Novel high range resolution approach in microwave imaging

Y.Wang D.G. Fang

indexing terms: Microwave imaging, Range resolution, Nonlinear spectral estimation, Deconvolution

Abstract: The authors propose a method of gaining range superresolution in a microwave imaging system. After frequency domain deconvolution of echo to the transmitted signal, the signal model is proved to be of the same form as that of DOA estimation. Hence, as in the case of DOA estimation, a nonlinear spectrum estimation algorithm can be applied to improve range resolution. Numerical simulation and experiments have shown the method to be cost- effective.

1 Introduction

In microwave imaging. the resolution of target scatter- ing centres involves problems of both angular and range resolution. Introduction of nonlinear spectral estimation into the direction-of-arrival (DOA) problem has made it possible to obtain superresolution in the angular coordinate. In the range coordinate, the high- range resolution of target scattering centres is carried out using a wideband signal.

Having the character of being simple, systems that send narrow-pulse signals are widely used. In practice, it is still very difficult to reduce the pulse width to less than several nanoseconds. Range resolution can be enhanced by the deconvolution of the received signal to the transmitted signal: this is to be analysed in this paper and in [l]. In the deconvolution method, the transmitted signal is a series of coherent pulse, whose spectrum is composed of discrete spectral lines. The deconvolution operation is carried out by recording the spectra of both transmitted and received signals and dividing of corresponding amplitude of each spectral line. The procedure is to equal the contribution of each spectral line to the received signal. After deconvolution, the discrete spectral lines can be viewed as echoes of a series of continuous signals, whose frequencies are f o , Jo + Aj, ..., f o + NAL respectively, with Af being the repe- tition frequency of pulses. Hence, the achievements of using nonlinear spectral estimation in the DOA prob- lem, which has received considerable attention, are also applicable to the improvement of range resolution.

0 IEE, 1997 IEE Proceedings online no. 19971074 Paper first received 20th March and in revised form 26th November 1996 The authors are with the Millimeter Wave Technique Laboratory, Department of Electronic Engineering, Nanjing University of Science & Technology, Nanjing 210094, People’s Republic of China

In practical applications, the deconvolution problem is mathematically classified as an ill-posed problem [2, 31. The zeros of the spectrum of the transmitted signal make the results of division indeterminate, conse- quently, the performance of the nonlinear spectral esti- mation algorithms is degraded. To solve this problem, an automated optimum compensation deconvolution [4] is adopted in this paper.

In our microwave imaging system, the transmitted signal is coherent pulse with the carrier frequency in the X-band. Spectral amplitudes of transmitted and echo signals are recorded using a sweeping spectrum analyser. Spectral phase measurement is realised by a single channel time-sharing phase detecting method [9] in which phase information is derived from amplitude measurement and no vector network analyser is required. This technique is detailed within; its particu- lar features are simplicity and low cost. This is borne out by experimental results which show that the pro- posed method is capable of cost-effectively gaining range superresolution.

2 Principles

2. I Signal model When a signal of form s(t) is transmitted into a target environment, the received echo will be a linear super- position of like signals. Assume that there are two point targets (scattering centres) of amplitudes A, and A2 at ranges rl and r2 respectively. The target function can be written as

where s(t) is a Dirac delta function and z1 = 2rl/c, z2 = 2r2/c. The received echo signal r(t) is

Using RCf), SCf) and A(f) to denote the frequency domain forms of r(t) , s(tj and a(t), eqn. 2 transforms to

a ( t ) = A,6(t - 71) + A26(t - 7 2 ) (1)

~ ( t ) = ~ ( t ) @ a ( t ) = Als(t - 7 1 ) + Azs(t - 7 2 ) (2)

R ( f ) = S ( f ) A ( f ) = S ( f ) . + A2ep32xfTz 1 (3)

With the measured bandwidth being limited, the fre- quency domain deconvolution is derived as

where B is the bandwidth and rect (xj is a rectangular function. After inverse Fourier transformation, the esti- mation of a(t) is derived from eqn. 4 as follows:

From eqn. 5, the range resolution is known to be cl2B.

I I1 IEE Prot.-Radur, Sonar Navig.. Vol. 144, No. 4, Alrgust I997

When signal s( t ) is a coherent-pulse and the repeti- tion frequency of these pulses is AJ S o is the domain of discrete spectral lines on frequency points fo, fo + Af. ..., f o + NAf and NAf = B. In eqn. 4, let ,f = J;, = f o + nAf . x, is defined as

".' (6) 27.

z, = A(f,) z Ale-J2n;fn + A2e-327ifn T

Let ulc = 4mk/c (k = 1, 2). So, 2, = Ale-J.fnul + Aae-3fnu2

(7) - - Ale-j,foo"l e - ~ n a f u ~ + Aze-3fn"ze-3nAfo"z

When Fk = Ake-fOUk (k = 1, 2), x, = F,e-i"AhI + F2e-jnAfU2. If there are p point targets,

P

(8) 5 , = Fke-JnAfuk

k=l It can be seen from eqn. 8 that x, is superposition of p complex sinusoidal signals with complex amplitude Fk and phase nAfulc (k = 1, 2, ..., p ) .

