PD Radar MTs and STs range and speed with sign 2 Triangle Wave FMCW MTs and STs range and speed with sign 2

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Lenard Watson
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1 I. Background Continuous Wave (CW) radar is coherent radar system, which coupling part of TX power as RX LO. CW radar typical is used to detect target speed (without or with sign), range. They are popular used in max 100m to 2km range detection with 1m-range resolution. The first generation CW radar only detect moving or static target. The second generation CW radar can detect both moving and static radar. The third generation CW radar has ability to detect angle. (This paper does not mention it) This paper is on mathematic formulas of radar. Radar Type Radar function Generation Doppler Moving target (MT) Speed without sign 1 I/Q Doppler MT Speed with sign 1 PD Doppler MT range and speed with sign 1 Saw-tooth FMCW Static Target (ST) range only 1 PD Radar MTs and STs range and speed with sign 2 Triangle Wave FMCW MTs and STs range and speed with sign 2 MP-TW-FMCW MTS and STs angle, range, sign speed 3 MP-PD radar MTS and STs angle, range, sign speed 3 DBF-TW-FMCW MTS and STs angle, range, sign speed 3 DBF-PD radar MTS and STs angle, range, sign speed 3 II. Mathematic Formula of Electron-magic Wave All traveling electron-magic wave at time-space (t, r,, ) is in the form of: A*EXP [j( *t + k*r*cos( )*cos( )] We only analyze radar beam points to target at range d, and then it is: A*EXP [j( *t + k*d] Where =2 *f is angle frequency of carrier f, k=2 / = /C is wave number =C/f is wavelength of carrier frequency f C is light speed (3*10^8m/s) A is amplitude, most analyzes normalize to 1. k*d is phase at d point Page 1 of 6

2 III. Doppler radar Doppler radar RX couples part of TX power as LO. LO position is at d = 0. Moving target at range R has added Doppler frequency to carrier in return RF signal. Round trips of radar wave (from radar to target and reflected to radar) distance d is 2R. Then: LO: EXP [j( *t + k*0)] RF: EXP [j( + d)*t - *(2R/C)] IF: EXP [j( d*t )] * EXP[-j* *(2R/C)] ( d>0) EXP [j( d*t )] * EXP[+j* *(2R/C)] ( d<0) Analogy RX only sends positive frequency out. Doppler radar only report moving target speeds, but has no ability to report moving direction (approaching or receding). IV. I/Q Doppler radar There are typical 3 kinds of I/Q Doppler RX. 1) Add 90degree delay line at LO of Q-CH; 2) add 90degree delay line at RF of Q-CH; 3) add one 45degree delay line at LO of Q-CH and another 45degree delay line at RF of I-CH. Approach 3 is better for RF circuit board design. It is popular used when insertion loss of 45-degree delay line is less than 0.1dB. I-CH LO: EXP [j( *t )] Q-CH LO: EXP [j( *t - /4)] I-CH RF: EXP [j( + d)*t - *(2R/C)- /4] Q-CH RF: EXP [j( + d)*t - *(2R/C)] Then when d > 0 I-CH IF: EXP [j( d*t)]*exp[-j( *(2R/C)-j /4] Q-CH IF: EXP [j( d*t)]*exp[-j( *(2R/C)+j /4] I-CH phase substrate Q-CH phase is - /2 And when d<0 I-CH IF: EXP [j( d*t)]*exp[j( *(2R/C)+j /4] Q-CH IF: EXP [j( d*t)]*exp[j( *(2R/C)-j /4] I-CH phase substrate Q-CH phase is /2 It is clearly to see that I/Q Doppler radar detects moving target speed by checking return frequency and does direction by comparing 2 CH phase difference. Anyway, one carrier frequency only detects moving target speed parameters. Not target range parameter. Page 2 of 6

3 V. PD Doppler radar Phase Detect (PD) Doppler radar sends two carrier frequencies in time period 2T. That is radar sends and receives F1 in time period (0,T) and F2 in (T, 2T). T should be selected carefully so that target speed in 2T time period is processed as constant, and the range gap dr=vt is smaller than designed range resolution dr, for example, 1/10 th of dr in engineering design. On the other hand, T must be larger than maximum delay time max=2rmax/c. In other words, RX gets right Doppler frequency is in time period T- max (which >2 max, or T>=3 max) Follow analyzes under condition d>0 (same result for d<0 ) and F2>F1, df=f2-f1 At time period (0, T) LO1: RF1: IF1: EXP [j( 1*t)] EXP [j( 1+ d)*t ( 1+ d)*(2r/c)] EXP [j( d*t )] * EXP[-j*( 1+ d)*(2r/c)] At time period (T, 2T) LO2: EXP [j( 2*t)] RF2: EXP [j( 2+ d)*t ( 2+ d)*(2r/c)] IF2: EXP [j( d*t )] * EXP[-j*( 2+ d)*(2r/c)] Phase difference between two time-slots dp at carrier Fd is: P1-P2=( 2-1)*(2R/C)] or dp=2 *df*(2r/c) Since maximum dp range is limited from 0 to, radar is designed to detect targets at max range Rmax with max Vmax in 2T time. The design formulas for PD Doppler radar are: PD Doppler Radar equation Range Resolution Limitation condition Maximum IF frequency: 4dF*Rmax < C dr = [C/(4 *df)]* P 3*(2Rmax/C)<T <=(dr/vmax)/10 Fd(max) = 2Vamx*fc2/C Where P is phase resolution of DSP. One 24.1GHz PD Doppler radar design is used as example, which has Rmax =1 mile (1610m), Vmax =100 m/sec. P is from 0.2degrre to 1 degree. Let T has at least 3 times of maximum delay time, and target position shift during is 1/10 th of range resolution Then df<c/(4rmax) [=3*10^8/(4*1610)]=46.58KHz. 40 KHz is a reasonable design data. Page 3 of 6

