sensors Article
Multi-Target Detection Method Based on Variable Carrier Frequency Chirp Sequence Wei Wang 1,2,3 , Jinsong Du 1,2, * and Jie Gao 1,2,3 1 2 3
*
Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China;
[email protected] (W.W.);
[email protected] (J.G.) Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences, Shenyang 110016, China University of Chinese Academy of Sciences, Beijing 100049, China Correspondence:
[email protected]; Tel.: +86-24-8360-1084
Received: 1 September 2018; Accepted: 5 October 2018; Published: 10 October 2018
Abstract: Continuous waveform (CW) radar is widely used in intelligent transportation systems, vehicle assisted driving, and other fields because of its simple structure, low cost and high integration. There are several waveforms which have been developed in the last years. The chirp sequence waveform has the ability to extract the range and velocity parameters of multiple targets. However, conventional chirp sequence waveforms suffer from the Doppler ambiguity problem. This paper proposes a new waveform that follows the practical application requirements, high precision requirements, and low system complexity requirements. The new waveform consists of two chirp sequences, which are intertwined to each other. Each chirp signal has the same frequency modulation, the same bandwidth and the same chirp duration. The carrier frequencies are different and there is a frequency shift which is large enough to ensure that the Doppler frequencies for the same moving target are different. According to the sign and numerical relationship of the Doppler frequencies (possibly frequency aliasing), the Doppler frequency ambiguity problem is solved in eight cases. Theoretical analysis and simulation results verify that the new radar waveform is capable of measuring range and radial velocity simultaneously and unambiguously, with high accuracy and resolution even in multi-target situations. Keywords: multi-target detection; continuous wave radar systems; variable carrier frequency chirp sequence; Doppler ambiguity
1. Introduction With the development of electronic technology, the application field of radar has expanded to the fields of intelligent transportation systems [1,2], vehicle assisted driving [3,4], positioning and navigation [5], health monitoring [6], pedestrian and obstacle detection [7,8], etc. The selection of waveform type and signal processing technology depends to a great extent on the specific tasks and functions of radar. Civilian fields, such as intelligent transportation and automotive collision detection, require radar to have the characteristics of simple structures, low cost and high integration [9,10], thus the continuous wave system is the preferred. There are several continuous waveform (CW) proposals which have been developed in the last years for new radar systems to measure range and radial velocity simultaneously. Linear Frequency Modulation (LFM) [11,12] and Frequency Shift Keying (FSK) [13] which are shown in Figure 1 are two typical waveforms.
Sensors 2018, 18, 3386; doi:10.3390/s18103386
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be distinguished by phase information, which is the inherent defect of FSK waveform. The combination of FSK and LFM waveform design principle offers the possibility of an unambiguous target range and velocity measurement simultaneously [14–17]. Multi Frequency Shift Keying (MFSK) consist in this case of two linear frequency modulated up-chirp signals. However, this Sensors 2018, suffers 18, 3386 from low estimation accuracy due to the phase measurement. 2 of 12 technique
(a)
(b)
Figure Figure 1. 1. Two Two typical typical waveforms waveforms (a) (a) Linear Linear Frequency Frequency Modulation Modulation (LFM) (LFM) signal signal and and (b) (b) Frequency Frequency Shift Shift Keying (FSK) signal.
