regarded as a random variable.",
"The total energy allocated to the transmit OCDM symbol for joint RadCom operation is consequently $\\mathcal {E}^\\mathrm {RadCom}=\\mathcal {E}^\\mathrm {Rad}+(N-2N_\\mathrm {CP}+1)\\mathcal {E}^\\mathrm {Com}$.",
"The last $N_\\mathrm {CP}-1$ inactive subchirps of each OCDM symbol are used as a guard interval to avoid interference of the $N-2N_\\mathrm {CP}+1$ communication subchirps onto the $N_\\mathrm {CP}$ radar subchirps, as the OCDM symbols experience a circular convolution with the radar CIR.",
"Due to the OCDM symbol structure from (REF ), the proposed OCDM-based RadCom system is named sector-modulated OCDM-based RadCom system.",
"After transmitted, the OCDM RadCom signal will not only reflect off radar targets and be received by the same OCDM RadCom system that originally transmitted it, but also propagate towards another OCDM communication or RadCom device.",
"The radar case is first explained as follows.",
"Assuming $n_{\\Delta ,\\eta }\\in \\mathbb {Z}$ and $k_{\\Delta ,\\eta }\\in \\mathbb {Z}$ for the sake of conciseness and using the result from (REF ), the elements of the matrix $\\mathbf {\\dot{Y}}^\\mathrm {rad}\\in \\mathbb {C}^{N\\times M}$ representing the radar receive frame after CP removal and transformation into the discrete-Fresnel domain are expressed as in (REF ).",
"Similarly, the result for $n_{\\Delta ,\\eta }\\in \\mathbb {R}$ and $k_{\\Delta ,\\eta }\\in \\mathbb {R}$ can be achieved based on the result from (REF ).",
"Since the length of the CIR in $\\hat{\\mathbf {h}}^\\mathrm {rad}$ is equal to or smaller than $N_\\mathrm {CP}$ , the aforementioned CIR matrix can be redefined as $\\hat{\\mathbf {h}}^\\mathrm {rad}\\in \\mathbb {C}^{N_\\mathrm {CP}\\times M}$.",
"Based on both (REF ) and the sufficient guard interval composed by null subchirps at the end of the discrete-Fresnel domain OCDM symbols in (REF ), the elements $\\hat{h}^\\mathrm {rad,corr}_{k^{\\prime \\prime \\prime },m}\\in \\mathbb {C}$, $k^{\\prime \\prime \\prime }\\in \\lbrace 0,1,\\dots ,N_\\mathrm {CP}-1\\rbrace $, of the aforementioned CIR matrix can be extracted from $\\dot{\\mathbf {Y}}^{\\mathrm {rad}}\\in \\mathbb {C}^{N\\times M}$ as $\\hat{h}^{\\mathrm {rad,corr}}_{k^{\\prime \\prime \\prime },m} \\approx \\dot{Y}^\\mathrm {rad}_{k^{\\prime \\prime \\prime },m}.$ Apart from the aforementioned maximum range, all other radar performance parameters from Table REF also hold for the considered sector-modulated OCDM-based RadCom system with discrete-Fresnel domain channel estimation.",
"As for the captured signal by the receiver of another OCDM communication or RadCom device, it is henceforth assumed that both timing and frequency synchronizations are ensured via techniques such as Schmidl & Cox's algorithm [26], [27].",
"After CP removal and transformation into the discrete-Fresnel domain, CIR estimates can be obtained from the same unmodulated pilot subchirps that used for radar sensing as previously mentioned.",
"After channel equalization, the modulation symbols contained in subchirps $k=N_\\mathrm {CP}$ to $k=N-N_\\mathrm {CP}+1$ can finally be extracted and demodulated following a typical OCDM processing chain."
],
[
"Numerical and Measurement Results",
"In this section, a performance analysis of discrete-Fresnel domain channel estimation for OCDM-based radar systems is performed.",
"Aiming at mid-range HAD applications [3], a carrier frequency $f_\\mathrm {c}={79}{}$ and a frequency bandwidth $B={1}{}$ are adopted.",
"Additionally, $N=2048$ subchirps, no CP, i.e., $N_\\mathrm {CP}=0$ , and $M=5120$ OCDM symbols are considered unless explicitly stated otherwise, which results in an OCDM symbol time duration of ${2.05}{}$ , an OCDM frame time duration of ${10.49}{}$ , and in the radar performance parameters listed in Table REF .",
"Table: Resulting radar performance parameters in the considered OCDM-based radar system.Figure: Simulated Doppler-shift tolerance: (a) PPLR, (b) PSLR, and (c) ISLR of a single target reflection with different n Δ =2RB/c 0 n_{\\Delta }=2RB/c_0 and k Δ =f D /Δfk_\\Delta =f_\\mathrm {D}/\\Delta f pairs obtained in an OCDM-based system with discrete-Fresnel domain radar processing.Figure: Obtained range-velocity radar images from measurements with OCDM-based radar systems with discrete-Fresnel domain channel estimation for a single target with relative radial velocities of (a) 0/{0}{/} (k Δ =0k_\\Delta =0), (b) 92.71/{92.71}{/} (k Δ =-0.1k_\\Delta =-0.1), (c) 231.78/{231.78}{/} (k Δ =-0.25k_\\Delta =-0.25), and (d) 463.56/{463.56}{/} (k Δ =-0.5k_\\Delta =-0.5).",
"In all radar images the actual target range is 30{30}{} (n Δ =200n_\\Delta =200).Based on the aforementioned parameters, simulated PPLR, PSLR, and ISLR [32], [3] results are shown in Fig.",
"REF to assess the distortion of the range mainlobe and sidelobes induced by Doppler shifts.",
"Since those calculations assume a single point target, the subindex $\\eta $ is henceforth omitted for $n_{\\Delta ,\\eta }$ and $k_{\\Delta ,\\eta }$ for the sake of simplicity.",
"In this figure, the whole unambiguous interval for range and relative radial velocity were considered, which results in $n_\\Delta \\in [0,2048]$ and $k_\\Delta \\in [-0.5,0.5]$.",
"The values for $k_\\Delta $ were defined based on the maximum unambiguous velocity expression from Table REF , the relationship $f_\\mathrm {D}=2v/\\lambda =2vf_\\mathrm {c}/c_0$ between Doppler shifts and relative radial velocities, and the frequency resolution $\\Delta f=B/N$ of the OCDM-based radar system.",
"Consequently, $k_\\Delta $ can be interpreted as a normalized Doppler shift expressed as $k_\\Delta =f_\\mathrm {D}/\\Delta f$.",
"Similarly, $n_\\Delta $ can be interpreted as a normalized range by the range resolution and expressed as $n_\\Delta =R/\\Delta R$.",
"Overall, degradations of PPLR, PSLR, and ISLR are mostly only observed for $\\left|k_\\Delta \\right|>0.1$ , which is also expected, e.g., in OFDM-based radar and RadCom systems where the maximum tolerable velocity is associated with this $k_\\Delta $ upper bound [31], [3].",
"The PPLR results in Fig.",
"REF (a) show a highest degradation of the processing gain $G_\\mathrm {p}$ of up to about ${4}{dB}$ , e.g., for the approximate region covered by $n_\\Delta \\in [0, 767]\\cup [1279,2048]$ and $k_\\Delta =-0.5$.",
"Next, Fig.",
"REF (b) shows the PSLR results, which indicate the ratio between the highest sidelobe and the mainlobe, while the ISLR results, which indicate the ratio between the integrated sidelobe level and the integrated mainlobe level, are presented in Fig.",
"REF (c).",
"Near the aforementioned region in the PPLR case, both PSLR and ISLR experience significant degradation, which can be explained by the results from (REF ) and (REF ).",
"Besides the range-Doppler coupling in these equations, the range and Doppler dependent phase terms, which are $^{\\frac{\\pi }{N}\\left(2nk_{\\Delta ,\\eta }-k_{\\Delta ,\\eta }^2+Nk_{\\Delta ,\\eta }\\right)}$ in (REF ), introduce changes in the shape of the estimated CIR w.r.t.",
"the ideally expected ones and is illustrated in more detail as follows.",
"Fig.",
"REF shows range-velocity radar images obtained from measurements with a Zynq UltraScale+ RFSoC ZCU111 from Xilinx, Inc.",
"The aforementioned SoC platform was used to emulate both the OCDM-based radar system with the adopted parameters in this section and transmit power $P_\\text{Tx}={0}{dBm}$, as well as the RTS described in [33] for a single radar target with RCS $\\sigma _\\mathrm {RCS}={30}{dBsm}$ at ${30}{}$ ($n_\\Delta =200$ ) and velocities ranging from ${00}{/}$ ($k_\\Delta =0$ ) to ${463.56}{/}$ ($k_\\Delta =-0.5$ ).",
"In these images, it is possible to see that, for increasing radial relative velocities and consequently Doppler shifts, the aforementioned range-Doppler coupling and the range and Doppler dependent phase terms may bias the target reflection from its actual range and duplicate the target's reflection in the range direction if $k_\\Delta =0$ , which explains the PPLR, PSLR, and ISLR results from Fig.",
"REF .",
"Conversely, the Doppler shift or relative radial velocity estimation is not affected by the aforementioned effects and only depends on the Doppler-induced phases $^{\\phi _{m,\\eta }}$ in (REF ) and (REF ).",
"As a consequence, similar performance for the velocity processing, e.g., to OFDM or PMCW radars is achieved.",
"For a more detailed visualization, Fig.",
"REF shows range cuts at the estimated target velocities of the measured radar images from Fig.",
