Adaptive pulse shaping for OFDM RadCom systems in highly dynamic scenarios

Abstract

Due to the spectrum and complexity efficiency, the integrated radar and communications (RadCom) systems have been widely favored, in which orthogonal frequency division multiplexing (OFDM) is the most popular signal to conduct the two functions simultaneously. However, an unoptimized pulse could suffer from severe inter-carrier interference (ICI) and high out-of-band emission (OOBE), which greatly degrades the system performance. In this paper, we introduce the pulse shaping scheme dedicated to RadCom systems, in which both transmitter and receiver can adaptively design pulses with the assistance of radar estimation. We first optimize the transmitting pulse with the weighted sum of signal-to-interference-plus-noise ratio (SINR) and OOBE by employing the popular genetic algorithm. Then, we design an improved-matched pulse at the receiver for maximizing the SINR with the fmincon solver. In this way, they both utilize the readily available radar information and keep the pulse optimal even in highly dynamic scenarios, which makes the most of RadCom systems while avoiding the overhead of channel estimation and feedback. Simulations prove the feasibility of proposed scheme and reveal that the radar image and communications SINR stay close to their optimum in most cases with much lower OOBE. An improved-matched pulse can further improve the communications performance when severe ICI occurs compared with a matched pulse.

1 Introduction

Radar and communications are two important technologies in modern radiofrequency systems, which have similarities in hardware implementation and signal processing [1]. In practical applications such as aircraft flying, autonomous driving, and environment monitoring, radar and communications are always designed separately, which will result in doubled hardware costs and inefficient resource utilization [2]. To this end, the integration of radar and communications (RadCom) systems is regarded as a promising technology to solve the above issues, which has been widely investigated in recent years.

Authors in [3] proposed the integration of radar and communications utilizing linear frequency modulation (LFM) waveforms. They used $\pi /4$ differential quadrature phase shift keying (DQPSK) modulated chirp signals to achieve a slightly higher data rate. The radar performance is satisfactory while the communications bit error rate (BER) deteriorates by 2 dB. Authors in [4] designed a RadCom system that could avoid the interference of radar and communications by using different pseudo-noise codes. However, the BER performance is as poor as the former one and the data rate is too low to meet the requirement of the fifth generation (5 G). Due to the high spectrum efficiency, robustness against multipath, and flexible modulation schemes, orthogonal frequency division multiplexing (OFDM) is the most popular multicarrier signal which has been implemented in many standards for 5 G service [5]. In [6], the authors first proposed a method for range and velocity estimation from the echoes of OFDM signals. They proved that the delay and Doppler shifts can be directly estimated simply by the Fourier transforms after proper data symbol cancellation. They also found that the OFDM-based RadCom system outperforms the one based on direct sequence spread spectrum (DSSS) in high dynamic range, high transmission rate, independence from data symbols, availability for velocity estimation, and simple implementation. In the following years, considerable improvements in estimation accuracy and system complexity were proposed. Authors in [7] derived a radar processing algorithm to achieve a higher resolution with the same processing gain. They further introduced a method that is able to find an unambiguous estimate for high-speed targets with low complexity in [8].

OFDM signals implement Nyquist pulse to maintain the orthogonality among sub-carriers. If there is any carrier frequency offset between the transmitter (TX) and receiver (RX) due to Doppler shifts or oscillator detuning, the sub-carriers are no longer orthogonal and the first few dominant side-lobes will cause significant inter-carrier interference (ICI), which drastically decreases the performance. Besides, the out-of-band emission (OOBE) caused by the low asymptotic decay rate (ADR) of the side-lobes requires a larger guard interval between users. The optimization of pulse shaping function is a key method to tackle those problems. In recent years, there were a variety of novel pulses derived, including the better than raised-cosine (BTRC) pulse [9], the second-order continuous window (SOCW) [10], and two types of double-jump (DJ) pulse [11], which have lower side-lobes and are able to suppress ICI in case of frequency offset more or less. The performances of the above pulses were compared in [12], and the results showed that the DJ1 pulse has outperformed others under the majority of circumstances. Authors in [13] provided an overview of pulse shaping methods in OFDM systems and proposed a new design method with good time-frequency localization property. Reference [14] proposed a systematic method to construct pulses with lower side-lobes and higher ADR by multiplying any two base pulses in frequency domain. In this way, the height of the side-lobes in the multiplied pulse is lower than that of either base pulse, and the ADR is much higher. Furthermore, it also creates at least four extra degrees of freedom on the pulse shaping function, i.e., the types of two base pulses and their own roll-off factors, which could have a positive impact on different criteria including radar image/communications signal-to-interference-plus-noise ratio (SINR) together with OOBE. Reference [15] and [16] used DJ1 and DJ2 to multiply Gaussian pulse in frequency domain, respectively. The BER performances of multiplied pulses are almost identical, but far superior to other traditional ones.

However, the aforementioned methods have two major shortcomings, which can be tackled effectively when combined with RadCom systems. First, the optimal parameters of the pulse are dependent on the SINR and frequency offset; a fixed pulse cannot perform well in all circumstances [12]. In RadCom systems, such information can be easily obtained from range and velocity of radar estimation, avoiding the cost of pilot redundancy and computational effort. Hence, the pulse can be adaptively optimized based on the motion of TX and RX. Second, previous works usually adopt matched pulses at each side, which could maximize the SINR in additive white Gaussian noise (AWGN) channels. However, when the transmitting pulse is distorted by Doppler shifts, matched pulse might not be the best choice [13]. We propose the improved-matched pulse at RX which may not have zero-crossings at integer multiples of the sub-carrier interval, but have lower ICI in the presence of Doppler shifts. With the radar estimation, RX can recover the Doppler-distorted transmitting pulse and adaptively derive an improved-matched pulse that further enhances the communications performance.

In this paper, we, for the first time, propose a pulse shaping framework dedicated to RadCom systems in highly dynamic scenarios. Pulse-shaped OFDM (PS-OFDM) signals are generated, which can be utilized for radar in both TX and RX as well as communications. With the assistance of radar, they each can adaptively derive an optimal pulse through the closed-loop pulse shaping framework. The transmitting pulse shaping functions are designed by multiplying two pulses in frequency domain, to simultaneously achieve satisfactory performance in three metrics, i.e., radar image/communications SINR and OOBE, whereas the RX derives an improved-matched pulse to maximize communications SINR, which is the only criterion that the RX needs to be concerned with. Simulation results validate the feasibility of our proposed scheme and reveal the performance superiority even in highly dynamic scenarios.

