International Journal on Electrical Engineering and Informatics - Volume 6, Number 3, September 2014

Improved Compressive Sampling SFCW Radar by Equipartition of Energy Sampling

Andriyan Bayu Suksmono

School of Electrical Engineering and Informatics, Institut Teknologi Bandung,

Jl. Ganesha No.10, Bandung, Indonesia

Abstract: A Stepped-Frequency Continuous Wave (SFCW) radar performs imaging by transmitting a number of electromagnetic tones whose frequency is increased step-wisely to obtain an equivalent representation of a signal in frequency domain, then Fourier inversion is conducted to obtain a range profile. A Compressive Sampling (CS) SFCW radar reduces measurement time by randomly selecting a small number of the tones, followed by CS reconstruction. This paper shows empirically, that the knowledge of typical magnitude spectrum of the radar’s signal can be used to improve the CS-SFCW radar system in term of either better reconstruction quality or less number of required samples. Instead of random selection, the knowledge of the spectrum is used to select the best set of frequencies, by applying equipartition of energy sampling (EES) principle. Three sampling schemes, i.e., Frequency Equidistant Sampling (FES), Uniform Random Sampling (URS), and the proposed EES, are used to obtain the samples of both simulated and actual A-scan data and then identical CS reconstruction methods are applied. Objective performance evaluation in term of PSNR (Peak Signal to Noise Ratio) shows that reconstructed result of URS outperforms the FES, while the EES outperforms both of them. Keywords: Compressive sampling, CS-SFCW radar, Non- Uniform Sampling, Ultra Wideband Radar

1. Introduction Compressive sampling (CS) is an emerging method with various practical applications [1, 2]. In contrast to the Shannon sampling theorem that put a minimum limit at 2∆Ω sampling rate for a ∆Ω band limited signal, the CS capable to reconstruct the signal exactly based on much lower rate or fewer number of samples. Currently, there are some efforts to improve the performance of CS by incorporating prior knowledge. Paper [3] proposes a method for sparse signal recovery that outperforms standard l1 algorithm in term of less number of required samples. The algorithm solves a sequence of l1 minimization problem where the weights used for the next iteration are computed from the value of the current solution. Related to this method, the authors of paper [4] propose an algorithm to recover sparse signal from system of underdetermined linear equations when there is prior information about the probability of each entry of the unknown signal being nonzero. While in [5], a method of modifying CS for a problem with partially known support is discussed. In practice, one is more interested to know how the modification of his/her measurement protocol improves the performance. This paper shows that a simple method to select the location of the samples in the projection domain can significantly improve the objective performance for a given number of samples. The problem can be formulated as follows: given the spectral energy distribution of an A-scan GPR signal and a restricted budget on the number of measurements, how to select a set of samples that best represents the signal in the sense of CS? This problem occurs especially in the compressive SFCW (Stepped-Frequency Continuous Wave) radar

Received: November 26th, 2013. Accepted: September 27th, 2014

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[6]. It should be noted that the knowledge on absolute values of the signals Fourier coefficients defining the spectral energy density alone cannot be used directly to recover the signal without any knowledge on their phase values. In an SFCW radar, an impulse is not-directly transmitted in time-domain. Instead, the Fourier coefficients representing the signal are collected by measuring responses of the observed objects within a frequency band. The A-scan, which is reflections of the attenuated and shifted impulses, usually is modeled by a monocycle, which is derivative of t h e Gaussian function. Normally, the magnitude of the signal spectrum will almost remain the same during the measurement. Therefore, the information of the signals spectral energy density can be used as a prior knowledge to select the best set of samples. When the number of samples can be further reduced, the acquisition speed of the compressive SFCW radar can be eventually increased. In this paper, a simulated one dimensional UWB (Ultra Wide Band) signal that consisting of shifted and attenuated monocycles and an actual A-scan signal generated by a SFCW-GPR (Ground Penetrating Radar) built from a Network Analyzer, are used for evaluating the method. The objective performance of CS reconstruction in term of PSNR (Peak Signal to Noise Ratio) of three different sampling schemes, namely, the uniform random sampling (URS), the frequency equidistant sampling (FES), and the energy equipartition sampling (EES) are compared. It has been shown in [7] that the EES performs better for direct FFT inversion representing the l2 reconstruction, compared to the uniform sampling scheme. The proposed method is actually the l1 extension of this scheme [8]. The rest of the paper is organized as follows. Section 2 explains briefly the principle of the standard CS and the modified CS when signal’s spectral density is known. This section also explains an algorithm to select a set of best samples in frequency domain for a given spectral energy density. Experiments and analysis are given in Section 3 and Section 4 concludes the paper. 2. CS Principle and Frequency Domain Sampling A. Principle of Compressive Sampling in Brief The CS deals with sampling and reconstruction of sparse or compressible signal. Consider an N -length discrete-time signal [ ]TNn sssss ......21=

