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Smoothed Phase-Coded FMCW: Waveform Properties and Transceiver Architecture

Smoothed phase-coded frequency modulated continuous waveform (SPC-FMCW), which is aimed to improve the coexistence of multiple radars operating within the same frequency bandwidth, is studied and the receiving strategy with a low ADC sampling requirement is investigated. The Gaussian filter is applied to obtain smooth phase transition, and the phase lag compensation is performed on transmitted phase code to enhance decoding. The proposed waveform is examined in different domains, and its waveform properties are analysed theoretically and demonstrated experimentally. Both the simulation and experimental results show that the introduced waveform with the investigated processing steps helps to combine all advantages of the FMCW waveform including hardware simplicity and small operational bandwidth of the receiver with advantages of the phase coding.

Abstract-Smoothed phase-coded frequency modulated continuous waveform (SPC-FMCW), which is aimed to improve the coexistence of multiple radars operating within the same frequency bandwidth, is studied and the receiving strategy with a low ADC sampling requirement is investigated. The Gaussian filter is applied to obtain smooth phase transition, and the phase lag compensation is performed on transmitted phase code to enhance decoding. The proposed waveform is examined in different domains, and its waveform properties are analysed theoretically and demonstrated experimentally. Both the simulation and experimental results show that the introduced waveform with the investigated processing steps helps to combine all advantages of the FMCW waveform including hardware simplicity and small operational bandwidth of the receiver with advantages of the phase coding. R ADARS provide detection, tracking, and classification of targets under various weather conditions. As a consequence, radars are utilised in many areas such as surveillance, meteorology, defence and automotive systems. The dramatic increase of radar sensors used for different applications has raised the concern about the spectral congestion and the coexistence of the radar sensors [1]- [4]. The mutual interference between multiple radar sensors downgrades the sensing performance of the radar and needs to be mitigated [5]- [8]. Moreover, the radar systems used in civil applications (e.g. automotive radar, indoor monitoring) generally have limited processing power, preventing them from using computationally heavy techniques. In order to cope with these issues, designing a robust waveform with a low sampling requirement and improving the independent operation of multiple radars within the same frequency bandwidth is of interest.
The linear frequency modulated continuous waveform (FMCW) has been widely used in civil radar applications [9]. In FMCW radar, the received signal is mixed with the complex conjugate of the transmitted signal for the stretch processing (also known as dechirping or deramping), which allows small analogue bandwidth of the receiver and a simple hardware structure [10]. Moreover, the FMCW radars can achieve good sensing performance with high resolution. However, discrimination of FMCW between multiple radars is limited, and the FMCW radars suffer from radar-to-radar interference [11]- [13]. For the purpose of unique waveform recognition, waveform coding has been widely used in radars [14]. In particular, the phase modulated continuous waveform (PMCW) provides high mutual orthogonality and, consequently, improves the radar's robustness against interference [15]- [17]. However, such coding spreads the waveform spectrum over a wide bandwidth and requires a dramatic increase in the receiver's analogue bandwidth.
Lately, the phase-coded frequency modulated continuous waveform (PC-FMCW) has attracted much attention due to taking advantage of both FMCW and PMCW [18]- [20]. Applying coding to FMCW improves the waveform diversity and ensures the discrimination of self-transmitted signals from the waveforms transmitted by other radars [21]. Furthermore, the phase-coded FMCW enables joint sensing and communication [22]- [24]. In [18], matched filtering is used as a receiver strategy to process the PC-FMCW. For the matched filtering operation, the received signal is convoluted with the complex conjugate of the transmitted signal. The conventional matched filtering operation in the digital domain demands acquisition of the received signal with total bandwidth. Hence, this processing approach could not reduce the analogue receive bandwidth, and the key disadvantage of PMCW has been transferred to PC-FMCW. To decrease the sampling requirement of the receiver, the decirping and decoding receiver strategy is adjusted for the PC-FMCW. However, the coded beat signals are needed to be aligned to perform appropriate decoding. In [21], [22], the alignment of the beat signals is realized by using an ideal group delay filter, and the range information of the target is extracted from the beat signals after decoding. However, the group delay filter causes a quadratic phase shift (group delay dispersion effect) on the phase-coded signal. This additional quadratic phase shift distorts the received code. Thus, the decoding becomes imperfect, which raises the sidelobes of the range profile.
The most popular phase coding scheme used for the PC-FMCW in the preliminary studies [21], [25] was the binary phase shift keying (BPSK). The BPSK coding causes abrupt phase changes that lead to large leakage of the BPSK spectrum outside of the 3-dB bandwidth. As a consequence, the BPSK only with a small bandwidth compared to the sampling frequency (a few chips per chirps) can be used for sensing [24]. In the case of the BPSK code bandwidth being comparable to the sampling frequency of the analog-to-digital converter (ADC), the sidelobe level significantly increases. Therefore, other phase modulation types with lower spectral leakage and thus better sensing performance are still of much interest.
In this paper, we have studied the smoothed phase-coded frequency modulated continuous waveform (SPC-FMCW) to enhance the coexistence of multiple radars, and we have investigated the receiving strategy with a low ADC sampling requirement. The phase smoothing operation is proposed to obtain a smooth phase transition that addresses the bandwidth limitations of BPSK, and the phase lag compensation is applied to the transmitted phase code to eliminate the undesired effect of the group delay filter. For analysis, we have used the Gaussian filter as a smoother and derived the waveform in different domains. Subsequently, we have investigated the waveform properties analytically. In addition, we have applied the proposed waveform to a real scenario and examined its sensing performance experimentally.
The rest of the paper is organised as follows. Section II describes signal model for the generic PC-FMCW and gives the transceiver structure for the signal. Section III presents a smoothing operation to improve the phase transition of the PC-FMCW. Section IV provides the phase lag compensation, and Section V investigates waveform properties of the resulting waveforms. Section VI demonstrates the application of the waveforms to a real scenario. Finally, Section VII highlights the concluding remarks.