The main contributions of this paper can be summarized as follows.

• To our best knowledge, this paper is the first investigation into optimizing the pulse shaping functions in RadCom systems, such that both TX and RX can optimize pulse according to the SINR and frequency offset that is readily available at radar. The pulse can be kept optimal even in highly dynamic scenarios, and the computational efforts in channel estimation and feedback are significantly reduced.

• The optimization framework for transmitting pulse is designed to achieve satisfactory performance in three criteria. A look-up table containing the pre-calculated optimal pulses can be formed, from which the TX can select according to each radar estimation. The RX can also easily conduct radar estimation with the same high accuracy, even if the recovered symbols have high BER. An improved-matched receiving pulse can be derived to further enhance communications SINR.

• We perform extensive simulations to demonstrate the effectiveness and superiority of our proposed scheme. In our investigated cases, the optimal pulses for TX are always the convolution of DJ1 and Gaussian pulse, which keeps radar image/communications SINR close to their optimum with much lower OOBE. Besides, an improved-matched receiving pulse can achieve better communications performance against Doppler shifts than a matched one.

The remainder of the paper is organized as follows. Section proposes the system and signal models, the radar and pulse shaping methods, as well as the optimization framework. Section presents the simulation results which validate the superiority of the proposed scheme. The conclusion is drawn in section.

2 Methods

2.1 System and signal model

As shown in Fig. 1, we consider a typical RadCom scenario, where a high-speed fighter and helicopter both need to transmit data to each other while estimating their relative range R and radial velocity v simultaneously. Without loss of generality, we denote the fighter as TX and the helicopter as RX. Provided that the duration of one OFDM frame is short enough, the channel can be regarded as quasi-static and radar estimation can be calculated. Since an improved-matched pulse might have better performance in the presence of Doppler, we formulate two independent closed-loop pulse shaping frameworks for TX and RX with no feedback required. For TX, the front wave is used for communications, while the echoes are utilized for radar estimation and the subsequent pulse optimization. Since it knows the transmitted symbols exactly, range and velocity can be estimated after OFDM demodulation and communications symbol cancellation. As for RX, it first uses the pulse derived from previous time slot to demodulate received signals into binary bits and then modulates again to complex symbols, which can be coarsely regarded as transmitted symbols. They might be high in BER, but still sufficient for radar estimation. Then, it can optimize the receiving pulse to maximize communications SINR based on the recovered transmitting pulse and radar estimation. The optimal pulse can once again be employed to demodulate the received signal to lower the BER. The optimization on both sides utilizes the radar information on hand, which makes the most of RadCom systems and avoids the overhead of channel estimation.

Fig. 1

The RadCom system model

A frame of PS-OFDM signals in baseband can be represented as

$$ s(t) = \sum \limits _{n = 0}^{{N_{\textrm{s}}} - 1} {\sum \limits _{m = 0}^{{N_{\textrm{c}}} - 1} {{d_{m,n}}{g_{\textrm{T}}}(t - nT\left( {1 + \alpha } \right) ){e^{j2\pi m\Delta f(t - nT\left( {1 + \alpha } \right) )}}} }, $$

(1)

where $N_{\textrm{c}}$ and $N_{\textrm{s}}$ denotes the number of sub-carriers and OFDM symbols, respectively, $d_{m,n}$ is the complex data symbol in the mth sub-carrier, nth OFDM symbol, $\Delta f=1/T$ denotes the sub-carrier interval, where T is the duration of the elementary OFDM symbol. The subscriptions ‘T’ and ‘R’ denote variables at the TX and RX side, respectively, and ${g_{\textrm{T}}}(t)$ stands for the transmitting pulse shaping function in the time domain that has the duration of $T\left( {1 + \alpha } \right)$ where $\alpha$ is the roll-off factor.

The received signal would be deteriorated by time delay, Doppler shifts, channel gain, and AWGN. Here, the channel gain is regarded the same among sub-carriers and thus can be simplified as path loss. With the normalized signal power, the channel gain can be embodied in the noise power. Hence, the signals received by RX can be presented as

$$ {r_{\textrm{R}}}(t) = \sum \limits _{n = 0}^{{N_{\textrm{s}}} - 1} {\sum \limits _{m = 0}^{{N_{\textrm{c}}} - 1} {{d_{m,n}}{g_{\textrm{T}}}(t - nT\left( {1 + \alpha } \right) - \tau ){e^{j2\pi m\Delta f(t - nT\left( {1 + \alpha } \right) - \tau )}}{e^{j2\pi {f_{\textrm{d}}} t }} + n(t)} }, $$

(2)

where $\tau = R/c$ is the one-way delay and c denotes the light speed, ${{f}_{\textrm{d}}}\approx {{v}}{{f}_{\textrm{c}}}/c$ is the Doppler shifts provided that ${{N}_{\textrm{c}}}\Delta f$ is much smaller than the carrier frequency $f_c$, $n(t)\sim {{\mathcal {C}}}{{\mathcal {N}}}(0,\sigma _{{\textrm{R}}}^2)$ represents the complex AWGN.

The noise power at RX in dB can be calculated from the range estimation using the propagation model, which can be configured for airborne scenarios with line-of-sight (LoS) millimeter-wave channels, expressed as [17]

$$ \sigma _{{\textrm{R}}}^2 = {P_{\textrm{n}}} - \left( {{P_{\textrm{t}}} - {\textrm{P}}{{\textrm{L}}_{{\textrm{1m}}}} - {\textrm{10}}k{{\log }_{10}}\left( R \right) + {X_{{{{\textrm{PL}}}}}} + G} \right), $$

(3)

where ${P_{{\textrm{n}}}}$ is the thermal noise power, $P_{\textrm{t}}$ is the transmitting power, ${\mathrm{PL_{1\,m}}}=32.4 + 20{\log 10}( f_c/10^9)$ denotes the carrier frequency related free space path loss at 1 m, ${\textrm{10}}k{\log _{10}}\left( {{R}{{}}} \right)$ describes the attenuation beyond the reference distance in which k is the path loss exponent, $X_{{{\textrm{PL}}} }$ is the log-normal random shadowing with mean $0~\textrm{dB}$ and standard deviation ${\sigma _{{\textrm{PL}}} }$, G is the antenna gain.

