Noise Radar Technology
4.1 Introduction
4.1.1.2 Constant radial velocity model
The most popular detection scheme is based on the linear, constant velocity motion model rðtÞ ¼r0þv0t and a band limited signal model in the form of low-pass noise upconverted to a carrier frequencyFc.
xTðtÞ ¼xBðtÞexpðj2pFctÞ (4.8)
where xBðtÞis low-pass (frequency limited) complex value noise, as depicted in Figure 4.5.
Spectral density
Fc F
Figure 4.5 Transmitted signal spectrum
Neglecting thetrterm in (4.3) one can obtain the equation yðr;vÞ ¼ Toþðti
t¼T0
xRðtÞexpðj2pFctÞxB t2rþ2vt c
exp j2p 2vFc
c
t
dt (4.9) This equation can be further simplified by neglecting the stretch effects caused by target motion.
yðr;vÞ ¼ Toþðti
t¼T0
xRBðtÞxB t2r c
exp j2p 2vFc
c
t
dt (4.10)
wherexRBðtÞ ¼xRðtÞexpðj2pFctÞis the received signal shifted to the baseband.
Again it is possible to change the variable from motion notation (range and velocity) to signal notation (delay and frequency) and obtain
yðt;fÞ ¼ ð
Toþti
t¼T0
xRBðtÞxBðttÞexpðj2pftÞdt (4.11) wheret¼2r=candf ¼2vFc=c.
In addition, to limit the velocity sidelobes, a time windowing functionwðtÞ such as a Hamming, Hanning or Blackman window is usually applied
yðt;fÞ ¼ Toþðti
t¼T0
wðtÞxRBðtÞxBðttÞexpðj2pftÞdt (4.12) This is a fundamental equation for noise radar signal processing. It is very similar to an ambiguity function [40–42] and again can be calculated using different computational schemas, directly from (4.11) for example, asyðt;vÞ ¼fft w ðtÞxRBðtÞxBðttÞ
or applying more advanced computational algorithms.
Equation (4.11) has to be calculated for all possible delays and frequency shifts in the rangest2 ð0;2RMAX=cÞandf 2 ð2VMAX=l;2VMAX=lÞwhereVMAXis the maximum velocity of the target.
An example of target echoes in a noise radar using rectangular and Hamming windows is presented in Figure 4.6.
The question is, what are the limits of application for the simplified target motion model? The stationary target model (4.5) can be applied if the phase shift due to the Doppler component is small, usually smaller thanp. Thus, the integra- tion time is limited to the value
ti< c
4VMAXFc (4.13)
These limits can be also expressed in terms of the wavelengthlof the transmitted signal instead of the carrier frequency.
ti< l
4VMAXc (4.14)
–400
125 120 115 110 105 100
90 95 –350
–300 –250 –200 –150
V [m/s]
–100 –50 0 50 100
10 15 20 25
Range [km]
(a)
(b)
Range-Doppler plane
30 35
85
120 115 110 105 100 95 90 85 80 75 80
–400 –350 –300 –250 –200 –150
V [m/s]
–100 –50 0 50 100
Range-Doppler plane
10 15 20 25
Range [km]
30 35
Figure 4.6 An example of the range-Doppler correlation function of noise radar return – rectangular time window (a) and Hamming window (b)
The constraint (4.14) has a very clear physical interpretation: the target displacement during observation time should be smaller than a quarter of the wavelength. The results are plotted in Figure 4.7. For an L-band noise radar (l¼20 cm) observing aircraft targets with a maximum target velocity of 1000 m/s (Mach 3), the maximum integration time is equal to 50ms and for humans with a maximum velocity of 10 m/s the maximum integration time is limited to 5 ms.
Constraint (4.14) is not applicable for radars working at baseband. Such radars do not have a carrier frequency and the direct application of constraint (4.14) will allow infinite integration time. In such cases, the limitation originates from range walk phenomena. The range resolution of the noise radar, defined as the width of the correlation peak (4.4), can be expressed as
DR¼ c
2B (4.15)
The range resolution versus signal bandwidth is plotted in Figure 4.8.
As the target has to remain in the correlation gate during the observation time, the second constraint is in the form
ti< DR
VMAX¼ c 2BVMAX
(4.16) The integration time limitation caused by the range walk is presented in Figure 4.9.
For supersonic targets and medium resolution radar (B¼10 MHz), the maximum integration time is limited to a few ms (15 ms).
102 101 100 10–1 10–2 10–3 10–4
10–1 100 101
Fc [GHz]
v = 10 [m/s]
v = 100 [m/s]
v = 1,000 [m/s]
ti [ms]
102 103
Figure 4.7 Maximum integration time for a noise radar using stationary target model (4.5)
Constraint (4.16) can also be re-written in the slightly modified form Bti< c
2VMAX (4.17)
The termBtiis the integration gain of the processing (the signal-to-noise ratio after processing to signal-to-noise ratio before processing). This ratio is then limited by
104 103 102 101 100 10–1
10–1 100 101 102
B [MHz]
ti [ms]
103 104
10–2
v = 10 [m/s]
v = 100 [m/s]
v = 1,000 [m/s]
Figure 4.9 Maximum integration time versus bandwidth – limited by range walk phenomena (4.16)
104 103 102 101 100 10–1
10–1 100 101 102
B [MHz]
R [m]
103 104
10–2
Figure 4.8 Radar range resolution versus signal bandwidth
range walk phenomena to the value predicted by constraint (4.17), which is depicted in Figure 4.10.
