Step Frequency Radar Using Compressed Sensing - wiki

Step Frequency Radar Using Compressed
Sensing
Ioannis Sarkas
I. INTRODUCTION
The current and advent progress in semiconductor technologies will allow the development
of integrated circuits operating in frequency ranges well above 100 GHz. This development will
the utilization of the unallocated bandwidth available in this frequency range for a multitude of
applications including high data rate wireless communications as well as radars for civilian applications.
For example, even today, the 77 GHz automotive collision avoidance radar represents
a landmark progress due to each potential to save thousands of lives every year, by providing to
the driver of a vehicle early warning of a possible collision with another vehicle or a pedestrian.
Following the collision avoidance example, radars for other commercial applications can be
developed. In this report, the possibility of a radar for non-contact respiration rate monitoring
as graphically illustrated in figure 1 is investigated. As shown in the figure, a radar transceiver
is placed in a position that has Line-of-Sight contact with the thorax of a sleeping patient. By
accurately measuring the distance to the patient’s thorax, several times a second, the minute
displacements caused due to respiration are recorded. These data are used in a later stage for
calculation and evaluation of the patient’s respiration pattern. By knowing this pattern for a
period of several days, a medical doctor can evaluate whether the patient suffers from a sleep
disorder such as sleep apnea.
The range accuracy (resolution) requirement for this application, as well as many other similar
applications, is of the order of 1 mm. As will be shown in the following sections, this accuracy
requirement is particularly challenging to meet using standard radar processing. Therefore, an
alternative technique using basis pursuit reconstruction is explored. Various target separation and
noise scenarios are explored and different reconstruction methods are evaluated.
II. STEP FREQUENCY RADAR
There are different types of radar that can be utilized for accurate distance measurement [1].
In this work, we selected the Step Frequency Radar (SFR), which was originally introduced
in [2], as a non-real-time Time-Domain-Reflectometer, and was later further analyzed in [3], [4].
This type of radar was selected due to its relatively waveform, i.e the signal waveform that the
radar transmits and processes in order to calculate the distance. The step frequency radar utilizes
continuous wave sinusoids of different frequencies i.e., during each measurement, a sinusoid of
only one frequency is transmitted, received and processed thus greatly simplifying the signal
generation, reception and digitization.
This operation is illustrated in figure 2. A wireless transmitter transmits a continuous wave
of constant frequency. The wave is reflected back by the N targets located inside the radar field
of view. The reflected signal is collected by the receiver which in turn, it frequency shifts the
signal and moves it into lower frequency where it can be digitized and processed by a digital
processor.
1

Radar
Transmitter
&
Receiver
Fig. 1: Non contact respiration rate monitoring.
Radar
Transmitter
&
Receiver
Fig. 2: Monostatic radar
Reflecting
Target
In order to analyze the radar operation, it is useful to assume first that there is only one target,
which is denoted as target n in figure 2. The transmitted signal by the radar is a essentially
electromagnetic plane wave of frequency f :
TX(t)=e 2πift
In the above equation, as well as in the remainder of this work, we will express sinusoidal
signals in the analytical representation, i.e. as complex exponentials. Furthermore, the transmit
signal amplitude is normalized to 1.
The signal is reflected back from the target before is collected by the receiver antenna. During
this process, it is subjected to attenuation, due to the free space loss, as well as time delay,
since it travels a total distance of 2Rn. Accounting for these effects, the signal at the input of
the receiver will be:
RX(t)= σn
A(Rn) e2πif(t−τn)
2
(1)
(2)

