*1.1. Related Works*

The OFDM waveform for sensing can be processed either by the conventional correlationbased approach [24,25], or by OFDM symbol-based processing [26]. Correlation-based sensing is usually performed by cross-correlation in the delay and Doppler domains between the transmitted and received pulses, and different schemes have been proposed to improve sensing performance. For example, a good approximation of the transmitted signal is generated at the receiver for removing clutter in the correlation-based target detection [15]. Work in [25] proposes to use the information of data symbols for ambiguity suppression, and circular correlation for range extension up to an OFDM symbol duration. Different correlation-based OFDM radar receiver schemes have been compared in [27], in terms of complexity, signal-to-interference-plus-noise-ratio, and robustness against ground clutter.

Alternatively, similar to OFDM-based communication, OFDM-based sensing can also use IFFT/FFT operations to extract range and speed information. Based on this approach, a 77 GHz OFDM-based sensing system with a bandwidth of 200 MHz demonstrated a sensing resolution of 0.75 m with the maximum range of 150 m [28]. Another OFDM-based radar at 77 GHz used a stepped carrier approach to achieve a sensing resolution of 0.146 m with a bandwidth of 1.024 GHz, while the maximum range is 60 m [29]. Moreover, the authors implemented OFDM-based radar processing for automotive scenario by using a relatively longer interval of 128 ms to achieve speed resolution of 0.22 m/s, while the range resolution was 1.87 m for a bandwidth of 80 MHz at 5.2 GHz [30].

These two sensing processing approaches were employed in the development of OFDM-based radars, while from the viewpoint of converging OFDM-based communication and sensing, OFDM symbol-based sensing processing is more attractive, provided that a sensing receiver is synchronized with the transmitter and the transmitted data are readily available for sensing processing. Some interesting research has been done on OFDM-based convergence in the microwave band. By using OFDM waveforms which are designed for 3GPP-LTE and 5G-NR at 2.4 GHz with a bandwidth of 98.28 MHz, OFDM-based sensing supports a sensing resolution of 1.5 m and a maximum range of 350 m and performs an algorithm for self-interference cancellation in the full-duplex mode [31]. Authors in [32] provide measurement results for the indoor mapping using a 28 GHz carrier frequency for the 5G-NR with a bandwidth of 400 MHz and achieve a sensing resolution of 0.4 m. Another work in [33] shows results of mmWave demonstration testbed for joint sensing and communication; measurements were performed at 26 GHz with a bandwidth of 10 MHz to identify the angular location of different targets using beamforming technique. The work in [34] also presents a range resolution of 1.61 m and a maximum range of 206 m within 93 MHz bandwidth at the 24 GHz band. In addition, authors in [35] provide a parameter selection criterion for joint OFDM radar and communication systems by considering vehicular communication scenarios, such as CPI, subcarriers spacing, and coherence time of the channel.

### *1.2. Motivation and Contribution*

Please note that enabling the sensing functionality of the OFDM waveform (which is designed for wireless communication) does not provide the flexibility of parameter adjustment according to the sensing requirements. Furthermore, the ISI cancellation/compensation techniques proposed for OFDM wireless communication are not differently applicable for OFDM-based sensing because the transformation or truncation-based equalization destructs the sensing information. Ideally, the delay of an echo for sensing should fall within

the CPI, and the Doppler frequency normalized over OFDM waveform interval should be an integer. However, in a real scenario for sensing, a target is located randomly and moves with an arbitrary speed. Consequently, an OFDM waveform designed for communication shows limitation in obtaining high sensing resolution and a large detection range.

As we know, the detection range of a single target is determined by the detectable OFDM signal strength and an adjustment of delay offset. In the case of multiple echoes with delay beyond the CPI, the OFDM-based sensing is mainly limited by the ISI, free-spacepath-loss (FSPL), and processing gain. Echoes outside the CPI cause ISI as previous OFDM symbols interfere with current OFDM symbol in the processing window, which increases the threshold for target detection. In addition, echoes with delay longer than the CPI will achieve less processing gain, which reduces linearly with the delay. This loss of processing gain along with the ISI makes it difficult for OFDM-based sensing to detect targets outside the CPI, particularly in the millimeter-wave region featuring large bandwidth and high FSPL. Therefore, the extension of sensing range beyond the conventional limit of CPI is one of the important issues in developing communication and sensing converged systems for applications such as indoor mapping, digital health monitoring, unmanned aerial vehicles, and residential security.