"REF with a focus on the target's range.",
"Results for three additional systems parameterized to yield the same frame duration are shown for each considered velocity in Fig.",
"REF for the sake of comparison.",
"The first are for the proposed sector-modulated OCDM-based RadCom system in Section REF with $N=2048$ subchirps, CP length of $N_\\mathrm {CP}=512$ , and $M=4096$ OCDM symbols.",
"The second set of results is for the conventional OCDM-based radar system from [13] with $N=2048$ subchirps, $N_\\mathrm {CP}=512$ CP length, and $M=4096$ OCDM symbols.",
"Finally, the third set of results is for an OFDM-based RadCom system with $N=2048$ subcarriers, CP length of $N_\\mathrm {CP}=512$ , and $M=4096$ OFDM symbols.",
"For all three aforementioned modulation schemes, the resulting maximum tolerable range defined by the CP length is $R_\\mathrm {max,CP} = {76.8}{}$, the achieved processing gain is $G_\\mathrm {p} = {69.24}{\\mathrm {dB}}$ , and QPSK modulation with uniform power allocation is adopted for the modulated subchirps or subcarriers.",
"As shown in Fig.",
"REF , the proposed OCDM-based radar system and its sector-modulated OCDM-based RadCom yield virtually equal radar range profiles.",
"At zero Doppler shift, i.e., $k_\\Delta =0$ , they yield the same range profile result as their OFDM counterpart and a more regular sidelobe trend than the conventional OCDM-based RadCom system from [13], whose range sidelobes are significantly influenced by the modulated data onto its subchirps due to its correlation-based range processing.",
"As for $k_\\Delta =-0.1$ , which corresponds to the maximum tolerable Doppler shift for OFDM [34], [31], the proposed OCDM-based radar and RadCom systems present comparable performance to their OFDM counterpart, while the conventional OCDM-based RadCom system from [13] still yields the worse range sidelobe pattern.",
"Relevant differences in the range profiles obtained with the proposed OCDM-based systems w.r.t.",
"OFDM are only observed for $k_\\Delta =-0.25$ and $k_\\Delta =-0.5$ , namely, peak splitting and noticeable range-Doppler coupling due to the effects described in (REF ) and (REF ).",
"However, it should be noted that these Doppler shift levels yield very high relative radial velocities and are therefore not to be expected in practical scenarios.",
"Additionally, the combination of these Doppler shifts with the range of ${30}{}$ ($n_\\Delta =200$ ) leads the proposed OCDM-based radar and sector-modulated OCDM-based RadCom systems into their most critical region in terms of PPLR, PSLR, and ISLR as shown in Fig.",
"REF .",
"The results for these range and Doppler shift pairs are therefore lower bounds on the radar sensing performance of the proposed OCDM-based schemes and better results are expected for less critical combinations of range and Doppler shift values.",
"Figure: Range cuts of radar images from measurements with SISO OCDM-based radar system (★\\bigstar ), sector-modulated OCDM-based RadCom system (▪\\blacksquare ), conventional OCDM-based RadCom system (⧫\\blacklozenge ), OFDM-based RadCom system () for a single target with relative radial velocity of (a) 0/{0}{/} (k Δ =0k_\\Delta =0), (b) 92.71/{92.71}{/} (k Δ =-0.1k_\\Delta =-0.1), (c) 231.78/{231.78}{/} (k Δ =-0.25k_\\Delta =-0.25), and (d) 463.56/{463.56}{/} (k Δ =-0.5k_\\Delta =-0.5).",
"In all radar images the actual target range is 30{30}{} (n Δ =200n_\\Delta =200).",
"Since the three latter modulation schemes yield processing gain around 1dB{1}{dB} lower than the OCDM-based radar system, the magnitudes of the range profiles obtained with each modulation scheme are normalized w.r.t.",
"their respective results at 0/{0}{/} (k Δ =0k_\\Delta =0) for a fairer comparison.Figure: Range cuts of radar images from measurements with transmit channels p=0p=0 (★\\bigstar ), p=1p=1 (▪\\blacksquare ), p=2p=2 (⧫\\blacklozenge ), and p=3p=3 () in the considered MIMO OCDM-based radar system for a single target with relative radial velocity of (a) 0/{0}{/} (k Δ =0k_\\Delta =0), (b) 92.71/{92.71}{/} (k Δ =-0.1k_\\Delta =-0.1), (c) 231.78/{231.78}{/} (k Δ =-0.25k_\\Delta =-0.25), and (d) 463.56/{463.56}{/} (k Δ =-0.5k_\\Delta =-0.5).",
"In all radar images the actual target range is 30{30}{} (n Δ =200n_\\Delta =200).To address the FrDM multiplexing strategy for MU or MIMO operation described in Section REF , similar measurement results to the ones in Fig.",
"REF are presented in Fig.",
"REF for signals transmitted by $P=4$ collocated transmitters and evaluated at a single receiver.",
"For the results in this figure, the same OCDM signal parameters adopted for the SISO OCDM-based radar system as mentioned at the beginning of this section are adopted, leading to the same radar performance parameters listed in Table REF .",
"The only exception is the maximum unambiguous range, which is reduced to $R^{\\mathrm {MU/MIMO},P}_\\mathrm {max,ua} = {76.8}{}$ due to the multiplexing among $P=4$ transmitters.",
"Compared to the results for the SISO OCDM-based radar system in Fig.",
"REF , which are virtually the same as for the transmitter $p=0$ in the MIMO case, the results from Fig.",
"REF only differ significantly for $k_\\Delta =-0.25$ and $k_\\Delta =-0.5$ .",
"This is due to the combined effect of the range-Doppler coupling and Doppler dependent phase terms in (REF ), which directly influence the peak splitting and range-Doppler coupling effects.",
"As already mentioned previously, however, such high Doppler shift levels are not to be expected in practice.",
"To assess the communication performance of the proposed OCDM-based RadCom in Section REF , measurements over a communication channel emulated by re-purposing the previously described RTS were performed with the same SoC platform from Xilinx, Inc, ensuring time and frequency synchronization between the communication transmitter and receiver with the SC algorithm as described in [35].",
"For the communication analysis, the proposed sector-modulated OCDM-based RadCom system was compared with the OFDM-based and conventional OCDM-based RadCom systems.",
"Additionally, the same signal parameters adopted for the radar performance analysis were kept for all three modulation schemes, resulting in a data rate of ${0.80}{{bit}/}$ for the sector-modulated OCDM-based RadCom system and a data rate of ${1.40}{{bit}/}$ for its OFDM and conventional OCDM counterparts, which is influenced by inserted pilot subcarriers in the OFDM case and use of FSP-inserted pilots in the conventional OCDM case [36].",
"The magnitude response of the experienced communication CFR, whose frequency-selectivity is due to the sinc-shaping resulting from the use of an IF-sampling architecture and uncalibrated effects of cables, baluns and filters, is shown in Fig.",
"REF .",
"To enable channel equalization and subsequent extraction of the receive QPSK symbols from the receive signal, the unmodulated subchirp in the sector-modulated OCDM-based RadCom system was used.",
"As for OFDM- and the conventional OCDM-based RadCom system from [12], channel estimation was performed using the comb-pilot-based techniques, namely the one described in [37] for OFDM and its FSP-based version for the conventional OCDM-based RadCom system [36].",
"For all three modulation schemes, multiple channel estimates were averaged to reduce noise effect before performing equalization.",
"Figure: Magnitude response of the experienced communication CFR.Figure: Normalized receive QPSK constellations: sector-modulated OCDM-based RadCom system (), conventional OCDM-based RadCom system (), and OFDM-based RadCom system ().",
"For reference, an unit-radius circle (– –) and a QPSK constellation with unit symbol energy () are shown.The obtained normalized receive QPSK constellations are shown in Fig.",
"REF .",
"Based on the achieved results, both sector-modulated and conventional OCDM-based RadCom systems yield virtually the same communication performance.",
"While the first has an estimate SNR of ${29.65}{dB}$ and an EVM with mean value of ${-26.27}{dB}$ and standard deviation of ${5.59}{dB}$ , an estimated input SNR of ${29.74}{dB}$ and EVM with mean value of ${-26.20}{dB}$ and standard deviation of ${5.53}{dB}$ for the latter.",
"As for the OFDM-based RadCom system, an estimated input SNR of ${29.65}{dB}$ and EVM with mean value of ${-27.64}{dB}$ and standard deviation of ${6.30}{dB}$ were obtained.",