r represented in a sparsity basis [ ]Nψψψ

rrr ...21=Ψ , i.e.,

∑=

=N

knnas

1ψrr (1)

The signal is compressible if the ordered magnitude of the coefficients decays rapidly and it is a sparse signal if it can be represented by K number of coefficients [9], i.e.,

∑=

=K

kkks

1ψαrr (2)

where K << N. According to CS, a small number of measurements M << N by a projection operator Φ will be sufficient to reconstruct the sparse signal, provided M ≥ C⋅μ2(Φ,Ψ)⋅K⋅log(N) (3) where C is a positive constant and ( ) ψφμ

ψφ,max,

, Ψ∈Φ∈=ΨΦ (4)

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denotes the coherence between t h e sparsity basis and the projection basis. When the signal is measured in DFT (Discrete Fourier Transform) basis as in the SFCW system, giving a small set of (complex-valued) Fourier coefficients [ ]TMm

sub SSSSS ......21=r

, the original signal can be recovered by the following TV minimization ( ) ( ) subSstosubjectsTV =ΨΦ −1min (5) where 1−Ψ denotes inverse DFT (IDFT) bases. Usually, the measurement by projection operator Φ is performed randomly. In this paper, three selection schemes of sampling points will be evaluated, which are

• FES, i.e., by selecting of M -number of samples spaced equally in frequency and is represented by FESΦ .

• URS as in the standard CS and is represented by URSΦ . • EES, i.e., by dividing the magnitude spectrum into M -bins and select the middle

of the bins as position of the samples and is represented by EESΦ . B. Frequency Domain Sampling Schemes In the FES method, the radar bandwidth is divided into N sub-bands in a regular/uniform manner, i.e, Ni ΔΩ==ΔΩ==ΔΩ ......1 (6) In this paper, a different approach to get a better time-domain reconstruction results is proposed; i.e., by proportionally counting the contribution of the spectral energy in each frequency sub-bands, which is illustrated in Fig.1.

Figure 1. Non uniform frequency spacing scheme based on equipartition of the spectral

energy

The left part of the figure shows a time-domain impulse s(t), such as a monocycle, while the right part is its spectral energy density |S(Ω)| obtained from the Fourier transform of s(t). The main idea in the new scheme is to select sub-bands of frequencies and its range [ )2,2 iiii ΔΩ+ΩΔΩ−Ω , so that the energies in the ∆Ωi intervals are identical. It is shown in the figure as dashed bars that have identical areas. The center of the sub bands Ωi will become the location of selected samples in frequency domain. For a given signal bandwidth bounded by ΩL, and ΩU, approximation of the absolute magnitude sum or the magnitude-energy in each dashed area, is given by:

NE

≡ε (7)

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where

( ) ΩΩ= ∫Ω

Ω

dSEU

L

ˆ (8)

A new scheme called the equipartition of the energy sampling (EES) divides the band into N sub bands whose magnitude energy are equals, i.e., ( ) ( ) ( )NNii SSS ΩΔΩ==ΩΔΩ==ΩΔΩ ......11 (9) Consequently, the ith frequency Ωi is obtained along with the corresponding range of frequency as

⎟⎠

⎞⎜⎝

⎛ ΔΩ+ΔΩ+Ω<Ω≤⎟

⎠

⎞⎜⎝

⎛ ΔΩ−ΔΩ+Ω ∑∑

== 22 11

ii

kkLi

ii

kkL

(10)

and the width of the ith subband is ( )ii S Ω=ΔΩ ε (11) According to (10) and (11), to determine the set of frequencies Ωi one needs the spectral energy density |S(Ω)| and the number of sample N. A simple algorithm to determine the sample locations in frequency domain according to energy equipartition sampling (EES) can be immediately formulated. Fig.2 displays the EES algorithm.