II. SIGNAL MODEL AND TRANSCEIVER STRUCTURE
This section introduces the signal model and the state of the art transceiver structure of the phase-coded frequency modulated continuous waveform (PC-FMCW) [21] as illustrated in Figure 1.
The transmitted signal for frequency modulated continues waveform (FMCW) can be written as: where f c is the carrier frequency, T is the sweep duration of the signal, k = B/T is the slope of the linear frequency modulated waveform, and B is the bandwidth. In PC-FMCW, the phase of the FMCW changes according to the code sequence. The transmitted signal for PC-FMCW radar can be represented as: where s(t) is a phase-coded 1 signal. The received signal reflected from a moving point-like target can be written as: where α 0 is a complex amplitude proportional to the target back-scattering coefficient and propagation effects. Hereinafter, we substitute all the constant terms in α 0 with no loss of generality. The round trip delay τ (t) for a target with constant velocity can be represented as: where R 0 is the range, v 0 is the velocity, and c is the speed of light. The range and velocity information of the target can be obtained by extracting the τ (t) from the received signal. In the state of the art transceiver structure, the received signal is mixed with the complex conjugate of uncoded transmit signal (1) for dechirping process [21]. The complex mixer output can be written as: where (·) * denotes the complex conjugate, f d = 2v0fc c is the Doppler frequency and f b is the beat frequency defined as: Since the velocity of the target is much smaller then the speed of light as v 0 c, the delayed code term can be approximated as s(t−τ 0 + 2v0t c ) ≈ s(t−τ 0 ). Moreover, e j(2πfcτ0) is constant phase term and thus incorporated into α 0 .
In automotive radars, the Doppler frequency of the target is typically negligible compared to the frequency resolution of the beat signal, i.e. f d f s /N , where f s is the sampling frequency of the beat signal and N is the number of fast-time samples. Thus, we can approximate the f d +f b ≈ f b and write the mixer output as: The frequency-domain representation of the mixer output can be obtained as: where for the second equality we used (6).
In the decoding process, the mixer output (7) is multiplied with the complex conjugate of the transmitted code to compensate the phase changes initiated by the phase code. For a short-range radar application e.g. indoor monitoring, the delay can be neglected s(t − τ 0 ) ≈ s(t), and the mixer output can be decoded by multiplying (7) with s * (t) directly [24]. However, this assumption does not hold for the applications with R ≥ c/(2B c ), where B c is the bandwidth of phase-coded signal s(t). For these applications, each coded beat signal (the response in all the range cells) are required to be aligned in fast-time to compensate the time delay before decoding. This  [21] or frequency-domain [22]. Assume we have a group delay filter with frequency response: and unity magnitude, |H g (f )| = 1, ∀f . The Taylor series expansion of the phase response θ g (f ) around f b can be found as: The resulting filter causes the group delay, τ g (f ), which is the first derivative of the phase response and shifts the envelope of the signal. To align coded beat signals, the group delay needs to eliminate τ 0 . Thus, the required group delay equals to: and consequently, the first derivative of the phase response can be written as: The group delay filter which gives the required group delay in (12) can be written as: We apply the group delay filter by multiplying the spectrum of mixer output as: For each beat frequency, the frequency characteristic of the group delay filter can be expressed by the Taylor series expansion of its phase response around f b as: Subsequently, multiplying the mixer output spectrum with the frequency characteristic of the filter gives: and, taking the inverse Fourier transform of the group delay filter output (16) gives: where ⊗ denotes the convolution operation and the constant phase term exp πf b 2 k is incorporated into α 0 . Note that the delay τ 0 is eliminated after the group delay filter for each coded beat signal. Moreover, the derived group delay filter has a quadratic frequency component within its phase response, and it applies different time delays to each frequency component. Consequently, the filter causes the so-called group delay dispersion effect shown as term π k (f −f b ) 2 , which leads to a non-linear shift in addition to its linear group delay.
The spectrogram of the BPSK phase-coded signal and the alignment of coded beat signals after using group delay filter are shown in Figure 2. It can be seen that the signals with lower frequency are shifted less compared to signal with higher frequency. As a result of coded beat signal alignment, the decoding can be performed by multiplying the group delay filter output with the complex conjugate of the transmitted phase code. However, the spectrum of the phase-coded signal is shifted non-linearly (Figure 2 b), and thus the decoding becomes imperfect. The decoded beat signal can be written as [22]: where (t) is the residual phase error due to the group delay filter dispersion that causes imperfection in decoding. The dispersion effect can be neglected for a narrow-band signal where the bandwidth of the phase-coded signal is very small compared to the sampling frequency B c f s . However, the dispersion effect becomes crucial for a signal with wide spectrum where B c is comparable to f s . One example of such a signal is the BPSK phase-coded beat signal. In the time instance of phase shifts, the BPSK signal has a widespread spectrum due to abrupt phase changes. Applying nonlinear phase shifts to its spectrum leads to huge imperfection in decoding. Consequently, the BPSK signal suffers from the distorted range profile after decoding, as demonstrated in Figure 3. The distortion of the range profile raises for the long code sequences (the bandwidth of the chips increase). We will address the compensation of the group delay dispersion effect (quadratic phase lag) in Section IV.