"The higher mean and value and lower standard deviation of the EVM in the OCDM-based systems can be explained by the fact that the degradation imposed by channel is spread over all subchirps.",
"Conversely, while a lower EVM mean value is experienced in the OFDM-based RadCom system, its higher standard deviation is due to different subcarriers experiencing different attenuation levels while propagating through the channel.",
"More specifically, the subcarriers on the rightmost side of the spectrum experience around ${10}{dB}$ more attenuation than the ones on the leftmost side (see Fig.",
"REF ), being therefore more severely degraded and having their receive QPSK symbols shifted away from the reference constellation in Fig.",
"REF .",
"This behavior agrees with the predicted higher robustness of the OCDM-based RadCom systems in frequency-selective channels, e.g., as reported in [9], [38], [15], [3], and leads to a better overall communication performance compared to their OFDM counterpart.",
"Finally, the simulated CCDF of the baseband PAPR for a single OCDM symbol is shown for the investigated OCDM-based radar system, its MU/MIMO variant, and the proposed sector-modulated OCDM-based RadCom system.",
"For the sake of comparison, the PAPR of considered OFDM- and conventional OCDM-based RadCom systems in the previous radar and communication analysis are also shown as a baseline.",
"As in [3], the PAPR was calculated assuming an oversampling factor of 20 to capture fast variations of time-domain signals.",
"The particularly low baseband PAPR for the considered OCDM-based radar system and its MU/MIMO variant is explained by the single active subchirp in (REF ) and (REF ), which yields a complex exponential signal with a virtually constant envelop in the discrete-time domain after the IDFnT in (REF ).",
"As for the sector-modulated OCDM-based RadCom system, the presence of active, modulated subchirps in (REF ) yields nearly ${6}{dB}$ higher average PAPR.",
"Compared to the OFDM- and conventional OCDM-based RadCom systems, which perform equally, the sector-modulated OCDM-based RadCom system presents reduced PAPR by around ${1.1}{dB}$ .",
"If lower communication data rates are aimed, this difference can be increased by deactivating some of the QPSK-modulated subchirps, which yields a similar effect to the PAPR reduction by tone reservation in OFDM-based systems.",
"Figure: Simulated baseband PAPR for the considered OCDM-based radar system and its MU/MIMO variant per transmit channel (▪\\blacksquare ), sector-modulated OCDM-based RadCom system (⧫\\blacklozenge ) and an OFDM- and conventional OCDM-based RadCom systems (), all with symbol length of N=2048N=2048."
],
[
"Conclusion",
"This article has investigated the use of discrete-Fresnel domain channel estimation for OCDM-based radar systems.",
"The proposed processing strategy relies on a strategic subchirp allocation for OCDM symbols in the discrete-Fresnel domain and exploits the pulse-compression-like effect of the DFnT.",
"After a thorough mathematical formulation of the individual and joint effects of time and frequency shifts on OCDM signals in the discrete-Fresnel domain, discrete-Fresnel domain radar channel estimation strategies have been proposed for both SISO and MIMO radar applications, as well as for RadCom operation.",
"Finally, a detailed performance analysis supported by simulation and measurements with a RTS has been carried out assuming a mid-range HAD scenario to validate the contributions of this article and compare the achieved performances with the ones of OFDM- and conventional OCDM-based radar/RadCom systems.",
"The achieved results have shown that the investigated discrete-Fresnel domain channel estimation for OCDM-based radar and RadCom systems yields comparable performance to OFDM-based radar/RadCom systems in all its forms, i.e., SISO and MIMO operations or sector-modulated RadCom operation.",
"Significant differences, namely peak splitting and range migration, are only observed for Doppler shifts associated with relative radial velocities that are not to be expected in practice.",
"Compared to the conventional OCDM-based RadCom system from a previous work, all of the proposed strategies yield improved radar sensing performance as their range sidelobes are not influenced by modulated data symbols onto subchirps.",
"Additionally, the proposed strategies for SISO and MIMO OCDM-based radar sensing yield significantly lower PAPR than their OFDM and conventional OCDM counterpart.",
"As for the sector-modulated OCDM-based RadCom system, the experienced PAPR reduction is lower, but can be improved in exchange for data rate reduction.",
"For radar-centric systems, where very high data rates are not necessarily a requirement, the proposed sector-modulated OCDM-based RadCom system also appears as an attractive alternative to its OFDM and conventional OCDM counterparts, since it shows robustness for both radar sensing and communication.",
"The finite geometric series and its solution used for the derivation of equations in Sections and are expressed as $\\sum \\limits _{\\eta =0}^{\\gamma }\\alpha ^\\eta &=& 1 + \\alpha + \\alpha ^2 + \\cdots + \\alpha ^\\gamma \\nonumber \\\\&=& \\frac{\\alpha ^{\\gamma +1}-1}{\\alpha -1}\\quad \\text{for}\\quad \\alpha \\ne 1$ for $\\eta \\in \\mathbb {N}$ , $\\gamma \\in \\mathbb {N}$ , $\\alpha \\in \\mathbb {C}$ .",
"[Figure: NO_CAPTION He was with the Laboratório de Comunicações (LCom), UFJF, from June 2014 to March 2019.",
"From April 2015 to March 2016, he was also with the Digital Communications Group, University of Duisburg-Essen, Duisburg, Germany.",
"Since April 2019, he has been with the Institute of Radio Frequency Engineering and Electronics (IHE), KIT, where he is currently a Research Associate.",
"His research interests are in the areas of signal processing, digital communication, and their applications to integrated radar sensing and communication systems and networks.",
"Mr. Giroto de Oliveira was a recipient of the Science without Borders (CsF) scholarship funded by Coordenação de Aperfeiçoamento de Pessoal de NÃvel Superior (CAPES), Brazil, from September 2014 to March 2016, and of the Research Grant for Doctoral Programmes in Germany from the German Academic Exchange Service (DAAD), Germany, from April 2019 to March 2021.",
"He was also a co-author of the Best Conference Paper award winning paper at the 2022 International Workshop on Antenna Technology (iWAT).",
"[Figure: NO_CAPTION He is currently working as a Group Leader for radar systems at the Institute of Radio Frequency Engineering and Electronics (IHE).",
"The focus of his work is on the development of efficient future radar waveforms and interference mitigation techniques for multi-user scenarios.",
"His current research interests include orthogonal frequency-division multiplexing-based multiple-input multiple-output radar systems for future automotive applications and drone detection.",
"He was also the author and co-author of the best publication of 2021 and 2022 within the ITG Society, which was awarded the VDE ITG Prize.",
"[Figure: NO_CAPTION He was a Research Associate with the Chair of Information Systems, Innovation & Value Creation (WI1), Friedrich-Alexander-Universität Erlangen-Nürnberg, Nuremberg, Germany.",
"From 2010 to 2014, he was with Syrian Telecommunication Company, Homs, and worked there in different positions.",
"He has been a Research Associate with the Institute of Radio Frequency Engineering and Electronics (IHE), Karlsruhe Institute of Technology, Karlsruhe, Germany, since 2017.",
"His current research interests include chirp sequence (CS) joint radar-communication and orthogonal frequency-division multiplexing-based multiple-input multiple-output radar systems for future automotive applications.",
"[Figure: NO_CAPTION His main research interests include digital radar target simulation for the purpose of automotive radar sensor validation and realistic target modeling.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION From 1994 to 2001, he was a Research Assistant with the Institut für Höchstfrequenztechnik und Elektronik (IHE), TH.",
"In February 2001, he joined IBM as Research Staff Member at the IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA.",
"From October 2004 to September 2007, he was with Siemens AG, Lindau, Germany.",
"During that period, he managed the RF development team for automotive radars.",
"In October 2007, he became a Full Professor with the Karlsruhe Institute of Technology (KIT), Karlsruhe.",
"He is currently the Director of the Institute of Radio Frequency Engineering and Electronics (IHE), KIT.",
