Step Frequency Radar Using Compressed Sensing

Transcription

1 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 00 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 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 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 [] 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, ie 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 ie, 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

2 2 Radar Transmitter & Receiver Fig : Non contact respiration rate monitoring Reflecting Target Radar Transmitter & Receiver Fig 2: Monostatic radar 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, ie as complex exponentials Furthermore, the transmit signal amplitude is normalized to 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 2R n Accounting for these effects, the signal at the input of the receiver will be: RX(t)= σ n A(R n ) e2πif(t τ n) (2)

3 3 where σ n is the target reflectivity, which assumes values in the range (0,) τ n 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 τ n = 2R n c A(R n) is the free-space propagation loss, which is a function of the distance R n In the case of direct Line-of-Sight propagation with no additional parasitic reflectors (such as the ground), A(R n ) can be expressed as: ( 4π(2Rn ) f ) 2 ( 8πRn f ) 2 A(R n )= = (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 n th target: r n = G RX(t) TX(t)= Gσ n A(R n ) e2πif(t τ n) e 2πift = Gσ n A(R n ) e 2πifτ n (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 r n = n= N n= Gσ n A(R n ) e 2πift n = N a n e 2πif 2R n c (5) n= where a n = Gσ n A(R n ) 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 R = m, R 2 = m + 5mm and R 3 = 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, ie R 2 R = 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 R 3 R = 05m The locations of the targets can be easily calculated, assuming knowledge of y( f ), by calculating the inverse Fourier transform: ) F {y( f )} = a n e 2πifτ n e 2πift df (6) ( N n= N = a n e 2πif(t τn) df = n= N N ( = a n δ(t τ n )= a n δ t 2R ) n n= n= 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 a n defined above For perfect conductors, σ n = e iπ, however, we will ignore the 80 phase shift, since it will be introduced in all frequencies and will not affect the detection

4 Coherence Bandwidth Fig 3: Example channel transfer function Sampling Point Sampling Bandwidth 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 f, f 2,, f K 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 0% of their center frequency For example, a system designed for operation at 20 GHz can typically operate in the 4 GHz - 26 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

5 5 frequencies of the sampling points f, f 2,, f K are uniformly distributed and the frequency spacing among two consecutive samples is Δ f = f l f l : { f, f 2, f 3,, f K } = { f, f + Δ f, f + 2Δ f,, f + KΔ f } (7) As a result, we can calculate the IDFT of the sampled data: t y( f ) t 2 = y( f F 2 ) K (8) y( f K ) t K 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,,,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 a n (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 = 2KΔ f = (9) 2 BW However, this period corresponds to a distance based on the formula: Δt = ΔR (0) c Therefore, the distance resolution is: ΔR = c 2KΔ f = c () 2 BW Furthermore, since we have K samples of distance ΔR, the maximum range of unambiguous detection is: R u = K ΔR = c (2) 2Δ f Equations () and (2) 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 (), 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 mm Equation () for this case predicts the necessary bandwidth 2 ΔR c of BW = = 50GHz 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

6 6 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 () If we assume that only one target is present, then equation (5) reduces to: y( f )=ae 2πifτ (3) ie 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( f ) = ae 2πif τ (4) y( f 2 ) = ae 2πif 2τ the phase of y( f ) y( f 2 ) can be easily found to be: y( f ) y( f 2 ) = e 2πi( f f 2 )τ = 2πΔ f τ (5) by simply dividing by 2πΔ f, τ and therefore R are calculated: R = τ c 2 = c 4πΔ f y( f ) (6) y( f 2 ) Therefore, in the single target case, using equation (6) 2, we can measure the distance with infinite resolution by conducting only two measurements, even though the range constraint of equation (2) still holds The above result looks surprising similar to L 0 reconstruction: Two measurements are conducted and the sparsest solution (ie 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 (ie 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 F K : t 0 y( f ) y( f 2 ) = F t K y( f K ) t K (7) In a second step, an arbitrary resolution ΔR a < ΔR is selected The number of elements of the vector [t 0,t,,t K ] T is increased from K to K a where K a = R u c = (8) ΔR a 2Δ f ΔR a 2 It is interesting to note that the US Patent No 6,856,28B2 deals with this simple formula

7 Fig 5: Matrix R operation Under these modifications, the linear system (7) assumes the well known overdetermined system form: y( f ) t 0 t 0 y( f 2 ) = RF t K a = Φ t (9) y( f K ) where the matrix R has dimensions of K K a and selects the first K a rows of F Ka : R = =[ }{{} I }{{} 0 ] (20) K K (K a K) (K a K) The operation of matrix R is graphically illustrated in figure 5 Based on the required resolution ΔR a, a bandwidth block of size 2R c would ideally need to be sampled Nevertheless, measurements 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 2R c a KΔ f bandwidth, a fact which is expressed by the zero submatrix 0 of size (K a K) (K a K) The overdetermined system (9) can now be solved by utilizing a basis pursuit (BP) minimization: min t L st y RF Ka t L2 < ε (2) 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 t Ka t Ka