While for echoes at TX, which experience a round-trip path loss, the noise power can be represented as

$$ \sigma _{\textrm{T}}^2 = {P_{\textrm{n}}} - \left( {{P_{\textrm{t}}} - {\textrm{P}}{{\textrm{L}}_{{\textrm{1m}}}} - {\textrm{20}}k{{\log }_{10}}\left( R \right) + {X_{{\textrm{PL}}}} + 2 G + {\sigma _{{\textrm{rcs}}}}} \right), $$

(4)

where ${\sigma _{\mathrm{{rcs}}}}$ denotes the radar cross section of the target.

Then, the OFDM demodulation is done by correlating the received signal ${r_{{\textrm{R}}}}(t)$ with the receiving pulse

$$ { d_{\mathrm{{R}},m,n}} = \int _{ - \infty }^\infty {{r_{\textrm{R}}}(t){g^*}_{{\textrm{R,}}m,n}(t){\textrm{d}}t}, $$

(5)

where $g_{{\textrm{R}},m,n}(t)$ is the time-frequency shifted version of the receiving pulse with

$${g_{{\textrm{R,}}m,n}}(t) = {g_{\textrm{R}}}\left( {t - nT\left( {1 + \alpha } \right) } \right) {e^{j2\pi m\Delta f\left( {t - nT\left( {1 + \alpha } \right) } \right) }}.$$

(6)

The overall pulse shaping function g(t) can be generated through the optimization problem derived in the following subsections. When the TX and RX both implement the square root of the pulse, i.e., ${g_T}(t) = {g_R}(t) = \sqrt{g(t)}$, a full pulse can be recovered through synchronization and matched filter, and (5) can be rewritten as

$$ \begin{aligned} {d_{{\textrm{R}},m,n}} & = \int _{ - \infty }^\infty {{d_{m,n}}{g_n}\left( t \right) {e^{j2\pi {f_{\textrm{d}}}t}} + n\left( t \right) {\textrm{d}}t} \\ =&{d_{m,n}}\int _{ - \infty }^{ + \infty } {{g_n}\left( t \right) {{\textrm{e}}^{j2\pi {f_{\textrm{d}}}t}}~{\textrm{d}}t}\\&+ \sum \limits _{\begin{array}{c} \scriptstyle m' \ne m \\ \scriptstyle m' = 0 \end{array}}^{{N_{\textrm{c}}} - 1} {{d_{m',n}}} \int _{ - \infty }^{ + \infty } {{g_n}\left( t \right) {{\textrm{e}}^{j2\pi \left( {\left( {m' - m} \right) \Delta f + {f_{\textrm{d}}}} \right) t}}~{\textrm{d}}t} + {\tilde{n}(t)}, \end{aligned}$$

(7)

where

$$ \begin{aligned} {g_{n}}\left( t \right) & ={g_{{\textrm{T}},m,n}}\left( {t - nT\left( {1 + \alpha } \right) } \right) \times {g^*}_{{\textrm{R}},m,n}\left( {t - nT\left( {1 + \alpha } \right) } \right) \\ & ={g}\left( {t - nT\left( {1 + \alpha } \right) } \right). \end{aligned} $$

(8)

The first term in (7) is the recovered data symbol whose power is degraded by the Doppler shifts, and the second is the ICI from other sub-carriers.

2.2 Radar estimation

For radar estimation, we can notice from the exponential terms in (2) that the delay brings a linear phase shift only along the frequency axis, while the Doppler does so only along the time axis [6]. They have an orthogonal influence on the matrix form of modulation symbols, and thus, the relationship between transmitted and received data can be written as

$$ {{\textbf {D}}_{{\textbf {R}}}} = {{\textbf {D}}}\odot \left( {{{{\textbf {k}}}_R}{{{\textbf {k}}}_v}} \right) +n(t),$$

(9)

where ${{{\textbf {D}}}_{\textrm{R}}} \in \mathbb {C} {^{{N_{\textrm{c}}} \times {N_{\textrm{s}}}}}$ is the output from the OFDM frame, ${{\textbf {D}}}\in \mathbb {C} {^{{N_{\textrm{c}}} \times {N_{\textrm{s}}}}}$ is the transmitted symbols, and $\odot$ represents the element-wise multiplication, ${{{\textbf {k}}}_R}$ and ${{{\textbf {k}}}_v}$ are delay and Doppler effects. According to (2), the delay $\tau$ introduces a negative phase shift with an increasing sub-carrier index, while the Doppler $f_d$ causes a positive one as time increases. Since the duration of an OFDM symbol is considerably short, the time t can be approximately discretized as $nT(1+\alpha )$. Therefore, ${{{\textbf {k}}}_R}$ and ${{{\textbf {k}}}_v}$ can be represented as

$$\begin{aligned}&{{{\textbf {k}}}_R} = {\left[ {\begin{array}{*{20}{c}} 1&{\exp \left( { - j2\pi \Delta f R/c } \right) }&\cdots&{\exp \left( { - j2\pi \left( {{N_{\textrm{c}}} - 1} \right) \Delta f R/c } \right) } \end{array}} \right] ^{\textrm{T}}}~, \\&{{{\textbf {k}}}_v} = \left[ {\begin{array}{*{20}{c}} 1&{\exp \left( {j2\pi T(1 + \alpha ){v {f_c}/c}} \right) }&\cdots&{\exp \left( {j2\pi \left( {{N_{\textrm{s}}} - 1} \right) T(1 + \alpha ){v {f_c}/c}} \right) } \end{array}} \right]. \end{aligned}$$

(10)

Then, we employ an element-wise division, a matrix containing the delay and Doppler information but without communications data can be obtained

$$ {{\tilde{{\textbf {D}}}}} = \frac{{{\textbf {D}}_{{\textbf {R}}}}}{{{\textbf {D}}}} = {{{{\textbf {k}}}_R}{{{\textbf {k}}}_v}} +n(t) . $$

(11)

For general radar systems, the signal is known to the receiver so that it can perform radar estimation by comparing the transmitted and received signals. However, in RadCom systems, all the symbols carry communication information, and RX cannot be made aware of the transmitted symbols in advance. Thus, the radar processing needs to be modified. The received signals can first be demodulated into binary bits and then modulated again into complex symbols ${\hat{{\textbf {D}}}}$, which can be a rough substitute for the transmitted symbols ${{\textbf {D}}}$.