III. SMOOTH PHASE TRANSITION WAVEFORMS
This section presents the smoothing operation to improve the phase transition of the state of the art waveform and enhance its spectral efficiency to obtain SPC-FMCW.

A. BPSK PC-FMCW
In BPSK, the phase changes φ bpsk (t) ∈ {0, π} as shown in Figure 4 and the transmitted code term can be represented as: where N c is the number of chips within one chirp, T c = T /N c is the chip duration, rect(t) = 1, t ∈ [−T c /2, T c /2] and zero otherwise is the rectangle function, and φ n denotes the phase corresponding to the n th bit of the N c bits sequence.
In addition, analyzing the spectrogram of the phase-coded signal is complementary as it provides an additional perspective that may not easily be seen on time or frequency domain representation of the signal. The instantaneous frequency for the BPSK code sequence can be written as [25]: The proof is given in Appendix A. The instantaneous frequency of the BPSK is demonstrated in Figure 5 a. It can be seen that the abrupt phase changes cause a short burst in the spectrum, which mathematically comes from the derivatives of unit step functions and is represented as Dirac delta in (20).
For the BPSK PC-FMCW, we can replace the s(t−τ 0 ) term with c(t − τ 0 ) and the mixer output becomes: and the frequency domain representation of the mixer output can be written as: The frequency-domain representation of the mixer output with τ 0 = 0 is shown in Figure 6 in blue color. The first null for the mixer output of BPSK PC-FMCW is defined by sinc function as shown in (22). Therefore, the first null location of the coded beat signal for the BPSK PC-FMCW can be calculated as: where B c = 1/T c is the bandwidth of chip. Assume we have an ideal Brick-wall low pass filter (LPF) as: The output of the LPF can be represented as: To include the k th null, the cut-off frequency of the LPF can be written as: where f bmax = kτ max is the maximum beat frequency and τ max = 2R max /c is the maximum round trip delay for the stationary target at the maximum range R max . The BPSK coding results in substantial leakage of the BPSK spectrum outside of the 3-dB bandwidth due to rapid phase shifts. Thus, the BPSK coding requires the sampling of a few multiples of code bandwidth. Using the BPSK code with large bandwidth comparable to the sampling frequency and filtering of the spectrum leads to increased sidelobe level. To cope with this problem, we can apply smoother.

