OTFS and LFM combination-based common-inductance integrated waveform design method

Abstract

The invention provides a combined waveform design method based on an orthogonal time frequency space (OTFS) and a linear frequency modulation (LFM) signal, and the combined waveform design method based on the OTFS and the LFM is characterized in that the combined waveform design method comprises the following steps of: designing a combined waveform based on the OTFS and the LFM; the method comprises the following steps of: firstly, embedding an LFM pulse signal into a delay-Doppler (DD) domain grid, deducing a distribution rule of the LFM pulse signal in the DD domain, and determining a main energy point track of the LFM pulse signal; secondly, a window matrix is designed according to the distribution position of the main energy point, and spectrum diffusion of the window matrix is inhibited; then, the remaining unoccupied DD domain grids are distributed to OTFS communication symbols, and resource multiplexing of sensing and communication signals is achieved. And finally, analyzing an LFM signal component at a receiving end to complete distance and speed estimation of a target, and demodulating an OTFS communication symbol at the same time to realize communication data recovery. According to the combined waveform design method based on the OTFS and the LFM, the communication performance of the OTFS can be maintained, the problem that the sensing distance of an existing OTFS system is limited is solved, and the sensing distance of the system is remarkably increased.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of waveform design, and particularly relates to an integrated sensing and communication waveform design method based on OTFS and LFM pulse combination under an integrated sensing and communication system. BACKGROUND

[0002] With the evolution of wireless communication systems to high frequency bands, large bandwidths and high mobility scenarios, integrated sensing and communication (ISAC) technology has gradually become a key development direction for new generation wireless systems. ISAC systems achieve the coordinated operation of target sensing and data communication by sharing software and hardware platforms and spectrum resources, and have wide application prospects in intelligent transportation, environmental monitoring, human-computer interaction, intelligent manufacturing and other fields. Waveform design plays a core role in ISAC systems, which designs dual-purpose waveforms with communication and sensing functions through unified scheduling and shared use of communication and sensing signaling resources, thereby achieving the integration gain of spectrum and hardware resources.

[0003] In the existing ISAC technology system, orthogonal time frequency space modulation (OTFS) as a modulation method based on Delay-Doppler (DD) domain modeling has attracted widespread attention because it can equivalent high-speed mobile channels to quasi-static channels. OTFS is particularly suitable for communication scenarios in high-speed motion environments, and is also used for target sensing research based on communication waveforms. However, due to its own subcarrier spacing and frame structure, the sensing system based on OTFS waveforms has inherent bottlenecks in the maximum unambiguous ranging capability, which is difficult to meet the demand for long-distance target detection. In contrast, as a classic radar waveform, linear frequency modulated (LFM) signal has significant advantages in long-distance target detection due to its large time-bandwidth product and good pulse compression characteristics, and its maximum ranging capability is mainly determined by the pulse width, which can naturally support long-distance and high-resolution sensing. Therefore, in actual systems, LFM pulse signals are often used to complete high-precision sensing tasks, and wireless communication systems use other modulation methods to achieve information transmission. This communication and sensing separated system architecture leads to low spectrum resource use efficiency and high hardware system complexity, which is difficult to meet the demand for highly integrated, low-power and high spectrum efficiency systems.

[0004] In summary, a waveform design method based on OTFS and LFM joint is designed and implemented in this paper. The LFM pulse signal is embedded in the grid structure of the DD domain, and a window function matrix is designed to limit the spectral leakage of the LFM pulse signal, thereby realizing the long-range perception of the system. At the same time, OTFS symbols are added to the remaining positions of the DD domain grid for communication. The proposed waveform design method can introduce radar signal structure with long-range perception capability while maintaining the performance of OTFS communication, breaking the problem that the perception distance of traditional OTFS modulation in existing ISAC systems is limited, significantly improving the ranging range of the system, and efficiently realizing the collaborative operation of communication and perception, giving the system the ability of long-range perception and communication. SUMMARY

[0005] The purpose of the present application is to provide an OTFS-LFM joint waveform design method based on the OTFS framework in the ISAC system, which can solve the problem of limited perception distance of traditional OTFS system and significantly improve the ranging range of the system without affecting the communication performance.