Different kinds of windows could be implemented in each line and column of ${{\tilde{{\textbf {D}}}}}$ to suppress the side-lobes to be caused by the subsequent Fourier transforms, which could avoid weak targets being overshadowed, while at the cost of a degraded SINR and broadened main-lobe. Calculate the inverse discrete Fourier transform for every column and the discrete Fourier transform of every row of ${{\tilde{{\textbf {D}}}}}$. Due to the cancellation of the exponential terms, a peak will occur at the index of $\left\lfloor { T(1+\alpha ) N_{\textrm{s}}}{ v} {f_{c}}/{c}\right\rfloor$ and $\left\lfloor { \Delta f N_{\textrm{c}}}R/{c}\right\rfloor$ in column and row, respectively, where $\left\lfloor \cdot \right\rfloor$ denotes rounding to negative infinity. Therefore, the resulting matrix directly represents a two-dimensional radar image in range and velocity with each peak representing a target.

Since the signal is added coherently, whereas the noise is done stochastically, an approximate processing gain of $N_{\textrm{c}}N_{\textrm{s}}$ can be achieved. In case of demodulation errors in the recovered symbols ${\hat{{\textbf {D}}}}$, a small portion of the previously coherent data symbols now becomes noise-like, which degrades the radar image SINR. However, the indexes of peaks remain the same and the estimation is accurate, since the correct symbols still sum up to a lower peak.

The resolution of the radar image is closely related to the signal length in the Fourier transforms, represented as

$$ \Delta v = \frac{c}{{{f_c}{N_{\textrm{s}}}T(1+\alpha )}}, $$

(12)

$$ \Delta R = \frac{c}{{{N_{\textrm{c}}}\Delta f}}. $$

(13)

In order to improve the efficiency while ensuring the quasi-static channel during the transmission, a frame of OFDM signal usually has much larger ${N_{\textrm{c}}}$ than ${N_{\textrm{s}}}$, which leads to a lower resolution and broader main-lobe in velocity than that of range. Therefore, zero-padding could be applied before the discrete Fourier transform to obtain a smoother radar image as well as a higher resolution in velocity, but the width of the main-lobe still stays constant.

Remark

The radar operation at the TX side is almost the same but differs in that the process of symbol recovery is not required as it knows the transmitted symbols quite well, which could achieve a higher processing gain. However, the echoes will experience much more severe path loss together with doubled delay and Doppler, and thus, radar image SINR at TX needs to be considered in the optimization.

2.3 Pulse construction

In order to ensure the orthogonality between sub-carriers, the Nyquist-I criterion should be fulfilled in the frequency domain, which is expressed as

$$\begin{aligned} G(f)= \left\{ {\begin{array}{*{20}{l}} {1,} &{} \quad {{\mathrm{}}f = 0} \\ {0,} &{} \quad {{\mathrm{}}f = m\Delta f{\mathrm{}}\left( {m \ne 0} \right) }~. \end{array}} \right. \end{aligned}$$

(14)

Suppose a Nyquist pulse is multiplied with a pulse of any kind in frequency domain, the original zero-crossings at $m\Delta f{\mathrm{}}\left( {m \ne 0} \right)$ remain the same, implying the sub-carriers are still orthogonal to each other. Furthermore, the height of the side-lobes in the multiplied pulse is lower than that of either the base pulses, with the much higher ADR [14]. Hence, it is possible to generate a pulse with better performances in this way.

For system implementation, the pulse is time-limited, and the duration is important in pulse construction. To enable better control of pulse duration, according to the convolution theorem, we use the convolution in the time domain instead, as

$$ g(t) = {g_1}(t) \otimes {g_2}(t), $$

(15)

where $\otimes$ denotes the linear convolution, the subscriptions ‘1’ and ‘2’ denote the primary and secondary base pulses, in which the former one is to keep orthogonality, while the latter is to suppress side-lobes.

The duration of g is $T\left( {1 + {\alpha }} \right)$, where $\alpha$ is the overall roll-off factor of the multiplied pulse. Since $g_1$ has the zero-crossings at $m\Delta f{\mathrm{}}\left( {m \ne 0} \right)$, ${T_1}=T$ should be satisfied and its duration is $T\left( {1 + {\alpha _1}} \right)$, while that of $g_2$ is ${T_2}\left( {1 + {\alpha _2}} \right)$. However, due to the property of convolution, the duration of $g_1$ and $g_2$ should add up to that of g. Thus, duration of $g_2$ can also be expressed as $T\left( {\alpha - {\alpha _1}} \right)$, and we have

$$\begin{aligned} {T_2}=\frac{T\left( \alpha -\alpha _1\right) }{1+\alpha _2}~. \end{aligned}$$

(16)

where ${\alpha _1}$ should be larger than 0 but less than ${\alpha }$, and ${\alpha _2}$ can be any value between 0 and 1.

We consider 4 Nyquist pulses in this paper, i.e., DJ1 and DJ2 [11], BTRC [9], plus SOCW [10]; their definitions in the time domain are expressed as follows

$$\begin{aligned} g_{\textrm{DJ1}}(t)&= \left\{ \begin{array}{ll}\frac{1}{T}, &{} |t|<\frac{T(1-\alpha )}{2} \\ \frac{1}{T}\left( 1-\frac{|t|}{T}\right) , &{} \frac{T(1-\alpha )}{2} \le |t|<\frac{T(1+\alpha )}{2} \\ 0, &{} \text{ otherwise } \end{array}\right. \end{aligned}$$

(17)

$$\begin{aligned} g_{\textrm{DJ2}}(t)&=\left\{ \begin{array}{ll}\frac{1}{T}, &{} 0 \le |t|<\frac{T(1-\alpha )}{2} \\ \frac{1}{2 T}, &{} \frac{T(1-\alpha )}{2} \le |t|<\frac{T(1+\alpha )}{2} \\ 0, &{} \text{ otherwise } \end{array}\right. \end{aligned}$$

(18)

$$\begin{aligned} g_{\mathrm{{BTRC }}}(t)&=\left\{ \begin{array}{ll}\frac{1}{T}, &{} |t|<\frac{T(1-\alpha )}{2} \\ \frac{1}{T} \textrm{e}^{\frac{-2 \ln 2}{\alpha T}\left[ |t|-\frac{T(1-\alpha )}{2}\right] ,} &{} \frac{T(1-\alpha )}{2} \le |t|<\frac{T}{2} \\ \frac{1}{T}\left\{ 1-\textrm{e}^{\frac{-2 \ln 2}{\alpha T}\left[ \frac{T(1+\alpha )}{2}-|t|\right] }\right\} , &{} \frac{T}{2} \le |t|<\frac{T(1+\alpha )}{2} \\ 0, &{} \text{ otherwise } \end{array}\right. \end{aligned}$$