4 Range resolution at df=40khz is close to 10.4 P, or 2.08m~10.04m. Maximum Doppler frequency is KHz. T is at time range from 32.2us to 2080us. I/Q PD Doppler is used to get moving target range, speed and speed direction parameter. Above 3 kinds of radar system are the first generation CW radar. Because of Doppler radar does not have ability to detect static target. VI. PD Radar The difference between PD radar and PD Doppler radar is that LO of PD radar is not coupling from TX. Its LO is coherent with TX frequency with fixed gap dfc1, and dfc2. Let F3 is LO frequency, and F2>F1 are two TX frequency sends out in adjacent time slot T. Let frequency design meets df1=f3-f1 > df2=f3-f2 > Fdmax. At time period (0, T) LO: EXP [j( 3*t)] RF1: EXP [j( 1+ d)*t ( 1+ d)*(2r/c)] IF1: EXP [j2 *(df1-fd)*t] * EXP [j*( 1+ d)*(2r/c)] At time period (T, 2T) LO: EXP [j( 3*t)] RF2: EXP [j( 2+ d)*t ( 2+ d)*(2r/c)] IF2: EXP [j2 *(df2-fd)*t * EXP [j*( 2+ d)*(2r/c)] PD radar gets moving and static targets information either from IF1 or IF2. No I/Q CH required. Compare phase of IF1 at df1-fd carrier with phase of IF2 at df2-fd carrier. The phase gap is: dp=2 *df*(2r/c) The design formulas of PD radar are: Where PD Radar equation Range Resolution Limitation condition Maximum IF frequency: 4dF*Rmax < C dr = [C/(4 *df)]* P 3*(2Rmax/C)<T <=(dr/vmax)/10 Fd(max) = 2Vamx*fc2/C df=f2-f1 df1=f3-f1 > df2=f3-f2>fdmax Still use 24.1GHz PD radar as example, which has: Rmax = 1mile(1610m), Vmax = 100m/sec. P is from 0.2degrre to 1 degree. Page 4 of 6

5 Let T has at least 3 times of maximum delay time, and target position shift during is 1/10 th of range resolution Then df<c/(4rmax) [=3*10^8/(4*1610)]=46.58KHz. 40 KHz is a reasonable design data. Range resolution at df=40khz is close to 10.4 P, or 2.08m~10.04m. Maximum Doppler frequency is KHz. F1= GHz, then F2= GHz, and F3>= GHz, RX band is from 1KHz to 57KHz. T is at time range from 32.2us to 2080us. VII. Saw-tooth Modulated FMCW radar Frequency Modulation Continuous Wave (FMCW) radar is coherent radar. Saw-tooth is one of modulation, which is used for static or slower moving targets. At frequency-time plane, saw-tooth is: ft(t)=fs+kt (nt<t<(n+1)t, n=0, 1, 2, ) Where: fs is start carrier frequency, k=b/t is the slop of saw, B is swept band B=fu-fs, fu is stop frequency T is the time period of saw wave Only consider n=0, the return signal fr(t): fr(t)=fs+k(t- ) Where: Or =2R/C is the time delay from target R is the distance of target C is light velocity 3x10^8m/s LO: RF: IF: EXP [j2 (fs+k*t)] EXP [j2 (fs+k*(t- )+fd]t j2 (fs+k*(t- )+fd)*(2r/c)] EXP [j(2 ( k* +fd)*t] * EXP [j2 (fs+k*(t- )+fd)*(2r/c)] The beat signal is: fb=k* +fd or fb=(2b/c) * (R/T) +fd Range is know as: R=(C/2B) T * (fb fd) The range resolution is know as: R=C/2B Saw tooth FMCW only works in maximum Doppler frequency Fdmax is much less than modulation frequency Fm (Fm= 1/T), and no speed testing. Page 5 of 6

6 VIII. Triangle Wave FMCW radar For moving target, Doppler frequency fd is included in return signal. Triangle-wave is used to pick up range and speed 2 parameters by 2 slops of triangle wave. Triangle wave is formed by two saw-tooth wave. Each of two saw-tooth wave periods T is equal to half of triangle wave period Tt (Tt=2T). They have equal swept band B. only slop is posited. Then TX are: ft(t)=fs+kt (nt<t<(n+1)t, n=0,1,2.) ft(t)=fu-kt ((n+1)t<t<2nt, n=0,1,2 ) Return signal are: Mixer output (k >fd): Doppler frequency is fr(t)=fs+k(t- )+fd fr(t)=fu-k(t- )+fd fb(+)=k* -fd fb(-)=k* +fd fd=2v*(fc/c) Where V is target radial velocity fc is the carrier frequency, (fc=(fs+fu)/2 ) Then 2 radar parameters are: R=(C/2B)T [fb(+)+fb(-)]/2 V=(C/2fc) [fb(-)-fb(+)]/2 Note: B, T, fc are parameters for radar design parameters fb(+) and fb(-) are parameters that radar output to DSP PD radar and Triangle Wave FMCW radar is the second-generation CW radar. They can detect both ranges, speeds (with sign) for moving and/or static targets Page 6 of 6

ELEC RADAR FRONT-END SUMMARY

ELEC RADAR FRONT-END SUMMARY ELEC Radar Front-End is designed for FMCW (including CW) radar application. The output frequency of each RX provides range, speed, and amplitude information to DSP. It will detect target azimuth angle