[0006] The OTFS-LFM integrated waveform design method based on the OTFS framework provided by the present application comprises the following steps:

[0007] Step one: modulate the LFM pulse signal to the DD domain, and observe the distribution rule in the DD domain. The specific algorithm process is as follows:

[0008] In the DD domain, define an N*M grid, where M and N represent the number of subcarriers and symbols respectively, and T represents the period of a single symbol, so the duration of the whole frame signal is NT. Define an LFM pulse signal with α pulses, where i represents the index of the i th pulse, 1≤i≤α, the signal bandwidth is B, and the pulse repetition interval is The frequency modulation slope is The pulse width is Where ζ is the duty cycle. Therefore, the time domain expression of the LFM pulse signal is:

[0009]

[0010] Where the expression of a single pulse is:

[0011]

[0012] Where, P c is the transmitted signal power, f c is the carrier center frequency. In the following transformation process, the carrier frequency term exp(j2πf ct), because when doing linear operations such as Fourier transform, the carrier frequency only causes the shift of the spectrum, and does not change the structure of the signal.

[0013] Firstly, the LFM pulse signal is transformed from time domain to time-frequency domain by Wigner Transform, and has:

[0014]

[0015] where g rx (t) represents a rectangular pulse limited in the interval [nT, (n+1)T], 0≤n≤N-1, 0≤m≤M-1, and △f is the subcarrier spacing, so the above formula can be changed to:

[0016]

[0017] that is,

[0018]

[0019] For the i-th pulse, there is only signal in its non-zero interval, so the integration interval of the above formula is the intersection of [nT, (n+1)T] and [(i-1)T c ,(i-1)T c +T w ]. Define the time variable: τ=t-(i-1)T c , and the new integration interval as [a i ,b i ], and the above integral is replaced by variable substitution, and has

[0020]

[0021] where [a i ,b i ]=[max(0, nT-(i-1)T c ), min(T w ,(n+1)T-(i-1)T c )].

[0022] Define the integral in the i-th pulse segment as

[0023]

[0024] For the integral I i (m; n), calculate its stationary point to observe the position where the main contribution item, i.e. the high energy point, appears, and its phase function is The stationary point satisfies Φ'(τ)=0, and considering that the time slot n in the time-frequency domain corresponds to the time t=nT, so there is also τ=nT-(i-1)T c, bring in Φ'(τ)=0, get

[0025]

[0026] Bring in and There are

[0027]

[0028] Modulus of the results, there are So the integral I i (m;n) high-energy points are distributed according to this rule.

[0029] Integral I i (m;n) is to calculate the Fourier component of LFM pulse at frequency mΔf:

[0030] I i (m;n) = ∫s i (τ)·e -j2πfτ dτ| f＝mΔf

[0031] The frequency of LFM signal changes linearly with time, and in a short enough time, its frequency change is very small, so in the duration of the window function, the LFM signal can be approximated as a single-frequency complex exponential signal, and its frequency can be approximated as:

[0032] f0=μ·τ=μ·(nT-(i-1)T c )

[0033] The corresponding subcarrier index is

[0034] So in the integral interval, there is an approximation: Its Fourier transform is δ(f-f0), and the Fourier transform of the rectangular window function is W(f)=T w ·sinc(T w f), time domain multiplication is equivalent to frequency domain convolution, and finally get

[0035]

[0036] So there is:

[0037]

[0038] Let be a constant 1, so finally:

[0039]

[0040] After transforming the LFM pulse signal into the time-frequency domain, its main energy point is It is distributed and diffuses near the main energy point.