The received signal LFMare is directly to baseband the transmitted signal. Therefore, Chirp Sequence (CS)of radars gainingmixed popularity because by of their inherent ability to extract the the resulting baseband beat frequency f B describes the difference betweenbased the transmitted frequency range and velocity parameters of multiple targets from the beat signals on two-dimensional andFourier the received frequency. the situation of stationary the also propagation delay leads to fast transform [18–21].InHowever, the chirp sequence target, waveform has a drawback; that is, frequency shiftunambiguously f R , and the beatmeasurable signal frequency is influenced target distance R only. However, the maximum velocity is limited by the repetition interval of the chirps. whenDoppler the target is moving,problem the beat signal f R and Doppler The ambiguity occursfrequency when thedepends velocityonofthea target targetrange exceeds thethe maximum frequency f Dvelocity , as shown Equation (1). unambiguous frequency can be increased by increasing the measurable [22].inThe maximum chirp repetition rate of the chirp sequence (reducing the duration of the chirp sequence). The signal B 2 2 needs to meet a certain number of sampling so the sampling R + vfrequency should be increased, and (1) f B = f R −points, fD = − Tchirp c λ the hardware should meet the data processing rate. Therefore, the above methods still cannot effectively solve problem.modulation Multiple approaches have developed to modulation resolve Doppler ambiguity wherethe B iscurrent the frequency bandwidth, T been is the frequency period and the chirp
distance and speed of the target can be solved by the up beat frequency and the down beat frequency. However, in the case of multiple targets, there are many errors in combinations of up beat frequencies and down beat frequencies. In order to solve the distance-velocity coupling problem, multiple sets of chirped continuous waves with different frequency modulations are needed. The extended measurement time is an important drawback of this LFM technique. FSK system transmits two discrete frequencies f 1 and f 2 sequentially with a duration of TCPI . The frequency step between the two carriers is represented as f Step = f 2 − f 1 . The received signal is converted to baseband signal by a homodyne receiver. The baseband outputs carry the Doppler frequencies generated by the moving target. By maintaining f Step very small in comparison to the transmitted signals, the Doppler frequencies extracted from the baseband outputs will be approximately the same. The peak phase difference is used to estimate the distance, as shown in Equation (2). R=−
c · ∆ϕ 4π · f Step
(2)
It can be realized easily, which is the advantage of FSK waveform. However, when multiple targets are moving at the same speed or when multiple targets are stationary, multiple targets cannot be distinguished by phase information, which is the inherent defect of FSK waveform. The combination of FSK and LFM waveform design principle offers the possibility of an unambiguous target range and velocity measurement simultaneously [14–17]. Multi Frequency Shift Keying (MFSK) consist in this case of two linear frequency modulated up-chirp signals. However, this technique suffers from low estimation accuracy due to the phase measurement. Chirp Sequence (CS) radars are gaining popularity because of their inherent ability to extract the range and velocity parameters of multiple targets from the beat signals based on two-dimensional fast Fourier transform [18–21]. However, the chirp sequence waveform also has a drawback; that is, the maximum unambiguously measurable velocity is limited by the repetition interval of the chirps. The Doppler ambiguity problem occurs when the velocity of a target exceeds the maximum measurable velocity [22]. The maximum unambiguous frequency can be increased by increasing the chirp repetition rate of the chirp sequence (reducing the duration of the chirp sequence). The signal needs to meet a certain number of sampling points, so the sampling frequency should be increased, and the hardware
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[23–26]. A modified chirp sequence waveform was developed by adding an additional frequency shift between every two adjacent chirps [23,24]. Doppler ambiguities can be resolved using the Sensors 2018, 18, 3386 3 of 12 additional phase information introduced by the frequency shift. Frequency shift is a key factor in determining the phase difference. It is difficult to maintain high-precision frequency shift. The frequency between every rate. two adjacent is changed delay. Theeffectively delay of solve the new should meetshift the data processing Therefore,chirps the above methodstostill cannot the improved waveform can beapproaches precisely controlled bydeveloped the systemtoclock. TheDoppler waveform implementation current problem. Multiple have been resolve ambiguity [23–26]. is modified simpler. chirp Moreover, the waveform accuracy of thedeveloped resolved by unambiguous Doppler frequency frequencies is between not well A sequence was adding an additional shift guaranteed since phase information is susceptible to noise interference. every two adjacent chirps [23,24]. Doppler ambiguities can be resolved using the additional phase Therefore, the principle waveform design for continuous wave radar the following information introduced by theof frequency shift. Frequency shift is a key factor in includes determining the phase points: difference. It is difficult to maintain high-precision frequency shift. The frequency shift between every two chirps is changed