Ambiguity Function Analysis of AFDM Signals for Integrated Sensing and Communications

Let $N$ denote the number of chirp subcarriers and $\mathbf{x}=\left[x[0],x[1],\dots,x[N-1]\right]^{T}$ represent the transmitted DAFT-domain vector with its entries consisting of pilot symbol, guard symbols or data symbols. Then, $N$-point inverse DAFT (IDAFT) is firstly implemented on $\mathbf{x}$ to modulate the $N$ DAFT-domain symbols onto the $N$ chirp subcarriers, respectively, which generates the discrete-time-domain AFDM-ISAC signal as

where $n$ and $m$ denote the discrete-time domain and DAFT domain indices, respectively, and $\phi_{m}[n]$ represents the $m$th subcarrier with an expression given by

$c_{1}$ and $c_{2}$ are two key parameters that determine the chirp slope and the initial phase of all AFDM chirp subcarriers, respectively. As we shall see later, these two parameters, especially $c_{1}$, significantly influence the AFs of AFDM signals. Moreover, a chirp-periodic prefix (CPP) given by

is appended to $s[n]$ to cope with the multipath effect of the communication channel, where $L_{\text{CPP}}$ is the length of the CPP and should be set larger than or equal to the maximum delay spread of the communication channel. Note that when $2N|c_{1}|\triangleq C$ is an integer and $N$ is even, as assumed throughout this paper, equation (3) simplifies to $s[n]=s[n+N],\ n=-L_{\text{CPP}},\dots,-1$, i.e., the CPP reduces to a conventional CP as in OFDM. The corresponding continuous-time representation of $\phi_{m}[n]$ can be obtained in a piecewise manner as [39]

where $t$ and $\Delta t$ denote the continuous-time domain and the Nyquist sampling interval, respectively, $\tilde{c}_{1}\triangleq\frac{c_{1}}{\Delta t^{2}}$, $t_{m,q}$ is the $q$th ($q=0,1,\dots,C$) spectrum wrapping point of the $m$th chirp subcarrier with a definition given by

where $T=N\Delta t$ is the duration of $\phi_{m}(t)$. This paper adopts the widely used rectangular pulse shaping, wherein all chirp subcarriers have finite support over the interval $[0,T]$ [22]. One can easily verify that the equation of $\phi_{m}(t)|_{t=n\Delta t}=\phi_{m}[n],\ n=0,1,\dots,N-1$ holds for all $m=0,1,\dots,N-1$, i.e., $\phi_{m}[n]$ can be acquired by sampling $\phi_{m}(t)$ with a sampling interval of $\Delta t$. For clarity, we illustrate the time-domain and time-frequency-domain representations of two AFDM chirp subcarriers in Fig. 2. Building on this, the CPP-free continuous-time transmitted AFDM-ISAC signal can be expressed as¹¹1 The transmitted AFDM-ISAC signal $s(t)$ in (9) serves as an “ideal” continuous-time counterpart of $s[n]$ in (1), enabling insightful close-form derivations of the AFs for AFDM signals. In practice, it can be well approximated by interpolating $s[n]$ and passing the result through a digital-to-analog filter [23].

As shown in Fig. 1, part of the AFDM-ISAC signal interacts with the communication channel and arrives at the communication receiver, while part of it reaches the targets and is reflected to a sensing receiver. Generally, both communication and sensing channels are modeled as DSC, whose impulse responses with delay $\tau$ and Doppler $\nu$ can be defined in the following way as [46]

where $P$ denotes the number of paths (targets), $h_{i}$, $\tau_{i}$, and $\nu_{i}$ denote the channel gain, delay shift, and Doppler shift of the $i$th path, respectively. For monostatic sensing, the round-trip delay and Doppler shift of the $i$th target are given by $\tau_{i}=\frac{2d_{i}}{v_{c}}$ and $\nu_{i}=\frac{2v_{i}f_{c}}{v_{c}}$, respectively, where $d_{i}$ and $v_{i}$ denote the distance and relative radial velocity of the target, and $v_{c}$ and $f_{c}$ represent the speed of light and the carrier frequency, respectively. In the bistatic sensing, these parameters become $\tau_{i}=\frac{d_{i,t}+d_{i,r}}{v_{c}}$ and $\nu_{i}=\frac{(v_{i,t}+v_{i,r})f_{c}}{v_{c}}$, where $d_{i,t}$ and $d_{i,r}$ are the distances from the transmitter and receiver to the target, and $v_{i,t}$ and $v_{i,r}$ are the corresponding relative radial velocities. Subsequently, the received AFDM-ISAC signal, for both the communication and sensing receivers, can be formulated as

where $w(t)\sim\mathcal{C}\mathcal{N}\left(0,N_{0}\right)$ is the time-domain additive white Gaussian noise (AWGN) process.