where σn is the target reflectivity1 , which assumes values in the range (0,1). τn is the total time
the signals needs to reach the target, be reflected, and travel back to the receiver. Since the EM
wave travels with the speed of light c, the propagation delay is τn = 2Rn
c . A(Rn) is the free-space
propagation loss, which is a function of the distance Rn. In the case of direct Line-of-Sight
propagation with no additional parasitic reflectors (such as the ground), A(Rn) can be expressed
as:
4π(2Rn) f
2
8πRn f
2 A(Rn)=
=
(3)
c
c
The receiver collects the signal of equation (2), amplifies it and multiplies by the complex
conjugate replica of the transmitted signal, yielding the output for the nth target:
rn = G · RX(t) · TX(t)= Gσn
A(Rn) e2πif(t−τn) −2πift Gσn
· e =
A(Rn) e−2πifτn (4)
Where G is the total system gain. It can be seen that the output of the receiver is a complex
number, representing the amplitude reduction and phase shift of the signal due to the target n.
In the case of N total targets, the output is the sum of all amplitude and phase shift terms:
y( f )=
N
∑ rn =
n=1
N
∑
n=1
Gσn
A(Rn) e−2πiftn =
N
2Rn −2πif
∑ ane c (5)
n=1
where an = Gσn
A(Rn) .
For now, let us assume that we have perfect knowledge of y( f ), which will be referred to as
channel transfer function, a term used extensively in mobile communications. As an example,
the magnitude of y( f ) over frequency is illustrated in figure 3 when 3 targets are present and
at distances of R1 = 1m, R2 = 1m + 5mm and R3 = 1.5m. As expected, since y( f ) is a sum of
sinusoids, it is a periodic function of frequency. The largest bandwidth block in the waveform
is known as the coherence bandwidth (a term also used extensively in mobile communications)
and depends upon the smallest target separation, i.e. R2 − R1 = 5mm in our case. Inside each
coherence bandwidth ”lobe”, as depicted in figure 3, the channel transfer function is also periodic,
this period now depends upon the larger separation of R3 − R1 = 0.5m.
The locations of the targets can be easily calculated, assuming knowledge of y( f ), by calcu-
lating the inverse Fourier transform:
F −1 {y( f )} =
=
=
∞
−∞
N
∑
n=1
N
∑
n=1
∞
an
−∞
N
∑ anδ(t − τn)=
n=1
ane −2πifτn
e 2πift df (6)
e 2πif(t−τn) df =
N
∑ anδ
n=1
t − 2Rn
c
Consequently, the inverse Fourier transform of the channel transfer function is a series of N
delta functions that ”sit” at the target locations. The amplitude of the delta function located at
target n is equal to the coefficient an defined above.
1For perfect conductors, σn = e−iπ , however, we will ignore the 180◦ phase shift, since it will be introduced in all frequencies
and will not affect the detection.
3

1
0.8
0.6
0.4
0.2
1
0.8
0.6
0.4
Coherence
Bandwidth
120 125
0 20 40 60 80 100 120 140 160 180 200
Fig. 3: Example channel transfer function
Sampling
Bandwidth
Sampling
Point
0.2
112 114 116 118 120 122 124 126 128
Fig. 4: Step frequency radar sampling procedure
The above analysis reveals that by conducting sinusoidal measurements in the free space,
in presence of reflective targets, we can determine the target distances by calculating a simple
inverse Fourier transform. Nevertheless, in practice the full channel transfer function y( f ) cannot
be exactly known. In the case of step frequency radar, we can only measure discrete samples
of y( f ) at a set of frequencies f1, f2,..., fK. Furthermore, these frequencies can only be located
inside a predetermined frequency block. This stems from the fact that at high frequencies, most
electronic systems exhibit narrowband characteristics and can typically operate inside a frequency
range that is typically 5 − 10% of their center frequency. For example, a system designed for
operation at 120 GHz can typically operate in the 114 GHz - 126 GHz frequency range. This
sampling procedure is illustrated in figure 4.
In order to calculate the distance to the targets based on the discrete samples of the channel,
we need to utilize the Inverse Discrete Fourier Transform (IDFT). First, it is assumed that the
4

