Signal Processing and Time Delay Resolution of Noise Radar - PIERS

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Signal Processing and Time Delay Resolution of Noise Radar System Based on Retrodirective Antennas V. V. Chapursky1 , V. A. Cherepenin2 , and V. I. Kalinin2 1

2

Bauman Moscow State Technical University, Russia Institute of Radioengineering and Electronics, Moscow, Russia

Abstract— Theoretical comparison results of basic noise signal processing methods for one channel retrodirective noise radar system are obtained. Three methods of noise signal processing were studied: cross-correlation processing with delayed reference transmitter signal; double spectrum processing of received signal; cross-correlation processing with recirculated reference signal at various reference time delays.

1. INTRODUCTION

One of the most interesting self-steering technologies for UWB radar is application of noise sounding signals. In recent years, investigations for the target observation method using wideband noise radar with retrodirective antennas have been resumed [1–3]. The retrodirective antenna principle of sounding is based on recirculation of a noise signal in the spatial feedback loop. The loop is closed by means of combining a portion of the received radar signal with the current signal of the transmitter and radiation of the obtained sum signal toward the target. The given method theoretically allows to improve radar resolution and detection ability, including at MIMO radar surveillance [4]. At the same time in the elaboration of noise signal processing methods in radar with spatial recirculation there is a number of unresolved questions to the part of which the contents of the given work is devoted. This report presents theoretical comparison results of basic noise signal processing methods for one channel retrodirective noise radar system: 1) cross-correlation processing with delayed reference signal of noise transmitter generator; 2) double spectrum processing of received recirculation noise signal; 3) cross-correlation processing with reference noise signal in special design recirculator based on various reference time delays. 2. CROSS CORRELATION PROCECCING WITH REFERENCE NOISE SIGNAL

A bloc diagram of the cross-correlation retrodirective noise radar system is shown in Fig. 1. A noise generator forms noise signal s(t) with rectangular envelope A(t) of duration T : s (t) = A (t) ξ (t) ,

(1)

where ξ (t) — stationary noise process with zero mean value, correlation function kξ (τ ) = M{ξ(t)ξ(t + τ )} (M is a symbol of mean value) and correlation time τξ . The recirculation of noise signal is performed in the spatial feedback loop with target time delay τt . In the case when envelope duration T and target delay τt are satisfied conditions T À τt À τξ then analysis is simplified and complex transmission coefficient of the spatial feedback loop with time delay τt can be calculated at stationary approximation as [3]: K˙ (ω) =

KΣ exp (−jωτt ) , 1 − γKΣ exp (−jωτt )

(2)

where KΣ is transmission coefficient of the unclosed spatial feedback loop without attenuator loss γ [5]. Total transmission coefficient of closed spatial feedback loop satisfies the important condition KΣ γ < 1 when self-oscillations are not excited in closed feedback loop including transmitter and receiver amplifiers. Consequently to this assumption noise waveform η (t) on the output of spatial feedback loop (or receiver output) can be written as η (t) =

∞ X k=0

KΣk+1 γ k ξ (t − (k + 1) τt ),

(3)

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The average value evaluation of cross-correlation function on the correlator output in Fig. 1 is expressed as ∞ X kξη (τ ) = M {ξ (t − τ ) η (t)} = KΣk+1 γ k kξ (τ − (k + 1) τt ). (4) k=0

The expression (4) of cross-correlation function contents a set of equidistantly components placed at multiple points τk = (k + 1) τt where k = 0, 1, 2, 3, . . .. The form of each component coincides with correlation function kξ (τ ) of initial noise signal ξ (t). The level of correlation components is decreased as law (KΣ γ)k when time delay τk is increased. Therefore the range (time delay) ambiguity of target detection is appeared as result of spatial recirculation of sounded signals. Analysis of correlation function (4) shown in Fig. 3 as solid curve confirms that space (time delay) resolution of retrodirective noise radar with delayed noise reference is only defined frequency bandwidth of noise signal ξ (t) as well as correlation noise radar without any recirculation. 3. DOUBLE SPECTRUM PROCESSING OF RECEIVED RECIRCULATION SIGNAL