[0041] Then, the signal is transformed from the time-frequency domain to the D domain using the Symptotic Finite Fourier Transform (SFFT):

[0042]

[0043] Substitute C TF [n,m],

[0044]

[0045] Where l is the index of the time delay dimension, k is the index of the Doppler dimension, 0≤l≤M-1, 0≤k≤N-1. First, sum the m in the above formula, and let m′=m-m0, then we have:

[0046]

[0047] definition Substituting back into the above equation, we have:

[0048]

[0049] Let γ = k - αl, then we have

[0050]

[0051] For S(γ), the common ratio is The sum of a geometric series is:

[0052]

[0053] Normalization yields the Dirichlet kernel:

[0054]

[0055] Finally, the expression in the DD domain is derived as follows:

[0056]

[0057] That is, the distribution of the principal energy points after the LFM pulse signal is transformed into the DD domain follows k = <αl> N It is accompanied by a diffusion phenomenon.

[0058] Step 2: Construct a window matrix in the DD domain to suppress the spectral spread of the LFM pulse signal, and add OTFS communication symbols to the remaining unoccupied DD domain grid. The specific algorithm flow is as follows:

[0059] After the LFM pulse signal is transformed into the DD domain, the principal energy point is concentrated at k = <αl>. N On this straight line, where l is the index of the time delay dimension and k is the index of the Doppler dimension, there are M principal energy points in the DD domain, distributed across M columns, with one principal energy point in each column. A window function is designed for each principal energy point in each column to suppress its energy diffusion. The window functions are constructed as follows:

[0060]

[0061] Where d k d represents the distance from other grid points to the principal energy point k0(l). k (k,l)=min(|k-k0(l)|,N-|k-k0(l)|), where w is the protection radius of the main energy point and β is the roll-off coefficient. Taking the main energy point of each column as the center, each column is multiplied by this roll-off window function, so the final representation of the LFM pulse signal in the DD domain is:

[0062] C′ DD [k,l]=C DD [k,l]·W DD [k,l]

[0063] After suppressing the spread of LFM pulse signals in the DD domain, OTFS symbols are added to the remaining unoccupied grid for communication, resulting in the following across the entire DD domain grid:

[0064]

[0065] Step 3: Transform the DD domain signal into the time domain and transmit it. The specific algorithm flow is as follows:

[0066] First, the signal is transformed from the DD domain to the time-frequency domain using the Inverse Symmetric Finite Fourier Transform (ISFFT).

[0067]

[0068] Then, through the Heisenberg transform, the signal is transformed from the time-frequency domain to the time domain before transmission, as follows:

[0069]

[0070] Among them, g tx (t) represents a rectangular pulse constrained by the interval [nT, (n+1)T].

[0071] Suppose there are p multipath components, where the path gain of the ψ-th path is h.ψ The time delay is θ ψ And the Doppler frequency shift is ν ψ Then the response of the wireless channel is:

[0072]

[0073] The signal received is represented as follows:

[0074]

[0075] Where n(t) is Gaussian white noise.

[0076] Step 4: Process the received signal. The specific algorithm flow is as follows:

[0077] At the communication receiver, the OTFS communication symbols are demodulated to recover the communication data. At the sensing receiver, matched filtering and Doppler frequency analysis are performed on the LFM pulse signal components to estimate the target's range and velocity. Matched filtering of the received signal is used to estimate the target's time delay, as follows:

[0078]

[0079] in, The output signal |y(t)| at t=θ ψ Obtain the maximum value, which corresponds to the time delay of the target, and then calculate the distance: Where c is the speed of light.

[0080] For target velocity estimation, observe each distance gate t = θ ψ The phase change of the corresponding slow-time series is estimated by performing a Fast Fourier Transform on the series to estimate its Doppler frequency shift.