frequencies of the sampling points f1, f2,..., fK are uniformly distributed and the frequency
spacing among two consecutive samples is Δ f = fl − fl−1:
{ f1, f2, f3,..., fK} = { f1, f1 + Δ f , f1 + 2Δ f ,..., f1 + KΔ f } (7)
As a result, we can calculate the IDFT of the sampled data:
⎡ ⎤ ⎡ ⎤
t1 y( f1)
⎢t2
⎥ ⎢y(
f2) ⎥
⎢ ⎥
⎣
.
⎦ = F−1 ⎢ ⎥
K ⎣
.
⎦
y( fK)
tK
As expected from equation (6), the l th entry of the vector on the left hand side of (8)
corresponds to a particular distance from the radar (l · ΔR), l = 0,1,...,K. If a target is not
present at distance (l · ΔR), then the l th entry of the vector will be zero. If a target is present,
the corresponding l th entry will be equal to the target an (defined in equation (5).
In order to calculate the resolution ΔR, we need to employ the Shannon-Nyquist sampling
theorem. The total sampled frequency bandwidth by the step frequency radar is BW = K Δ f .As
a result, the sampling period in time domain is:
Δt = 1
2KΔ f
= 1
2 BW
However, this period corresponds to a distance based on the formula:
Δt = ΔR
(10)
c
Therefore, the distance resolution is:
ΔR = c c
= (11)
2KΔ f 2 BW
Furthermore, since we have K samples of distance ΔR, the maximum range of unambiguous
detection is:
Ru = K ΔR = c
(12)
2Δ f
Equations (11) and (12) represent the two most fundamental limits of the step frequency radar,
namely, the resolution is proportional to the bandwidth while the range is proportional to the
frequency step. Moreover, equation (11), which was derived from the fundamental properties of
the Fourier transform, can be extended to every continuous wave radar [5].
For respiration rate monitoring, as well as other applications such as texture evaluation, the
required resolution is at least 1mm. Equation (11) for this case predicts the necessary bandwidth
of BW = c
2 ΔR = 150GHz. This bandwidth requirement will be impossible to be met using stateof-the
art electronics.
Driven by this limitation, an alternative method to IDFT for calculating the target distances
from the available channels transfer function samples needs to be explored. In this work, we
will evaluate the performance of reconstruction using compressed sensing. The accuracy of the
method will be evaluated based on different target placement scenarios, as well as different kinds
and variances of noise.
5
(8)
(9)

III. STEP FREQUENCY RADAR USING COMPRESSED SENSING
The author’s interest for using compressed sensing was driven by a well known, but confusing
result that contradicts equation (11). If we assume that only one target is present, then equation
(5) reduces to:
y( f )=ae −2πifτ
(13)
i.e. there is only a single complex exponential. Furthermore, the amplitude in now constant
over frequency and there is only a linear phase change with frequency. Assuming that only two
frequency measurements are conducted:
y( f1) = ae −2πif1τ
(14)
y( f2) = ae −2πif2τ
the phase of
y( f1)
y( f2)
can be easily found to be:
y( f1)
∠
y( f2) = ∠e−2πi( f1− f2)τ
= 2πΔ f τ (15)
by simply dividing by 2πΔ f , τ and therefore R are calculated:
R =
τ c
2
= c
4πΔ f
· ∠y( f1)
y( f2)
Therefore, in the single target case, using equation (16) 2 , we can measure the distance with
infinite resolution by conducting only two measurements, even though the range constraint of
equation (12) still holds.
The above result looks surprising similar to L0 reconstruction: Two measurements are conducted
and the sparsest solution (i.e. one target) that satisfies the measured data is evaluated.
Since there is actually only one target, perfect reconstruction is achieved. Driven by this result,
we will further investigate the reconstruction from step-frequency data using compressed sensing.
By examining equation (8), we observe that the SFR reconstruction problem is a good
candidate to apply compressed sensing. First, the measurements are collected in an orthogonal
basis (i.e. Fourier basis) and second, the vector on the left hand side of (8) is N-sparse, N being
the the number of targets. Compressed sensing has been applied before for reconstruction in SFR
in [6], [7]. However, although both of these papers show that this reconstruction is better than
the IDFT approach, they don’t discuss the resolution limits of the compressed sensing approach.
In order to formulate the reconstruction problem for compressed sensing reconstruction, we
first multiply both sides of equation (8) with FK:
⎡ ⎤
y( f1)
⎢y(
f2) ⎥
⎢ ⎥
⎣
.
⎦
y( fK)
= FK
⎡
t0
⎢t1
⎥
⎢ ⎥
⎣
.
⎦
In a second step, an arbitrary resolution ΔRa < ΔR is selected. The number of elements of the
vector [t0,t1,...,tK] T is increased from K to Ka where
Ka = Ru
=
ΔRa
c
tK
⎤
2Δ f ΔRa
2 It is interesting to note that the US Patent No. 6,856,281B2 deals with this simple formula.
6
(16)
(17)
(18)