"He is a co-editor of three books and the author or a co-author of 120 journal articles, over 400 contributions at international conferences, and 15 granted patents.",
"His research topics include wave propagation, stochastic channel modeling, channel measurement techniques, material measurements, microwave techniques, millimeter-wave antenna design, wireless communication, and radar system design.",
"Dr. Zwick has been a member of the Heidelberg Academy of Sciences and Humanities since 2017.",
"Since 2019, he has been a member of acatech (German National Academy of Science and Engineering).",
"His research team received over ten best paper awards at international conferences.",
"In 2013, he was the General Chair of the International Workshop on Antenna Technology (iWAT 2013) in Karlsruhe and the IEEE MTT-S International Conference on Microwaves for Intelligent Mobility (ICMIM) in Heidelberg in 2015.",
"He was the TPC Chair of the European Microwave Conference (EuMC) 2013 and the General TPC Chair of the European Microwave Week (EuMW) 2017.",
"In 2023, he will be the General Chair of EuMW in Berlin.",
"He has served on the technical program committees (TPC) of several scientific conferences.",
"From 2008 to 2015, he was the president of the Institute for Microwaves and Antennas (IMA).",
"He was selected as a Distinguished IEEE Microwave Lecturer for the 2013-2015 period with his lecture on “QFN Based Packaging Concepts for Millimeter-Wave Transceivers.” In 2019, he became the Editor-in-Chief of the IEEE Microwave and Wireless Components Letters.",
"In 2022 he was awarded an honorary doctorate of the Faculty of Electrical Engineering and Informatics at the Budapest University of Technology and Economics in Hungary."
]
] | 2207.10521 |
[
[
"Spin-phonon dispersion in magnetic materials"
],
[
"Abstract Microscopic coupling between the electron spin and the lattice vibration is responsible for an array of exotic properties from morphic effects in simple magnets to magnetodielectric coupling in multiferroic spinels and hematites.",
"Traditionally, a single spin-phonon coupling constant is used to characterize how effectively the lattice can affect the spin, but it is hardly enough to capture novel electromagnetic behaviors to the full extent.",
"Here, we introduce a concept of spin-phonon dispersion to project the spin moment change along the phonon crystal momentum direction, so the entire spin change can be mapped out.",
"Different from the phonon dispersion, the spin-phonon dispersion has both positive and negative frequency branches {even in the equilibrium ground state}, which correspond to the spin enhancement and spin reduction, respectively.",
"Our study of bcc Fe and hcp Co reveals that the spin force matrix, that is, the second-order spatial derivative of spin moment, is similar to the vibrational force matrix, but its diagonal elements are smaller than the off-diagonal ones.",
"This leads to the distinctive spin-phonon dispersion.",
"The concept of spin-phonon dispersion expands the traditional Elliott-Yafet theory in nonmagnetic materials to the entire Brillouin zone in magnetic materials, thus opening the door to excited states in systems such as CoF$_2$ and NiO, where a strong spin-lattice coupling is detected in the THz regime."
],
[
"Introduction",
"Using the lattice to control the spin degree of freedom has garnered a significant attention worldwide, as a new way to alter spintronics.",
"But long before the current surging research activities [1], [2], [3], phonon excitation is known to be essential to demagnetization in a magnetic material by transferring energy to the spin subsystem, inducing magnetic phase transition.",
"Even in nonmagnetic crystals, the magnetic field affects photon-phonon interactions, which is known as morphic effects [4].",
"The discovery of strong magnetodielectric coupling in Mn$_3$ O$_4$ [5] demonstrated that the lattice involvement is more than just energy transfer.",
"Hirai et al.",
"[6] reported a large atomic displacement at a magnetic phase transition, with structural change detected [7].",
"Extreme modification on the crystal lattice is expected to change the magnetic ordering and has been demonstrated experimentally with multiferroic (La,Ca)MnO$_3$ [8].",
"The Raman scattering experiment [9] revealed that there is an opposite effect on the lattice from an external magnetic field, or magnetostructure change.",
"A real-space magnetic imaging of multiferroic spinel was also obtained experimentally [10].",
"When lattice vibrates, it alters the exchange interaction [11].",
"Theoretically, Lizarraga et al.",
"[12] showed the magnetism itself may play a role in the phase transition from hcp Co to fcc Co. Kuang et al.",
"[13] suggested that Co may have magneto-elastic effects.",
"Verstraete [14] showed that the electron-phonon coupling is also spin-dependent and the spin minority band has a larger coupling than the majority one.",
"Lefkidis et al.",
"[15] demonstrated for the first time that the phononic effects also appear on an ultrashort time scale, through the electron-phonon coupling [16], [17].",
"The effect becomes stronger in the rare-earth Gd(0001) [18], [19].",
"Henighan et al.",
"[20] directly detected the signature of THz coherent acoustic phonons in Fe.",
"To characterize the coupling between the phonon and spin excitation, a common method is to use the phonon-magnon coupling or magnetoelastic coupling [21], [22], [23], [24], [25], [26], [27], [28].",
"Pattanayak et al.",
"[29] recently detected some of the key signatures of spin-phonon coupling in hematite crystallites through dielectric and Raman spectroscopy.",
"However, a single spin-phonon coupling constant is hardly enough to capture the complexity as how spin and phonon interact with each other.",
"The lattice vibration has $3N-6$ degrees of freedom for a system of $N$ atoms, which is likely to limit the scope of applications of a single spin-phonon coupling constant, especially true in magnetostructure and magnetodielectric coupling.",
"In this paper, we go beyond a single spin-phonon coupling constant and introduce a concept of the spin-phonon dispersion.",
"We take the hcp Co as our first example to show that there is a well-defined nearly linear dependence of the spin moment on the volume of the cell for a fixed $c/a$ ratio, and on the $c/a$ ratio for a fixed volume.",
"The second order in the spin moment change is dispersed along the phonon crystal momentum.",
"A new picture emerges.",
"Different from the phonon counterpart, the eigenvalues of the spin-phonon dispersion can be either positive or negative, which corresponds to the spin moment increase or decrease, respectively.",
"The orders of the spin-phonon bands are reversed with respect to the phonon dispersion; and at the $\\Gamma $ point the single-degenerate band appears at the bottom and the triple-degenerate band at the top.",
"Along the high symmetry lines in the Brillouin zone, both spin reduction and enhancement are found.",
"The situation in bcc Fe is different.",
"The major reduction is around the N point, and there is only spin enhancement along the $\\Gamma $ -H line.",
"We attribute the main differences between the spin-phonon dispersion and the regular phonon dispersion to their respective force matrices.",
"While the matrix for second derivative of spin with respect to atomic displacement, i.e., the spin force matrix, has a similar structure as the vibrational force matrix, its diagonal elements are smaller than its off-diagonal elements, so the spin dynamical matrix has both negative and positive eigenvalues.",
"The spin-phonon dispersion proposed here is expected to be applicable across all magnetic materials; it points out regions where the spin properties can be tailored through the phonon excitation.",
"The rest of the paper is arranged as follows.",
"In Sec.",
"II, we present the theoretical formalism, where we first define the spin dynamical matrix and then present the details of the spin-phonon dispersion calculation.",
"Section III provides our results, where the spin change in hcp Co for two phonon modes is first examined, followed by the spin-phonon dispersions of hcp Co and bcc Fe.",
"We explain these main differences using the vibrational force and spin force matrices.",
"Section IV is devoted to the discussion of our results and their relation to the ongoing experimental research and the Elliot-Yafet theory.",
"We conclude this paper in Sec.",
"V."