When i = 0, 1, ..., N, from eqn. 8 1 1 1

(9)

eqn. 8 can be written in the form

X = U F (10) In the above equations, X is an ( N x 1) vector repre- senting the received frequency domain data and Ul, is 0, x 1) range vector. Obviously, these equations take the same form of the model used in the direction-of- arrival estimation. Application of nonlinear spectral estimation algorithms in the DOA estimation to improve angular resolution has been extensively researched. So, it is natural to consider using them in the range co-ordinate problem based on the same mathematical model. The only difference lies in that, in the DOA problem, U is a matrix made of direction vec- tors while here U is composed of range vectors. Thus, nonlinear spectral estimation can be utilised to obtain high resolution estimation of U,<, which contains range information, and consequently range superresolution, beyond the limitation of bandwidth (cl2B).

Among these algorithms, the eigendecomposition- based or singular-value-decomposition-based algo- rithms (e.g. MUSIC) has demonstrated the ability to achieve high resolution. At the same time, effective approaches have been developed for solving the prob- lems of coherent sources in DOA estimation [5, 61. Hence, the MUSIC method is adopted in this paper [7], although other nonlinear algorithms are also applica- ble.

2.2 Review of the automated optimum compensation deconvolution technique [3, 41 In general, the zeros of S(~U) are also zeros of R(ju). The deconvolution results at these frequency points are indeterminate because of the lack of detecting signals at

178

these points. In the region where S(jco) is fairly small or zero, the division of eqn. 4 will be large or infinite, which can be viewed as &U) function. Upon the appli- cation of the inverse Fourier transformation to R(~u)/ SQo), the SGU) error spikes which were concentrated in the frequency regions of small SGw) produce error con- tributions spread over the entire time-domain trans- form epoch and may hide most of the details. When nonlinear spectral estimation algorithms are applied instead of inverse Fourier transformation, the perform- ance of these algorithms is degraded, especially when the pulse width is large.

Hence, the deconvolution operation needs to be modified. Such a modification is carried out by apply- ing a properly designed filter F(jco) before the deconvo- lution operation R(jco)/SGco). FGu) is defined as

In this paper, the automated optimum compensation deconvolution [4] is adopted, where q5 (U) = A. This technique is robust and uses an adaptive filter with passbands that coincide with the frequency bands where lS(j~)o> is significant. The deconvolution of eqn. 4 is transformed to

In this technique, the criterion for selecting the opti- mum value of the parameter A is based on a compro- mise between deconvolution accuracy and noise content indicators [8].

3 Numerical simulation

Two target problem is chosen for the simulation. The transmitted signal is assumed to be a coherent pulse with pulse repetition frequency Af = 5MHz and band- width B = 500MHz. The series {x ' l , d2, ... x ' ~ } is cal- culated by transformation of transmitted and received time domain signals to frequency domain and deconvo- lution operation by using eqns. 2, 3 and 4, in which

z: = A'(f,) (n = 1, . . . , N ) (13) White noise with normal distribution of different lev-

els is added to the simulation. The sample number of each datum is 100. In the numerical simulation, factors affecting range resolution are studied as follows: (i) Table 1 shows the range resolution of applying inverse Fourier transformation and MUSIC algorithm to the series { x ' ~ , x$, ... xfN} when noise of different levels is added. Effectiveness of application of nonlin- ear spectral estimation algorithm after deconvolution is clearly demonstrated. From Table 1, the method using the MUSIC algorithm is more sensitive to noise. How- ever, range resolution is improved at least three times under higher SNR in comparison to the FFT method. This improvement is equivalent to broadening the measuring bandwidth three times without complicating the whole system.

Table 1: Comparison of range resolution (RR) using FFT and MUSIC algorithm after deconvolution under differ- ent SNR

SNR, dB CO 20 0 -5

RR of FFT, m 0.46 0.46 0.46 0.52

RR of MUSIC, m 0.1 1 0.11 0.16 0.30

IEE ProcRadar, Sonar Navig , Vol. 144, No. 4, August 1997

(ii) In Fig. la, range resolution under different band- widths obtained by the two methods is compared when SNR = OdB. From eqn. 5, range resolution is deter- mined by cl2B when FFT is applied as is shown in Fig. la. For the method using MUSIC, range resolu- tion is also improved as bandwidth is enlarged.

2 -

1.8

1.6

E l . L - C' 0 1.2

51 P g 0.8-

e 0.6-

3 L -

c

0.L

0.2

2.2

-

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1. ' '. '\

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. - - - - I-- - - - _

--. - - ---.. I - ' . '.