B. Gaussian PC-FMCW
A smoother can be applied to the phase of the signal to improve spectral efficiency. In this paper, we have used the Gaussian filter as a smoother for analysis. However, a different smoothing filter can be selected depending on the required where η = and B s is the 3-dB bandwidth of the Gaussian filter. The Gaussian filter in frequency domain can be written as [26]: Applying Gaussian filter to the binary code, we obtained the Gaussian binary code φ gauss (t) = φ bpsk (t) ⊗ h(t) as demonstrated in Figure 4. The instantaneous frequency for the Gaussian binary code can be written as: The proof is given in Appendix A. The equation (29) shows that the abrupt phase changes are smoothed by the Gaussian filter, and the term e −η 2 (t−nTc) 2 is expected when the phase changes with respect to time. This can be seen in Figure 5 b that the phase changes cause Gaussian shape in the instantaneous frequency of the Gaussian binary code.
In addition, the mixer output of Gaussian PC-FMCW can be represented as: and its frequency spectrum can be written as: As seen in (31), the first null of the mixer output for the Gaussian PC-FMCW is decided by sinc function bounded by the Gaussian filter H(f ). Consequently, the first null location becomes f b +B s , and the required cut-off frequency to include the main lobe becomes: Using a Gaussian filter improves the spectral efficiency of the mixer output as shown in Figure 6 where the frequencydomain representation of the mixer output for Gaussian PC-FMCW is demonstrated in red color. It can be observed that the spectral leakage in the spectrum outside of the 3-dB bandwidth is lowered for Gaussian PC-FMCW compared to BPSK coded signal ( Figure 6).

C. GMSK PC-FMCW
The Gaussian minimum shift keying (GMSK) is a popular modulation scheme in communication due to its spectral efficiency. In GMSK, the binary code signal is filtered by a Gaussian filter, and the filtered code is integrated over time [27], [28]. The resulting phase for GMSK becomes Figure 4. The instantaneous frequency for the GMSK phase code can be obtained as: where erf(t) represents the error function. The proof is given in Appendix A. The (33) demonstrates that the Dirac delta term seen in BPSK coding due to abrupt phase change is replaced by the term erf (η(t − nT c )) for the GMSK phase code. This behaviour can be observed in Figure 5 c, where the phase changes lead error functions (combination of left and right parts gives a smoothed rectangle shape) in the instantaneous frequency of the GMSK phase code. Subsequently, the mixer output of GMSK PC-FMCW can be represented as: and its frequency-domain representation can be written as [26]: Similar to the Gaussian case, the Gaussian filter H(f ) bounds the frequency components of the mixer output for GMSK PC-FMCW as shown in (35). Thus the first null location becomes f b + B s , and the required cut-off frequency to include the main lobe is the same as (32). In addition, GMSK PC-FMCW has a sinc 2 term instead of a sinc function, which is seen in the BPSK code. This is because the GMSK phase code has a smoothed triangular shape while the BPSK has rectangular, as illustrated in Figure 4. The spectral behaviour of the GMSK PC-FMCW is shown in Figure 6 in yellow color. It can be seen that the taking square of sinc function and bounding it by the Gaussian filter further reduces the spectral leakage in the spectrum outside of the 3-dB bandwidth compared to both BPSK and Gaussian code. Consequently, better sensing performance (i.e. lower sidelobe level) is expected for a GMSK PC-FMCW when the bandwidth of the code increases and becomes comparable to ADC sampling.