1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25 30
Fig. 5: Matrix R operation
Under these modifications, the linear system (17) assumes the well known overdetermined
system form:
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
y( f1)
t0 t0
⎢y(
f2) ⎥ ⎢ t1 ⎥ ⎢ t1 ⎥
⎢ ⎥
⎣
.
⎦ = RFKa
⎢ ⎥
⎣
.
⎦ = Φ⎢
⎥
⎣
.
⎦
y( fK)
(19)
where the matrix R has dimensions of K × Ka and selects the first Ka rows of FKa :
⎡
⎤
1 0 ··· 0 0 ··· 0
⎢ 0
R = ⎢
⎣
.
0
1
0
···
...
···
0
.
1
0
.
0
···
···
···
0 ⎥
.
⎦ =[ I | 0 ]
K×K (Ka−K)×(Ka−K)
0
(20)
The operation of matrix R is graphically illustrated in figure 5. Based on the required resolution
ΔRa, a bandwidth block of size c would ideally need to be sampled. Nevertheless, measure-
2Ra
ments were conducted only inside a smaller block of size KΔ f . This block is expressed by the
unity submatrix I of size K ×K. There is no available information from the remaining c −KΔ f
2Ra
bandwidth, a fact which is expressed by the zero submatrix 0 of size (Ka − K) × (Ka − K).
The overdetermined system (19) can now be solved by utilizing a basis pursuit (BP) minimization:
min tL1 s.t. y − RFKa tL2 < ε (21)
It is interesting to note that neither matrix R nor Φ satisfy the Restricted Isometry Property
(RIP). For example, if we form a test matrix T that contains the first column of the I submatrix
tKa
tKa
7

and the first column of the 0 submatrix and calculate the singular values of TT ⋆ , there will always
be one singular value that is equal to zero. However, the RIP is a sufficient, but not necessary
condition for reconstruction. Since there is no rigorous result to guarantee reconstruction or
give the conditions for perfect reconstruction, the reconstruction performance attained solving
problem (21) will be evaluated numerically.
IV. SIMULATION RESULTS
The basis pursuit problem (21) was solved numerically using NESTA [8], which was preferred
over its widely used alternative, L1-Magic, due to its speed. The main advantage of NESTA is that
the solution to problem (21) can be significantly speeded up by observing that FKa t = FFT{t}.
Furthermore the fact that FKa F⋆ Ka = F⋆ Ka
FKa = I allows NESTA to solve the problem by calculating
only two FFTs per iteration.
The parameter ε in (21) is usually set based on the noise variance σ. However, in this work, we
assume no prior knowledge of σ, which is the case in most practical measurement systems. An
interesting method for selecting ε using cross-validation was developed in [9]. In this approach,
a set of measurements is used for reconstruction while another set is used for cross-validation:
Initially, the basis pursuit problem is solved with the reconstruction set and an arbitrary value
for ε. Then, the result is validated with the cross-validation set and ε is updated accordingly
for the new iteration. The drawback of this approach is the fact that precious measurements are
sacrificed for validation, as well as the time consuming iteration that is required.
In our case, ε is permanently set to 0.01. This value was found to work well in practice in
all scenarios and noise conditions. Moreover, the sampling bandwidth was set to 117 GHz -
123 GHz, in 50 MHz steps, i.e. K = 121 sampling points. Under these conditions, the maximum
range is Ru = 3m and resolution ΔR = 25mm.
In the scenarios presented below, targets at various positions are assumed; then, the ideal
values of the samples [y( f1),y( f2),...,y( fK)] T are calculated and noise is added if required.
These samples are then fed to NESTA for reconstruction and the reconstructed positions are
compared with the original.
A. Two targets, Noiseless data
In the first scenario, we assume that two targets exist in locations R1 = 1.233m and R2 =
1.658m. The reconstruction approach using IDFT and basis pursuit are compared on noiseless
data and their accuracy is evaluated. The required resolution Ra for the BP reconstruction was
set to 0.5mm. Figure 6 illustrates the entries of vector t =[t0,t1,...] T , for the two reconstructions
methods, versus the distance the entries correspond to. The BP approach achieves perfect reconstruction
exhibiting, as expected, two sharp peaks at the target locations. On the contrary, the
IDFT data are more spread, showing a maximum reconstruction error of 7mm. Consequently,
IDFT reconstruction would fail to meet our specs, even without measurement error, due to finite
resolution.
This example alone shows the potential of compressed sensing as a powerful reconstruction
method, achieving resolutions much higher than the theoretically predicted from IDFT. Nevertheless,
it should be noted that the computational cost of solving the BP problem was significantly
higher than the IDFT case; namely, NESTA calculated 460 FFTs of vectors consisting 6000
elements. Compared with the one FFT of 121 elements, used by the IDFT reconstruction, the
computational cost for solving the BP problem was approximately 20,000 times higher!.
8