(19)

$$\begin{aligned} {g_{{\textrm{SOCW}}}}(t)&= \left\{ {\begin{array}{*{20}{l}} {\frac{1}{T},} &{} {|t|< \frac{{T(1 - \alpha )}}{2}} \\ {\frac{1}{T}\left[ {1 - f\left( { - \frac{{2|t|}}{{\alpha T}} + \frac{1}{\alpha }} \right) } \right] ,} &{} {\frac{{T(1 - \alpha )}}{2} \le |t|< \frac{T}{2}} \\ {\frac{1}{T}f\left( {\frac{{2|t|}}{{\alpha T}} - \frac{1}{\alpha }} \right) ,} &{} {\frac{T}{2} \le |t| < \frac{{T(1 + \alpha )}}{2}} \\ {0,} &{} {{\textrm{otherwise }}} \end{array}} \right. \end{aligned}$$

(20)

where $f(t) = 0.5 + {\beta _{\textrm{S}}}t - \left( {0.5 + {\beta _{\textrm{S}} }} \right) {t^2}$ and $\beta _{\textrm{S}}$ is a parameter to be optimized.

Aside from the aforementioned Nyquist pulses, the Gaussian pulse is favored due to its high ADR and is usually employed to improve the existing pulses. It can be expressed as¹

$$\begin{aligned} {g_{{\textrm{Gaussian}}}}(t) = \left\{ \begin{array}{l} \exp \left( { - \frac{t}{{\beta _{\textrm{G}} {T_2}}}} \right) ,{\mathrm{\hspace{20.0pt}}}|t| < \frac{{{T_2}}}{2} \\ 0,{\mathrm{\hspace{70.0pt}otherwise}} \end{array} \right. \end{aligned}$$

(21)

where $\beta _{\textrm{G}}$ is a parameter that controls the pulse width, which is also to be optimized.

The Nyquist pulses can be either primary or secondary pulses, while the Gaussian pulse can only be the secondary pulse since it does not have zero-crossings to maintain orthogonality. With the type of two base pulses and three or four parameters, i.e., $\alpha ,~{\alpha _1}$, ${\alpha _2}$ and possible $\beta _{\textrm{S}}$ and $\beta _{\textrm{G}}$ brought by SOCW or Gaussian pulse, the multiplied pulse can be fully determined.

2.4 Performance indicators

In this subsection, we focus the performance indicators on the radar image/communications SINR and OOBE, which are highly relevant to pulse shaping functions.

The radar image SINR, denoted as ${\Psi _{\textrm{r}}}$, is the ratio of peak power with the presence of target to the average noise power without target in the range-velocity image. The higher SINR allows weak targets to be easily detected among clutters. Due to the additional radar signal processing including Fourier transforms and windowing, it differs from the ambiguity function that describes the effects of time delays and Doppler shifts on the output of a matched filter. Therefore, the closed form of radar image SINR cannot be obtained directly. The SINR is measured at the TX side only, since it is quite adequate for RX due to the high signal power and the processing gain resulting from Fourier transforms.

The communications SINR is calculated at RX after synchronization and correlation, which could directly influence the BER. The power matrix can be represented as

$$ {{\textbf {M}}} = {{\textbf {G}}}_{\textrm{R}}^{\textrm{H}}{{{\tilde{{\textbf {G}}}}}_{\textrm{T}}}{{\tilde{{\textbf {G}}}}}_{\textrm{T}}^{\textrm{H}}{{{\textbf {G}}}_{\textrm{R}}}, $$

(22)

where each column of the matrix ${{{\tilde{{\textbf {G}}}}}_{\textrm{T}}}$ is composed of the frequency-shifted pulses with Doppler shifts, while ${{{\textbf {G}}}_{\textrm{R}}}$ are composed of those without Doppler shifts, shown

$$\begin{aligned}&{{{\tilde{{\textbf {G}}}}}_{\textrm{T}}} = \left[ {\begin{array}{*{20}{c}} {{{\tilde{{\textbf {g}}}_{\textrm{T}}}}},&{{{\tilde{{\textbf {g}}}_{\textrm{T}}}}{e^{j2\pi {\Delta f }t}}},&\cdots&{{{\tilde{{\textbf {g}}}_{\textrm{T}}}}{e^{j2\pi {{N_{\textrm{c}}}\Delta f } t}}} \end{array}} \right] ~, \end{aligned}$$

(23)

$$\begin{aligned}&{{{\textbf {G}}}_{\textrm{R}}} = \left[ {\begin{array}{*{20}{c}} {{\textbf {g}}_{\textrm{R}}},&{{\textbf {g}}_{{\textrm{R}}}{e^{j2\pi \Delta {f}t}}},&\cdots&{{\textbf {g}}_{{\textrm{R}}}{e^{j2\pi {N_{\textrm{c}}}\Delta ft}}} \end{array}} \right] ~, \end{aligned}$$

(24)

where ${{\tilde{{\textbf {g}}}_{\textrm{T}}}}={{\textbf {g}}_{{\textrm{T}}}{e^{j2\pi { {f_{\textrm{d}}}} t}}}$ is the Doppler shifted transmitted pulse.

The diagonal elements of ${\textbf {M}}$ contain the signal power in each sub-carrier, while the others are ICI power. The ICI varies among different sub-carriers with the ones near the edges being less affected. Therefore, the average communications SINR in dB can be described as

$$ {\Psi _{\textrm{c}}} = \frac{1}{{{N_{\textrm{c}}}}}\left( {2\sum {{\textrm{diag}}\left( {\textbf {M}} \right) } - \sum {\textbf {M}} } - \sigma _{\textrm{R}}^{\textrm{2}}\right). $$

(25)

where $\textrm{diag}({\cdot })$ represents the diagonal elements of a matrix, and $\sum$ denotes the summation of all the elements.