Wideband noise radar with double spectrum processing (DSP) was proposed in basic work [5] and in detail investigated on the object of signal/noise ratio in [6]. Here retrodirective antenna-based noise radar with double spectrum processing is considered. Its block diagram shown in Fig. 2 contents new elements: second linear summator, first SA1 and second SA2 spectrum analyzers unlike correlation noise radar in Fig. 1. Double spectrum processing is performed by means of successive connected the first SA1 and second SA2 spectrum analyzers. Three different radar schemes with double spectrum processing are examined: 1) received noise waveform is applied on the input of SA1 when switch is placed at position 1; 2) sum of received and noise generator waveforms are applied on the input of SA1 when switch is placed at position 2; 3) sum of received noise and inner feed-back loop reference waveforms are applied on the input of SA1 when switch is placed at position 3. Atr Noise generator

Target

Power amp

+

τt Var delay

Arec

Atten

τ Correlator

Receiver

Figure 1: Bloc diagram of retrodirective noise radar with cross correlation processing.

Atr

2 1

τt

Arec

3

SA1

SA2

Figure 2: Bloc diagram of retrodirective noise radar with double spectral processing.

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Figure 3: Simulated correlation functions for three DSP schemes.

Complex transmission coefficients are determined for the three above mentioned cases as KΣ exp (−jωτt ) , 1 − γKΣ exp (−jωτt ) KΣ exp (−jωτt ) K˙ 2 (ω) = 1 + , 1 − γKΣ exp (−jωτt ) (1 + γ) KΣ exp (−jωτt ) K˙ 3 (ω) = 1 + . 1 − γKΣ exp (−jωτt ) K˙ 1 (ω) = K˙ (ω) =

(5) (6) (7)

In the first case results of double spectrum and cross correlation processing from Section 2 exactly coincide. If power spectrum Sξ (ω) is rectangular approximated by Sξ0 in a frequency bandwidth (ωL , ωH ) then because symmetric properties of Fourier transform the output of DSP can be written as ZωH¯ ¯2 ¯ ¯ ki (τ ) = 2Sξ0 ¯K˙ i (ω)¯ cos (ωτ ) dω,

i = 1, 2, 3,

(8)

ωL

Formula (8) is used for calculation of output correlation functions in the three DSP schemes. It is let following parameters for DSP noise radar based on retrodirective antenna: fL = 100 MHz; fH = 500 MHz; KΣ = 1; γ = 0.5. The time delay τt of reflected signal is established as τt = 15 ns. Output correlation functions simulated for all DSP schemes are presented in Fig. 3. It can be clearly seen in this figure that all DSP schemes and cross correlation processing from Section 2 have alike correlation functions with the same width of correlation peaks. Therefore, all DSP and cross correlation schemes are characterized by the same range (time delay) resolution. 4. CROSS CORRELATION PROCESSING BASED ON RECIRCULATED REFERENCE

In this paper we present new method of retrodirective radar detection with cross-correlation processing based on recirculation noise reference. A block diagram of that noise radar is shown in Fig. 4. Radar transmitter contents additional recirculation loop with variable time delay τ and recirculation coefficient γ0 < 1. The output of the recirculation reference loop is fed to first input of a correlator. Its second input is connected to one of the receiver output. The other output of a receiver closes the recirculation loop with the spatial feedback loop having total time delay τt . The proposed method has the highest theoretical and practical interest and therefore correlation signal processing is analyzed in detail here. It is assumed that noise signal s(t) = A(t)ξ(t) has rectangular envelope A(t) of finite duration T . Received waveforms η(t, τt ) and recirculation reference η0 (t, τ0 ) can be expressed analogically (3) as η (t, τt ) =

∞ X k=0

KΣk+1 γ k A (t − (k + 1) τt ) ξ (t − (k + 1) τt ),

(9)

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Atr

τ

Arec γ0

τ

Figure 4: Bloc diagram of retrodirective noise radar based on recirculation reference.

Figure 5: Comparision of classic and retrodirective noise radars resolution.

η0 (t, τ0 ) =

∞ X

Figure 6: The view of grating and combination lobes in noise radars ambiguity function.

γ0k A (t − (k + 1) τ0 ) ξ (t − (k + 1) τ0 ).