1
0.8
0.6
0.4
0.2
CS Error = 0
IDFT Error = 7mm
CS
IDFT
CS Error = 0
IDFT Error = 3mm
0
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
Distance (m)
Accuracy (mm)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Fig. 6: Comparison of BP and IDFT.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Phase Noise σ (Degrees)
Fig. 7: Single target reconstruction accuracy for different values of phase noise variance.
B. One target, Noisy data
In the second scenario, we assume the existence of a single target at a random location.
Furthermore, the measurement data vector is corrupted with phase noise i.e., the data provided
to NESTA are: ˆy( fk)=y( fk)ei φk,
¯
where φk ¯ is a random variable, assuming values from a gaussian
distribution of zero mean and variance σφ . This form of cyclo-stationary noise typically occurs
in electronic systems due to instability of the frequency fk over time.
In this experiment, for different values of the noise variance σφ , 200 basis pursuit reconstruction
runs are conducted with random target location and noise samples φk ¯ for each run.
The reconstruction accuracy for each value of σφ is assumed equal to the worst case accuracy
among the 200 runs.
The results of this experiment are depicted in figure 7. The accuracy of BP reconstruction is
surprisingly accurate even under extreme phase noise conditions. For example when σ = 10◦ ,
the accuracy remains below 1mm proving the fact the BP reconstruction handles phase noise
very graciously.
9

1
0.8
0.6
0.4
0.2
C. Three targets, Two closely spaced
Error=0
Error=2mm
Error=1.5mm
0
0.4 0.5 0.6 0.7
Distance (m)
0.8 0.9 1
Fig. 8: Reconstruction in the presence of three targets
The last scenario under investigation is when three targets are present at locations R1 = 0.5m
and R2 = 0.774m, R3 = 0.774m. ThetargetsatR2 and R3 are deliberately closely spaced in order
to evaluate the target separation performance of the BP reconstruction approach. Separation of
closely spaced targets (especially from noisy data) is an important criterion for radar perforce [1].
In our case, phase noise of σφ = 10 ◦ and regular amplitude noise of σA = 0.1 was added to the
data.
Figure 8 shows the reconstructed data for the above scenario. The two closely spaced targets
were separated, even in the presence of large phase noise, albeit with relatively low accuracy
(the error is 2mm). Furthermore, in the area between the two closely spaced targets, the vector
t attains high values, risking the detection of a spurious target (i.e. a non-existent target).
Trying to improve on the above results, basis pursuit reconstruction with reweighting [10] is
attempted. In this approach, each iteration solves the following problem:
where Wi = diag
1
t (i)
1 +εr
min Wi t (i+1) L1 s.t. y − RFKa t(i+1) L2 < ε (22)
. Consequently, the values of vector t of the ith , 1
t (i)
2 +εr
,..., 1
t (i)
Ka +εr
iteration are used to normalize the entries calculated by the next iteration to 1. The factor
εr is added for numerical stability, especially under noise.
Figure 9 illustrates the reconstructed data for the two closely spaced targets when reweighting
is utilized with two different values of εr. In both cases the target separation is more clear
compared with the case without reweighting. Nevertheless, in the case of εr = 0.01, the noise in
the data is magnified since the algorithm tries to normalize the small spurs due to noise (seen
in both figures 8 and 9) to 1. In this case, the value of εr acts as a threshold which in case it is
tootight,asintheεr = 0.01 case, the reconstructed signal starts getting corrupted.
In the last experiment, the distance between R2 and R3 is reduced to 20 mm. The results
of the reconstruction are shown in figure 10. It can observed that in the case when the target
separation is less then 25 mm, which is the maximum resolution predicted by equation (11), the
two targets appear merged in the reconstruction in both cases (with and without reweighting).
10

