Photonic generation of arbitrarily phase-modulated microwave signals based on a single DDMZM

Abstract: We propose and demonstrate a compact and cost-effective photonic approach to generate arbitrarily phase-modulated microwave signals using a conventional dual-drive Mach-Zehnder modulator (DDMZM). One arm (arm1) of the DDMZM is driven by a sinusoidal microwave signal whose power is optimized to suppress the optical carrier, while the other arm (arm2) of the DDMZM is driven by a coding signal. In this way, the phase-modulated optical carrier from the arm2 and the sidebands from the arm1 are combined together at the output of the DDMZM. Binary phase-coded microwave pulses which are free from the baseband frequency components can be generated when the coding signal is a three-level signal. In this case, the precise π phase shift of the microwave signal is independent of the amplitude of the coding signal. Moreover, arbitrarily phase-modulated microwave signals can be generated when an optical bandpass filter is attached after the DDMZM to achieve optical single-sideband modulation. The proposed approach is theoretically analyzed and experimentally verified. The binary phase-coded microwave pulses, quaternary phase-coded microwave signal, and linearly frequencychirped microwave signal are experimentally generated. The simulated and the experimental results agree very well with each other.


Introduction
In modern radars, pulse compression is widely used to increase the range resolution.This technique permits the transmission of long phase-modulated microwave signals, leading to an efficient use of the average power capability of the radars and avoiding the generation of high peak power microwave pulses [1].To enlarge the pulse compression ratio, a phase-coded or frequency-chirped microwave signal is usually involved.Conventionally, phase-modulated microwave signals are generated using electronic devices, e.g.direct digital synthesizer and analog mixer.However, they encounter some limitations in terms of small bandwidth and low carrier frequency (up to a few GHz).Photonic generation of phase-modulated microwave signal is therefore proposed to overcome these limitations [2].
The phase-modulated microwave signal can be generated based on wavelength-to-time mapping [2][3][4][5], where the desired waveform is preprogrammed on the optical spectrum of a mode-locked laser using such as a Wave Shaper.Afterwards, a dispersive element transfers the preprogrammed waveform to the time domain.The merit of this technique lies in the fact that the microwave signal can be finely predefined even with cycle-to-cycle phase management.However, the drawback is that the reconfiguration of the waveform is not flexible, the pulse rate is related to the repetition rate of the mode-locked laser, and the pulse duration depends on both the width of the optical spectrum and the dispersive element.In [6], the generation of reconfigurable phase-modulated microwave signal has been reported using free-spaced structure, where a spatial light modulator is used to update the waveform in real time.However, the fiber-to-space and space-to-fiber connections make the system bulky and complicated.
Other techniques based on optical heterodyning have attracted great attentions during the past few years.The phase modulated microwave signal is generated by beating two optical carriers in a photodetector (PD).The relative phase of the two optical carriers is modulated by an optical phase modulator.Generally, the two optical carriers can be generated using a mode-locked laser with optical filters [7,8] or using the optical components from an external modulation scheme [9][10][11][12].The two optical carriers are spatially separated using a Mach-Zehnder structure [8] or a Sagnac loop [9] with one of the branches incorporated a phase modulator.However, they suffer from the stability problems because both setups are sensitive to the environmental influence.Alternatively, the two optical carriers can be separated into two orthogonal polarization states using a polarization maintaining fiber [10], a polarization modulator (PolM) [11], or a polarization beam combiner [12].In this way, arbitrarily phasemodulated microwave signals can be generated.Nevertheless, these techniques usually require two external modulators which make the system complicated and costly.On the other hand, the most widely used phase-modulated microwave signal employs two phase states, 0 and π, which is the so-called binary phase-coding.The precise π phase shift of the microwave signal has to be ensured to maximize the pulse compression ratio.It is noted that the amplitude of the coding signal should be the half-wave voltage of the modulator to obtain an accurate π phase shift [7][8][9][10][11][12], which is power consuming and inconvenient.To overcome this problem, a phase modulator together with a polarization controller and a polarizer was used to generate binary phase-coded microwave signal [13].The precise π phase shift is independent of the amplitude of the coding signal.Recently, the generation of phase-coded microwave signal using a single dual-parallel Mach-Zehnder modulator (DPMZM) [14] or dual-drive MZM (DDMZM) [15] has also been reported.However, both schemes are only capable of generating binary phase-coded microwave signal.Moreover, the precise π phase shift of the microwave signal is determined by the amplitude of the coding signal.
It is also worth noting that most of the reported methods can only generate phasemodulated microwave signal in continuous wave (CW) mode [9][10][11][12][13][14][15].In order to generate phase-modulated microwave pulses.The other optical intensity modulator was used in [10] to truncate the optical signal into pulses.Moreover, the additional intensity modulation generates baseband frequency components which have to be filtered out using an electrical bandpass filter [8].To eliminate the baseband frequency components, we have reported approaches using two cascaded PolMs [16,17].Again, the system is complicated and costly due to the use of multiple modulators and polarization controls.
In this paper, we report a novel, compact, and cost-effective approach to generate arbitrarily phase-modulated microwave signal using a single DDMZM.One arm (arm1) of the DDMZM is driven by a sinusoidal microwave signal.The microwave power is optimized to achieve optical carrier suppressed modulation.The other arm (arm2) of the DDMZM is driven by a coding signal.The phase-modulated optical carrier provided by the arm2 and the sidebands derived from the arm1 are combined together at the output of the DDMZM.The previous scheme based on the DDMZM [15] can only generate binary phase-coded microwave signal in a CW mode.Moreover, the precise π phase shift of the microwave signal is determined by the amplitude of the coding signal and an electrical bandpass filter is required to remove the baseband frequency components [15].However, in our scheme, the system is highly reconfigurable to generate variety kinds of phase-modulated microwave signals.It is possible to generate not only the CW binary phase-coded microwave signals but also binary phase-coded microwave pulses which are free from baseband frequency components.Thus, the electrical bandpass filter [8] is no longer required.In this case, the precise π phase shift is independent of the amplitude of the coding signal which is totally different from [15].A more remarkable advantage of the proposed approach is that arbitrarily phase-modulated microwave signal can be generated by attaching an optical bandpass filter (OBPF) after the DDMZM to realize optical single-sideband modulation.The proposed scheme is theoretically analyzed and experimentally verified.Binary phase-coded microwave pulses, quaternary phase-coded, and linearly frequency-chirped microwave signals are experimentally generated, which fit well with the simulated results.