8 8 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 (2) will be evaluated numerically IV SIMULATION RESULTS The basis pursuit problem (2) was solved numerically using NESTA [8], which was preferred over its widely used alternative, L-Magic, due to its speed The main advantage of NESTA is that the solution to problem (2) can be significantly speeded up by observing that F Ka t = FFT{t} Furthermore the fact that F Ka FK a = FK a F Ka = I allows NESTA to solve the problem by calculating only two FFTs per iteration The parameter ε in (2) 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 00 This value was found to work well in practice in all scenarios and noise conditions Moreover, the sampling bandwidth was set to 7 GHz - 23 GHz, in 50 MHz steps, ie K = 2 sampling points Under these conditions, the maximum range is R u = 3m and resolution Δ R = 25mm In the scenarios presented below, targets at various positions are assumed; then, the ideal values of the samples [y( f ),y( f 2 ),,y( f K )] 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 R = 233m and R 2 = 658m The reconstruction approach using IDFT and basis pursuit are compared on noiseless data and their accuracy is evaluated The required resolution R a for the BP reconstruction was set to 05mm Figure 6 illustrates the entries of vector t =[t 0,t,] 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 2 elements, used by the IDFT reconstruction, the computational cost for solving the BP problem was approximately 20,000 times higher!

9 9 08 CS Error = 0 IDFT Error = 7mm CS Error = 0 IDFT Error = 3mm CS IDFT Distance (m) Fig 6: Comparison of BP and IDFT Accuracy (mm) 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 ie, the data provided to NESTA are: ŷ( f k )=y( f k )e i φ k, where φ k is a random variable, assuming values from a gaussian distribution of zero mean and variance σ φ This form of cyclo-stationary noise typically occurs in electronic systems due to instability of the frequency f k over time In this experiment, for different values of the noise variance σ φ, 200 basis pursuit reconstruction runs are conducted with random target location and noise samples φ k for each run The reconstruction accuracy for each value of σ φ 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 = 0, the accuracy remains below mm proving the fact the BP reconstruction handles phase noise very graciously

10 Error=0 Error=2mm Error=5mm Distance (m) Fig 8: Reconstruction in the presence of three targets C Three targets, Two closely spaced The last scenario under investigation is when three targets are present at locations R = 05m and R 2 = 0774m, R 3 = 0774m ThetargetsatR 2 and R 3 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 [] In our case, phase noise of σ φ = 0 and regular amplitude noise of σ A = 0 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 (ie a non-existent target) Trying to improve on the above results, basis pursuit reconstruction with reweighting [0] is attempted In this approach, each iteration solves the following problem: ( where W i = diag t (i) +ε r min W i t (i+) L st y RF Ka t (i+) L2 < ε (22) ),, Consequently, the values of vector t of the i th t (i) Ka +ε r, t (i) 2 +ε r iteration are used to normalize the entries calculated by the next iteration to The factor ε r 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 ε r In both cases the target separation is more clear compared with the case without reweighting Nevertheless, in the case of ε r = 00, 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 In this case, the value of ε r acts as a threshold which in case it is tootight,asintheε r = 00 case, the reconstructed signal starts getting corrupted In the last experiment, the distance between R 2 and R 3 is reduced to 20 mm The results of the reconstruction are shown in figure 0 It can observed that in the case when the target separation is less then 25 mm, which is the maximum resolution predicted by equation (), the two targets appear merged in the reconstruction in both cases (with and without reweighting)

11 03 02 ε r = ε r =00 Reweighting No Reweighting Fig 9: Separation of closely spaced with and without reweighting Fig 0: Reconstruction when the target separation is 20 mm As a result, the limit of equation () 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 (), 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 [] 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 2, pp , dec 972 [3] K Iizuka and A Freundorfer, Detection of nonmetallic buried objects by a step frequency radar, Proceedings of the IEEE, vol 7, no 2, pp , feb 983 [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 , 984 [5] J A Scheer, Coherent Radar Performance Estimation Artech House, 993

12 [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 , july 2009 [7] S Shah, Y Yu, and A Petropulu, Step-Frequency Radar with Compressive Sampling (SFR-CS), in Proc ICASSP 200, March 200 [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 4th Workshop on, aug 2007, pp [0] E Candes, M Wakin, and S Boyd, Enhancing sparsity by reweighted l minimization, J Fourier Anal Appl, vol 4, pp ,