to delay. The delay for of the new improved waveform precisely (1) adjacent Waveforms must have probing capabilities multi-target scenes to meet can the be application controlled by the system clock. transportation, The waveform automotive implementation is simpler. Moreover, the accuracy requirements of intelligent collision detection, etc. of the resolved unambiguous Doppler frequencies is not well guaranteed since phase information is (2) Under current conditions, it is difficult to meet the accuracy requirements by extracting the susceptible to noise interference. phase information to solve the parameters such as velocity and range. Waveforms must be Therefore, the and principle waveform design for continuous wave radar includes the simple in form easy to of implement in hardware. following points: (3) It is not possible to increase the computational complexity and increase the cost of hardware
because of the special signal form.capabilities for multi-target scenes to meet the application Waveforms must have probing requirements of intelligent transportation, automotive collision detection, The main contribution of this paper is to design a waveform that satisfiesetc. the above principles. (2) Under current conditions, it isisdifficult to meet thechirp accuracy requirements by extracting the phase The new radar waveform, which improved on the sequence, is capable of measuring range information to simultaneously solve the parameters such as velocitywith and high range.accuracy Waveforms be simple and radial velocity and unambiguously, and must resolution even in in form and easy to implement in hardware. multi-target situations. The remainder of this paper is organized as follows. Section 2 introduces the chirpItsequence waveform and its the corresponding signal model, and analyzes (3) is not possible to increase computational complexity and increasethe theDoppler cost of ambiguity hardware problem. In Section 3, the variable carrier frequency chirp sequence waveform and its corresponding because of the special signal form. signal processing method are proposed. Simulation results, which are shown in Section 4, verify that The main of this paper is to design a waveform that satisfies the above principles. the new radar contribution waveform has the adaptability for multi target situations. Section 5 provides some The new radar waveform, which is improved on the chirp sequence, is capable of measuring conclusions. range and radial velocity simultaneously and unambiguously, with high accuracy and resolution even in multi-target situations. The remainder of this paper is organized as follows. Section 2 2. Chirp Sequence Waveform introduces the chirp sequence waveform and its corresponding signal model, and analyzes the Doppler The classical chirp sequence waveform shownfrequency in Figurechirp 2. The waveform consistsand of its L ambiguity problem. In Section 3, the variableiscarrier sequence waveform equispaced chirps. Each single chirp has a are duration of Tp .Simulation This baseband signal is described by the corresponding signal processing method proposed. results, which are shown in following equation: Section 4, verify that the new radar waveform has the adaptability for multi target situations. Section 5 provides some conclusions. s t , l exp j 2 f t f l T (3) (1)
B
D
p
where describes the chirp number. The parameter describes the phase. The beat 2. Chirpparameter Sequence lWaveform f B contains f D (see Equation frequency the target range isf R shown and Doppler frequency (1)). The classical chirp both sequence waveform in Figure 2. The waveform consists of L
This baseband signalsingle is sampled and FFTofprocessed each individual chirp signal, s t , l chirp equispaced chirps. Each has a duration Tp . This for baseband signal is described by the following equation: which splits the echo signal into N FFT adjacent range gates. Then the Doppler FFT is performed in s t, l = exp j2π f · t − f · l · T + ϕ (3) ( ) p B stored D as a matrix S each range gate. The range–Doppler spectrum is of N rows and L RD
FFT
FFT
N FFTtheand columns. The parameter thenumber. parameter represent the FFT length. The process is where parameter l describes chirp TheLFFT parameter ϕ describes the phase. The beat frequency f B contains shown in Figure 2. both the target range f R and Doppler frequency f D (see Equation (1)).
Tp
B
FFT
FFT
FFT
FFT Range
FFT FFT
FFT FFT Doppler
Figure 2. 2D-FFT processing based on chirp sequence.
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This signal s(t, l ) is sampled and FFT processed for each individual chirp signal, which Sensors 2018,baseband 18, x FOR PEER REVIEW 4 of 13 splits the echo signal into NFFT adjacent range gates. Then the Doppler FFT is performed in each range gate. The range–Doppler is stored as a matrix SRDsequence. of NFFT rows and L FFT columns. Figure spectrum 2. 2D-FFT processing based on chirp The parameter NFFT and the parameter L FFT represent the FFT length. The process is shown in The Figure 2. target information can be extracted from the matrix S RD . The beat frequency f B and the The frequency target information can be in extracted from the matrix SRD . The beat frequency f B and the f D are shown Doppler the following equation: Doppler frequency f D are shown in the following equation: B 2 v fB fR fD R2 (4) Tp c B 2 v fB = fR − fD = − ·R+2 (4) v λ f D 2 Tp c (5)
Based on this chirp sequence waveform, radar targets v are resolved in range and in radial velocity (5) f D = −2 separately. λ c Based on this chirp sequence waveform, radarTptargets are resolved in range and in radial (6) R fB fD B 2 velocity separately.