In this work, we propose and experimentally demonstrate a converged communication and sensing system operating at 97 GHz using the same 16-QAM (quardrature amplitude modulation) OFDM waveform. An approach based on zero-delay shift is proposed to extend the detectable range by compensating for the IFFT processing gain for echoes outside the CPI. In the proposed method, we extended the range of an OFDM-based sensing, while the simplicity of operations for range and speed estimation is achieved using IFFT/FFT operations. The proposed method uses delay-shifts in the received signal before processing a received OFDM symbol. Active subcarriers in the received OFDM symbol are divided by the active subcarriers in the current and previous transmitted OFDM symbols (employed number of transmitted OFDM symbols determine the rang extension), and IFFT operations are used after each delay-shift to generate matrices in the delay and delay-shift domains (delay domain is the result of IFFT operation). Delay-shift rows at delay zero are concatenated to extend the delay-shift domain. Concatenation of delay-shift rows for different received OFDM symbols provides a matrix in delay-shift and time domain, and FFT operations over time domain provide the speed estimation. An experiment with a heterodyne W-band transmitter/receiver is performed, and both sensing and communication performance are measured in terms of range/speed profile and bit-error-rate (BER). The proposed approach for range extension is verified for distances well beyond the CPI and provides a range resolution of 0.042 m, and speed resolution of 0.79 m/s using a single OFDM waveform, which is promising in driving OFDM-based converged systems for future applications.

The rest of this paper is organized as follows. Section 2 presents the model for the OFDM-based converged system to provide the details of extracting sensing information from the received OFDM waveform. Section 3 details the proposed method for range extension in an OFDM-based converged system. Section 4 provides simulation results, while Section 5 is dedicated to experimental measurement results and discussions. Section 6 provides the conclusion of this work.

#### **2. Communication and Sensing Convergence Using OFDM Waveforms**

Motivated by the OFDM-based sensing presented in [26,34], a reference system model for the convergence of communication and sensing is presented here. An OFDM waveform for communication purposes consists of several OFDM symbols, each with orthogonal subcarriers modulated by data symbols and cyclically extended by appending the last part of the signal at the beginning called cyclic prefix (CP). If Δ*f* represents the subcarriers spacing, *N* the number of orthogonal subcarriers, *T* the OFDM symbol duration, *T*cp as the CPI, *T*<sup>s</sup> = *T*cp + *T* the effective duration of the OFDM symbol, and *M* the number of OFDM symbols, then the analytical expression of the transmitted OFDM waveform is [34],

$$s(t) = \sum\_{\mu=0}^{M-1} \sum\_{n=0}^{N-1} S(\mu N + n) e^{\left(j2\pi n \Delta f (t - \mu T\_\mathrm{s})\right)} e^{\left(-j2\pi n \Delta f T\_\mathrm{cp}\right)} \text{rect}\left(\frac{t - \mu T\_\mathrm{s}}{T\_\mathrm{s}}\right),\tag{1}$$

where *S*(*μN* + *n*) is the data symbol at *n*th subcarrier of *μ*th OFDM symbol. The rect(*t*) in (1) is the rectangular pulse shape, such that rect(*t*) = 1 for *t* ∈ [0 1] and 0 otherwise. The term exp <sup>−</sup>*j*2*πn*Δ*f T*cp appears due to the cyclic extension of OFDM symbols by the CP.

In order to fulfill the orthogonality among subcarriers, over the interval *T*, the following condition must be held:

$$
\Delta f = \frac{1}{T'} \tag{2}
$$

and *T*cp should accommodate the maximum expected delay caused by the radio channel. The baseband signal *<sup>s</sup>*(*t*) is up-converted by a carrier frequency *<sup>f</sup>*<sup>c</sup> to form *s*(*t*) for transmission,

$$
\widetilde{s}(t) = s(t)e^{j2\pi f\_\xi t}.\tag{3}
$$

The received signal *a*(*t*) at the sensing receiver is the sum of echoes from different targets. Using point-target channel model for *L* number of targets,