The OOBE is the power leakage outside the desired spectrum of the OFDM signal, i.e., $\left[ { - {N_{\textrm{c}}}\Delta f/2,{N_{\textrm{c}}}\Delta f/2} \right]$. It reflects the inter-user interference and, therefore, can be used as a subsidiary optimization objective. Since the signal is time-limited, its spectrum will extend throughout the entire frequency axis. However, the power spectrum density (PSD) outside the sampling frequency cannot be obtained. Thus, we employ an OFDM signal with $2{N_{\textrm{c}}}$ sub-carriers, whose sampling rate is twice of the original signal. The ${N_{\textrm{c}}}/2$ sub-carriers at each edge are allocated zero power, and the PSD of the remaining ${N_{\textrm{c}}}$ sub-carriers in the middle is calculated. The PSD covers a frequency band of $\left[ { - {N_{\textrm{c}}}\Delta f,{N_{\textrm{c}}}\Delta f} \right]$, in which the frequency within $\left[ { - {N_{\textrm{c}}}\Delta f/2,{N_{\textrm{c}}}\Delta f/2} \right]$ is the signal PSD, and that within $\left[ { - {N_{\textrm{c}}}\Delta f, - {N_{\textrm{c}}}\Delta f/2} \right) \cup \left( {{N_{\textrm{c}}}\Delta f/2,{N_{\textrm{c}}}\Delta f} \right]$ is OOBE. In order to make the OOBE quantified in optimization problems, we select three representative frequency points, i.e., $0.55{{{N_{\textrm{c}}}\Delta f}}$, $0.65{{{N_{\textrm{c}}}\Delta f}}$, and $0.8{{{N_{\textrm{c}}}\Delta f}}$, and the final OOBE is defined as the weighted mean of the PSD envelope at these frequency points as

$$ {\varphi _{\textrm{o}}} = 0.2\rho \left( 0.55{{{{N_{\textrm{c}}}\Delta f}} } \right) + 0.4\rho \left( 0.65{{{N_{\textrm{c}}}\Delta f}}{} \right) + 0.4\rho \left( 0.8{{{{N_{\textrm{c}}}\Delta f}}} \right), $$

(26)

where $\rho (\cdot )$ is the envelope of PSD.

Fig. 2

The PSD of DJ2 with $\alpha =0.45$ and DJ1$\times$Gaussian with $\alpha =0.45$, ${\alpha _1}=0.1$, and ${\beta _{\textrm{G}}}=0.3$

Due to the envelope extraction, the OOBE is also calculated through simulation. As we can see from Fig. 2, the variation of OOBE between different pulses is so high that proper compression is required to highlight the improvements in radar image and communications SINR. The OOBE used in optimization objective is defined by a segmentation function as

$$\begin{aligned} {\Psi _{\textrm{o}}} = \left\{ \begin{aligned}&- {\varphi _{\textrm{o}}}/20,&{\mathrm{for~}}{\varphi _{\textrm{o}}} > - 100 \\&5,&{\mathrm{for~}}{\varphi _{\textrm{o}}} \ge - 100 \end{aligned} \right. \end{aligned}$$

(27)

2.5 Pulse shaping framework

The pulse shaping framework aims to obtain satisfactory performances in radar image/communications SINR, as well as low OOBE. The optimization problem is formulated as

$$ \begin{aligned} {{{\mathcal {P}}}_{\textrm{T}}}:~&{\mathop {{\mathrm{min.}}}\limits _{\alpha ,~{\alpha _1},~{\alpha _2},~\beta _{\textrm{G}},~\beta _{\textrm{S}} } }{\mathrm{~ - }}\left( {{\Psi _{\textrm{r}}} + {\Psi _{\textrm{c}}} + {\Psi _{\textrm{o}}}} \right) \\&{\hspace{12.0pt}\mathrm{s.t.}\hspace{15.0pt}}\alpha ,~{\alpha _1},~{\alpha _2},~\beta _{\textrm{G}},~\beta _{\textrm{S}}\in \Xi, \end{aligned} $$

(28)

where $\alpha ,~{\alpha _1},~{\alpha _2},~\beta _{\textrm{G}},~\beta _{\textrm{S}}$ is the configuration of the pulse which is constrained by a set $\Xi$ shown in Table 1, and the transmitting pulse ${\textbf {g}}_{{\textrm{T}}}$ can be constructed according to Sect. .

Table 1 The constraint set $\Xi$ of pulse configurations

Since the radar image SINR and OOBE can only be obtained by numerical simulations, the closed form of the optimal pulse cannot be obtained. Thus, we utilize the genetic algorithm (GA) to solve the problem and use the simulation time to reflect the complexity. GA is a heuristic search algorithm that mimics the process of natural evolution and has been shown to achieve superior performance in various optimization problems. We set the population size, maximum iteration number, crossover rate, and mutation rate to 100, 500, 0.8, and 0.05, respectively. Besides, the parameters in $\Xi$ are discretized, and the interval is set to 0.02. The optimization is processed in MATLAB with Intel Core i9-10850k CPU and 64 GB RAM. The average simulation time is listed in Table 2.

Table 2 The average simulation time (sec.) for each pulse type

To make it suitable for real-time processing, proper sampling in the range-velocity grid is conducted, and a look-up codebook for a handful of pre-calculated results can be formed accordingly. During the flight, the TX can estimate the range and velocity independently, and select the optimal pulse from the codebook, which could be efficiently done in one second.

As mentioned before, the improved-matched pulse could outperform the matched one, since a small amount of power loss in the main-lobe, as well as a certain level of interference, is tolerable, in exchange for greater communications SINR with the influence of Doppler shifts. With the codebook of transmitting pulse and the estimation of range and velocity from radar, the RX can derive an optimal receiving pulse that maximizes the communications SINR. In order to reduce the computational complexity, we use the SINR of one sub-carrier to approximate the overall SINR, expressed as

$$\begin{aligned} {{\tilde{\Psi }} _{\textrm{c}}} = \frac{{{{\textbf {g}}}_{\textrm{R}}^{\textrm{H}}{{{\tilde{{\textbf {g}}}}}_{\textrm{T}}}{{\tilde{{\textbf {g}}}}}_{\textrm{T}}^{\textrm{H}}{{{\textbf {g}}}_{\textrm{R}}}}}{{{{\textbf {g}}}_{\textrm{R}}^{\textrm{H}}\left( {{{{{\tilde{{\textbf {G}}}}}}_{\textrm{T, ICI}}}{{\tilde{{\textbf {G}}}}}_{\textrm{T, ICI}}^{\textrm{H}} + \sigma _{\textrm{R}}^2{{\textbf {I}}}} \right) {{{\textbf {g}}}_{\textrm{R}}}}}~, \end{aligned}$$

(29)

where ${{{\tilde{{\textbf {G}}}}}_{\textrm{T, ICI}}} = \left[ {\begin{array}{*{20}{c}} {{{\tilde{{\textbf {g}}}_{\textrm{T}}}}{e^{j2\pi {\Delta f }t}}},&\cdots&{{{\tilde{{\textbf {g}}}_{\textrm{T}}}}{e^{j2\pi {{N_{\textrm{c}}}\Delta f } t}}} \end{array}} \right]$ is the interference sub-carriers and ${{\textbf {I}}}$ is the identity matrix.

