FMCW Radar 100 - Radar Imaging
Radar vs Camera Imaging
Summary
Radar elements are combined coherently to form a sparse TOI matrix.
CMOS imager are combined indepentently to form a dense TOI matrix.
Parameter |
Radar imaging formulas |
Camera field of view and angular resolution |
|---|---|---|
FOV w/o lense |
\(6843AOP_{FOV} \approx 140^o\) |
\(fov \approx 180^o\) |
FOV w/ lense |
TBD |
\(fov = 2 \cdot tan^{-1}(\frac{H}{2 \cdot f})\) |
MIMO FOV w/o lense (3) |
\(\theta_{FOV} = +/- arcsin(\frac{\lambda}{2d})\) |
$ fov = 2 \cdot `tan^{-1}(:nbsphinx-math:frac{H}{2 cdot f}`)$ |
single element angular resolution w/o lense |
\(\Delta \theta_1 \approx \frac{1}{L \cdot \lvert cos \theta \rvert}\) |
N/A |
single element angular resolution w/ lense |
\(TBD\) |
\(fov = 2 \cdot tan^{-1}(\frac{u}{2 \cdot f})\) |
sensing element size |
\(\approx \lambda \approx 5mm\) |
1 to 10 \(\mu\)m * |
N (number sensing element) |
6843 or 2243: 12 |
|
M*2243: \(N = 12*4^{M-1} \implies M>8 \iff N>10^6\) |
VGA (~300k) up to MPix |
|
array angular resolution (1) |
\(\Delta \theta_N \approx \frac{1}{N \cdot \lvert cos \theta \rvert}\), with N number element in azimuth or elevation |
\(fov = 2 \cdot tan^{-1}(\frac{u}{2 \cdot f})\) |
array angular resolution (2) |
\(\Delta \theta_N \approx \frac{\lambda}{N \cdot d \cdot \lvert cos \theta \rvert}\), with N number element in azimuth or elevation |
$ fov = 2 \cdot `tan^{-1}(:nbsphinx-math:frac{u}{2 cdot f}`)$ |
sensor size |
\(\approx \lambda ^2 * 4 \approx 10mm \cdot 10mm\) or $ 100mm^2 *4^{M-1}$ |
~1 mm x 1mm |
TX Beam Forming angle width |
\(\theta_{width} = \frac{sin(N\pi d_x (s-s_0)/\lambda_0)}{N sin(\pi d_x (s-s_0)/\lambda_0)}\) (Eq 1.57 p17 [1]) |
N/A |
Depth of field |
N/A |
\(\approx \frac{2D^2FC}{f^2}\) |
raw data processing |
Range, CFAR, AoA (LFB, CAPON), Doppler |
Demosaic, YUV, Gamma, Sharpnes, color LPF, ALC, … |
temporal resolution |
?10ms / frame ? |
60fps ~ 16ms/frame |
where:
C is the circle of confusion (the diameter of which is expressed in m)
D is distance to TOI (m)
f is the focal of the lense (\(m^{-1}\))
F is the lense
fnumber (unit-less)H is the sensor size:
H = N * u
N is the number of sensors
u is the sensor size
(1), (2), (3) apperture is a function of N*d on angular resolution where λ is the wavelength of the radar signal and d is the spatial spacing between the channels. The field of view (FOV, maximum unambiguous angle without grating lobes) can be calculated by θFOV so that a diminished FOV of (< 180◦) can be achieved by a spacing of λ/2 or below
Notes:
radar angular resolution [4.] Section 6.6.5
for more on beyond simple pinhole here fig 5
camera angular resolution limited by Airy disk or diffraction where
- C: circle of confusion\(sin \theta \approx 1.22 \frac{\lambda}{d}\)
pixel sensor size source > Pixel size ranges from 1.1 microns in the smallest smartphone sensor, to 8.4 microns in a Full-Frame sensor. As an example, the 8 megapixel sensor above has a resolution of 3,264 x 2,448 pixels, with 327,184 pixels in an area just 1mm x 1mm in size
Considerations on coherent sampling
aka Spatial filtering
From the above we can see the clear analogy FIR filtering in time domain and spatial filtering by the antenna array.