$$
\widetilde{a}(t) = \sum\_{l=1}^{L} b\_l \widetilde{s}(t - \tau\_l) \, . \tag{4}
$$

where *τ<sup>l</sup>* and *bl* represent delay and attenuation related to the *l*th target, respectively. If *l*th target is located at a distance *Rl* and moving with a speed of *vl*, delay *τ<sup>l</sup>* in the received echo can be expressed as

$$
\tau\_l = \frac{\mathbf{2}(R\_l - v\_l t)}{c\_0},
\tag{5}
$$

and *bl* [34],

$$b\_l = \sqrt{\frac{c\_0^2 G\_{\text{TX}} G\_{\text{RX}} \sigma\_{\text{RCS}}}{(4\pi)^3 R\_l^4 f\_\text{c}^2}},\tag{6}$$

where in (6), *c*<sup>0</sup> is the speed of light in free space; *σ*RCS*<sup>l</sup>* is the radar cross-section of the *l*th target; and *G*Tx, *G*Rx represent transmitting and receiving antenna gain, respectively.

For the communication link, the signal attenuation *b*com is

$$b\_{\rm com} = \sqrt{\frac{c\_0^2 G\_{\rm Tx} G\_{\rm Rx}}{(4\pi)^2 R\_{\rm com}^2 f\_{\rm c}^2}}\tag{7}$$

where *R*com indicates the distance of the communication link.

For sensing processing, a single target is sufficient for mathematical derivations due to the linear operation in (1). The analytical expression for the received echo from a target, located at a distance *R*, moving with the speed of *v*, and attenuated by ˆ *b* (assuming constant attenuation factor for frequencies within the bandwidth) is obtained by using delay *τ* = (2*R* − 2*vt*)/*c*<sup>0</sup> in (1), i.e.,

$$\begin{split} \widetilde{a}(t) &= \sum\_{\mu=0}^{M-1} \sum\_{n=0}^{N-1} \widehat{b}S(\mu N + n) e^{\left(j2\pi n \Delta f \left(t - \frac{(2R - 2vt)}{c\_0} - \mu T\_\mathrm{s} - T\_\mathrm{cp}\right)\right)} \\ &\cdot e^{\left(j2\pi f\_c \left(t - \frac{(2R - 2vt)}{c\_0}\right)\right)} \mathrm{rect}\left(\frac{t - \left(\frac{2R - 2vt}{c\_0}\right) - \mu T\_\mathrm{s}}{T\_\mathrm{s}}\right) + \mathcal{2}(t) \end{split} \tag{8}$$

where *z*ˆ(*t*) is to account for the additive white Gaussian noise (AWGN). Since *f*<sup>c</sup> is usually very high compared to the bandwidth of the signal, in particular in the millimeter-wave band, the Doppler shift (*n*Δ*f* 2*v*)/*c*<sup>0</sup> is negligible for the subcarriers and the overall Doppler shift appears only caused by (2*v f*c)/*c*0.

The received signal is down-converted to baseband, which is equivalent to

$$a(t) = \sum\_{\mu=0}^{M-1} \sum\_{n=0}^{N-1} bS(\mu N + n)e^{\left(j2\pi n\Lambda f(t - \frac{(2R)}{c\_0} - \mu T\_\text{s} - T\_\text{cp})\right)} e^{\left(j2\pi \frac{2\mu \xi\_c}{c\_0} t\right)} \text{rect}\left(\frac{t - \left(\frac{2R}{c\_0}\right) - \mu T\_\text{s}}{T\_\text{s}}\right) + z(t), \tag{9}$$

where *b* and *z*(*t*) represent ˆ *b*exp(−*j*2*π f*c2*R*/*c*0) and *z*ˆ(*t*)exp(−*j*2*π f*c*t*), respectively.

Finally, the signal *a*(*t*) is sampled, the CP part is removed before it is converted into the frequency domain by using FFT operation,

$$\begin{split} A(\mu N + n) &= bS(\mu N + n)e^{\left(-j2\pi n \Delta f \frac{2R}{\epsilon\_0}\right)} e^{\left(j2\pi \frac{2\eta f\_c}{\epsilon\_0} \mu T\_8\right)} + Z(\mu N + n), \\ n &\in [0, N - 1], \quad \mu \in [0, M - 1] \end{split} \tag{10}$$

where *A*(*μN* + *n*) and *Z*(*μN* + *n*) are frequency domain equivalents of *a*(*t*) and *z*(*t*).