Equation (29) is typically a generalized Rayleigh quotient. Its maximum $\overset{\frown }{\xi }\$, the maximal generalized eigenvalue of the ${{{\tilde{{\textbf {g}}}}}_{\textrm{T}}}{{\tilde{{\textbf {g}}}}}_{\textrm{T}}^{\textrm{H}}$ and $\left( {{{{{\tilde{{\textbf {G}}}}}}_{\textrm{T}}}{{\tilde{{\textbf {G}}}}}_{\textrm{T}}^{\textrm{H}} + \sigma _{\textrm{R}}^2{{\textbf {I}}}} \right)$, is achieved when ${{{\textbf {g}}}_{\textrm{R}}}$ is the corresponding eigenvector ${{\overset{\frown }{{\textbf {g}}}}_{\textrm{R}}}$, i.e., ${{{\tilde{{\textbf {g}}}}}_{\textrm{T}}}{{\tilde{{\textbf {g}}}}}_{\textrm{T}}^{\textrm{H}}{{\overset{\frown }{{\textbf {g}}}}_{\textrm{R}}} = \overset{\frown }{\xi }\ \left( {{{{{\tilde{{\textbf {G}}}}}}_{\textrm{T}}}{{\tilde{{\textbf {G}}}}}_{\textrm{T}}^{\textrm{H}} + \sigma _{\textrm{R}}^2{{\textbf {I}}}} \right) {{\overset{\frown }{{\textbf {g}}}}_{\textrm{R}}}$. However, the optimal pulse ${{\overset{\frown }{{\textbf {g}}}}_{\textrm{R}}}$ is complex-valued due to the Doppler shifts, while the pulse shaping functions implemented in real systems are usually real-valued. Therefore, we adopt the iteration-based method to derive a real-valued sub-optimal pulse. The optimization problem at RX can be written as

$$ \begin{aligned} {{{\mathcal {P}}}_{\textrm{R}}}:~&{\mathop {{\mathrm{min.}}}\limits _{{{{\textbf {g}}}_{\textrm{R}}}} }{\mathrm{~ - }}{{\tilde{\Psi }} _{\textrm{c}}} \\&{\hspace{4.0pt}\mathrm{s.t.\hspace{6.0pt}}} 0 \prec {{{\textbf {g}}}_{\textrm{R}}} \prec 1. \end{aligned} $$

(30)

Unlike (28) where the transmitting pulse is calculated by several parameters, the receiving pulse here is much more flexible, with the variable number equaling to the pulse length and each bounded between 0 and 1. After the estimation of range and velocity, the RX could recover the transmitted pulse with Doppler shifts as well as the noise power. It then chooses ${{{\textbf {g}}}_{\textrm{T}}}$ as the initial point and iterates toward the optimum. The computation complexity for each evaluation is $\mathcal {O}(N_{p}^3+N_{p}^2+2N_{p})$, where $N_{p}=N_{c}(1+\alpha )$ represents the pulse length. However, the term ${{{{\tilde{{\textbf {G}}}}}}_{\textrm{T}}}{{\tilde{{\textbf {G}}}}}_{\textrm{T}}^{\textrm{H}}$ is constant during the optimization process and can be computed in advance. Therefore, the complexity is actually $\mathcal {O}(N_{p}^2+2N_{p})$. We use the fmincon solver in MATLAB, which has relatively lower complexity, to iteratively solve this nonlinear and nonconvex optimization problem.

The resulting pulse is then employed again to demodulate the communications data to reduce BER. The operations at TX and RX are summarized in Algorithm 1 and , respectively.

Algorithm 1

The pulse shaping framework at TX.

Algorithm 2

The pulse shaping framework at RX.

2.6 Simulation parameters and feasibility study

We present the simulation parameters in Table 3. In this case, the maximal duration of an OFDM frame is 1.28 ms when $\alpha =1$, in which the range variation is 0.41 m at a velocity of 320 m/s. According to [6], if the range variation is less than the range resolution, which is 1.46 m from (13), the effects of delay and Doppler are still orthogonal and radar estimation can be conducted. The path loss parameters are adopted from [17].

Table 3 The simulation parameters

3 Results and discussion

3.1 Radar processing

Due to the relatively large number of sub-carriers, the performance of range estimation is satisfying. Hence, we mainly focus on the improvements in velocity estimation. Figure 3 shows the radar image in velocity with different processing configurations. The blue line is the original one with length $N_{\textrm{p}}=128$ for the inverse discrete Fourier transform, which is equal to the number of OFDM symbols, and no additional windowing is applied. We can see that there is a peak indicating the target, but the estimated velocity deviates from the actual value ($v=142.7~\mathrm m/s$) quite a bit. This is because shorter signal length would lead to a lower resolution, which is only $5.2~\mathrm m/s$ when $\alpha =0.5$. Then, we pad trailing zeros to increase $N_{\textrm{p}}$ to 1024, reaching a good balance between accuracy and complexity. In this way, the resolution of velocity is $0.65~\mathrm m/s$. However, as shown in the red line, there are strong side-lobes which could overshadow weak targets, resulting in a missed detection. Therefore, hamming window is implemented after data symbol cancellation in (11), due to its ability to suppress side-lobes with the smallest SINR loss among different windows. We can see the side-lobes of the yellow line are well suppressed, whereas at the cost of $2.7~\mathrm dB$ lower in SINR and 1.5 times broader in main-lobe.

Fig. 3

The radar image in velocity with different processing configurations

The transmitted symbols need to be recovered by RX first and then used in radar processing. Thus, the BER is inevitable, especially when the range and velocity change drastically and the previous receiving pulse is not optimal anymore. Hence, the radar performance needs to be investigated when there are errors in the recovered symbols. From Fig. 4, we can find that even with high BER, the peak is still distinguishable and the estimation is still accurate, with some degradation in peak height. When BER is 0.19, the SINR decreases by 4.2 dB. This is because Fourier transforms make data symbols superpose coherently, while errors, together with noise, add up stochastically. Thus, it would not influence the index but the value of peak, which proves that the proposed scheme of radar estimation followed by communications demodulation is feasible.

Fig. 4

The radar image on velocity with respect to BER

3.2 Pulse generating

We present an example of the multiplied pulse in time and frequency domain in Fig. 5. It was generated by multiplying DJ1 and Gaussian pulse, with $\alpha =0.45,~{\alpha _1}=0.1,~\mathrm{and ~}{\beta _{\textrm{G}}}=0.3$. We could notice that the Gaussian pulse is truncated by rectangular pulse due to the time limit in (16). Thus, in the frequency domain, it is no longer Gaussian but the convolution of Gaussian and Sinc function, which brings minor side-lobes just like other Nyquist pulses. The convolution in time domain makes the primary pulse smoother. While multiplication in frequency domain makes the side-lobes quite small, with the first three dominant ones suppressed by 1.6, 4.5, and 9 dB, respectively. However, the suppression cannot be too aggressive since it may narrow the main-lobe and the signal power would decline significantly with the presence of frequency offset.