(10)

k=0

Mean value of correlation integral for signals (9) and (10) is defined as result of integral calculating at current time t and statistic averaging:  ∞   Z ¯ (τt , τ0 ) = M Q η (t, τt ) η (t, τ0 ) dt   =

−∞ ∞ X

KΣk1 +1 γ k1 γ0k2 [b (∆k1 ,k2 (τ0 , τt )) − a (∆k1 ,k2 (τ0 , τt ))]

k1 ,k2 =0

×h (b (∆k1 ,k2 (τ0 , τt )) − a (∆k1 ,k2 (τ0 , τt ))) · kξ (∆k1 ,k2 (τ0 , τt )) ,

(11)

where ∆k1 ,k2 (τ0 , τt ) = (k2 + 1) τ0 − (k1 + 1) τt , a (∆) = max (0, ∆), b (∆) = min (T, T + ∆). Expression (11) for averaged cross-correlation functions of recirculation signals (9), (10) is general and valid for arbitrary duration T of noise impulse s(t) = A(t)ξ(t). Averaged crosscorrelation (11) is evaluated when large duration is assumed T À τt À τξ with the purpose of a comparison between previously obtained results. It is let the same parameters of wideband noise signal fL = 100 MHz, fH = 500 MHz as in Sections 2, 3. Other parameters are equal KΣ = 1, γ0 = 1, γ = 0.9, τt = 50 ns. Reference delay τ0 is varied from 40 ns to 60 ns in the time interval centered on target delay τt = 50 ns. In Fig. 5 output cross correlations are shown for noise radar based on retrodirective antennas in case with delayed convention noise reference according to Section 2 (dash curve) and in case with recirculation noise reference according to Section 4 (solid

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curve). A comparison between solid curve and dash curve indicates that solid curve width is some times narrow as dash curve width. Consequently space resolution of retrodirective noise radar with recirculation reference exceeds some times then in case with delayed reference. Output cross correlation functions (solid and dash in Fig. 6) simulated on large time delay interval content a set of intensive combination lobes which can cause target detection ambiguity on time delays multiple of target delay τt = 50 ns. This ambiguity can be eliminated by means of time strobing for example decreasing noise impulse duration. 5. CONCLUSION

The design principles of wideband noise radar based on retrodirective antennas are described. The correlation and double spectrum processing methods are discussed. Analytical relationships for output cross-correlations calculation are derived. Numerical results have allowed to reveal presence of recirculation maxima on delays, multiple to a basic target delay and also to estimate and to compare resolution of the considered methods on a delay (range). There was, that only in cross correlation processing with recirculated transmitter reference on an interval of reference delays it is probable an achieving resolution ability on range in some times surpassing resolution of classical noise radar and actually reaching the superresolution effect. By virtue of high resolution this method demands the further profound studying, especially regarding for the development of combinational response rejection algorithms. ACKNOWLEDGMENT

Researches were carried out at the financial support of the Russian Foundation of Basic Research, grants 07-07-00195-a, and 07-02-00351-a. REFERENCES

1. Gupta, S. and T. R. Brown, “Noice-correlating radar based on retrodirective antennas,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 43, No. 2, 472–479, April 2007. 2. Chapursky, V. V. and V. I. Kalinin, “Ultrawideband noise radiolocation on basis of retrodirective antenna arrays with signal recirculation,” Radio Engineering and Electronics, Vol. 53, No. 10, 1266–1277, 2008 (in Russian). 3. Kalinkevich, A. A., M. S. Krylova, and M. S. Turygin, “Ultrawideband signals in a radars, communications and acoustics,” Record of All Russian Science Conf., 415, Murom, Russia, July 1–3, 2003 (in Russian). 4. Lesturgie, M., J. P. Eglizeaud, D. Muller, B. Olivier, and C. Delhote, “The last decades and the future of low frequency radar concepts in France,” RADAR 2004, International Conference on Radar Systems, 1SE-PLEN-3, Toulouse, France, 2004. 5. Poirier, J. L., “Quasi-monochromatic scattering and some possible radar applications,” Radio Science, Vol. 3, No. 9, 881–886, 1968. 6. Chapursky, V. V. and V. I. Kalinin, “Efficiency of the double spectral analysis in noise radiolocation at presence of local subjects reflections,” Radio Engineering and Electronics, Vol. 51, No. 3, 303–313, 2006 (in Russian).