Theory and principle
The schematic diagram of the proposed phase-modulated microwave signal generator is shown in Fig. 1(a), which consists of a laser diode (LD), a DDMZM, a tunable OBPF and a PD.Each arm of the DDMZM is a phase modulator.In our scheme, one arm (arm1) of the DDMZM is driven by a sinusoidal microwave signal.The other arm (arm2) of the DDMZM is driven by a coding signal.The normalized optical field at the output of the DDMZM is given by where ω 0 is the angular frequency of the optical carrier.J n (•) is the Bessel function of the first kind of order n.β 1 and β 2 are the phase modulation indices of the phase modulators on each arm of the DDMZM, respectively, which can be expressed as β 1 = πV m /V π and β 2 = πV c /V π .V π is the half-wave voltage of the phase modulator.V m and ω m are the amplitude and the angular frequency of the sinusoidal microwave signal, respectively.V c is the peak-to-peak amplitude of the coding signal c(t).φ is the static phase difference between the two arms which is controlled by the bias of the DDMZM.

1) Binary phase-coded microwave pulse generation
As can be seen from Eq. ( 1), a series of sidebands are generated around the optical carrier due to the nonlinearity of the DDMZM.They will beat with each other as well as with the optical carrier in the PD, generating the fundamental microwave signal and the undesired harmonic components.In order to eliminate the undesired harmonics, an electrical lowpass filter can be attached after the PD.Alternatively, an OBPF after the DDMZM can be used to remove the undesired optical sidebands in the optical domain.To generate binary phase-coded microwave pulse, the optical sidebands higher than the first-order ones are removed by the OBPF, leaving the optical carrier and the two first-order sidebands.In this case, Eq. ( 1) can by simplified as If the optical signal is detected by the PD, the photocurrent is given by As can be seen from Eq. ( 3), the photocurrent consists of dc and ac parts.The dc part varies with the coding signal c(t).Thus, the baseband frequency components will be detected in the PD.The generation of phase-modulated microwave requires that the average amplitude of the detected photocurrent keeps constant all the time.Therefore, J 0 (β 1 ) = 0 and J 1 (β 1 )≠0 has to be satisfied, which means that the optical carrier at the arm1 of the DDMZM is fully suppressed, leaving only the sidebands [18,19].
Figure 2 shows the variations of J 0 (β) and J 1 (β) versus β.For β = 2.4 rad, we have J 0 (β 1 ) = 0 and J 1 (β 1 ) = 0.5.If we let φ = 0, Eq. ( 3) is rewritten as It can be seen from Eq. ( 4) that the sinusoidal microwave signal is generated, whose amplitude is shaped by a sine term.If c(t)>0, the microwave signal has a phase of 0. If c(t)<0, the detected microwave signal has a phase of π because the sine term is an odd function.Therefore, CW binary phase-coded microwave signal is generated by applying a two-level rectangular coding signal, c(t) = 1 or -1, to the arm2 of the DDMZM.Moreover, the precise π phase shift is only determined by the sign of the coding signal and is independent of the amplitude of the coding signal.As can be seen from Eq. ( 4), the ac part is zero for c(t) = 0.It means that the microwave signal cannot be recovered for c(t) = 0. Therefore, binary phasecoded microwave pulses can be generated if the coding signal is a three-level signal, c(t) = 0, 1, or -1.If c(t) or the bias voltage of the DDMZM is not accurately equal to 0 due to the bias drifting problem, the microwave signal with phase of 0 or π will be generated.It means that the CW microwave signal cannot be effectively truncated into microwave pulses.Therefore, sophisticated voltage control is required in practical systems.

2) Arbitrarily phase-modulated microwave signals generation
For arbitrarily phase-modulated microwave signal generation, the OBPF is used to realize single-sideband modulation by removing the sideband at either side of the optical carrier.Only the phase-modulated optical carrier and one of the first-order sidebands are left.Thus, Eq. ( 1) can be rewritten as When the optical signal is detected in the PD, the photocurrent can be expressed as ] .

( ) ( ) cos( ) ( ) cos[ ( ) ]
To eliminate the baseband frequency components, we again have J 0 (β 1 ) = 0 and J 1 (β 1 ) = 0.5.If we let φ = 0, Eq. ( 6) is rewritten as As can be seen from Eq. ( 7), the microwave signal at the angular frequency of ω m is recovered.Moreover, its phase is modulated by the coding signal c(t).It therefore provides us an opportunity to generate arbitrarily phase-modulated microwave signals by applying an arbitrarily user-defined coding signal to the DDMZM.For example, if c(t) is a N-step stair wave coding signal, a N-step polyphase-coded microwave signal is obtained.Similarly, a linearly frequency-chirped microwave signal can be generated if c(t) is a parabolic signal.