1) Communication receiver: After performing Nyquist sampling on $r(t)$ and discarding the CPP component, the received discrete-time communication signal $r_{\text{com}}[n]$ can be obtained, which is then demodulated via DAFT as

enabling DAFT-domain channel estimation [19, 20, 21] and signal detection [26, 27].

2) Monostatic sensing receiver (MSR): At the MSR, the transmitted AFDM-ISAC signal $s(t)$ is fully known despite its randomness introduced by the random data symbols given that the MSR is collocated with the AFDM-ISAC transmitter. Therefore, one common operation is performing MF on the echo signal, i.e., the received AFDM-ISAC signal $r_{\text{MSR}}(t)$ with $s(t)$, which can be formulated as

Subsitituting (11) into (13), we have

where $\tilde{w}(t)=\int_{-\infty}^{\infty}w(t)s^{*}(t-\tau)e^{-j2\pi\nu t}\text{d}t$ is the noise after MF, $\tilde{h}_{i}=h_{i}e^{-j2\pi(\nu-\nu_{i})\tau_{i}}$, $A_{s,s}(\tau,\nu)$ is the aperiodic AF with a definition given by

In particular, AF is known as AAF and CAF when $b(t)=a(t)$ and $b(t)\neq a(t)$, respectively. Moreover, as shown in (14), the output of MF of $r_{\text{MSR}}(t)$ and $s(t)$ is the superposition of $P$ scaled and shifted AAF of $s(t)$, where the shifts along the delay domain and the Doppler domain are exactly the delay shifts and the Doppler shifts associated with the $P$ sensing targets, respectively²²2Although the AF analysis in this paper focuses on a single-antenna setting, it can be readily extended to various multi-antenna configurations by incorporating the spatial domain into the AF through the use of appropriate spatial steering vectors..

3) Bistatic sensing receiver (BSR): The BSR has a different location from the AFDM-ISAC transmitter, which means only the pilot component of $s(t)$, i.e., one pilot chirp subcarrier, is known³³3Here, we assume that there is no fronthaul connection between the ISAC transmitter and the BSR; otherwise, the BSR could utilize the full signal $s(t)$ to perform MF as in (13), rather than relying solely on its pilot component.. Therefore, the BSR usually performs MF on the echo signal $r_{\text{BSR}}(t)$ with the pilot component of $s(t)$, which is deeply determined by the CAF between $s(t)$ and one chirp subcarrier.

The above discussion highlights the close connection between the MF operation in sensing and the AF. Building on these insights, we proceed with the analysis of the AFs of AFDM-ISAC signals in the following. For clarity, we focus on the AFs of CPP-free AFDM-ISAC signals, which comprise $\phi_{m}(t)$ in (4). The length of the CPP is determined by the communication channel and is typically independent of the sensing channel. However, as we shall demonstrate later, all conclusions presented in this paper remain valid for CPP-appended AFDM-ISAC signals.

We first derive the AAF of the $m$th chirp subcarrier, which can be formulated as

For ease of derivation, we rewrite $\phi_{m}(t)$ in (4) as

where

is a step function that indicates the intentional spectrum wrapping in chirp subcarriers. Substituting (17) into (16), we have (19) (shown at the top of the next page), which shows no relevance to $c_{2}$.

Note that owing to the piecewise property of AFDM chirp subcarriers, the integral in (19) should also be calculated in a piecewise way correspondingly, whose exact expression shows a slight difference with respect to the value of $\tau$. Due to space limitations, we only demonstrate one representative case, from which the remaining cases can be obtained with minor modifications. To proceed, we define an indicator function $\tilde{q}_{m}(\tau)$ as

and assume that

where $k=0,1,\dots,C-1$. Then, (19) can be derived as (22), which is based on the fact that $\phi_{m}(t)$ has finite support over the interval $[0,T]$ and it can be further decomposed by substituting (8) and (18), as shown in (23).