Once the received signal is translated back into the frequency domain, the elementwise division of the received OFDM symbol by the respective transmitted OFDM symbol is performed to construct the channel matrix **H**, i.e.,

$$H(\mu N + n) = be^{\left(-j2\pi n \Delta f \frac{2R}{c\_0}\right)} e^{\left(j2\pi \frac{2vf}{c\_0} \mu T\_s\right)} + \frac{Z(\mu N + n)}{S(\mu N + n)},\tag{11}$$

where *H*(*μN* + *n*) represents the *μ*th column and *n*th row of the channel matrix **H**, and *Z*(*μN* + *n*)/*S*(*μN* + *n*) defines the noise floor that depends on the digital modulation, e.g., a 16-QAM mapping affects the noise floor by approx 2.7 dB [36]. The IFFT of **H**, along subcarriers, provides the range information,

$$r(d) = e^{\left(j2\pi \frac{2vf\_c}{c\_0} \mu T\_s\right)} \frac{b}{N} \sum\_{n=0}^{N-1} e^{\left(-j2\pi n \Delta f \frac{2R}{c\_0}\right)} e^{\left(j\frac{2\pi}{N} nd\right)} + \mathfrak{z}(d), \tag{12}$$
  $d \in [0, N-1]$ 

where *z*ˇ(*d*) represents the noise part.

The |*r*(*d*)| shows a peak value under the following condition:

$$d = \left\lfloor \frac{2R\Delta fN}{c\_0} \right\rfloor,\tag{13}$$

i.e., the value of *d* corresponding to the maximum of |*r*(*d*)| holds the information of the target range, and the range resolution Δ*R* (minimum distinguishable distance between the two targets) is defined as

$$
\Delta R = \frac{c\_0}{2\Delta fN}.\tag{14}
$$

Similarly, the FFT operation over different OFDM symbols in **H** (over *μ* domain in (11)) provides the information about the speed of the target and can be recognized by using

$$p = \left\lfloor \frac{2vf\_{\text{c}}T\_{\text{s}}M}{c\_{0}} \right\rfloor, \quad p \in [0, M-1] \tag{15}$$

whereas the speed resolution Δ*v* can be calculated by setting *p* = 1,

$$
\Delta v = \frac{\mathcal{L}\_0}{2f\_\text{c}T\_\text{s}M}.\tag{16}
$$

The IFFT/FFT operations on **H** provide processing gain due to coherent addition of signals, and the overall processing gain is

$$G = NM.\tag{17}$$

If there are *N*´ guardband subcarriers on each side of the OFDM symbol, then the processing gain reduces to (*<sup>N</sup>* <sup>−</sup> <sup>2</sup>*N*´ )*M*. Here, it is important to note that the guardband subcarriers reduce the bandwidth of the OFDM waveform, and consequently the value of <sup>Δ</sup>*<sup>R</sup>* increases in (14) when *<sup>N</sup>* is replaced by (*<sup>N</sup>* <sup>−</sup> <sup>2</sup>*N*´ ). Although <sup>Δ</sup>*<sup>R</sup>* increases due to guardband subcarriers, improvement in the range accuracy (resolution of IFFT) is linked to the size of IFFT [37].

Figure 1 highlights the sensing processing using **H**. Figure 1a shows the real part of the channel matrix **H**, which has sinusoidal variations due to the range and speed of a single target. The IFFT operation, along with subcarriers, identifies the delay associated with the range, as shown in Figure 1b. Afterwards, the FFT operation provides the sensing information in the delay-Doppler profile, as shown in Figure 1c,d.

It is clear that the sensing performance depends on the OFDM waveform parameters because bandwidth defines the range resolution, and the duration of the OFDM waveform determines speed resolution. For the speed, the upper limit is selected as Δ*f* > 20 *f*c*v*/*c*<sup>0</sup> to maintain the acceptable level of orthogonality among the subcarriers [35].

**Figure 1.** OFDM–based sensing from the channel matrix **H**. (**a**) IFFT operation over subcarriers. (**b**) FFT operation over OFDM symbols. (**c**) 3–D plot of the delay–Doppler profile. (**d**) Radar image, indicating the delay associated to the range and the Doppler frequency related to the speed of the target.