[0081] For target ψ, after determining its range gate, the echo signal of the i-th pulse is:

[0082]

[0083] Along its slow time series of length N FFT Fast Fourier transform,

[0084]

[0085] Find the largest index: Calculate the Doppler frequency shift corresponding to this index: Its corresponding speed is:

[0086] Beneficial effects

[0087] This method first constructs a fused waveform framework in the DD domain, embedding the LFM pulse signal into the DD domain grid of the OTFS, and deriving the distribution pattern of the LFM pulse signal in the DD domain to determine its corresponding principal energy point trajectory. Secondly, based on the distribution position of the derived principal energy point trajectory in the DD domain, a window function matrix is ​​designed to window the LFM signal near the principal energy point to suppress its spectral spread. Then, the remaining unused DD domain grid resources are allocated to the OTFS communication symbols, achieving collaborative mapping and resource reuse of sensing and communication signals within the same DD domain grid. Finally, at the receiving end, matched filtering and Doppler frequency analysis are performed on the extracted LFM signal components to estimate the target's distance and velocity. Simultaneously, at the communication receiving end, the OTFS communication symbols are demodulated to recover communication data, thus completing the joint processing flow of the integrated sensing system.

[0088] The joint waveform design method based on OTFS and LFM designed in this invention can introduce a radar signal structure with long-range sensing capability while maintaining the communication performance of OTFS. This overcomes the problem of limited sensing range in existing ISAC systems, significantly improves the ranging range of the system, realizes long-range sensing and communication, and provides a good foundation for the wide application of ISAC. Attached Figure Description

[0089] Figure 1 This is the algorithm flowchart. Detailed Implementation Plan

[0090] The purpose of this invention is to provide a joint waveform design method for OTFS-LFM based on the OTFS framework under the ISAC system. This method can solve the problem of limited sensing distance in traditional OTFS systems and significantly improve the ranging range of the system without affecting communication performance.

[0091] The OTFS-LFM integrated waveform design method based on the OTFS framework described in this invention specifically includes the following steps:

[0092] Step 1: Modulate the LFM pulse signal into the DD domain and observe the distribution pattern in the DD domain. The specific algorithm flow is as follows:

[0093] In the DD domain, an N×M grid is defined, where M and N represent the number of subcarriers and symbols, respectively, and T represents the period of a single symbol. Therefore, the duration of the entire frame signal is NT. An LFM pulse signal with α pulses is defined, where i represents the index of the i-th pulse, 1 ≤ i ≤ α, the signal bandwidth is B, and its pulse repetition interval is... The frequency modulation slope is Pulse width is Where ζ is the duty cycle. Therefore, the time-domain expression of the LFM pulse signal is:

[0094]

[0095] The expression for a single pulse is:

[0096]

[0097] Among them, P c f is the transmitted signal power. c The carrier center frequency is given. The carrier frequency term exp(j2πf) does not need to be considered in the following transformation process. c t), because when performing linear operations such as Fourier transform, the carrier frequency only causes a shift in the spectrum, but does not change the structure of the signal.

[0098] First, the LFM pulse signal is transformed from the time domain to the time-frequency domain using the Wigner Transform:

[0099]

[0100] Among them, g rx (t) represents a rectangular pulse constrained by the interval [nT, (n+1)T], 0≤n≤N-1, 0≤m≤M-1, and Δf is the subcarrier spacing. Therefore, the above equation can be transformed into:

[0101]

[0102] Right now

[0103]

[0104] For the i-th pulse, there is a signal only in its non-zero interval, so the interval for integrating the above equation is [nT, (n+1)T] and [(i-1)T]. c ,(i-1)T c +T w The intersection of ] is defined as the time variable: τ = t - (i-1)T c Define the new integration interval as [a i ,b i By substituting variables into the above integrals, we have:

[0105]

[0106] Among them, [a i ,b i ]=[max(0,nT-(i-1)T c ),min(T w ,(n+1)T-(i-1)Tc )).

[0107] Define the integral within the i-th pulse segment as

[0108]

[0109] For integral I i (m;n), calculate its stationary point to observe the location of its main contribution, i.e., the high-energy point, and its phase function is: The stationary phase point satisfies Φ′(τ)=0. Considering that the time slot n in the time-frequency domain corresponds to the time t=nT, we also have τ=nT-(i-1)T. c Substituting Φ′(τ)=0, we get

[0110]

[0111] Substitute and have

[0112]

[0113] Taking the modulo of the result, we have Therefore, integral I i The high-energy points of (m;n) are distributed according to this pattern.

