S f s t j ft dt. S f s t j ft dt S f. s t = S f j ft df = ( ) ( ) exp( 2π

Transcription

1 Reconfigurable stepped-frequency GPR prototype for civil-engineering and archaeological prospection, developed at the National Research Council of Italy. Examples of application and case studies. Raffaele Persico

2 Stepped frequency: underlying concepts

3 The delta function

4 Reminds ( ) = ( ) exp( 2π ) ( ) = ( ) exp( 2π ) = ( ) + ( ) ( ) exp( 2π ) s t = S f j ft df = + S f s t j ft dt + S f s t j ft dt S f = + 0 ( ) ( π ) 2 Re S f exp j2 ft df

5 Convolution Reminds + ( ) ( ) ( ) f f = g t = f f t d τ τ τ Convolution and Fourier Transform ( ) = ( ) ( ) FT g g G f G f ( ) FT g g = G G

6 Reminds Sampling property of the delta + + ( ) f ( ) d = f ( 0) δ τ τ τ ( ) f ( ) d = f ( ) δ τ τ τ τ τ + o ( ) ( ) ( ) f δ = f τ δ t τ dτ = f t o

7 Ideal sampling of the spectrum Re S ( f ) exp( j2π ft ) δ ( f f n f + o ) df = 0 n= { ( ) ( ( ) )} N S f n f j π f n f t o o = 2 Re + exp 2 + = N n = 1 n= 1 ( ) ( ) ( π ( ) ) ( ) π ( ) 2 Re S f n f cos 2 f n f t Im S f n f sin 2 f n f t = o o o o N n = 1 ( ) π ( ) ( ) 2 = S f + n f cos 2 f n f t S f n f o o o f

8 Expression of the received sinthetic pulse for 2N+1 samples, constant radiated spectrum, constant attenuation and phase of the reflected signals (( N + ) π ft) sin 2 1 s( t) 2 f cos 2π ft+ θ c sin π ft ( ) ( ) The synthetic pulse is replicated, but the replicas are in general not equal to each other

9 Flux diagram A sequence of sinusoids is transmitted ( π ) = π ( + ) ( ) min cos 2 ft cos 2 f n f t n A sequence of sinusoids is retrieved ( ) ( ) ( ) min Acos 2π ft+ ϕ = Acos 2π f + n f t+ ϕ n n n n n Amplitude and phase A n and ϕ n are extracted The received harmonics are summed

10 In synthesis but the pulse is replicated with pseudo-replicas at distance 1/ f

11 Trigonometric reminds 1 1 ( a) = + ( a) cos 2 cos ( a) = ( a) sin 2 cos sin cos sin 2 2 ( a) ( a) = ( a)

12 Homodyne demodulation A cos 2 ( π ft+ ϕ ) n n n 2cos 2 ( π ft) n

13 ( π + ϕ ) ( π ) = ( π ) ( π ) ( ϕ ) ( π ) ( ϕ ) 2 ( ϕ ) ( π ) ( π ) ( π ) ( ϕ ) 2A cos 2 ft cos 2 ft Upper branch (in phase component) n n n n = 2A cos 2 ft cos 2 ft cos sin 2 ft sin = n n n n n n = 2A cos cos 2 ft 2A sin 2 ft cos 2 ft sin = n n n n n n n 1 1 = 2A cos( ϕ ) cos( 4 ) sin ( ) sin ( 4 n n + π ft A ϕ π ft n n n n ) 2 2 ( ϕ ) after filtering we achieve A cos = I n n n

14 Lower branch (quadrature component) ( π + ϕ ) ( π ) = ( π ) ( π ) ( ϕ ) ( π ) ( ϕ ) 2 ( ϕ) ( π ) ( π ) ( ϕ ) 2A cos 2 ft sin 2 ft n n n n = 2A sin 2 ft cos 2 ft cos sin 2 ft sin = n n n n n n = Acos sin 4 ft 2A sin 2 ft sin = n n n n n 1 1 = Acos( ϕ ) sin ( 4π ft) 2A cos( 4 ft) sin n n n n π n ( ϕn) 2 2 ( ϕ ) after filtering we achieve A sin = Q n n

15 Final result for each harmonic signal ( ) cos( 2π ) sin ( 2π ) h t = I ft + Q ft = n n n n n ( π ft ϕ ) = Acos 2 + n n n Synthetic pulse N ( ) ( ) s t = h t n n= 1

16 Drawbacks of the homodyne demodulation The baseband signal is subject to more noise, in particular the flicker noise, approximately decreasing as 1/f or as a roof function. It is difficult to translate in baseband N signals from N different frequencies keeping uniformly low the noise. For this reason it usually preferred an heterodyne demodulation scheme.