0.3
0.2
0.1
ε r=1 ε r=0.01
0
0.65 0.7 0.8 0.9
Reweighting
No Reweighting
0.65 0.7 0.8 0.9
Fig. 9: Separation of closely spaced with and without reweighting.
1
0.8
0.6
0.4
0.2
0
0.35
0
0.7 0.8 0.9
0 0.5 1 1.5 2 2.5 3
Fig. 10: Reconstruction when the target separation is 20 mm.
As a result, the limit of equation (11) still holds for the SFR with compressed sensing with
respect to maximum possible target separation.
V. CONCLUSIONS
Compressed sensing through basis pursuit was utilized as a reconstruction method for step
frequency radar data, achieving very good performance. Numerical experiments revealed that
for a given sampling bandwidth, as long as the target spacing remains larger than the limit
predicted by equation (11), perfect reconstruction can be achieved. Apart from this fundamental
limitation, basis pursuit reconstruction exhibits excellent accuracy, much higher than IDFT, even
under conditions of extreme noise levels.
REFERENCES
[1] M. Skolnik, Introduction to Radar Systems. McGraw-Hill, 2002.
[2] L. Robinson, W. Weir, and L. Young, “An RF Time-Domain Reflectometer Not in Real Time,” Microwave Theory and
Techniques, IEEE Transactions on, vol. 20, no. 12, pp. 855 – 857, dec 1972.
[3] K. Iizuka and A. Freundorfer, “Detection of nonmetallic buried objects by a step frequency radar,” Proceedings of the
IEEE, vol. 71, no. 2, pp. 276 – 279, feb. 1983.
[4] K. Iizuka, A. P. Freundorfer, K. H. Wu, H. Mori, H. Ogura, and V.-K. Nguyen, “Step-frequency radar,” Journal of Applied
Physics, vol. 56, no. 9, pp. 2572–2583, 1984.
[5] J. A. Scheer, Coherent Radar Performance Estimation. Artech House, 1993.
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[6] A. Gurbuz, J. McClellan, and W. Scott, “A Compressive Sensing Data Acquisition and Imaging Method for Stepped
Frequency GPRs,” Signal Processing, IEEE Transactions on, vol. 57, no. 7, pp. 2640 –2650, july 2009.
[7] S. Shah, Y. Yu, and A. Petropulu, “Step-Frequency Radar with Compressive Sampling (SFR-CS),” in Proc. ICASSP 2010,
March 2010.
[8] S. B. Jerome Bobin and E. Candes, “NESTA: A Fast and Accurate First-order Method for Sparse Recovery,” California
Institute of Technology, Tech. Rep., 2009.
[9] P. Boufounos, M. F. Duarte, and R. G. Baraniuk, “Sparse Signal Reconstruction from Noisy Compressive Measurements
using Cross Validation,” in Statistical Signal Processing, 2007. SSP ’07. IEEE/SP 14th Workshop on, aug. 2007, pp. 299
–303.
[10] E. Candes, M. Wakin, and S. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl., vol. 14,
pp. 877–905, 2007.
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