[0114] Integral I i The essence of (m;n) is to calculate the Fourier component of the LFM pulse at frequency mΔf:

[0115] I i (m;n)=∫s i (τ)·e -j2πfτ dτ| f＝mΔf

[0116] The frequency of the LFM signal changes linearly with time. Within a sufficiently short time interval, its frequency change is very small. Therefore, within the duration of the window function, the LFM signal can be approximated as a single-frequency complex exponential signal, and its frequency can be approximated as:

[0117] f0=μ·τ=μ·(nT-(i-1)T c )

[0118] The corresponding subcarrier index is

[0119] Therefore, within the integration interval, we have approximately: Its Fourier transform is δ(f-f0), and the Fourier transform of the rectangular window function is W(f) = T. w ·sinc(T w f), time-domain multiplication is equivalent to frequency-domain convolution, finally yielding

[0120]

[0121] Therefore:

[0122]

[0123] Will Let it be a constant 1, so we ultimately have:

[0124]

[0125] After transforming the LFM pulse signal to the time-frequency domain, its principal energy point is determined according to... It is distributed and diffuses near the main energy point.

[0126] Then, the signal is transformed from the time-frequency domain to the D domain using the Symptotic Finite Fourier Transform (SFFT):

[0127]

[0128] Substitute C TF [n,m],

[0129]

[0130] Where l is the index of the time delay dimension, k is the index of the Doppler dimension, 0≤l≤M-1, 0≤k≤N-1. First, sum the m in the above formula, and let m′=m-m0, then we have:

[0131]

[0132] definition Substituting back into the above equation, we have:

[0133]

[0134] Let Υ = k - αl, then we have

[0135]

[0136] For S(Υ), the common ratio is The sum of a geometric series is:

[0137]

[0138] Normalization yields the Dirichlet kernel:

[0139]

[0140] Finally, the expression in the DD domain is derived as follows:

[0141]

[0142] That is, the distribution of the principal energy points after the LFM pulse signal is transformed into the DD domain follows k = <αl> N It is accompanied by a diffusion phenomenon.

[0143] Step 2: Construct a window matrix in the DD domain to suppress the spectral spread of the LFM pulse signal, and add OTFS communication symbols to the remaining unoccupied DD domain grid. The specific algorithm flow is as follows:

[0144] After the LFM pulse signal is transformed into the DD domain, the principal energy point is concentrated at k = <αl>. N On this straight line, where l is the index of the time delay dimension and k is the index of the Doppler dimension, there are M principal energy points in the DD domain, distributed across M columns, with one principal energy point in each column. A window function is designed for each principal energy point in each column to suppress its energy diffusion. The window functions are constructed as follows:

[0145]

[0146] Where d k d represents the distance from other grid points to the principal energy point k0(l). k (k,l)=min(|k-k0(l)|,N-|k-k0(l)|), where w is the protection radius of the main energy point and β is the roll-off coefficient. Taking the main energy point of each column as the center, each column is multiplied by this roll-off window function, so the final representation of the LFM pulse signal in the DD domain is:

[0147] C′ DD [k,l]=C DD [k,l]·W DD [k,l]

[0148] After suppressing the spread of LFM pulse signals in the DD domain, OTFS symbols are added to the remaining unoccupied grid for communication, resulting in the following across the entire DD domain grid:

[0149]

[0150] Step 3: Transform the DD domain signal into the time domain and transmit it. The specific algorithm flow is as follows:

[0151] First, the signal is transformed from the DD domain to the time-frequency domain using the Inverse Symmetric Finite Fourier Transform (ISFFT).

[0152]

[0153] Then, through the Heisenberg transform, the signal is transformed from the time-frequency domain to the time domain before transmission, as follows:

[0154]

[0155] Among them, g tx (t) represents a rectangular pulse constrained by the interval [nT, (n+1)T].