Experiment and simulation
The proposed scheme was experimentally verified based on the setup shown in Fig. 1.The parameters of the key devices used in the experiment are summarized as follows: the wavelength and the power of the LD are ~1550 nm and 15 dBm, respectively.The DDMZM has a 3-dB bandwidth of 40 GHz and a measured half-wave voltage V π of 2.28 V at 15 GHz.The sinusoidal microwave signal driven to the arm1 of the DDMZM was pre-amplified by an electrical amplifier (EA1) which has a 3-dB bandwidth from 40 kHz to 38 GHz, a gain of 26 dB, and a maximum output power of 22 dBm.The coding signal was provided by an arbitrary waveform generator (Tektronix AWG70002A) which has a maximum sampling rate of 25 Gs/s and a maximum peak-to-peak output voltage of 0.5 V.The signal from the AWG was pre-amplified by the other EA2 before driving to the arm2 of the DDMZM.The EA2 has the same performance as EA1.The OBPF (Yenista XTA-50 Ultrafine) is tunable in terms of the center wavelength and the bandwidth.The center wavelength of the OBPF can be tunable from 1480 to 1620 nm and the bandwidth is tunable from 32 to 650 pm.The OBPF has a typical flatness of 0.2 dB and an edge roll-off of 800 dB/nm.The PD has a 3-dB bandwidth of 18 GHz.The generated phase-modulated microwave waveforms were recorded by a sampling oscilloscope (OSC).The optical spectrum was measured by an optical spectrum analyzer (OSA) with a resolution bandwidth of 0.01 nm.First of all, we will show the generation of binary phase-coded microwave signal.As the generation of binary phase-coded microwave pulses is more challenging, we will directly show it rather than the binary phase-coded microwave signal in the CW mode.To do so, a sinusoidal microwave signal at a frequency of 15 GHz was applied to the arm1 of the DDMZM.The microwave power was adjusted to be ~18 dBm, which corresponds to β = 2.4 rad as described in Section 2. On the other hand, a 13-bit Barker code that is commonly used in radars was applied to the arm2 of the DDMZM.The coding signal was defined as a threelevel 32-bit fixed pattern "-1, -1, -1, -1, -1, +1, +1, -1, -1, +1, -1, +1, -1, 0,0,…,0" (i.e.13bit Barker code plus 19-bit "0") operated at a data rate of 3.75 Gb/s with a peak-to-peak amplitude of ~0.5 V, and a duty cycle of 13/32.As predicted in Section 2, the precise π phase shift is independent of the amplitude of the coding signal.Therefore, the amplitude of the coding signal is chosen to be ~0.5 V which is less than the V π (2.28 V) of the DDMZM.The measured optical spectrum at the output of the DDMZM is shown in Fig. 3.A series of sidebands are generated around the optical carrier.As described in Section 2, the optical carrier provided by the arm2 of the DDMZM is phase-modulated by the 13-bit Barker code.Thus, the optical carrier is wider than the sidebands.The simulated optical spectrum at the output of the arm1 of the DDMZM is also shown in Fig. 3.The simulated sidebands agree well with the measured ones.The suppressed optical carrier cannot be directly observed in the experiment because the phase-modulated optical carrier at the arm2 overlaps with the suppressed optical carrier at the arm1 of the DDMZM.Nevertheless, the excellent fit between the measured and simulated sidebands implies that the carrier-suppressed modulation was successfully achieved at the arm1 of the DDMZM.