IV. PHASE LAG COMPENSATED WAVEFORMS
This section introduces the phase lag compensated waveforms and the proposed block diagram is shown in Figure 7.
The group delay filter applies different time delays to each frequency component and causes a dispersion effect on the phase-coded signal, which leads to a distorted range profile as explained in Section II. To eliminate the undesired effect of the group delay filter, we perform quadratic phase lag compensation on the transmitted code by multiplying its spectrum with the quadratic phase term as [29]: Then the mixer output (7) becomes: x and for the time and frequency domain representation, respectively. Subsequently, the output of the group delay filter in frequencydomain becomes: Note that the undesired term π k (f − f b ) 2 caused by the phase response of the filter (15) is eliminated with the phase lag compensation. After taking the inverse Fourier transform and incorporated the constant phase term exp πf b 2 k into α 0 , the time-domain representation of the new group delay filter output (39) becomes: In addition, we can shift the group delay filter output to the maximum beat frequency (defined with sampling frequency) by multiplying its spectrum with linear phase delay exp −j2πf f b max k for physical correctness and guarding the beginning of next chirp. Consequently, the (40) is shifted to the maximum time delay as: An example of the resulting signal can be seen in Figure 8 a, where the spectrogram of the BPSK phase-coded beat signals using both phase lag compensation and group delay filter Fig. 9. The recovered range profile of decoded signal after using phase lag compensation and group delay filter is illustrated. Note that each coded beat signal is perfectly aligned, and we can apply decoding signal which is the complex conjugate of the transmitted phase code shifted to maximum beat frequency. Subsequently, the decoded beat signal becomes: It can be seen in (42) that the code term is removed properly, and the residual phase error caused by the imperfection in decoding is eliminated by using the phase lag compensation. As a consequence, the distorted range profile shown in Figure 3 is recovered for a wide-band signal where B c is comparable to f s as illustrated in Figure 9. Moreover, the beat signals are obtained similar to the dechirped signal of conventional FMCW radar as demonstrated in Figure 8 b. This aids in re-utilise all software algorithms previously developed for FMCW radar with the proposed waveform and transceiver structure.

V. WAVEFORM PROPERTIES
This section provides the properties of the phase lag compensated PC-FMCW. For the numerical simulations, we consider a radar operating with a carrier frequency f c = 3.315 GHz and transmitting the investigated waveforms with the chirp duration T = 1 ms and the chirp bandwidth B = 200 MHz. The phase lag compensated signals(t) is used for phase coding, and we have used the random code sequence for all three PC-FMCW. The duration of the chip T c is controlled with the number of chips per chirp N c as T c = T /N c . To achieve a smoothed phase transition, the 3-dB bandwidth of the Gaussian filter is set to two times the chip bandwidth B s = 2B c . On receiver side, (37) is low-pass filtered with the the cut-off frequency f cut = 40 MHz and sampled with f s = 40 MHz. As a consequence, we have N = 40000 range cells (fast-time samples) for this setting. The group delay filter is applied to the sampled signal to align the beat signals of different targets. Before decoding, the same LPF is applied to the reference phase-coded signal to prevent a signal mismatch. To focus on the waveform properties, we assume a noise-free scenario in the numerical simulations.

A. Sensing
The sensing performance of the phase lag compensated waveforms are assessed by using the investigated processing method and compared with the FMCW. After proper decoding, the code term is removed, and the beat signal is recovered similar to the dechirped signal of traditional FMCW as explained in Section IV.
In Figure 10, we investigate the Doppler tolerance of the waveforms and proposed receiver strategy. In order to do that, we simulate the received signal after dechirping (5) as a function of Doppler frequency shift and plot it in a form similar to the ambiguity function. Note that we do not apply the Matched filter receiver, and thus the plots in Figure 10 are not the ambiguity functions. The presented plots show the behaviour of FMCW and three phase lag compensated PC-FMCW with N c = 1024. It can be seen that all considered waveforms have the Doppler tolerance of the FMCW and exhibit the range-Doppler coupling, determined by the slope of the carrier chirp. Note that, in the vicinity of the mainlobe, they all have an identical response, determined by 100 dB Chebyshev window, applied to the signals before range FFT. The sidelobes of three phase lag compensated PC-FMCW The bandwidth of the chip B c raises as the number of chips per chirp N c increases. Consequently, the bandwidth of the chip becomes comparable to ADC sampling frequency and sidelobe level increases with filtering of the spectrum. However, the spectral leakage outside of the 3-dB bandwidth are different for the three phase lag compensated PC-FMCW as explained in Section III, and thus they provide different PSL. The PSL of investigated waveforms at normalized target range R/R max = 0.2 as a function of the number of chips per chip is demonstrated in Figure 11 where the maximum range R max = cf bmax 2k and the maximum beat signal is determined as f bmax = f cut /2 = f s /2. It can be seen that the PSL of the BPSK PC-FMCW rapidly increases while the GMSK PC-FMCW can provide PSL similar to FMCW up to N c = 1024. In addition, we illustrated the PSL of investigated waveforms with N c = 1024 as a function of normalized target range in Figure 12. Note that the spectral leakage and filtering of the spectrum become crucial for the PC-FMCW as the target approaches the maximum range. Among phase lag compensated waveforms, the GMSK PC-FMCW has favorable sensing performance and provides lower PSL.