17 Trigonometric reminds 2 cos a cos b = cos a+ b + cos a b 2sin ( a) cos( b) = sin ( a+ b) + sin ( a b) ( ) ( ) ( ) ( )

18 Translation of the radiated signal on the frequency axis 4cos 2 ( π ft) L cos 2 ( π ft) n ( ) ( π f f t) 2cos 2 n L

19 Translation of the received signal on the frequency axis ( ) ( π f f t) 2cos 2 n L A cos 2 ( π ft+ ϕ ) Acos( 2π ft+ ϕ n L n) n n n

20 Heterodyne demodulation I n A cos 2 ( π ft+ ϕ ) n L n 2cos 2 ( π ft) L Q n

21 Averaged measurement of the in-phase and quadrature components I = m Q = m I + I +... I 1m 2m Nm N Q + Q +... Q 1m 2m Nm N (N is chosen by the manufacturer)

22 Power of the noise (white noise) on each received harmonic signal N = K B K TB B = J K o SNR S ( f ) 2 K TB B

23 Truncated sinusoid

24 Integration time T int

25 Spectrum of any radiated truncated n harmonic signal ( ) cos( 2π ) s t ft = Π n T int t FT Π = T sinc π ft T int int int t ( ) 1 FT f t f f f f 2 ( cos( 2π )) = δ ( ) + δ ( + ) n n n { } T ( ) int S f = T ( f f ) + T f + f 2 { sinc( π ) sinc( π ( ))} int int n n n

26 Let us consider the pieces sinc ( πt ( f f )), sinc πt ( f + f ) int n int n ( ) The main lobe is wide 1/T int for both pieces SNR ( ) 2 ( ) 2 S f T S f n int n KTB KT B n B Prolonging the integration time the SNR increases, but the measurement requires more time

27 The integration time is a default parameter chosen by the manufacturer. Possibly, there is the possibility to extend the default integration time times a factor. The integration time is usually the same for all the harmonics spanned by the stepped frequency system.

28 NON-AMBIGUOUS TIME INTERVAL The time windows where we can reliably examine the synthetic pulses is given by T max 1 = f The depth investigated cannot exceed the non-ambiguous level D max ct = max = f 2 2 c

29 The frequency step cannot exceed the maximum value: c f = 2D 2 max r c o εµ r D max N.B.: D max depends on the penetration of the signal, not on the maximum depth of interest.

30 Effect of possible imperfections on the demodulation chain A cos 2 ( π ft+ ϕ ) n L n 2cos 2 ( π ft) L

31 r Received synthetic pulse for a target at depth level t o (the spectrum of the signal is considered flat in its band, sampled with 2N+1 frequencies) ( ) s t K ( N ) π f ( t to ) π f ( t t ) 2 2 ε α sin α + 1 cos 2π fc( t to) θ tg ( ) ( ) o ε + sin ( N + ) π f ( t+ to ) π f ( t+ t ) ( ) ( ) o 2 2 ε α sin α + cos 2π fc( t to) θ tg ε sin 1 Hermitian image at f t o

32 Signal to Hermitian image ratio SHR r ( ) s t K ( N ) π f ( t t ) π f ( t t ) ( ) ( ) 2 2 ε α sin α + 1 cos 2π fc( t to) θ tg ε + sin ( N + ) π f ( t+ t ) π f ( t+ t ) ( ) ( ) 2 2 ε α sin α + cos 2π fc( t to) θ tg ε sin SHR 2 2 ε α ε α ε + α + 4 4

33 Counteraction against Hermitian images: halving the frequency step at parity of maximum investigated time depth t 0 t 1 o o f c f = 4D 4 T max max 1 = 2 f r c o εµ D r max 1 f t o

34 The reconfigurable GPR system (50 MHz-1 GHz)

35 Reconfigurable antennas

36 The matrix church of Parabita

37

38 SLICE ACHIEVED WITH A RIS-HI MODE PULSED SYSTEM

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41 The co-cathedral of St. John in Malta

42 Chapel of Aragon

43 Chapel of the Sacristy

44 The Tomb of the Pillar

45 Depth 1.26 m

46 Depth 1.61m

47 Depth 1.75m

48 Depth 2.20m

49 Depth 2.45m Apparent thickness of the tomb 49 cm, real thickness 2.1 m

50 The area of the cryptoporticus in Egnazia 30 m 0 m Depth 40 cm

51 30 m 0 m Depth 94 cm

52 30 m 0 m Depth 150 cm

53 The church of Santa Croce in Gravina in Puglia Area 3 Area 1 Area m

54 Depth 22 cm

55 Depth 77 cm

56 Depth 158 cm

57 The Archaic Ditch of Manduria Pulsed GPR Prototype Depth 360 cm

58 Reconfigurable integration times f n T int n Reconfigurable radiated power at each frequency

59 The reconfiguration of the integration times as strategy again interferences in-phase components Interferences in-quadrature components samples

60 Variance of the samples = = N Q Q Q N Q Q Q N I I I N I I I N N Q N N I σ σ

61 An experiment

62 Interference of the transceivers

63 Signal with the default integration times

64 Signal with the reconfigured integration times

65 An experiment in the field

66 Interference of the repeater

67 Signal with the default integration times

68 Signal with the reconfigured integration times Increasing of the comprehensive measurement time less than 5% in both cases.