In our experiment, the bandwidth of the PD is 18 GHz, which is much lower than the second-order harmonic of the fundamental microwave, i.e. 30 GHz.Therefore, the secondorder sidebands are no longer needed to be removed using the OBPF.In this experiment, the OBPF was not used.The optical signal at the output of the DDMZM was directly sent to the PD. Figure 4(a) shows the electrical 13-bit Barker code signal applied to the arm2 of the DDMZM which was measured by the OSC.The generated binary phase-coded microwave pulse is shown in Fig. 4(b).The CW microwave signal is truncated into microwave pulses with pulse duration of 3.46 ns.The phase jump in the pulse duration can be clearly seen.The simulated binary phase-coded microwave pulse is shown in Fig. 4(c), which is calculated based on Eq. ( 4) and the measured 13-bit Barker coded shown in Fig. 3(a).The simulated result agrees well with the measured one.The phase information extracted from Fig. 4(b) using Hilbert transform is shown in Fig. 4(d).The phase shift of π is confirmed.Figure 4(e) shows the phase information extracted from Fig. 4(c), which fits well with that shown in Fig. 4(d).To clearly show the generated microwave pulse, the waveform was measured in a larger time duration of 70 ns as shown in Fig. 4(f).A series of microwave pulses are generated which has a duty cycle of 13/32 as expected.The measured electrical spectrum of the generated binary phase-coded microwave pulse is shown in Fig. 5.The generated electrical spectrum centered at the frequency of 15 GHz consists of a mainlobe and a series of sidelobes.This is due to the Fourier transform of the rectangular pulse.The key advantage of the generated spectrum lies in the fact that there are no baseband electrical components, which means that an additional electrical bandpass filter usually required in other schemes [8] is no longer needed in our case.
The autocorrelation of the measured 13-bit Barker coded microwave pulse (see Fig. 4(b)) was calculated using MATLAB.The autocorrelation is shown in Fig. 6(a).The inset shows the zoom-in view.The peak-to-sidelobe ratio (PSR) is 9.9 dB.The full width at halfmaximum (FWHM) of the autocorrelation is 0.28 ns.The pulse compression ratio (PCR) is calculated to be 12.36, which is very close to the ideal value of 13 for the 13-bit Barker code. Figure 6(b) shows the autocorrelation of the simulated 13-bit Barker coded microwave pulse shown in Fig. 4(c), where the inset shows the zoom-in view.The PSR is 11.3 dB and the FWHM of the autocorrelation is 0.27 ns, corresponding to a PCR of 12.81.It should be noted that binary phase-coded microwave signal in the CW mode can be simply generated by applying a two level rectangular coding signal to the arm2 of the DDMZM.In the following part, we will show the system is capable of generating arbitrarily phasemodulated microwave signal.First, the system was reconfigured to generate polyphase-coded microwave signals.In this case, a sinusoidal microwave signal at a frequency of 10 GHz was applied to the arm1 of the DDMZM.The microwave power was adjusted to be around 18 dBm to suppress the optical carrier at the arm1 of the DDMZM.A four-level stair wave coding signal from the AWG was applied to the arm2 of the DDMZM.The bit rate of the coding signal is 2 Gb/s.The peak-to-peak voltage of the coding signal is amplified to be ~2.424V, corresponding to a maximum phase shift of 3.34 rad.This value is limited by the maximum output power of the EA.The optical spectrum at the output of the DDMZM is shown in Fig. 7.A series of sidebands are generated on both sides of the phase-modulated optical carrier.The simulated optical spectrum at the arm1 of the DDMZM shown in Fig. 7 agrees well with the measured sidebands.As discussed in Section 2, the undesired sidebands have to be removed to realize single-sideband modulation.A tunable OBPF was attached after the DDMZM.The optical spectrum after filtering is shown in Fig. 7.It can be seen that the single-sideband modulation is achieved.The left sideband is 32 dB lower than the right sideband.The response of the OBPF