B. Peak to Average Power Ratio
The quadratic phase lag compensation is applied to the spectrum of the transmitted code to eliminate the dispersion effect of the group delay filter. This quadratic phase lag compensation filter can be represented as: and the phase lag compensated code term in time-domain can be written as: To analyse the effect of quadratic phase lag compensation on phase-coded signal, let ξ = j π k , then the quadratic phase lag compensation filter in time-domain can be written as: Subsequently, the result of the convolution for the BPSK code sequence becomes: The proof is given in Appendix B. The amplitude of the phase lag compensated BPSK code is shown in Figure 13 a. The quadratic phase lag compensation applies different time delays to each frequency component of the transmitted phase code. During phase changes, the phase-coded signal has wide spectrum and shifting the frequency components nonlinearly creates ripples in the time-domain signal (Figure 13 a). Moreover, the time interval between phase changes becomes shorter for a long code sequence and ripples in the timedomain collectively summed up as the adjacent phase shifts interfering with each other. Therefore, the amplitude of the transmitted waveform is not constant anymore (Figure 13 b). The time-varying amplitude initiated by the phase lag compensation leads to a high peak-to-average power ratio (PAPR). The PAPR of the signal can be represented as: The PAPR of the investigated waveforms is compared as a function of N c in Figure 14. It can be seen that the PAPR of the three PC-FMCW without phase lag compensation are constant and equal to 1. The PAPR increases for all three phase lag compensated PC-FMCW as N c raises. However, the effect of the phase lag compensation and the resulting amplitude variation decrease as the phase transition becomes smoother. Note that the differences of PAPR between BPSK, Gaussian, and GMSK are comparable up to N c = 64 thereafter, PAPR varies notably. For long code sequences, the GMSK PC-FMCW provides the lowest PAPR while the BPSK PC-FMCW has the highest PAPR since the abrupt phase changes on the BPSK coding are affected more by the frequency-dependent shifts.