Finally, after some algebraic operations, we arrive at (24), where the integral indicator

and

Based on the above-derived results, we visualize $A_{\phi_{m},\phi_{m}}(\tau,\nu)$ in Fig. 3 and demonstrate its prominent properties in the following.

The “spike-like” local property of the AAF of AFDM chirp subcarriers can be attributed to their time-frequency spanning nature. This characteristic allows them to fully occupy the time and frequency resource of an AFDM frame, providing delay perceptibility with a resolution of $\frac{1}{B}$ and Doppler perceptibility with a resolution of $\frac{1}{T}$ simultaneously. This is a significant difference that distinguishes AFDM from conventional SCM and OFDM, where each SCM symbol occupies a bandwidth of $B_{\text{SCM}}^{\text{symbol}}=B$ and has a duration of $T_{\text{SCM}}^{\text{symbol}}=\frac{2T}{N}$, while an OFDM subcarrier occupies a bandwidth of $B_{\text{OFDM}}^{\text{subcarrier}}=\frac{2B}{N}$ and has a duration of $T_{\text{OFDM}}^{\text{subcarrier}}=T$ approximately, as shown in Fig. 4. Furthermore, the AAFs of a SCM symbol and an OFDM subcarrier are provided in Fig. 3(f) and Fig. 3(g), respectively. We can see that the delay mainlobe width of the AAF of a SCM symbol is $\frac{1}{B}$, while its Doppler mainlobe width is expected to be $B$, which means a SCM symbol only possesses delay perceptibility. Similarly, the Doppler mainlobe width of the AAF of an OFDM subcarrier is $\frac{1}{T}$ approximately, while its delay mainlobe width is expected to be $T$, which means that an OFDM subcarrier only possesses Doppler perceptibility [10]. This clearly demonstrates the advantage of AFDM chirp subcarriers over SCM symbols and OFDM subcarriers in radar sensing. Note that due to the spectrum wrapping phenomenon in AFDM chirp subcarriers, as discussed in Observation 1, AFDM exhibits relatively high out-of-band emission (OOBE), as shown in Fig. 4. This requires stricter spectrum management in practice, placing higher demands on digital-to-analog filters.

The “periodic-like” global property can be explained by considering the physical essence of the AAF in (16) and the subchirp splicing feature of AFDM chirp subcarriers, as summarized in Observation 1. Specifically, the AAF quantifies the degree of correlation between $\phi_{m}(t)$ and $\phi_{m}(t-\tau)e^{j2\pi\nu t}$. For AFDM chirp subcarriers, which are composed of multiple subchirps, the strong correlation requirement leads to the following two conditions:

1) Frequency alignment (FA) between the matched subchirps from $\phi_{m}(t)$ and $\phi_{m}(t-\tau)e^{j2\pi\nu t}$. Fig. 5(a) and Fig. 5(c) illustrate the two cases of frequency alignment for a given delay shift. For ease of demonstration, we define the frequency difference indicator as

which indicates the instant frequency difference between $\phi_{m}(t)$ and $\phi_{m}(t-\tau)e^{j2\pi\nu t}$ at time $t$. Then, the FA condition requires that $\mathcal{F}^{\tau,\nu}_{m,\tilde{c}_{1}}(t)=1$, which means that for a given delay shift $\tau$, frequency alignment condition can be satisfied at time $t$ when

which takes two different values for the two cases shown in Fig. 5(a) and Fig. 5(c), respectively. We can observe that, in each case, a portion of the matched subchirps from $\phi_{m}(t)$ and $\phi_{m}(t-\tau)e^{j2\pi\nu t}$ are overlapped with aligned instantaneous frequencies, while the remaining portions exhibit misaligned instantaneous frequencies. Insightfully, we associate the two parts of the matched subchirps with the decomposed form of the AAF by labeling them as follows: $\mathcal{S}_{a}^{i}$ ($i=0,1,\dots,C-\tilde{q}_{m}(\tau)+1$) and $\mathcal{S}_{b}^{i}$ ($i=0,1,\dots,C-\tilde{q}_{m}(\tau)$) in (23), respectively, and identify them in Fig. 5(a) and Fig. 5(c). Moreover, let $\mathcal{T}_{\mathcal{S}_{a}^{i}}$ and $\mathcal{T}_{\mathcal{S}_{b}^{i}}$ represent the time period occupied by $\mathcal{S}_{a}^{i}$ and $\mathcal{S}_{b}^{i}$, respectively, and