is also shown in Fig. 7.The optical signal at the output of the OBPF was then sent to the PD to generate quaternary phase-coded microwave signal.8(c).The phase shift has four levels of 0, 1.15, 2.23, and 3.34 rad. Figure 8(d) shows the simulated quaternary phase-coded microwave signal based on Eq. ( 7) and the measured electrical fourlevel stair wave coding signal shown in Fig. 8(a).The extracted phase information from Fig. 8(d) is shown in Fig. 8(e).As can be seen, the simulated results fit well with the measured ones.The calculated autocorrelation of the measured quaternary phase-coded microwave signal is shown in Fig. 9(a).The inset shows the zoom-in view.The waveform used to calculate the autocorrelation is shown in Fig. 9(b) which is a zoom-in view of Fig. 8(b).The PSR of the autocorrelation is 3.8 dB.The FWHM of the autocorrelation is 0.586 ns, corresponding to a PCR of 4.26.The autocorrelation of the simulated quaternary phase-coded microwave signal is shown in Fig. 9(c).The inset shows the zoom-in view.The waveform used to calculate the autocorrelation is shown in Fig. 9(d), which is a zoom-in view of Fig. 8(d).The autocorrelation has a PSR of 3.01 dB, a FWHM of 0.593 ns, and a PCR of 4.21 dB.The simulated results again match the measured ones.
Finally, we will show that the system is also capable of generating linearly frequencychirped microwave signal.To generate a linearly frequency-chirped microwave signal, the microwave carrier should be quadratic phase-modulated.In this case, the experimental conditions are the same as the generation of quaternary phase-coded microwave signal except that the four-level stair wave coding signal was replaced by a parabolic pulse which is expressed as where K = -β 2 /T 0 2 and β 2 = πV c /V π .V c is the peak-to-peak amplitude of the coding signal.The chirp rate is given by CR = β 2 /(πT 0 2 ) = V c /(V π T 0 2 ). Figure 10(a) shows the electrical parabolic coding signal generated by the AWG with parabolic pulse duration (2T 0 ) of 1.34 ns and a V c of 2.424 V. Thus, the chirp rate is calculated to be 2.37 GHz/ns.The generated linearly frequency-chirped microwave signal is shown in Fig. 10(b).The extracted phase information from Fig. 10(b) is shown in Fig. 10(c).As can be seen, the phase shift is also a parabolic pulse.Figure 10(d) shows the recovered instantaneous frequency shift from Fig. 10(b), which has a chirp rate of 2.42 GHz/ns.This value is very close to the theoretical one.Figure 10(e) shows the simulated linearly frequency-chirped microwave signal based on Eq. ( 7) and the measured electrical parabolic coding signal shown in Fig. 10(a).The extracted phase information and the recovered instantaneous frequency shift are shown in Figs.10(f) and 10(g), respectively.The simulated result shows a chirp rate of 2.44 GHz/ns, which agree well with the calculated result of 2.37 GHz/ns.
The pulse compression capability of the system is also evaluated.The calculated autocorrelation of the measured linearly frequency-chirped microwave signal is shown in Fig. 11(a).The inset shows the zoom-in view.Figure 11(b) shows the waveform used to calculate the autocorrelation.This waveform is the zoom-in view of Fig. 10(b).The PSR of the autocorrelation is 3.7 dB.The FWHM of the autocorrelation is 0.61 ns, corresponding to a PCR of 4.92.The autocorrelation of the simulated frequency-chirped microwave signal is shown in Fig. 11(c), where the inset shows the zoom-in view.The waveform used to calculate the autocorrelation is shown in Fig. 11(d), which is a zoom-in view of Fig. 10(e).The autocorrelation has a PSR of 3.74 dB, a FWHM of 0.606 ns, and a PCR of 4.95 dB.A good match between the simulated and the measured results is achieved.