Figure 2. The EES Algoritm

An illustration of EES sub band divisions compared to the FES for a 2GHz impulse of monocycle signal divided into 14 sub-bands is presented in Figure 3. The selected sample is located in the centre of each sub band for corresponding method. The figure shows that the sub band becomes wider when the spectral energy density is lower, yields non-uniformly distributed frequency-domain sample positions.

1. For a given spectral energy distribution |S(Ω)|, define the frequency range [ΩL, ΩU]and the number of sample N .

2. Calculate the total sum of spectral energy ( )∫Ω

ΩΩΩ= U

L

dSE and the

average energy in the sub band NE≡ε . 3. Starting from the lowest to the highest frequency:

[a] Integrate E(Ω) over an interval ∆Ω such that the total energy in the interval equal to ε. The middle of the interval is the location of selected sample.

[b] Repeat Step 3.a until all of the sampling points in the set ,..., Nii 1=Ω are found

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Figure 3. Comparison of EES with FES for 14 samples

Figure 4. CS reconstruction results based on 14 sub-samples produced by three different

sampling schemes: EES, URS, and FES. The top part (ORG) is the original signal.

C. CS Reconstruction of Three Different Sampling Schemes The CS reconstruction by the three different sampling will be evaluated and discussed in the next section. By denoting each of the sampling schemes in the projection operator as ΦURS, ΦFES, and ΦEES, the CS reconstruction will be performed as: sub

URSURS SstosubjectsTV =ΨΦ − )()(min 1 (12) sub

FESFES SstosubjectsTV =ΨΦ − )()(min 1 (13) sub

EESEES SstosubjectsTV =ΨΦ − )()(min 1 (14)

Improved Compressive Sampling SFCW Radar

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where subURSS , sub

FESS , , and subEESS denote subsamples collected by URS, FES, and EES,

subsequently. The performance will be measured in term of PSNR (Peak Signal to Noise Ratio). 3. Experiments and Analysis A. Experiments with A Simulated A-Scan Data In the first experiment, a 256 length discrete time signal representing a 2 GHz bandwidth monocycle signal of the GPR A-scan is generated. The A-scan consists of shifted and attenuated monocycle impulses, depending on the number of reflections and their range or distance from the antenna. In CS terminology, the number of the impulse defines the DoF (Degree of Freedom) or the sparsity (K) of the signal. Therefore, minimum number of required samples given in (3) will change according to the value of K. For the present case, one and three random reflections are simulated.

Figure 5. The spectrum of the original (ORG) and reconstructed signals (EES, URS, FES)

of the simulated A-scan

Figure 4 shows the reconstruction results of monocycle signal based on 14 samples selected by three sampling methods. Top part of the figure displays the original signal, while the next lower ones are reconstructed signal based on samples obtained by the EES, URS, and the FES, subsequently. The PSNR values of reconstructed signal for the present case by the FES method is -12.9 dB, URS gives -6.2 dB, and the EES yields 19.6 dB. Therefore, the EES gives the best results compared to both of the URS and the FES. Figure 5 shows the spectrum of original signal and the reconstructed ones. The spectrum also shows that the reconstructed signal from EES best fits the original magnitude spectrum. Figure 6 shows PSNR performance of various numbers of samples with one and three (attenuated and shifted) monocycles, displayed at the upper- and lower-parts of the figure, respectively. Each data point is an average of seven times signal generation, sampling, and reconstruction. This figure shows that the EES consistently outperforms both of the FES and random sampling in term of PSNR and demonstrates that higher DoF requires more sample to achieve the same PSNR at similar level as the lower one.

Andriyan Bayu Suksmono

558

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sub sampling corresponds to 13.57 times compression relative to the signal length or 6.74 times compression to actual number of measurements). The reconstruction results are shown in Fig. 7. Visual inspection show that the results are consistent with the simulated case, whereas PSNR measurements confirm quantitatively where FES gives -17.00 dB, URS gives -4.53 dB, and EES yields 9.44 dB. Figure 8 shows residual signal for the three sampling schemes, which also indicates the capability of each methods. In the frequency domain, Fig.9 shows that the EES spectrum is capable to follow the original spectrum. URS can also approximate the spectrum, but not as good as the EES; while the FES method does not consistent with the original spectrum.

Figure 8. Comparison of residual signals produced by the three methods

Figure 9: The spectrum of the original and reconstructed signals of

an actual A-scan data

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