[0156] Suppose there are p multipath components, where the path gain of the ψ-th path is h. ψ The time delay is θ ψ And the Doppler frequency shift is ν ψ Then the response of the wireless channel is:

[0157]

[0158] The signal received is represented as follows:

[0159]

[0160] Where n(t) is Gaussian white noise.

[0161] Step 4: Process the received signal. The specific algorithm flow is as follows:

[0162] At the communication receiver, the OTFS communication symbols are demodulated to recover the communication data. At the sensing receiver, matched filtering and Doppler frequency analysis are performed on the LFM pulse signal components to estimate the target's range and velocity. Matched filtering of the received signal is used to estimate the target's time delay, as follows:

[0163]

[0164] in, The output signal |y(t)| at t=θ ψ Obtain the maximum value, which corresponds to the time delay of the target, and then calculate the distance: Where c is the speed of light.

[0165] For target velocity estimation, observe each distance gate t = θ ψ The phase change of the corresponding slow-time series is estimated by performing a Fast Fourier Transform on the series to estimate its Doppler frequency shift.

[0166] For target ψ, after determining its range gate, the echo signal of the i-th pulse is:

[0167]

[0168] Along its slow time series of length NFFT Fast Fourier transform,

[0169]

[0170] Find the largest index: Calculate the Doppler frequency shift corresponding to this index: Its corresponding speed is:

Claims

1. A method for integrated waveform design based on OTFS and LFM, comprising the following steps: Step 1: Derive the distribution law of the LFM pulse signal in the DD domain and determine that its corresponding principal energy point trajectory follows k = <αl> N distributed; Step 2: Based on the distribution of the derived principal energy point trajectories in the DD domain, design the window function matrix W. DD [k,l], windowing is applied to the LFM signal near the principal energy point to suppress its spectral spread, ultimately resulting in the LFM signal occupying a grid cell C′ in the DD domain. DD [k,l], allocate the remaining unused DD domain grid resources to OTFS communication symbols to achieve collaborative mapping and resource reuse of sensing signals and communication signals within the same DD domain grid: Step 3: Transform the DD domain signal to the time domain for transmission; Step 4: At the sensing receiver, perform matched filtering and Doppler frequency analysis on the extracted LFM signal components to calculate the target's time delay and Doppler frequency shift (θ). ψ ,ν ψ This is used to estimate the distance and speed of the target, and at the receiving end, the OTFS communication symbols are demodulated to recover the communication data.