C. Mutual Orthogonality
The coding spreads the power of the signals in the beat frequency domain. As each transmitted PC-FMCW uses its phase-coded signal, only the correct signal passes through the received signal, which is matched to this code. The signals with other code sequences are not matched to these received signal, leading to the spread of the power over range. The theoretical limits of the suppression equal to the spreading factor and can be written as [10]: Assume the first radar (victim) transmits PC-FMCW with the phase lag compensated codes 1 (t) to detect a target. The received signal reflected from the target with complex coefficient α 1 can be written as: To illustrate the mutual orthogonality assessment, consider the worst-case scenario when a second radar is perfectly synchronized with the first radar and transmits a PC-FMCW with the phase lag compensated codes 2 (t). The signal transmitted from the second radar is delayed in time and captured by the first radar with complex coefficient α 2 as: where τ 2 is the round trip delay between the first and second radars. Subsequently, the total received signal on the first radar is the combination of received signals and can be written as: The total received signal is mixed and dechirped with the uncoded transmit signal of the first radar. The mixer output gives the summation of two coded beat signals. Subsequently, the group delay filter is applied to the mixer output and align coded beat signals at the maximum delay as discussed in Section IV. The output of the group delay filter can be represented as: (52) During decoding, the group delay filter output is decoded with the complex conjugate of the first code shifted to the maximum delay s 1 (t − τ max ). After decoding, the beat signal reflected from the target is obtained similar to the dechirped signal of conventional FMCW, while the beat signal initiated by the second radar remains coded as: (53) In Figure 15, we compare the cross-isolation between the two PC-FMCW using different random code sequences. We consider the number of chips per chirp N c = 1024 and the number of chirp pulses N p = 512. It is shown in Figure 15 that the second radar causes a beat signal according to the f b2 = kτ 2 which can be seen as a ghost target for a perfectly synchronized case (which is very difficult to generate in a reallife scenario and just used for the proof of mutual orthogonality concept), and it can not be distinguished from the target in the traditional FMCW (Figure 15 a). However, in the phasecoded FMCW cases, the beat signal initiated by the second radar f b2 remains coded, and thus its power is spread over both fast-time and slow-time. This cross-isolation between two BPSK PC-FMCW, Gaussian PC-FMCW, and GMSK PC-FMCW are given in Figure 15 b, c, and d, respectively. The theoretical upper-boundary limit regarding the suppression of the beat signal f b2 is 10 log 10 (512) + 10 log 10 (1024) = 57 dB for a perfectly orthogonal code (combined with the suppression in both slow-time and fast-time). However, the three PC-FMCW are not perfectly orthogonal, and their suppression behaviours in the fast-time are different according to their phase modulation type as shown in Figure 15. In particular, the Gaussian PC-FMCW has the local peaks between phase-coded signals, and it gives the worst suppression performance. The BPSK PC-FMCW spreads the power of f b2 to all range cells as the spectrum of BPSK has a significant leakage outside of the 3-dB bandwidth. The GMSK PC-FMCW spreads the power of the f b2 as a Gaussian shape (Triangular in dB scale) over the range cells defined by the 3-dB bandwidth of the Gaussian filter (smoother bandwidth) B s . Consequently, the GMSK PC-FMCW has a narrower spreading characteristic than the BPSK, which might help to avoid masking of targets with weak radar cross section (RCS) outside of the 3-dB bandwidth. Moreover, it can be seen in Figure 16 that the GMSK PC-FMCW can provide high cross-isolation while achieving low PSL. These facts favour the usage of GMSK PC-FMCW over the BPSK PC-FMCW.

VI. EXPERIMENTS
This section demonstrates the experimental results related to the sensing performance of the phase lag compensated PC-FMCW. The experimental investigation of the waveforms has been done using PARSAX radar [30]. We use the proposed transceiver structure for each PC-FMCW, and we apply the traditional dechipring transceiver structure for the FMCW waveform which is used as a benchmark. We use zerocorrelation-zone (ZCZ) code family [31] for the three phase lag compensated PC-FMCW and choose the system parameters as given in Table I. The resulting waveforms are performed in the real scenario to detect both stationary and moving targets. Note that the experimental environment is dynamic for the moving target experiment. To detect the same car and compare the sensing performance of the waveforms, we transmit four waveforms sequentially with 128 chirp pulses in each waveform. Moreover, we applied Chebyshev windowing with 80 dB suppression and compared it with a rectangle windowing case to highlight the sensing performance of the waveforms.

A. Stationary Target Experiment
For the stationary target experiment, we look at the chimney located at 1185 m away from the radar, as shown in Figure 17. We conduct two measurements with different number of chips as N c = 64 and N c = 1024 for the PC-FMCW. The range profiles of the four different waveforms are demonstrated in Figure 18. At the chimney location, the noise-clutter level of the range profile is around ∼ 140 dB for the FMCW, and it provides 60 dB dynamic range after applying Chebyshev windowing. Between the three PC-FMCW cases with N c = 64, the GMSK provides the best and closest performance to the FMCW by providing ∼ 60 dB dynamic range while the BPSK and Gaussian have increased sidelobe levels. The sensing performance of all three PC-FMCW degrades and only provide ∼ 40 dB dynamic range as we increase the N c = 1024. This can be explained by the time-varying amplitude of the transmitted waveform leading to high PAPR as the number of chips per chirp increases, and the high power amplifier (HPA) of the radar operates in saturation mode for chosen system parameters. As a consequence, the HPA clips and distorts the transmitted signal. Such effect can be lowered by systems in which the HPA operating mode is well controlled. In addition, Fig. 19. Comparison of matched filter response with the response of proposed transceiver structure for GMSK PC-FMCW we demonstrate the range profile of the GMSK PC-FMCW by using Matched Filter receiving strategy and compared it with the response of the proposed transceiver structure in Figure 19. It can be seen that the proposed transceiver structure gives a very similar result with the matched filter response for the GMSK PC-FMCW. Similar behaviour is seen for other investigated waveforms but not presented here due to the lack of space.