A FIR filter is defined by
\(y_F (t) = \sum_{k=0}^{m-1}h_ku(t-k)\triangleq h*y(t)\) (6.3.4)
Where:
\(h_k\) are the filter weights
u(t) is the input to the filter
Similarly for the spatial filter
\(y_F (t) = \lbrack h*a(\theta) \rbrack \cdot s(t)\)
Where \(a(\theta) = \lbrack 1 e^{j2π·d sin \theta/λ} e^{j2π·2\cdot d sin \theta/λ} e^{j2π·3\cdot d sin \theta/λ} .. e^{j2π·N\cdot d sin \theta/λ} \rbrack\)
Which clearly shows that the spatial filter can be selecte to enhance (attenuate) the signals coming from a given direction \(\theta\) by making \(h * a(\theta)\) large (or small)
The CAPON DOA are the locations of the largest peaks of:
\(\frac{1}{a(\theta) R^{-1} a(\theta)}\) (6.3.26)
AoA Compute Complexity
Algorithm |
Setup Operations |
Setup Operations per Solution |
Operations per θ Hypothesis |
# Operations per θ Hypothesis |
|---|---|---|---|---|
PSBF |
N/A |
N/A |
2 matrix mult |
42 |
CAPON |
1 Matrix inversion |
216 |
2 matrix mult2, 1 div |
43 |
MUSIC |
1 SVD, 1 Matrix mult, 1 threshold |
402 |
2 matrix mult2, 1 div |
43 |
MLE |
N/A |
N/A |
N/A |
630-6300* |
In our experiments, MUSIC/ESPRIT has fairly good performance when the number of ‘virtual antennas’ is high (say 40). For small number of antennas (say 4) we haven’t seen good performance. source: E2E
[1.] pp35
CAPON
Capon DOA estimation has been empirically found to possess superior performance as compared with beamforming. source [4 p 300]
\(E {|y_F (t) |^2 } = h* R h\) (5.4.3) or (6.3.8)
Which is the classical formula for the Energy output of a linear filter. Where:
E: energy output of the linear filter
h: filter
R: the covariance matrix
Which for spatial filter can be rewritten as :
\(E {|y_F (t) |^2 } = a^*(\theta) \cdot R \cdot a(\theta)\) (Eq. [4.] 5.4.3)
Given that the covariance matrix R cannot be exactly determined, it can be estimated by
\(\hat R = \frac{ \sum_{t=1}^{N} y(t) \cdot y^*(t) }{N}\)
From which, the problem defined by CAPON is 1969 is :
\(\underset{h}{min} (h*Rh)\) subject to \(h^*a(\theta) =1\) (Eq. [4.]6.3.23)
The solution to which is proven in 5.4 and is
\(h = \frac{R^{-1}a(\theta)}{a^*(\theta)R^{-1}a(\theta)}\) (Eq. [4.]6.3.24)
which when inserted in the output power formula (Eq. [4.]6.3.8)
\(E \left\{ \lvert y_F (t) \rvert ^2 \right\} = \frac{1}{a^*(\theta)R^{-1}a(\theta)}\)
From which we can derive that the CAPON DOA estimates are the locations of the largest peaks for the function
\(\frac{1}{a^*(\theta)R^{-1}a(\theta)}\) (Eq. [4.] 6.3.26)
Sensing elements
Radar Antenna patch element
linear feed vs b. corporate feed patch antenna
2 series feed antenna with elements of different size (but each antenna is the same)
Radar Array
Physical array
Source E2E `link <>`__
source E2E
Virtual array
Imaging sensor
While patch antenna for visible light is an active topic of research not industrial sensor uses this principle today
REFERENCES
3. simple introduction to BF as LTI filters
SPECTRAL ANALYSIS OF SIGNALS - Petre Stoica
maximum field of view for \(d = \frac{\lambda}{2}\)