],
[
"Theoretical formalism",
"Even in nonmagnetic metals, the traditional Elliot-Yafet (EY) theory [30], [31] has shown that phonons play a significant role in spin relaxation.",
"Spin hot spots were identified in polyvalent metals [32], [33].",
"In magnetic materials, states are spin-polarized and with the presence of spin-orbit coupling, one has to introduce the $SD$ factor to characterize the spin hot spots in laser-induced demagnetization [34], which highlights the fact that a single spin-phonon coupling parameter is not enough.",
"Here we employ the density functional theory as implemented in the Wien2k code [35] and VASP code [36], with the details presented elsewhere [37], [38], [39], [40], [41].",
"In brief, we solve the Kohn-Sham equation self-consistently [40], $\\left[-\\frac{\\hbar ^2\\nabla ^2}{2m_e}+v_{eff}({\\bf r}) \\right]\\psi _{i{\\bf k}}({\\bf r})=E_{i{\\bf k}} \\psi _{i{\\bf k}} ({\\bf r}),$ where $ \\psi _{i{\\bf k}}({\\bf r})$ and $E_{i{\\bf k}}$ are the eigenstate and eigenenergy of band $i$ and ${\\bf k}$ point, respectively.",
"$v_{eff}$ is determined by $v_{eff}({\\bf r})=v({\\bf r})+\\int \\frac{n({\\bf r}^{\\prime })}{|{\\bf r}-{\\bf r}^{\\prime }|} d{\\bf r}^{\\prime }+v_{xc}({\\bf r}), $ where $v_{xc}({\\bf r})$ is the exchange-correlation potential, $v_{xc}({\\bf r})=\\delta E_{xc}[n]/\\delta n({\\bf r})$ .",
"The spin-orbit coupling is included through the second variational principle [35].",
"We use the generalized gradient approximation for the exchange-correlation energy functional.",
"The density $n({\\bf r})$ is computed from $n({\\bf r})=\\sum _{i{\\bf k}}\\rho _{i,i,{\\bf k}}|\\psi _{i{\\bf k}}({\\bf r})|^2$ , where $\\rho _{i,i,{\\bf k}}$ is the electron occupation.",
"The lattice dynamics is governed by the potential energy change around the equilibrium position of lattice points.",
"The expansion of the potential energy $\\Phi $ is $\\Phi ({\\bf R})=\\Phi _0+\\sum _{i,\\alpha }\\left( \\frac{\\partial \\Phi }{\\partial { R}_{i,\\alpha }}\\right)_0 \\Delta { R}_{i,\\alpha }+\\frac{1}{2}\\sum _{i,\\alpha ;j,\\beta }\\left(\\frac{\\partial ^2\\Phi }{\\partial { R}_{i,\\alpha } \\partial {R}_{j,\\beta }}\\right)_0 \\Delta { R}_{i,\\alpha } \\Delta {R}_{j,\\beta } +\\cdots $ where ${ R}_{i,\\alpha }$ denotes the position of atom $i$ along the $\\alpha $ direction.",
"At equilibrium, the second term is zero because the net forces on all atoms should be zero.",
"So the lowest order in the potential is the second order (the third term in the equation).",
"The second-order potential derivative matrix ${\\cal P}$ , the vibration force matrix, is defined as ${\\cal P}_{ij}^{\\alpha \\beta }=\\left(\\frac{\\partial ^2\\Phi ({\\bf L}_i-{\\bf L}_j)}{\\partial { R}_{i,\\alpha } \\partial {R}_{j,\\beta }}\\right)_0, $ which is used to compute the phonon spectrum.",
"Here ${\\bf L}_i$ and ${\\bf L}_j$ are the lattice vectors for atoms $i$ and $j$ , respectively.",
"Terms higher than the second order are ignored.",
"Similar to the lattice vibration, the total magnetic spin moment of a crystal can be expanded as a function of the displacements of the atoms, i.e., $M_s=M_0+\\displaystyle \\sum _{i\\alpha }\\left(\\frac{\\partial M_s}{\\partial R_{i,\\alpha }}\\right)_0\\Delta R_{i,\\alpha }+\\frac{1}{2}\\displaystyle \\sum _{ij,\\alpha \\beta }\\left( \\frac{\\partial ^2 M_s}{\\partial R_{i,\\alpha }\\partial R_{j,\\beta }}\\right)_0 \\Delta R_{i,\\alpha } \\Delta R_{j,\\beta }+... ,$ where $M_s$ is the magnetic moment of the crystal, $M_0$ is the magnetic moment at the lattice equilibrium, ${\\bf R}$ is the position of the atom, $i(j)$ is the atomic index, and $\\alpha (\\beta )$ is the direction index, i.e., $\\alpha = x,$ $y, \\text{ or } z$ .",
"Different from the phonon calculation above, the linear term (the second term) $\\left(\\frac{\\partial M_s}{\\partial R_{i,\\alpha }}\\right)_0 $ is not zero, because the energy minimum is not the spin moment minimum in a magnetic material.",
"The second-order derivative of the spin moment ${\\cal S}$ , spin force matrix, is defined similarly as, ${\\cal S}_{ij}^{\\alpha \\beta }({\\bf L}_i-{\\bf L}_j)= \\left( \\frac{\\partial ^2M_s({\\bf L}_i-{\\bf L}_j)}{\\partial R_{i,\\alpha }\\partial R_{j,\\beta }}\\right)_0, $ which only depends on the relative lattice vector L. In this work, we use the finite difference method in a $(2\\times 2\\times 2)$ supercell to find the second order spin moment change.",
"Figure REF shows a two-cell structure for bcc Fe, where the first nearest neighbor and the second nearest neighbor atoms are denoted by the dashed and dotted lines, respectively.",
"The spin dynamical matrix is ${\\cal C}^{\\alpha \\beta }_{ij}({\\bf k}) = \\sum _{\\bf L} {\\cal S}_{ij}^{\\alpha \\beta }({\\bf L}) e^{-i{\\bf k}\\cdot {\\bf L}},$ where ${\\bf k}$ is the reciprocal lattice vector.",
"We diagonalize it to get the spin-phonon dispersion spectrum.",
"The first order change in spin moment, which is already taken into account in the EY theory [30], [31], only contributes a shift in the displacement, so it is not included under our current theory."
],
[
"Results",
"Despite the apparent importance of spin-lattice interaction in many materials [42], [43], except a short presentation [44], there has been no prior study on how the spin moment changes with the lattice.",
"Our study is also different from prior studies on magnon-phonon coupling [21] where the spin wave, i.e.",
"the spatial orientation of the spins, at each lattice site is coupled to the vibration but the spin amplitude remains the same.",
"Here, we consider the spin amplitude change when the atoms vibrate.",
"For this reason, it is necessary to develop a simple picture before the full scale of calculation."