2. The method for integrated sensing waveform design based on OTFS and LFM according to claim 1, wherein steps one and two are as follows: First, the LFM pulse signal is embedded into the DD domain grid of OTFS, and the distribution law of the LFM pulse signal in the DD domain is derived to determine its corresponding main energy point trajectory. Second, based on the distribution position of the main energy point trajectory in the DD domain, a window function matrix is ​​designed to window the LFM signal near the main energy point to suppress its spectral spread. Then, the remaining unoccupied DD domain grid resources are allocated to OTFS communication symbols to achieve collaborative mapping and resource reuse of sensing signals and communication signals in the same DD domain grid, including the following steps: In the DD domain, an N×M grid is defined, where M and N represent the number of subcarriers and symbols, respectively, and T represents the period of a single symbol. Therefore, the duration of the entire frame signal is NT. An LFM pulse signal with α pulses is defined, where i represents the index of the i-th pulse, 1 ≤ i ≤ α, and the signal bandwidth is B. Its pulse repetition interval is... The frequency modulation slope is Pulse width is Where λ is the duty cycle, therefore, the time-domain expression of the LFM pulse signal is: The expression for a single pulse is: Among them, P c f is the transmitted signal power. c For the carrier center frequency, the carrier frequency term exp(j2πf) does not need to be considered in the following transformation process. c t), because when performing linear operations such as Fourier transform, the carrier frequency only causes a shift in the spectrum and does not change the structure of the signal; First, the LFM pulse signal is transformed from the time domain to the time-frequency domain using the Wigner Transform: Among them, g rx (t) represents a rectangular pulse constrained by the interval [nT, (n+1)T], 0≤n≤N-1, 0≤m≤M-1, and Δf is the subcarrier spacing. Therefore, the above equation can be transformed into: Right now For the i-th pulse, there is a signal only in its non-zero interval, so the interval for integrating the above equation is [nT, (n+1)T] and [(i-1)T]. c ,(i-1)T c +T w The intersection of ] is defined as the time variable: τ = t - (i-1)T c Define the new integration interval as [a i ,b i By substituting variables into the above integrals, we have: where, [a i , b i = [max(0, nT - (i - 1)T c ), min(T w , (n + 1)T - (i - 1)T c )]; Define the integral within the i-th pulse segment as For integral I i (m;n), calculate its stationary point to observe the location of its main contribution, i.e., the high-energy point, and its phase function is: The stationary phase point satisfies Φ′(τ)=0. Considering that the time slot n in the time-frequency domain corresponds to the time t=nT, we also have τ=nT-(i-1)T. c Substituting Φ′(τ)=0, we get Substitute and have Taking the modulo of the result, we have Therefore, integral I i The high-energy points of (m;n) are distributed according to this pattern; Integral I i The essence of (m;n) is to calculate the Fourier component of the LFM pulse at frequency mΔf: I i (m;n)=∫s i (t)·e -j2πfτ dτ| f＝mΔf The frequency of the LFM signal changes linearly with time. Within a sufficiently short time interval, its frequency change is very small. Therefore, within the duration of the window function, the LFM signal can be approximated as a single-frequency complex exponential signal, and its frequency can be approximated as: f0=μ·τ=μ·(nT-(i-1)T c ) The corresponding subcarrier index is Therefore, within the integration interval, we have approximately: Its Fourier transform is δ(f-f0), and the Fourier transform of the rectangular window function is W(f) = T. w ·sinc(T w f), time-domain multiplication is equivalent to frequency-domain convolution, finally yielding Therefore: Will Let it be a constant 1, so we ultimately have: After transforming the LFM pulse signal to the time-frequency domain, its principal energy point is determined according to... The distribution shows a diffusion phenomenon, particularly near the main energy point; Then, the signal is transformed from the time-frequency domain to the D domain using the Symptotic Finite Fourier Transform (SFFT): Substitute C TF [n,m], Where l is the index of the time delay dimension, k is the index of the Doppler dimension, 0≤l≤M-1, 0≤k≤N-1. First, sum the m in the above formula, and let m′=m-m0, then we have: definition Substituting back into the above equation, we have: Let Υ = k - αl, then we have For S(Υ), the common ratio is The sum of a geometric series is: Normalization yields the Dirichlet kernel: Finally, the expression in the DD domain is derived as follows: That is, the distribution of the principal energy points after the LFM pulse signal is transformed into the DD domain follows k = <αl> N This is accompanied by a diffusion phenomenon; After the LFM pulse signal is transformed into the DD domain, the principal energy point is concentrated at k = <αl>. N On this straight line, where l is the index of the time delay dimension and k is the index of the Doppler dimension, there are M principal energy points in the DD domain, distributed across M columns, with one principal energy point in each column. A window function is designed for each principal energy point in each column to suppress its energy diffusion. The window functions are constructed as follows: Where d k d represents the distance from other grid points to the principal energy point k0(l). k (k,l)=min(|k-k0(l)|,N-|k-k0(l)|), where w is the protection radius of the main energy point, and β is the roll-off coefficient. Taking the main energy point of each column as the center, each column is multiplied by this roll-off window function. Therefore, the final representation of the LFM pulse signal in the DD domain is: C′ DD [k,l]=C DD [k,l]·W DD [k,l] After suppressing the spread of LFM pulse signals in the DD domain, OTFS communication symbols are added to the remaining unoccupied grid areas, resulting in the following final grid area across the entire DD domain:

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