B. Moving Target Experiment
The Doppler tolerance of the investigated waveforms is validated by the moving target experiment where we observe the road and detect a moving car located at 1148 m with a radial velocity ∼ 14 m/s as illustrated in Figure 20. We use N c = 64 for the three PC-FMCW. The range-Doppler profiles of the waveforms are demonstrated in Figure 21 where the peak location of the target is obtained at 1148 m for each waveform. Similar to the stationary target scenario, the noise level of the range profile is around ∼ 140 dB for the FMCW, and it has 60 dB dynamic range after windowing (Figure 21  a). The range profile of the BPSK PC-FMCW has increased sidelobe level due to abrupt phase changes, and it provides a dynamic range around 40 dB (Figure 21 b) Figure 21 d. Consequently, the GMSK PC-FMCW with N c = 64 can provide similar sensing performance that is offered by the FMCW, and it can also ensure the ability to distinguish different signals due to coding as discussed in Section V-C.

VII. CONCLUSION
The smoothing of the phase-coded frequency modulated continuous waveform has been introduced as an efficient tool to enhance the coexistence of multiple radars within the same spectrum. The impact of the wide spectrum spreading due to the abrupt phase changes of the BPSK is investigated, and the Gaussian filter is proposed to smooth the phase transition of the PC-FMCW. We have suggested receiving strategy with a low sampling requirement and analysed the group delay filter effect on the coded beat signals. In addition, the phase lag compensation is applied on the transmitted phase-coded signal to eliminate the undesired effect of the group delay filter and recover the beat signals properly after the decoding.
The properties of the investigated waveforms for the first time are analysed theoretically and verified experimentally. It is shown that the PSL, PAPR and the cross-isolation between signals increase as the bandwidth of the code raises for the three phase lag compensated PC-FMCW. The simulations and the experimental results demonstrate that the phase lag compensated GMSK PC-FMCW can provide sensing performance similar to that of uncoded FMCW. At the same time, it can provide high mutual orthogonality that can be used to improve cross-isolation between multiple radars.

APPENDIX A DERIVATIVES OF DIFFERENT TYPES OF PHASE CODE
In this proof, we demonstrate the taking derivative of the different types of phase code with respect to time. Recall that the rectangle function can be written as: where u is a unit step function. Similarly, the code term can be written as: and for the n th and (n + 1) th elements, respectively. Note that the φ n ∈ {0, π} denotes the phase corresponding to the n th bit of the N c bits sequence. To take the summation of unit step functions, we have to consider a junction point in which the adjacent elements are linked. Thus, the relevant junction point includes the right part of the n th and left part of the (n + 1) th elements and the phase of the BPSK code can be represented as: where the amplitude of the unit step function varies between π and −π depending on the value (φ n+1 − φ n ), and the summation of the unit step functions gives the phase of the BPSK code sequence φ bpsk (t) ∈ {0, π}. In the following subsections, we derive the instantaneous frequency of the different types of phase code.

A. BPSK
Taking the derivative of the φ bpsk (t) gives: where δ is the Dirac delta function. Same result can be seen in [25].

B. Gaussian
The convolution of the unit step function with filter h 0 (t) = e −t 2 where t ≥ 0 can be represented as: where erf(t) represents the error function as: Subsequently, the convolution of the unit step function and the Gaussian filter h(t) = η √ π e −η 2 t 2 can be written as: Replacing the γ = ητ and dγ = ηdτ , the equation becomes: Consequently, the phase of the Gaussian binary code can be written as: φ gauss (t) = φ bpsk (t) ⊗ h(t) The derivative of the error function can be obtained as: Subsequently, taking the derivative of (63) with respect to time gives:

APPENDIX B CONVOLUTION WITH PHASE LAG COMPENSATION
The convolution of unit step function and the signal h 1 (t) = e t 2 with t ≥ 0 can be written as: where the erfi(t) is the imaginary error function defined as: Let β = πk j , then the (45) becomes: Subsequently, the convolution of the unit step function and phase lag compensation filter can be found as: Replacing the ζ = βτ and dζ = βdτ , the equation becomes: Subsequently, the result of the convolution for the BPSK code sequence becomes: Nc n=1 e j(φn+1−φn) erfi πk j (t − nT c ) . (73)