## **3. Proposed Method for Range Extension**

In OFDM-based sensing, the maximum range is limited by the CPI [28,35] as echoes falling outside the CPI cause ISI and suffer in processing gain. We propose a zero-delay shift method to compensate for delay *τ* in an echo to maintain its processing gain *G* during IFFT operation for OFDM-based sensing.

Using sampling intervals Δ*T* = 1/*F*<sup>s</sup> and Δ*f* = 1/(*N*Δ*T*), the sampled version of (9) is

$$a(k\Delta T) = \sum\_{\mu=0}^{M-1} \sum\_{n=0}^{N-1} b\mathcal{S}(\mu N + n) e^{\left(j2\pi n \frac{1}{\left(N\Delta T\right)} \left(k\Delta T - \frac{2R}{c\_0\Delta T} - \mu \frac{T\_8}{\Delta T} - \frac{T\_{\text{CP}}}{\Delta T}\right)\right)} \tag{18}$$

$$\cdot e^{\left(j2\pi \frac{2vf\_c}{c\_0}k\Delta T\right)} \text{rect}\left(\frac{k\Delta T - \frac{2R}{c\_0} - \mu \frac{T\_8}{\Delta T}}{\frac{T\_8}{\Delta T}}\right) + z(k\Delta T)\_s$$

where *k* is the sampled time index. Using *F*s/Δ*f* = *T*/Δ*T* = *N*, *T*s/Δ*T* = *N*s, *T*cp/Δ*T* = *N*cp, *m* = (2*R*)/(*c*0Δ*T*), and *a*(*k*), *z*(*k*) to represent *a*(*k*Δ*T*) and *z*(*k*Δ*T*) respectively,

$$a(k) = \sum\_{\mu=0}^{M-1} \sum\_{n=0}^{N-1} bS(\mu N + n)e^{\left(j2\pi \frac{n}{N}(k - m - \mu N\_{\rm s} - N\_{\rm cp})\right)}$$

$$\cdot e^{\left(j2\pi \frac{2vf\_{\rm s}}{c}k\Lambda T\right)} \text{rect}\left(\frac{k - m - \mu N\_{\rm s}}{N\_{\rm s}}\right) + z(k). \tag{19}$$

$$k \in [0, MN\_{\rm s} - 1]$$

A delay-shift in *k* by *m* samples shifts the target at zero on the delay axis. Since *m* is unknown, sequentially increasing the delay-shift in *k* identifies *m* when a peak appears at delay zero. This process can identify echoes with delay longer than the CP, provided they arrive with detectable signal strength. If we extend the sensing range up to *Q* number of OFDM symbols, the proposed method can be described in following steps:


Using the above process, a matrix (in the delay-shift and time domains) of size *QN*<sup>s</sup> × *M* is constructed, which requires an *M*-point FFT operation to complete the range/speed plot.

Figure 2 shows different steps in the proposed method to detect two targets separated by more than one OFDM symbol duration; *Q* = 2 is used, and the current OFDM symbol number is *q*. Figure 2a,b are obtained using steps 1–4 of the proposed method; Figure 2c is the plot of step 5 and using *M* = 12 for step 6; Figure 2d is the final range/speed plot, where the range is extended up to two OFDM symbols. A schematic diagram of the proposed method is presented in Figure 3. Delay shifts are used in the sampled version of the incoming OFDM waveform to get the frequency domain signal *Yq*. Element-wise division of *Yq* is performed with the current OFDM symbol *Sq* and previous OFDM symbols for sensing and combining.

We use periodogram to compare the performance of the proposed method with conventional OFDM-based sensing. Periodogram of the conventional OFDM-based sensing is defined as [31],

$$\begin{aligned} D\_{\text{conv}(d,p)} &= \left| \sum\_{\mu=0}^{M-1} r(d) e^{\frac{-j2\pi\mu p}{M}} \right|^2 \\ &\quad d \in [0, N-1] \quad p \in [0, M-1] \end{aligned} \tag{20}$$

where *r*(*d*) is defined in (12). A target is detected if the peak in *D*conv(*d*,*p*) is above a threshold level (usually defined by the minimum detectable signal strength). For the

range/speed plot, **D** is often transformed to normalized power, and in dB scale using 10 log10 (**D**/*max*[**D**]), where *max*[**D**] represents the maximum value of **D**.