Conclusion
We have theoretically and experimentally demonstrated a photonic approach to generate arbitrarily phase-modulated microwave signal using a conventional DDMZM.The power of the sinusoidal microwave signal applied to one arm (arm1) of the DDMZM has been optimized to realize carrier-suppressed modulation, while the other of the DDMZM is driven by a coding signal.The phase-modulated optical carrier from the arm2 and the carrier-suppressed optical signal from the arm1 are combined at the of the DDMZM.We have experimentally generated binary phase-coded microwave pulses with a 13-bit Barker code pattern at the carrier frequency of GHz and a bit rate of 3.75 Gb/s, quaternary phase-coded microwave signal at the carrier frequency of 10 GHz and a bit rate of 2 Gb/s, and linearly frequency-chirped microwave signal at the carrier frequency of 10 GHz and a chirp rate of 2.42 GHz/ns.The pulse compression capability of the system has also been evaluated for different kinds of phase-modulated microwave signals.In addition, the phasemodulated microwave signals have also been simulated.The simulated results agree very well with the measured ones.
In our experiment, each type of the phase modulated microwave signal was generated at one carrier frequency.However, it is worth noting that the carrier frequency is widely tunable without affecting the phase modulating performance.Moreover, it is noted that the phase modulated optical carrier and the first-order sideband are selected by the OBPF in our experiments.The beating between the optical carrier and the first-order sideband generates the fundamental microwave signal.If a Wave Shaper is used instead of the OBPF to select the phase modulated optical carrier and the higher-order sideband, phase modulated microwave signal with frequency multiplying can be generated.Since an optical filter is involved in our approach, the proposed configuration is not transparent to the optical carrier wavelength.For optical carrier at other wavelength, the center frequency of the optical filter has to be tuned.