v fD (7) Tp c R = −( f B +2f D ) · (6) B 2aliasing occurs when the sampling rate According to the Shannon/Nyquist sampling theorem, λ is less than two times the maximum frequency. the Doppler ambiguity problem will occur (7) v =Therefore, − fD 2 when the chirp repetition rate of the chirp sequence waveform is lower than two times that of the According to thefrequency. Shannon/Nyquist sampling theorem, aliasing occurs when the ratethe is maximum Doppler The maximum unambiguous Doppler frequency is sampling denoted by less than two times the maximum frequency. Therefore, the Doppler ambiguity problem will occur following equation: when the chirp repetition rate of the chirp sequence waveform is lower than two times that of the f D ,max f r unambiguous 2 1 2 Tp (8) maximum Doppler frequency. The maximum Doppler frequency is denoted by the following equation: The Doppler frequency detected by the range–Doppler spectrum is f D ,amb , and the actual f D,max = f r /2 = 1/ 2 · Tp (8) Doppler frequency f D can be obtained by adding a multiple q of the maximum unambiguous The Doppler frequency detected by the range–Doppler spectrum is f D,amb , and the actual Doppler Doppler frequency 2 f D ,max to f D ,amb : frequency f D can be obtained by adding a multiple q of the maximum unambiguous Doppler frequency 2 f D,max to f D,amb : f D f D , amb q 2 f D ,max (9) f D = f D,amb + q · (2 f D,max ) (9)
where represents the number of Doppler frequency aliasing. where q qrepresents the number of Doppler frequency aliasing. The unknown Doppler frequency measurement, as The unknown factor factor of of qqrepresents representsthe theambiguity ambiguityofofthe the Doppler frequency measurement, depicted in in Figure 3. 3.Assuming that the repetition repetition as depicted Figure Assuming thatthe theduration durationofofthe thechirp chirpsequence sequence isis 0.1 0.1 ms, ms, the f frequency of the chirp signal = 1000 Hz. This means that for the slow time domain, the sampling frequency of the chirp signal f r r= 1000 Hz. This means that for the slow time domain, the sampling frequency is Doppler frequency rangerange by theby second FFT is from Hz to−500 frequency is 1000 1000Hz. Hz.TheThe Doppler frequency the second FFT −500 is from 500Hz. HzWe to f assumed that the carrier frequency is 24 GHz. According to the relationship between velocity and 500 Hz. We assumed that the carrier frequency f 0 is 24 GHz. According to the relationship between 0 velocity Dopplerthe frequency, theunambiguous maximum unambiguous can be calculated the Dopplerand frequency, maximum velocity canvelocity be calculated using the using following following equation: equation: f D,max c vunambmax =f − c (10) D ,max 2 f 0 vunamb max = (10) 2 f0
Figure 3. 3. Doppler Doppler ambiguity ambiguity schematic. schematic. Figure
The unambiguous velocity is from −11.25 km/h to 11.25 km/h. Obviously, it does not meet the needs of intelligent transportation and automotive collision avoidance. The maximum unambiguous frequency can be increased by increasing the chirp repetition rate of the chirp sequence (reducing the duration of the chirp sequence). Assuming that the duration of the chirp sequence is 0.01 ms, the
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The unambiguous velocity is from −11.25 km/h to 11.25 km/h. Obviously, it does not meet the maximum unambiguous frequencyand is 5000 Hz. Thecollision maximum unambiguous velocity is from −112.5 needs of intelligent transportation automotive avoidance. The maximum unambiguous km/h to 112.5 km/h. The signal needs to meet a certain number of sampling points, so the sampling frequency can be increased by increasing the chirp repetition rate of the chirp sequence (reducing frequency should be increased, and the hardware should meet the data processing rate. Therefore, the duration of the chirp sequence). Assuming that the duration of the chirp sequence is 0.01 ms, the above methods still cannot effectively solve problem. Doppler velocity ambiguity is a the maximum unambiguous frequency is 5000 Hz.the Thecurrent maximum unambiguous is from significant disadvantage of the chirp sequence. −112.5 km/h to 112.5 km/h. The signal needs to meet a certain number of sampling points, so the sampling frequency should be increased, and the hardware should meet the data processing rate. 3. Variablethe Carrier Chirp Sequence Therefore, aboveFrequency methods still cannot effectively solve the current problem. Doppler ambiguity is a significant of the chirp sequence. The newdisadvantage waveform consists of two chirp sequences, which are intertwined to each other. The