],
[
"Effects of breathing and stretching modes on the spin\nmoment in hcp Co",
"We strategically choose hcp Co as our first example, because without changing the unit cell size we can already investigate two vibrational modes: one is the breathing mode, where the volume is changed while keeping the ratio $c/a$ fixed, and the other is the stretching mode, where the ratio $c/a$ is changed by keeping $V$ fixed.",
"Our in-plane lattice constant $a$ is 4.713924 bohr, and out-of-the-plane one $c$ is 7.651595 bohr.",
"The unit cell volume is $V=\\sqrt{3}a^2c/2$ [45].",
"Here we use the Wien2k code to carry out our study.",
"We choose the Muffin-tin radius $R_{\\rm MT}$ of 2.26 bohr, and its product with the planewave cutoff $K_{\\rm cutoff}$ is 9.",
"The $k$ mesh is $33\\times 33\\times 17$ , which is sufficient to converge our results as tested before [45].",
"Figure REF (a) shows the total energy change $\\Delta E=E-E_{\\rm min}$ as a function of volume change $\\Delta V/V$ for the breathing mode, where $E_{\\rm min}$ is the energy minimum.",
"We see that the entire change is very smooth and forms a typical parabola with a minimum around -0.2%, which shows that our starting lattice constant is slightly too large, but this does not affect our conclusion.",
"By contrast, the spin moment ($M_S$ ) change increases linearly with $\\Delta V/V$ (see Fig.",
"REF (b)).",
"For every 1% change in $\\Delta V/V$ , the spin changes by about 0.01 $\\mu _B$ , which can be fitted to $M_{S}=\\left[ 1.73+0.00965 \\Delta V/V-0.0001469 (\\Delta V/V)^2 \\right] ~\\mu _B$ , where the second term is the linear contribution.",
"Figure REF (c) shows the orbital moment change with $\\Delta V/V$ , which can be fitted to $M_{O}=(0.0778+0.00189\\Delta V/V)\\mu _B$ .",
"The orbital change is only 1/5 the spin change.",
"The results from the stretching mode are shown in Figs.",
"REF (d), (e), and (f).",
"Figure REF (d) shows that the stretching mode has a similar smooth energy change as a function of $c/a$ .",
"Its spin change (Fig.",
"REF (e)) is smaller than that in the volume change; and for one percentile change in $c/a$ , the spin changes by 0.001895 $\\mu _B$ .",
"The second order coefficient is below $10^{-6}$ .",
"The orbital $M_O$ behaves very differently from the spin: it decreases with $c/a$ (Fig.",
"REF (f)), and the change is not strictly linear.",
"So far the spin changes are limited to two lattice modulations.",
"To go beyond this, we need to introduce the spin-phonon dispersion."
],
[
"Spin-phonon dispersion",
"We start with hcp-Co.",
"There have been many studies on the phonon dispersion of hcp Co. For instance, Pun and Mishin et al.",
"[46] employed the embedded-atom potential to compute the phonon spectrum.",
"We employ the VASP code [36] and the frozen-phonon method and our computed phonon dispersion is shown in Fig.",
"REF (a).",
"The frequencies of the two optical modes at $\\Gamma $ point are 4.45 (doublet) and 7.12 (singlet) THz, respectively, which match the neutron scattering results [47] of 4.30 and 7.60 THz, respectively.",
"This agreement gives us confidence that our theory can give an accurate prediction on the lattice dynamics.",
"We then move on to the spin moment change as a function of lattice vibration.",
"As predicted by Eq.",
"REF , the spin has a nonzero second-order term in the lattice displacement.",
"We make the same Born-Oppenheimer approximation for the spin where the lattice responds much slower than the electron spin, an assumption that needs experimental verification.",
"Then we can apply the same frozen-phonon method to the spin moment change as a function of atomic displacement.",
"By diagonalizing the spin dynamical matrix (Eq.",
"REF ), we obtain the spin dispersion as a function of the crystal momentum.",
"Physically, this represents the spin moment change as the atoms vibrate according to the normal modes of phonons.",
"Figure REF (b) is our spin dispersion for hcp Co.",
"The high symmetry points are highlighted with the dashed vertical lines.",
"The general structure follows the phonon (Fig.",
"REF (a)), but the order of the spectrum is different.",
"At the $\\Gamma $ point, the phonon spectrum has three acoustic bands at the bottom but the spin has them at the top.",
"In addition, the degeneracy increases from 1 to 3 from the bottom to the top.",
"The horizontal line sets the zero value for the spin dispersion.",
"This feature persists from $\\Gamma $ to K, but at the M point, similar to the phonon spectrum, no degeneracy is found.",
"At the A point, all the bands are close to a single point.",
"The major difference from the phonon spectrum is that the spin dispersion has several branches with “imaginary modes” (negative eigenvalues).",
"Just as the imaginary phonon modes leading to the structural instability, the spin imaginary modes lead to spin reduction or demagnetization.",
"The activation of those modes reduce the magnitude of the total spin moment.",
"This phonon-induced demagnetization process is different from the conventional magnon picture or thermal-based demagnetization mechanism where the spin spatial orientation change reduces the spin moment for the entire sample.",
"In our case, the magnitude of the local spin moment is changed as the atom vibrates along the eigenvector of a phonon mode.",
"Quantitatively, if we examine Fig.",
"REF (b) closely, we notice that the LO mode at the $\\Gamma $ point has a larger contribution to the reduction of spin moment than that of the two fold degenerate TO modes.",
"The reason why these modes behave differently can be explained by examining the Co-Co bond.",
"Physically, the longitudinal mode shortens the bond length on one side, while elongates it on the other.",
"When the distance between neighboring atoms decreases, the increased electron overlap broadens the 3d bandwidth and reduces the density of state at the Fermi level.",
"This mechanism is also the origin of the suppression of the magnetization when the lattice constant of Co reduces.",
"By contrast, the transverse mode acts differently.",
"Instead of changing the bond length, it distorts the angles between lattice vectors, i.e.",
"changing the bond angles.",
"In other parts of the Brillouin zone, both the spin increase and decrease are found.",
"This is our first major finding.",
"This spin moment change is not limited to hcp Co.",
"In bcc Fe, we find a similar pattern.",
"Figure REF (c) is our phonon dispersion.",
"With fewer atoms in the cell, the number of bands is reduced.",
"Our spin-phonon dispersion is shown in Fig.",
"REF (d).",
"Between the $\\Gamma $ and H points, the spin-phonon and the phonon dispersions are almost identical, but the major difference appears around the N point, where we see a major negative eigenvalue branch.",
"This corresponds to the spin moment reduction.",
"However, the spin enhancement dominates along these high symmetry lines.",
"The $k$ -point convergence test has been performed up to $(32\\times 32\\times 32)$ .",
"The data in the paper uses the $k$ -point mesh of $(16\\times 16\\times 16)$ ."
],
[
"Insights into the spin-phonon dispersion",
"To have a deeper understanding of the phonon-induced spin moment change, we compare the vibrational force matrix ${\\cal P}_{ij}^{\\alpha \\beta }$ and the spin force matrix ${\\cal S}_{ij}^{\\alpha \\beta }$ , where $i$ and $j$ are the atom indices and $\\alpha \\beta $ are the Cartesian coordinate indices.",
"These two matrices are used to compute the phonon and spin-phonon dispersions.",
"We take bcc Fe as an example.",
"For the supercell $2\\times 2\\times 2$ , eight atoms are at positions, $(0,0,0), (\\frac{1}{2},\\frac{1}{2},0),(\\frac{1}{2},0,\\frac{1}{2}), (0,\\frac{1}{2},\\frac{1}{2}),(\\frac{1}{2},0,0),$ $ (0,\\frac{1}{2},0), (0,0,\\frac{1}{2}),(\\frac{1}{2},\\frac{1}{2},\\frac{1}{2})$ .",
"So there are 64 combinations, leading to sixty-four $3\\times 3$ matrices along the $x$ , $y$ and $z$ axes.",
"The simplest matrix is the on-site one.",
"${\\cal S}_{1,1}=\\left(\\begin{array}{rrr}\\gamma & 0 & 0 \\\\0 & \\gamma & 0 \\\\0 & 0 & \\gamma \\\\\\end{array}\\right);\\hspace{28.45274pt}{\\cal P}_{1,1}=\\left(\\begin{array}{rrr}g & 0 & 0 \\\\0 & g & 0 \\\\0 & 0 & g \\\\\\end{array}\\right).$ It is clear that the spin and vibrational force matrices have the same structure.",
"This is true for four matrices for the nearest neighbors in the same primitive cell, ${\\cal S}_{1,5}=\\left(\\begin{array}{rrr}-\\alpha & \\beta &\\beta \\\\\\beta & -\\alpha &-\\beta \\\\\\beta &-\\beta & -\\alpha \\\\\\end{array}\\right);\\hspace{28.45274pt}{\\cal P}_{1,5}=\\left(\\begin{array}{rrr}-a & b &b \\\\b & -a &-b \\\\b&-b & -a \\\\\\end{array}\\right),$ but we notice a crucial difference.",
"In the spin force matrix, the diagonal element $\\alpha $ is always smaller than the off-diagonal element $\\beta $ , where $\\alpha $ and $\\beta $ are matrix elements, not to be confused with the direction indices above.",
"Numerically, if we use those ${\\cal S}_{1,5}$ matrices alone, we end up to have a negative eigenvalue.",
"This explains why the spin-phonon dispersion has a negative eigenvalue, but the phonon one does not.",
"The other three matrices are found by cyclically exchanging the off-diagonal elements.",
"Another major difference is in the matrices for the next nearest neighbor cell.",
"${\\cal S}_{1,2}=\\left(\\begin{array}{rrr}\\delta & 0 & 0 \\\\0 & \\delta & 0 \\\\0 & 0 & \\epsilon \\\\\\end{array}\\right);\\hspace{28.45274pt}{\\cal P}_{1,2}=\\left(\\begin{array}{rrr}-d & 0 & 0 \\\\0 & -d & 0 \\\\0 & 0 & -e \\\\\\end{array}\\right).$ The other two matrices can be reproduced by the cyclic relation.",
"One sees that the spin force matrix has a positive value, but the vibrational force matrix is all negative.",
"Since both the vibrational force matrix and the spin force matrix must obey the sum rule, where the summation over the cells and atoms is zero, we find that for the spin, $\\gamma +2\\delta +1\\epsilon -4\\alpha =0$ , but for the vibration, $g-2d-1e-4a=0$ .",
"All the parameters are given in Table REF ."