Then, for the case shown in Fig. 5(a), the frequency-aligned parts are $t\in\mathcal{T}_{\mathcal{S}_{a}}$, where

holds and hence (30) turns into

The frequency-misaligned parts are $t\in\mathcal{T}_{\mathcal{S}_{b}}$, where the instant frequency difference is $-\frac{1}{\Delta t}$, i.e.,

Similarly, for the case in Fig. 5(c), the frequency-aligned parts are $t\in\mathcal{T}_{\mathcal{S}_{b}}$, where

holds and hence (30) turns into

Correspondingly, the frequency-misaligned parts are $t\in\mathcal{T}_{\mathcal{S}_{a}}$, where the instant frequency difference is $\frac{1}{\Delta t}$, i.e.,

2) Residue phase difference alignment (RPDA) among the matched subchirp pairs from $\phi_{m}(t)$ and $\phi_{m}(t-\tau)e^{j2\pi\nu t}$. We further define the residue phase difference indicator as

which indicates the instant residue phase difference between $\phi_{m}(t)$ and $\phi_{m}(t-\tau)e^{j2\pi\nu t}$ at time $t$. Then, the RPDA condition suggests that the frequency-aligned parts of all matched subchirp pairs share the same instant residue phase difference, i.e., $\mathcal{P}^{\tau}_{m,\tilde{c}_{1}}(t)$ maintains the same value when $t\in\mathcal{T}_{\mathcal{S}_{a}}$ for case Fig. 5(a) or $t\in\mathcal{T}_{\mathcal{S}_{b}}$ for case Fig. 5(c). This means that the influence of variable $t$ in $\frac{q_{m}(t-\tau)}{\Delta t}\tau$ should be eliminated from $\mathcal{P}^{\tau}_{m,\tilde{c}_{1}}(t)$, which can be achieved by letting $\frac{q_{m}(t-\tau)}{\Delta t}\tau$ be an integer. According to (18), $q_{m}(t-\tau)$ is always an integer, meaning that the elimination of $t$ from $\mathcal{P}^{\tau}_{m,\tilde{c}_{1}}(t)$ can only hold when

Therefore, when (30) and (40) are satisfied, the frequency alignment and the residue phase difference alignment are achieved at the frequency-aligned part of the matched subchirps simultaneously. Subsequently, for the case in Fig. 5(a), substituting (34) and (40) into $\mathcal{S}_{a}^{i}$ in (23) arrives at

which yields

indicating energy accumulation, which in turn boosts a pulse in $A_{\phi_{m},\phi_{m}}(\tau,\nu)$. In contrast, the frequency-misaligned parts can be calculated by substituting (35) and (40) into $\mathcal{S}_{b}^{i}$ in (23) as

which has a small magnitude compared to $\mathcal{S}_{a}^{i}$ given its integrand is a complex-exponential function with a period of $\Delta t$. A similar result can be obtained for case Fig. 5(c) by substituting (37) and (40) into $\mathcal{S}_{b}^{i}$ in (23), where an energy accumulation process will be generated by $\sum_{i=0}^{C-\tilde{q}_{m}(\tau)}\mathcal{S}_{b}^{i}$.

Consequently, a parallelogram can be virtually outlined by any four adjacent pulses in the DD plane of the AAF, either in the form of Fig. 3(b) or Fig. 3(c), effectively forming a virtual grid. Building on this analysis, we arrive at the following interesting conclusion, stated as Corollary 1.

Corollary 1 suggests a tradeoff between the ranges of delay estimation and Doppler estimation when using an AFDM chirp subcarrier for unambiguous sensing, a topic that will be further discussed in the next subsection.

Due to the “periodic-like” global property discussed above and the MF process introduced in Sec. II-C, ambiguous sensing may arise when the pulses in $R(\tau,\nu)$ cannot be accurately associated with their corresponding target’s AAF. To address this issue, we analyze the conditions necessary for unambiguous sensing in Proposition 1.