length of the two intertwined waveforms is 2 LTp . Each chirp signal has the same frequency 3. Variable Carrier Frequency Chirp Sequence modulation, the same bandwidth B and the same chirp duration T p . The carrier frequencies are The new waveform consists of two chirp sequences, which are intertwined to each other. f 01 and f 02 respectively. The frequency shift f shift is the difference between frequency f 01 and The length of the two intertwined waveforms is 2LTp . Each chirp signal has the same frequency f shift =same f 02 fbandwidth f shift isfrequencies f 02 , that is, the ).and In the frequency shiftcarrier so small that modulation, thepaper same [24], chirpthe duration Tp . The arethe f 01 01 ( f 02 f 01B and frequency f 02 respectively. The frequency shift f shi f tDoppler is the difference between frequency f 02 , thatby is, beat measurement and ambiguous frequency measurement are fnot influenced 01 and f shi f frequency = f − f ( f > f ). In the paper [24], the frequency shift f is so small that the beat this shift waveform parameter. The frequency shift in this paper is large enough to ensure 02 02 01 01 t shi f t frequency measurement and in ambiguous Doppler frequency measurement are not influenced by this that the Doppler frequencies the two-chirp sequence for the same moving target are different. frequency shift waveform Thebyfrequency shift equations: in this paper is large enough to ensure that The baseband signals parameter. are described the following the Doppler frequencies in the two-chirp sequence for the same moving target are different. s1 t, l exp j 2following f B1 t f Dequations: (11) 1 2l Tp 1 The baseband signals are described by the
s2 t, l exp j 2 f B 2 t f D 2 2l 1 Tp 2
s1 (t, l ) = exp j2π f B1 · t − f D1 · 2l · Tp + ϕ1
(11) (12)
s2 (t, l ) = exp j2π f B2 · t − f D2 · (2l + 1) · Tp + ϕ2 (12) The two time-discrete signals are processed by a first FFT to calculate the beat frequency f B Thea two time-discrete signalsthe areDoppler processed by a first FFT the beat frequency f B and f D , ambto. calculate and by second FFT to measure frequency The range–Doppler spectrums are by a second FFT to measure the Doppler frequency f D,amb . The range–Doppler spectrums are stored as stored as matrix S RD1 and S RD 2 . The process flow of the variable carrier frequency chirp sequence matrix SRD1 and SRD2 . The process flow of the variable carrier frequency chirp sequence is similar to is similar the chirp shown the chirp to sequence, assequence, shown in as Figure 4. in Figure 4.
Tp
B
f 02 f 01
f FFT
FFT
FFT
FFT
FFT
FFT
Range
FFT FFT FFT FFT
Range
FFT FFT FFT FFT Doppler
Figure Figure 4. 4. 2D-FFT 2D-FFT processing processing based based on on variable variable carrier carrier frequency frequency chirp sequence.
For the the range–Doppler spectrum SSRD1 , For RD1 f B1 = − f B1f R − f R fD1f D= 1
BB 22 vv · R R +22 TTpp cc λ1 1
(13) (13)
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f D1 = f D1,amb + q1 · (2 f D,max ) = −2
v v = −2 f 01 λ1 c
(14)
For the range–Doppler spectrum SRD2 , f B2 = f R − f D2 = −
B 2 v ·R+2 Tp c λ2
f D2 = f D2,amb + q2 · (2 f D,max ) = −2
(15)
v v = −2 f 02 λ2 c
(16)
For the same moving target, the Doppler frequencies in the two-chirp sequence are different due to the different carrier frequencies. v v ∆ f D = f D2 − f D1 = − 2 ( f 02 − f 01 ) = −2 f shi f t c c
(17)
For the target P, the row and column indices of the matrix SRD1 and the matrix SRD2 are (n1P ,l1P ) and Where n1P and n2P correspond to the beat frequencies f B1 and f B2 , l1P and l2P correspond to the Doppler frequencies f D1,amb and f D2,amb (possibly frequency aliasing). The Doppler frequencies f D1 and f D2 are very close, so the influence on the beat frequencies f B1 and f B2 is small. The frequency resolution of the range domain is on the order of 102 Hz. Therefore, n2P is usually equal to n1P . The frequency resolution of the Doppler domain is on the order of 0.1 Hz. Then, l1P and l2P show differences. The velocity of the target can be calculated through the following equation:
(n2P ,l2P ).