Fig. 5

An example of the multiplied pulse in time (a) and frequency (b) domain

3.3 Optimizing

We present the optimal pulses at TX first. One of the most representative parameters is roll-off factor. Aside from the integrated optimization, we also present the results when only radar image or communications SINR is considered in optimization for comparison. From Fig. 6, we can see that the optimal roll-off factors for each case are distinctly different. For radar image SINR only, the roll-off factor is pretty small, which is mostly 0.05. With such a low $\alpha$, the pulse is almost rectangular and hence, there are only slight variations among different kinds of pulses, and the effect of the secondary pulse is very limited. As for communications only, $\alpha$ is almost 1, making the lowest side-lobes and suppressing interference as much as possible. The optimal pulse type is mostly SOCW, with some DJ1 or BTRC at close range and high velocity. Therefore, the optimal pulses for radar and communications are completely contradictory, and a good balance needs to be discovered. For integrated optimization, the optimal pulses are always DJ1 $\times$ Gaussian, and the roll-off factor is about 0.3, which is a bit larger at close range and high velocity. Besides, the $\alpha _1$ is relatively small to save more time for Gaussian pulses, and make the effect of the secondary pulse more noticeable.

Fig. 6

The diagram of roll-off factors of the optimized pulse

Unlike TX, which requires taking into account multiple performance metrics, the optimization at RX (30) is purer but more time-sensitive. Hence, the performance improvement with respect to the iteration number is investigated here in Fig. 7. The simulation is done at the range of 90 m and velocity of 320 m/s. As we can see, when matched pulse is employed in RX, the SINR is merely 4.65 dB due to the severe Doppler shifts. However, after only 10 iterations, the improved-matched pulse can increase the communications SINR by 26.4$\%$, reaching 5.86 dB. Therefore, we set the iteration number to 10 to make a tradeoff between performance and complexity.

Fig. 7

The performance improvement with respect to iteration numbers at RX

3.4 Performance evaluation

The performance of communications SINR at RX is shown in Fig. 8, in which the discontinuity of the line is due to the pulse switching during the flight. Compared with the optimum, the proposed integrated optimization is not much worse at long range or low velocity, with an average gap of only 1 dB. However, it degrades fast with respect to velocity and has a maximal gap of 4.5 dB at $v=320~\mathrm m/s$ and $R=90~\mathrm m$ when the impact of ICI is much greater than AWGN. The performance of optimal radar is close to that of integrated optimization at long range, but quite poor when it is close. To demonstrate the benefit of the optimization at RX side, we add the performance of the integrated optimization when improved-matched pulse is employed. We can see that there is a significant improvement at close range and high velocity when the influence of ICI is prominent. From the cumulative distribution function (CDF), we can see that in the majority of cases (80%), the SINR loss with respect to the optimal does not exceed 2.1 dB, while that of matched pulse is 2.6 dB. Besides, the maximal loss is reduced from 4.5 to 3.6 dB. In order to highlight the superiority of the radar-assisted pulse shaping, we also add a line that adopts a fixed pulse throughout the simulation. It is the result of integrated optimization at $v=170~\mathrm m/s$ and $R=160~\mathrm m$, which is DJ1 $\times$ Gaussian with $\alpha =0.3$, $\alpha _1=0.06$, and $\beta _{\textrm{G}}=0.12$. As we can see, the purple line performs worst at close range and high velocity, meaning the optimal pulse is highly range- and velocity-dependent, and radar estimation acts as a vital part of the proposed pulse shaping framework scheme.

Fig. 8

The performance of communications SINR at RX (a), the CDF for the performance loss (b), and the sectional drawing at $v=220~\mathrm{m/s}$ (c) and $R=100~\textrm{m}$ (d)

Figure 9 depicts the performance of radar image SINR at TX. The integrated optimization is close to optimal, with an interval of less than only 1 dB in the majority of cases. While for optimizing communications SINR only, the minimal gap is 1.7 dB when the velocity is low, and can be up to 11 dB at $v=320~\mathrm m/s$.

Fig. 9

The performance of radar image SINR at TX (a), the CDF for the performance loss (b), and the sectional drawing at $v=220~\mathrm{m/s}$ (c) and $R=100~\textrm{m}$ (d)

The OOBE performance is presented in Fig. 10. The differences between the three optimization criteria are quite evident, with the integrated optimization reaching -70 dB, while the other two are much higher. From the CDF diagram, the OOBE gain in most cases can be up to 53 and 34 dB, respectively. Therefore, the integrated optimization can have a smaller guard interval between adjacent users.

Fig. 10

The performance of OOBE (a), the CDF for the performance gain (b), and the sectional drawing at $v=220~\mathrm{m/s}$ (c) and $R=100~\textrm{m}$ (d)

4 Conclusion

In this paper, we proposed an adaptive pulse shaping framework dedicated to RadCom systems, where the radar estimation significantly contributes to pulse optimization both at TX and RX. For TX, we select the optimal pulse that has satisfactory performances in the three metrics with the assistance of radar estimation. As for RX, it first coarsely recovers data symbols and conducts radar estimation accordingly. Then, it reconstructs the transmitting pulse with the effect of Doppler shifts and derives an improved-matched receiving pulse to enhance communications SINR. According to the results obtained from simulations, we make the following conclusions: First, we proved that the RX is still able to accurately estimate the range and velocity even with the presence of high BER, whereas at the cost of lower image SINR. Then, an example of pulse design indicated that the method of convolving two pulses in the time domain to obtain a new pulse can greatly suppress the side-lobes, with the first three dominant ones suppressed by up to 9 dB. Besides, it is shown that an improved-matched pulse can enhance the communications SINR by up to $26.4\%$ with only 10 iterations. Simulations also revealed that the optimal transmitting pulses in our investigated cases are always DJ1 $\times$ Gaussian, which achieve satisfactory performance in each criterion simultaneously, with mostly less than 2.1 dB loss in communications, 1 dB loss in radar, and 34 dB better in OOBE. Therefore, the proposed pulse shaping framework does make the most of RadCom systems and significantly improves the performance in different aspects.