**Figure 2.** Explanation of the proposed method when range extension is up to two OFDM symbols. (**a**,**b**) Steps 1–4 of the proposed method provide two matrices. (**c**) Delay–zero rows of the matrices in (**a**,**b**) are concatenated according to the step 5. (**d**) Range/speed plot is completed using *M*–point FFT operation over time domain.

**Figure 3.** Schematic of the proposed range extension method to detect targets beyond the range limit in the conventional OFDM–based sensing.

Similarly, the periodogram of the proposed method is

$$D\_{\text{pro}(\mathcal{J},p)} = \left| \sum\_{\mu=0}^{M-1} \hat{r}\_{\mathcal{J}}(0,\mu) e^{\frac{-j2\pi\mu p}{M}} \right|^2, \quad \hat{d} \in [0,(Q-1)N\_s - 1] \quad p \in [0,M-1] \tag{21}$$

where <sup>ˆ</sup>*<sup>d</sup>* represents the delay-shift domain and *<sup>r</sup>*<sup>ˆ</sup> <sup>ˆ</sup>*d*(0, *<sup>μ</sup>*) is obtained by concatenation of *<sup>Q</sup>* segments as defined in step 5 of the proposed method, i.e.,

$$\mathcal{F}\_{\vec{d}}(0,\mu) = [r\_{\vec{d}}(0,\mu), r\_{\vec{d}}(0,\mu - 1), \dots, r\_{\vec{d}}(0,\mu - (Q - 1))], \qquad \vec{d} \in [0, N\_{\rm s} - 1] \tag{22}$$

where *r* ´*d*(0, *μ*) is obtained by using *d* = 0 in *r* ´*d*(*d*, *μ*),

$$r\_{\vec{d}}(d,\mu) = \frac{1}{N} \sum\_{n=0}^{N-1} H\_{\vec{d}}(\mu N + n) e^{\frac{j2\pi nd}{N}}, \qquad d \in [0, N\_{\sf s} - 1] \tag{23}$$

i.e.,

$$r\_d(0, \mu) = \frac{1}{N} \sum\_{n=0}^{N-1} H\_d(\mu N + n). \tag{24}$$

If we represent *m* = *gN*<sup>s</sup> + *m*ˆ where *g* ∈ [0, *Q* − 2] and *m*ˆ ∈ [0, *N*cp − 1], received *q*th OFDM symbol and transmitted OFDM symbols provide

$$\mathbf{H}\_{\vec{d},(q-i)} = \frac{\mathbf{Y}\_{d,q}}{\mathbf{S}\_{(q-i)}},\tag{25}$$

where **<sup>H</sup>** ´*d*,(*q*−*i*) is taken as simplified notation for *<sup>H</sup>* ´*d*((*<sup>q</sup>* <sup>−</sup> *<sup>i</sup>*)*<sup>N</sup>* <sup>+</sup> *<sup>n</sup>*), *<sup>i</sup>* <sup>∈</sup> [0, *<sup>Q</sup>* <sup>−</sup> <sup>1</sup>], and **<sup>Y</sup>** ´*d*,*<sup>q</sup>* <sup>=</sup> FFT(*<sup>A</sup>* ´*d*(*k*)), where *<sup>A</sup>* ´*d*(*k*) is the delay-shift of *<sup>A</sup>*(*k*) by ´*<sup>d</sup>* and *<sup>k</sup>* <sup>∈</sup> [ ´*<sup>d</sup>* + (*q*)*N*<sup>s</sup> <sup>−</sup> *gN*<sup>s</sup> <sup>−</sup> *<sup>m</sup>*<sup>ˆ</sup> <sup>+</sup>, ´*<sup>d</sup>* + (*<sup>q</sup>* <sup>+</sup> <sup>1</sup>)*N*<sup>s</sup> <sup>−</sup> *gN*<sup>s</sup> <sup>−</sup> *<sup>m</sup>*<sup>ˆ</sup> <sup>−</sup> *<sup>N</sup>*cp] (interval of the N-samples of the waveform is selected at the initial step of the proposed method). It is clear that at ´*<sup>d</sup>* <sup>=</sup> ´*d*<sup>0</sup> <sup>=</sup> *<sup>m</sup>*<sup>ˆ</sup> <sup>+</sup> *<sup>N</sup>*cp, *<sup>A</sup>* ´*d*<sup>0</sup> (*k*) represents (*q* − *g*)th OFDM symbol without *N*cp; hence, (25) changes to