v = −[ f D2,amb − f D1,amb + (q2 − q1 )(±2 f D,max )]c/ 2 f shi f t
(18)
where f D,max = 1/ 2 · 2Tp represents the largest unambiguous frequency, because the repetition frequency of the new signal f r is 1/ 2Tp . The symbol of f D,max is determined according to the aliasing direction. q2 and q1 indicate the number of Doppler frequency aliasing. According to the sign and numerical relationship of the Doppler frequency f D1,amb and f D2,amb , the calculation of the Doppler frequency difference is divided into eight cases, which are shown in Table 1. Table 1. Discussion on the Doppler frequency difference. Sign
Numerical Relationship
Doppler Frequency Difference
Velocity
Case
f D1,amb ≥ 0 f D2,amb ≥ 0
f D1,amb ≤ f D2,amb f D1,amb > f D2,amb
q2 = q1 ,∆ f D = f D2,amb − f D1,amb q2 = q1 ,∆ f D = f D2,amb − f D1,amb
v≤0 v>0
1 2
f D1,amb < 0 f D2,amb ≤ 0
f D1,amb < f D2,amb f D1,amb ≥ f D2,amb
q2 = q1 ,∆ f D = f D2,amb − f D1,amb q2 = q1 ,∆ f D = f D2,amb − f D1,amb
v0
3 4
f D1,amb ≥ 0 f D2,amb ≤ 0
f D1,amb − f D2,amb > f D,max f D1,amb − f D2,amb < f D,max
q2 = q1 + 1,∆ f D = f D2,amb + 2 f D,max − f D1,amb q2 = q1 ,∆ f D = f D2,amb − f D1,amb
v0
5 6
f D1,amb ≤ 0 f D2,amb ≥ 0
f D2,amb − f D1,amb > f D,max f D2,amb − f D1,amb < f D,max
q2 = q1 + 1,∆ f D = f D2,amb + (−2 f D,max ) − f D1,amb q2 = q1 ,∆ f D = f D2,amb − f D1,amb
v>0 v f D1 > 0. simulation results verify of the new waveform． When the velocity v < 0, then f D2 < f D1 < 0.
(a)
(b)
(c)
(d) Figure 8. Cont.
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(e)
(f)
(g)
(h)
Figure 8. The Doppler spectrum for 8 typical cases. (a) f D1,amb ≥ 0, f D2,amb ≥ 0, f D1,amb ≤ f D2,amb 0 , f D 2,≤ f Df 2, amb (b) The for>8 typical cases. (a) f 0 ,≤ f0, (d) f Df D1,amb f D2,amb f D1,amb f D1,amb 1, amb 0 f0D,max f ≤ f0, f D2,amb 0 , −f f D2,amb f 0 , f 0< , f (h)f D1,amb (g) ≥f 0, f D1,amb , f f D,max. f (h) f D 2, amb
D1, amb
D 2, amb
D ,max
D1, amb
D 2, amb
D1, amb
D 2, amb
D ,max
D1, amb
f D,max , f Dsigns . f f D 2, amb 0the 1, amb fof D 2, amb Although f D1,amb and D2,amb may be different, the numerical relationship of f D1,amb and f D2,amb is consistent with the numerical relationship of f D1 and f D2 . Normally, the number of 5. Conclusions Doppler frequency aliasing is q2 = q1 . The velocity the target (such as vehicles) is less than m/s.waveform According to Equation (19), This paper of focuses on the research of continuous wave50radar design, analyzes the the Doppler frequency difference ∆ f is less than 50 Hz. Two special cases shown in two-dimensional FFT processing method and the Doppler ambiguity problem for chirp sequence. Figure 8e,g. proposes When f aD1,amb f D2,amb which > f D,max > 0,chirp f D2,amb < 0) or which f D2,ambare − fintertwined ( f D1,amb D1,amb > This paper new − waveform consists of two sequences, f D,max 0, f D1,amb 0), it has means number modulation, of Doppler frequency aliasing q2 and 6= qthe ( f D2,amb 1. to each other.>Each chirp