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- - . - . - . __ -.-_ - . -__ . .. .. -

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01 I 1 I

100 200 300 LOO 500 bandwidth, MHz

b Fig. 1

- - MUSIC a Simulation results b Experimental results

Range resolution under d$erent bandwidths FFT - - - -

(iii) Range resolution using the MUSIC algorithm after deconvolution under different pulse width is listed in Table 2. Numerical simulation shows that without opti- mum deconvolution technique (A = 0 in eqn. 12), errors involved by deconvolution severely degrade the per- formance of the algorithm. Especially when the pulse width is large, targets cannot be resolved correctly. When A is properly selected, the pulse width will not affect range resolution as remarkably as in Table 2. Here, A is selected to be according to the numeri- cal simulation results.

Table 2: Range resolution using MUSIC algorithm after deconvolution under different pulse-width (SNR = 20 dB, A

Pulse width, ms 10 30 70 100

Range resolution, m 0.13 0.1 0.11 0.14

IEE Proc.-Radar, Sonur Navig., Vol. 144, No. 4, August 1997

4 Experimental verification

Experimental verification is performed by the high res- olution microwave imaging system with deconvolution processing shown in Fig. 2. Two rectangular horns are used in the measurement system, one for transmitting pulse modulated signal and the other for receiving returned echo. Transmitted coherent pulse is generated by oscillator, high-speed PIN switch and pulse modula- tor. The carrier frequency is 9.36GHz with pulse repeti- tion frequency 6MHz and pulse width 6811s. The system also include power amplifier and low noise amplifier.

T

r I I I 1 I I I I I I I

A targets

&

transmit t I ng antenna

receiving antenna

i P J 9 I R . - - - I - - - , I * attenuator v

phase shifter .'el matched

magic T

t spectrum analyser

h data sampling

power amplif ier

t I

1 p q modulator

t c\, oscillator 0

Fig. 2 Microwave imaging system with deconvolution processing

Measurement is carried out in spectral domain. Amplitude of spectra of transmitted and echo signals are recorded by sweeping spectrum analyser. Phase measurement of the spectra is realised by single channel time sharing phase detecting method [9] described as follows.

As shown by the dashed line block of Fig. 2, the main elements used in this phase detecting method are phase shifter and magic T. The attenuator is applied to shut off the reference channel so as to measure ampli- tude of the echo signal only. Labels 1 to 4 represent four ends of the magic T with 3 being the E entrance and 4 being the H entrance. Assuming that amplitude of reference and echo signals are R and A and phase difference is p, for an ideal magic T, output power of end 3 can be written as

1 2

P = -IR + Aejv12

When the amount of phase shifting of the reference channel is e,, P is changed to

P - -IReJez + Ae"'"j2 = 1 1

[R2 + A2 + 2RAcos(cp - OZ)] % - 2

(15)

179

The phase shifter is set to three different states and, respectively, P I , P2, P3 are detected. From eqn. 15, the following equations can be derived

t w = (16) K(COS& - C O S & ) - (COSOS - C O S O ~ ) -K(sin& -s in&) + (sin& - s in&)

1 p 2 - Pl cosy = - x RA (cos 8 2 - cos&) + tgy(sin 8 2 - sin el)

(17) where K = (P3 - P2)/(P2 - PI) .

Phase of the echo signal can be decided by eqns. 16 and 17. O , ( i = 1, 2, 3) are chosen to be O", 120" and 240" to minimise phase measurement error.

The measurement and signal processing procedure are as follows: (i) Record amplitude of the reference signal. (ii) Set phase shifter to three different states and detect PI, p2, p3. (iii) Shut off the reference channel and measure ampli- tude of the echo signal. (iv) Calculate phase of the echo signal by eqns. 16 and 17. (v) Carry out deconvolution processing of eqn. 12. (vi) Apply FFT or MUSIC algorithm to realise range resolution.

Fig. 3 shows the results of experiments for which the targets are two corner reflectors separated by 0.15m in the range co-ordinate, and the measured bandwidth is 50OMHz. Figs. 3a and b represent the results of using FFT and the MUSIC algorithm after deconvolution, respectively. In Fig. 3a, the two targets are not isolated because of bandwidth limitation in the method using FFT. It is clearly shown that only by nonlinear spectral estimation algorithm can the two closely placed targets be correctly resolved.

In Fig. lb, the range resolution, using FFT and the MUSIC algorithms, is tested when the measured band- width is changed in the range 100 - SOOMHz. The experimental results are in good agreement with the simulation results of Fig. la, which indicates the valid- ity of the above analysis and simulation.

5 Conclusion

A method of gaining range superresolution by applying a nonlinear spectrum estimation algorithm after fre- quency domain deconvolution of echo to the transmit- ted signal has been proposed. If the same range of resolution is desired by alternatively using the Fourier transformation, the bandwidth of the measurement sys- tem will need broadening several times. The price paid in this novel method is an increase of computer time which, hopefully, is not significant if compared to the resolution gained and the simplicity of the microwave imaging system. Numerical simulation and experimen- tal results have proved the method to be cost-effective.

0 1 2 3 1, 5 6 range, m

b Experimental results of applying FFT and the MUSIC algorithm Fig.3

after deconvolution Target separation = 0.15m in range co-ordinate Measured bandwidth = 500MHz a FFT b MUSIC

6

1

2

3

4

5

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8

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