Fig. 3 .
Fig.3.Measured optical spectrum at the output of the DDMZM and the simulated optical spectrum at the output of the arm1 of the DDMZM.The frequency of the microwave signal driven to the arm1 is 15 GHz and the coding signal applied to the arm2 is a 13-bit Barker code.

Fig. 4 .
Fig. 4. (a) The electrical 13-bit Barker coded signal applied to the arm2 of the DDMZM.(b) The measured binary phase-coded microwave pulse.(c) The simulated binary phase-coded microwave pulse.(d) The phase information extracted from Fig. 4(b).(e) The phase information extracted from Fig. 4(c).(f) The measured binary phase-coded microwave pulses in a time duration of 70 ns.

Fig. 5 .
Fig. 5. Measured electrical spectrum of the generated binary phase-coded microwave pulse.

Fig. 6 .
Fig. 6.(a) Autocorrelation of the measured 13-bit Barker coded microwave pulse shown in Fig. 4(b).Inset shows a zoom-in view of the autocorrelation.(b) Autocorrelation of the simulated 13-bit Barker coded microwave pulse shown in Fig. 4(c).Inset shows a zoom-in view of the autocorrelation.

Fig. 7 .Fig. 8 .
Fig.7.Measured optical spectra at the output of the DDMZM before filtering, after filtering by the OBPF, and the response of the OBPF as well as the simulated optical spectrum at the output of the arm1 of the DDMZM.The frequency of the microwave signal applied to the arm1 of the DDMZM is 10 GHz.The coding signal is a four-level stair wave signal.

Figure 8 (
Figure 8(a) shows the electrical four-level stair wave coding signal generated by the AWG.The measured quaternary phase-coded microwave signal is shown in Fig. 8(b) and the phase information extracted from Fig. 8(b) using Hilbert transform is shown in Fig.8(c).The phase shift has four levels of 0, 1.15, 2.23, and 3.34 rad.Figure8(d)shows the simulated quaternary phase-coded microwave signal based on Eq. (7) and the measured electrical fourlevel stair wave coding signal shown in Fig.8(a).The extracted phase information from Fig.8(d) is shown in Fig.8(e).As can be seen, the simulated results fit well with the measured ones.

Fig. 9 .
Fig. 9. (a) Autocorrelation of the measured quaternary phase-coded microwave signal.Inset shows a zoom-in view of the autocorrelation.(b) The zoom-in view of Fig. 8(b) used to calculate the autocorrelation in Fig. 9(a).(c) Autocorrelation of the simulated quaternary phase-coded microwave signal.Inset shows a zoom-in view of the autocorrelation.(d) The zoom-in view of Fig. 8(d) used to calculate the autocorrelation in Fig. 9(c).

Fig. 10 .
Fig. 10.(a) The electrical parabolic coding signal generated by the AWG.(b) The measured linearly frequency-chirped microwave signal.(c) The phase information extracted from Fig. 10(b).(d) The recovered instantaneous frequency shift from Fig. 10(b).(e) The simulated linearly frequency-chirped microwave signal.(f) The phase information extracted from Fig. 10(e).(g) The recovered instantaneous frequency shift from Fig. 10(e).

Fig. 11 .
Fig. 11.(a) Autocorrelation of the measured frequency-chirped microwave signal.Inset shows a zoom-in view of the autocorrelation.(b) The zoom-in view of Fig. 10(b) used to calculate the autocorrelation in Fig. 11(a).(c) Autocorrelation of the simulated frequency-chirped microwave signal.Inset shows a zoom-in view of the autocorrelation.(d) The zoom-in view of Fig. 10(e) used to calculate the autocorrelation in Fig. 11(c).