$$\mathbf{H}\_{\dot{d}\_{0\prime}(q-i)} = \frac{b\mathbf{S}\_{(q-g)} + \mathbf{Z}\_{(q-g)}}{\mathbf{S}\_{(q-i)}},\tag{26}$$

where **<sup>Z</sup>**(*q*−*g*) represents noise part in the (*<sup>q</sup>* <sup>−</sup> *<sup>g</sup>*)th OFDM symbol. Similarly, at ´*<sup>d</sup>* <sup>=</sup> ´*d*<sup>1</sup> <sup>=</sup> (*m*ˆ + *N*), *A* ´*d*(*k*) consists of last *N*cp samples of the (*q* − *g*)th OFDM symbol and *N* − *N*cp samples of (*q* − *g* + 1)th OFDM symbol; therefore,

$$\mathbf{H}\_{\mathbf{d}\_{1},(q-i)} = \frac{N - N\_{\rm cp}}{N \mathbf{S}\_{(q-i)}} b \mathbf{S}\_{(q-g)} e^{-\frac{j2\pi n}{N} \mathbf{N}\_{\rm cp}} + \frac{N\_{\rm cp}}{N \mathbf{S}\_{(q-i)}} b \mathbf{S}\_{(q-g+1)} + \frac{\mathbf{Z}\_{(q-g)}}{\mathbf{S}\_{(q-i)}}.\tag{27}$$

Using (26) in (24) provides the maximum of *r* ´*d*(0, *μ*), which is same as *r*(*d*) in (20), whereas (27) indicates the additional peak with height reduced by a factor of *N*cp/*N* and affected by the ISI. Similar to *D*conv(*d*,*p*), where a processing of *NM* is assigned to a peak, *D* pro( <sup>ˆ</sup>*d*,*p*) also provides the same processing gain when *<sup>i</sup>* <sup>=</sup> *<sup>g</sup>* in (26); otherwise, **<sup>H</sup>** ´*d*,(*q*−*i*) in (26) is interference. In (27), contrary to interference term, *N*cp samples are coherently added when used in (24) and the processing gain is 10log10(*N*<sup>2</sup> cp/(*N* − *N*cp)).

In a generalized scenario, there can be *L* echoes with delays not limited to CPI; the received signal *y*(*k*) is the summation of all echoes, each represented by the (19),

$$\begin{split} y(k) &= \sum\_{l=0}^{L-1} \sum\_{\mu=0}^{M-1} \sum\_{n=0}^{N-1} b\_l \mathcal{S} \left( \mu N + n \right) e^{\left( j2\pi \frac{n}{N} \left( k - m\_l - \mu N\_s - N\_{\rm cp} \right) \right)} \\ &\cdot e^{\left( j2\pi \frac{2v\_l \zeta\_c}{c} k \Delta T \right)} \text{rect} \left( \frac{k - m\_l - \mu N\_s}{N\_{\rm s}} \right) + z(k), \end{split} \tag{28}$$

where *ml* represents the delay associated with *l*th echo. Based on the delay *ml*, we split the *y*(*k*) into three portions such as *y*1(*k*) for *ml* ≤ *N*cp, *y*2(*k*) for *N*cp < *ml* ≤ *N*s, and *y*3(*k*) for *ml* > *N*s, i.e.,

$$y(k) = y\_1(k) + y\_2(k) + y\_3(k) + z(k). \tag{29}$$

Since *y*<sup>3</sup> is formed by the summation of echoes that are outside the current OFDM symbol, therefore this part is only ISI. Unlike *y*3(*k*), the ISI part of *y*2(*k*) increases as *ml* approaches to *N*s. The detection of the echoes in *y*2(*k*) and *y*3(*k*) is possible if the processing gain *G* is sufficient to overcome the related ISI and noise.