],
[
"Discussions and implication for experiments",
"In nonmagnetic metals, the spin lifetime $\\tau _s$ is equal to the electron momentum relaxation time multiplied by the EY constant $\\beta $ , which is the ratio of the energy gap $\\Delta E$ to the spin-orbit coupling $\\lambda _{soc}$ [30], [31]: $\\tau _s=\\beta \\tau _e =\\left(\\frac{\\Delta E}{\\lambda _{soc}}\\right)^2\\tau _e $ where the energy gap $\\Delta E$ is between two scattering states.",
"For multiple channels of scattering, the inverse spin lifetime $1/\\tau _s$ , or the rate of spin relaxation, is computed from $\\frac{1}{\\tau _s} =\\sum _i \\frac{1}{\\beta _i\\tau _{e,i}}.",
"$ This relation is controlled by the smallest $\\beta _i$ , which is often the phonon, $\\beta _{phonon}$ .",
"Fabian and Das Sarma [32] further showed in polyvalent metals, such as Al, that the spin relaxation is significantly increased around spin hot spots [33], where a Fermi surface cuts through the Brillouin zone boundaries and special symmetry points and lines.",
"Experiments showed that $\\beta $ is not even a constant and depends on surfaces and interfaces [48].",
"In magnetic materials, this is much more complicated [34] and also more important for spintronics.",
"Here, we make a moderate attempt to show that the lattice distortion affects the spin and spin flip among the band states, with a detailed study left for the future due to the enormous complications in magnetic materials.",
"We choose two states close to the Fermi level, band states 38 and 39.",
"Due to the band dispersion, we cannot guarantee that they always stay close to the Fermi level.",
"All the conclusions drawn here are specific to these two states, and should not be applied to others without a detailed calculation.",
"Initially, we aim to choose a special symmetry line, from $\\Gamma $ to A, along the $k_z$ axis, but it turns out that due to the momentum locking in the spin-orbit coupling, along this direction there is almost no spin flip.",
"Since showing results for a $k$ mesh of $(53\\times 53\\times 28)$ is overly excessive, we decide to choose five segments which are not along any high symmetry lines.",
"The $k$ only changes along the $z$ axis from 0 to 0.5 in the units of the reciprocal lattice vector, so we can monitor the pattern of change.",
"Other than this, the $k$ choice is arbitrary.",
"Segment 1, from 361 to 375, has the $k$ change from $k_0=(0,0.45,0)$ to $k_1=(0,0.45,0.5)$ , where the numbers 361 to 375, as well as those on the $x$ axis of Fig.",
"REF , are the $k$ index of the $k$ list out of $(53\\times 53\\times 28)$ .",
"Segment 2, from 376 to 390, has the $k$ change from $(0,0.4716,0)$ to $k_2=(0,0.4716,0.5)$ ; segment 3, from 391 to 405, has the $k$ change from $(0,0.4905,0)$ to $k_3=(0,0.4905,0.5)$ ; segment 4, from 406 to 420, has the $k$ change from $(0,0.01886,0)$ to $k_4=(0,0.01886,0.5)$ ; and segment 5, from 421 to 435, has the $k$ change from $(0.001886,0.0377,0)$ to $k_5=(0.001886,0.0377,0.5)$ .",
"Figures REF (a) and (b) show the spin moments $\\langle nk|s_z|nk\\rangle $ of states 38 and 39 and the spin flipping $\\langle nk|s^+|mk\\rangle $ and $\\langle nk|s^-|mk\\rangle $ between them, where $s_z$ is the $z$ -component of the spin operator, and $s^\\pm $ are the raising and lowering spin operators, respectively.",
"Figure REF (a) reveals that most of the $k$ points have spin moment close to either $+1\\mu _B$ or $-1\\mu _B$ .",
"These spin moments are not exactly at $+1\\mu _B$ or $-1\\mu _B$ ; with the spin-orbit coupling, either $+1\\mu _B$ or $-1\\mu _B$ is not allowed for states with a nonzero orbital angular momentum.",
"In the first segment between $k_0$ and $k_1$ , state 38 has spin moment close to $-1$ $ \\mu _B$ , and state 39 has spin moment close to $+1$ $\\mu _B$ , but this is not enough to have a spin flip.",
"Figure REF (b) shows that the spin flip between them is small because the spin flip, the expectation value of $\\langle 38|s^+|39\\rangle $ , critically depends on the product of the small spin component of one state's wavefunction with the large spin component of another state.",
"$\\langle 38 |s^-|39\\rangle $ is not shown since it is very small for this pair of states.",
"The first maximum occurs at the end of the first segment $k_2$ , where the spin of state 28 drops to $-0.69$ $\\mu _B$ .",
"A smaller spin flip is at $k_3$ and $k_4$ .",
"If we slightly distort the lattice in the same fashion as Figs.",
"REF (d), (e) and (f) (the lattice change is -5%), we can investigate the effect of the lattice vibration.",
"Figure REF (c) shows that while the majority of the spins remain similar to Fig.",
"REF (a), the hot spin pockets [34] are relocated to different $k$ points.",
"A bigger change is in the spin flip, with the maximum shifted to $k_3$ .",
"This shows that the spin flipping is very sensitive to the lattice perturbation.",
"Numerically we find that the initial wavefunctions that form the basis of the spin-orbit coupling in the second-variational step have a significant impact on the phase of the matrix elements of $s^+$ and $s^-$ , but the effect on the $s_z$ is tiny.",
"Our results may change if the wavefunction is converged at a different criterion.",
"In our calculation, we use an extremely low charge convergence of 10$^{-7}$ .",
"We should point out that Fig.",
"REF only samples a small portion of the Brillouin zone and one should not conclude that the distorted structure has a smaller spin flip, since a significant variation of the spin flip in comparison to the undistorted structure is found across the entire Brillouin zone.",
"Our goal here is to show that the phonons do affect the spin flip, highlighting the importance of the spin-phonon dispersion.",
"The negative eigenvalues found in the spin-phonon dispersion identify a demagnetization channel through the nearest neighbor vibration, which could have some implications on the recent debate on the mechanism of femtomagnetism [49], [50], [51].",
"It has been argued that the phonon plays a significant role in demagnetization [52], [53].",
"While we do not consider the electronic excited state, the spin-phonon dispersion demonstrates unambiguously that not every phonon excitation leads to demagnetization.",
"If we assume a harmonic oscillation of lattice with time as shown by Henighan et al.",
"[20], we expect that the spin oscillates as $M_{S}(t)=M_{S}^{(0)}+M^{(1)}_{S}\\sin (\\Omega t)$ , where $t$ is the time, $M_{S}^{(0)}$ and $M_{S}^{(1)}$ are two constants, and $\\Omega $ is the phonon frequency.",
"This does not necessarily correspond to demagnetization.",
"In hcp Co, Fig.",
"REF (b) shows that the maximum negative eigenvalue reaches $-3.7$ $\\mu _B/\\rm Å^2$ at $\\Gamma $ , which corresponds to the optical phonon mode in Fig.",
"REF (a).",
"If we suppose the phonon mode displacement to be 0.01 $\\rm Å$ , which is typical in solids, this leads to the spin moment reduction of $-3.7 \\times 10^{-4}$ $ \\mu _B$ , which is quite small.",
"However, there are at least two cases that the spin reduction can be boosted.",
"One is the multiphonon generation which is featured by the THz anharmonic frequency shift [54], [55].",
"Under the same temperature, lower-frequency phonon modes are generated more.",