The application of a simple stationary target model significantly limits the integration time in a noise radar. The application of a constant velocity model leads to the significant extension of possible integration time. Now, constraints come from two phenomena. As using a constant velocity model, noise radar estimates two parameters – range and range velocity – both the range and velocity walk limit the integration time.
The target detection is performed in this case using (4.12). For a selected velocity, this equation is identical to (4.5), so the range resolution for the constant velocity model is the same as for the stationary target model as described by (4.15).
As a result, the integration time is limited by constraint (4.16).
The target range velocity is calculated using Doppler frequency estimation. For constant delay formula (4.12) is equivalent to FFT, with frequency resolution Dfd ¼1=ti. As Dfd ¼2Dv=l, the velocity resolution defined as the width of the peak of the ambiguity function (4.10) in the velocity dimension is equal to
Dv¼ l 2ti¼ c
2tiFc (4.18)
The velocity resolution versus integration time for different carrier frequencies is plotted in Figure 4.11.
The target velocity should remain within the velocity cell during the integra- tion time. If a more advanced target model was assumed – a constant acceleration model for example – then the velocity changes linearly with timevðtÞ ¼voþa0t.
85 80 75 70 65
Integration gain Bt [dB]
60 55 50 45 40
100 101 102
Vmax [m/s]
103 104
Figure 4.10 Maximum processing gain versus maximum target velocity – limited by range walk phenomena (4.17)
If the target velocity is to remain in the velocity cell, then a constraint appears for the integration time in the form
ti< Dv aMAX ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi l 2aMAX
s
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 2aMAXFc
r
(4.19) The integration time constraints due to velocity migration for different accelera- tions versus carrier frequencies are plotted in Figure 4.12.
It is worth mentioning that (4.9) is simplified as the term tr in (4.3) was omitted. Additionally, in (4.10), a simplification is made by skipping the time stretch component soxBðt ð2r=cÞÞis used instead ofxBðt ðð2rþ2vtÞ=cÞÞ. The skipped term 2vt=cis responsible for signal stretch, while
xB t2rþ2vt c
¼xB t 12vt c
2r c
(4.20) The reference signal has to be scaled in time (stretch) by the factorð1 ð2vt=cÞÞ depending on the actual target radial velocity. Linear stretch processing is not a new idea in radar signal processing. It was introduced in the FMCW radar in [43].
The applications of stretch processing were also presented in [44–47].
In noise radars, stretch processing can be implemented in many different ways.
Among others, the three most popular algorithms that are used are as follows: linear interpolation between samples, cubic spline interpolation and re-sampling based on the pairing of chirp transform and inverse FFT. The results obtained by using the algorithms mentioned above are similar to each other.
Fc = 100 [MHz]
Fc = 1 [GHz]
Fc = 10 [GHz]
Fc= 100 [GHz]
102
101
100
10–1
10–2
10–2 100 102 104
ti [ms]
dV [m/2]
10–3
Figure 4.11 Velocity resolution versus integration time for different carrier frequencies
It should be pointed out that stretch processing has to be performed inde- pendently for each velocity resolution cell. Such an approach, however, requires very high computational power. In most cases, it is sufficient to stretch the reference signal not for each Doppler resolution cell but for the group of velocity cells occupying the velocity strip of widthc=2Bti. Thus, in order to calculate the whole range-Doppler correlation plane, it is necessary to performN ¼4Btivmax=c stretching operations and calculate the range-Doppler correlation in each narrow velocity interval. The details of stretch processing and its impact on the processing gain losses, processing complexity and implementation can be found in [45].
The application of stretch processing allows for extension of integration time, while the range migration due to the constant velocity component is mitigated. For such processing schema limitation (4.16) is not valid, and the integration time is limited by target acceleration. In such cases, the integration time is limited by acceleration introduced velocity walk – see constraint (4.19) – and also by acceleration introduced range walk. Assuming constant target acceleration, the range walk caused by the acceleration is equal toat2=2, so the integration time is limited to
ti<
ffiffiffiffiffiffiffiffiffiffiffi 2DR aMAX
r
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi c aMAXB r
(4.21) The time limits introduced by acceleration range walks are presented in Figure 4.13. It is worth noticing that the constraint (4.21) is less restrictive than (4.19), while bandwidth is usually smaller than carrier frequency.
101
100
10–1
10–2
a [m/s2] 10–3
10–1 100 101 102
ti [s]
Fc=100 [MHz]
Fc = 1 [GHz]
Fc=10 [GHz]
Fc=100 [GHz]
Figure 4.12 Integration time constraints due to velocity migration for different accelerations versus carrier frequencies
The omitted termtr in (4.3) for the constant velocity model is equal to tr¼ r0
cv0 (4.22)
As a result the full echo model takes the form xRðtÞ ¼xT t2r0þ2v0t2r0v0=ðcv0Þ
c
þxRðtÞ (4.23)
After straightforward simplification, one can obtain a more convenient expression in the form
xRðtÞ ¼xT t 12v0 c
2r0
c 1 v0 ðcv0Þ
þxRðtÞ (4.24) As a result, not only time is scaled by the factor 1 ð2v0=cÞbut also the range is scaled by the factor 1 ðv0=ðcv0ÞÞ.