"At an elevated temperature, we expect that the lattice vibration anharmonicity appears since the coupling between the phonon and demagnetization becomes highly nonlinear.",
"This may contribute additional amounts of demagnetization.",
"The second case is if one employs a strong laser pulse, where the displacement can reach 0.5 $\\rm Å$ .",
"In bcc Fe, even though only one pocket has a larger negative eigenvalue (around $N$ ), its value, $-4.4$ $\\mu _B\\rm /Å^2$ , is more negative than that in hcp Co.",
"This suggests a scenario for future experiments.",
"If one can selectively excite a special phonon mode, through Raman or neutron scattering [56], and then detect the spin moment change, one may be able to completely map out the entire spin-phonon dispersion.",
"Finally, we note that the phonon coupling to other degrees of freedom has been observed in nanomagnets [25], Gd [19] and other magnetic systems [23], [24], [14], [27], [57].",
"Shin et al.",
"[58] showed even a quasi-static strain plays an important role in ultrafast spin dynamics.",
"Rongione et al.",
"[59] detected a bipolar strain wave-induced THz torque in the (001) oriented NiO/Pt film where the stress from the Pt layer launches the strain wave.",
"Recently, Mashkovich et al.",
"[60] employed the strong spin-lattice coupling in antiferromagnetic CoF$_2$ to excite phonons via magnons.",
"For the same CoF$_2$ system, Disa et al.",
"[61] did the opposite.",
"They excited two degenerate $E_u$ IR modes to displace the lattice along the optically inaccessible $B_{2g}$ Raman mode, which transiently breaks the site symmetry between two Co atoms.",
"The spin moments on two Co atoms do not compensate each other, and CoF$_2$ temporally transitions to a ferrimagnet, with a net spin moment of 0.21 $\\mu _B$ .",
"The spin-phonon dispersion introduced here is expected to capture many opportunities in the entire Brillouin zone.",
"This points out a new direction – the spin-phonon dispersion in excited states.",
"Excited states often accommodate strong THz anharmonicity, leading to anharmonic frequency shifts through multiple phonon scattering."
],
[
"Conclusions",
"We have introduced a much needed concept of the spin-phonon dispersion, where the second-derivative of the spin moment change is dispersed along the phonon crystal momentum.",
"Our spin-phonon dispersion, with spin and phonon information both present, captures the complexity of the dependence of spin on the phonon, which goes beyond the spin-phonon coupling constant.",
"The spin-phonon dispersion in hcp Co shows that a large portion of the Brillouin zone accommodates both spin reduction and increase, but this is not the case for bcc Fe.",
"We only find a small pocket around the N point in bcc Fe that has spin moment reduction.",
"This may help us understand the higher Curie temperature in Fe.",
"Microscopically, we also understand why the spin-phonon dispersion has negative eigenvalues.",
"In contrast to the vibrational force matrix, the off-diagonal matrix elements in the spin force matrix are larger than its diagonal counterparts, so the spin dynamical matrix contains the negative eigenvalues.",
"Our spin-phonon dispersion is also different from the magnon-phonon coupling [21] where the spin orientation, not the spin magnitude, is taken into account.",
"Finally, our approach can be easily extended to systems where spin-orbit coupling should be taken into account.",
"Thus, the spin-phonon dispersion represents an important paradigm shift and will have a significant impact on future research in both simple and complex quantum magnetic materials in spatial and time domains [49].",
"We acknowledge the helpful comments on our paper from Dr. Qihang Liu (SUST, China).",
"MG was supported by Foundation for Distinguished Young Talents in Higher Education of Guangdong under Grant No.",
"2020KQNCX064, Shenzhen Science and Technology Innovation Council under Grant No.",
"JCYJ20210324104812034, and Natural Science Foundation of Guangdong Province under Grant No.",
"2021A1515110389.",
"GPZ and YHB were supported by the U.S. Department of Energy under Contract No.",
"DE-FG02-06ER46304.",
"Part of the work was done on Indiana State University's quantum and obsidian clusters.",
"The research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No.",
"DE-AC02-05CH11231.",
"$^*$ guo-ping.zhang@outlook.com https://orcid.org/0000-0002-1792-2701 Table: Spin (α,β,γ,δ,ϵ\\alpha ,\\beta ,\\gamma ,\\delta ,\\epsilon ) and phonon(a,b,g,d,ea,b,g,d,e) parameters used for bcc Fe to compute the spin-phononand phonon spectra, respectively, withα,β,γ,δ,ϵ\\alpha ,\\beta ,\\gamma ,\\delta ,\\epsilon in the units ofμ B /Å 2 \\mu _B/\\rm Å^2.Figure: An example of the two-cell bcc structure to explain how wecompute the spin-phonon dispersion.",
"Each atom's spin (empty arrows)is initialized along the zz axis.",
"The reference atom 1 is at thebody center, with the eight first neighboring atoms denoted (shadedspheres), where atom 5 is just an example.",
"The reference atom hassix second-neighbor atoms, where atom 2 is just an example.",
"For bothbcc Fe and hcp Co, we adopt a 2×2×22\\times 2\\times 2 supercell.",
"Thefigure here is just a part of it, without an overly cumbersomediagram.",
"Both the vibrational force and spin force matrices arecomputed by displacing one atom at one time.",
"Two filled arrowsdenote two unique displacement directions.Figure: (a) Total energy change ΔE\\Delta E as a function of ΔV/V\\Delta V/V for the breathing mode.",
"ΔE\\Delta E is referenced withrespect to the lowest total energy.",
"The k mesh is 33×33×1733\\times 33 \\times 17.",
"(b) and (c) are the spin and orbital momentchanges as a function of ΔV/V\\Delta V/V, respectively.",
"(d) Totalenergy change ΔE\\Delta E for the stretching mode.",
"The k meshis much larger, 53×53×2853\\times 53 \\times 28.",
"(e) and (f) are the spinand orbital moment changes as a function of Δ(c/a)/(c/a)\\Delta (c/a)/(c/a),respectively.Figure: (a) Phonon spectrum of hcp Co. (b) Spin-phonon dispersionfor hcp Co.",
"The total spin moment is expanded up to the secondorder.",
"Both the positive and negative spin moment changes arenoticed.",
"Numbers around the Γ\\Gamma point in (a) and (b) denote thedegeneracy of the bands.",
"(c) Phonon spectrum of bcc Fe.",
"(d)Spin-phonon dispersion for bcc Fe.Figure: (a) Spin expectation values of states i=38i=38 (circles)and i=39i=39 (boxes) along the crystal momentum for five segments withk 1 k_1 through k 5 k_5, whose coordinates are given in the maintext.",
"The vertical dashed lines denote their locations.",
"The numberson the xx axis are the kk index of the kk list, with the kk mesh(53×53×28)(53\\times 53\\times 28).",
"(b) Spin flip matrix element s + s^+ 〈38|s + |39〉\\langle 38|s^+|39\\rangle between states 38 and 39.",
"The element for s - s^- istiny, not shown.",
"The solid and dashed lines denote the real andimaginary parts, respectively.",
"(c) and (d) are the same as (a) and(b), but for the distorted structure, with -0.5%-0.5\\% contraction asFigs.",
"(d), (e) and (f).",
"This mimics the temperatureeffect.",
"(c) shows the spin change is concentrated in some hot spinspots.",
"(d) shows a larger spin flip change due to the latticedistortion.",
"Even the location of the larger spin flip ischanged.",
"The largest change is now at k 3 k_3."
]
] | 2207.10450 |