Research Orientation

This paper introduces firstly the basic principle behind one-dimensional range imaging via utilization of a radar bandwidth in chapter 2. Result shows that the algorithm to matched filtering imaging is very good by analyzed, because the signal matched to the echo of targets includes not only phase but amplitude information. Basing on this, the algorithm about azimuth imaging via utilization of a synthetic aperture is analogized in chapter 3. These basic principles, and their associated data acquisition and digital signal processing issues, will assist us in developing more concrete high resolution SAR.

Chapter 4 discusses system modeling and imaging for broadside strip-map SAR within the framework of the spectral properties of the SAR radiation pattern that is developed in chapter 3. At the same time, the conclusion is validated by use of computer simulation. Then, the relationship between the trajectory and resolution can be analyzed as a result of the strip-map SAR modeling.

Chapter 2 Procedure of Range Processing

For older radar systems where no pulse compression was possible, the range resolution was improved by reducing the pulse length. The disadvantage is that the transmitted power is also decreased which is directly linked to the signal to noise ratio (SNR). By instead transmitting frequency modulated pulses and using pulse compression techniques, the pulse length can be increased and still achieve fine range resolution.

2.1 Basics of Range Processing 2.1.1 Range Resolution

Range resolution, denotes as ΔR, is a radar metric that describes its ability to detect targets in lose proximity to each other as distinct objects. Targets separated by at least ΔR will be completely resolved in range.

Assume that the two targets are separated by cT_p / 4_{, where} T_{p is the} pulsewidth. In this case, when the pulse trailing edge strikes target 2 the leading edge would have traveled backwards a distance cTp, and the returned pulse would be composed of returns from both targets, as shown in Fig. 2-1a.

However if the two targets are at least cT_p / 2 apart, then as the pulse trailing edge strikes the first target the leading edge will start to return from target 2, and two distinct returned pulses will be produced, as illustrated by Fig. 2-1b. Thus,

ΔR should be greater or equal to cT_p / 2. And since the radar bandwidth B

2.1.2 Linear Frequency Modulation Waveforms

Frequency or phase modulated waveforms can be used to achieve much wider operating bandwidths. Linear Frequency Modulation (LFM) ^[1] is commonly used. In this case, the frequency is swept linearly across the pulsewidth, either upward (up-chirp) or downward (down-chirp). The matched filter bandwidth is proportional to the sweep bandwidth, and is independent of the pulsewidth. Fig. 2-2 shows a typical example of an LFM waveform. The pulsewidth is T_p, and the bandwidth is B.

The LFM up-chirp instantaneous phase can be expressed by

( ) 2 ( 2) coefficient. Thus, the instantaneous frequency is

1 d ψ

= = + T^p T^p

− ≤ ≤

Atypical LFM waveform can be expressed by

2.1.3 Correlation and Convolution

Correlation^[2] is the process of matching two waveforms, usually in the time domain, to determine their degree of fit and to determine the time at which the maximum correlation coefficient, or best fit, occurs. Correlation can occur in either the continuous or discrete realms.

This correlation from involves signals which are continuous and periodic.

A mathematical description of continuous correlation is

( ) ( ) ( ) ( ) ( ) z t x t h t ^∞ xτ h t τ τd

= ⊗ =

∫

−∞ + (2-5)

In the process, one signal [x( )τ ] is hold stationary in time and the other [h t( +τ) ] is displaced in time and slides across it. At each point in the displacement, or sliding, process, the product of x and h is taken and the area under the product found. This area is the correlation of x and h at time.

The variable tau ( )τ is time for purposes of finding the area under the product.

The parameter t represents the amount of displacement of h from its normal time position where t =0. Note that on the right of the equation, t is a parameter; on the left, it is a variable. Thus the process is truly a transform in that it changes the independent variable. The correlation process is shown in Fig.

2-3.

Two types of correlation exist. If the two waves being matched are different, x t( )≠h t( ), a cross-correlation is performed. If they are the same,

( ) ( )

x t =h t , it is an autocorrelation. Autocorrelation are used to evaluation the suitability of waveforms to certain radar tasks and cross-correlation are used primarily in pulse compression.

Convolution is a process by which multiplications are transferred from one domain to the other. The relationship between multiplication and convolution is given below and in Fig. 2-4.

[ ( ) ( )] [ ( )] [ ( )] ( ) ( )

FT f t w ti = FT f t ∗FT w t = F f ∗W f (2-6)

The process of convolution is almost identical to that of correlation. The only difference is that one of the signals (it makes no difference which one) is reversed, or folded, in time before the displace-multiply-integrate operations.

Convolution of continuous functions is given below

( ) ( ) * ( ) ( ) ( ) z t x t h t ^∞ x τ h t τ τd

= =

∫

−∞ − (2-7) Graphically, convolution is shown in Fig. 2-5. Note its similarity to correlation as in Fig. 2-3; the only difference is time-reversal of h.

2.2 The Matched Filter

In radar applications we generally utilize the reflected known signal to detect the existence of a reflecting target. The probability of detection is related to the signal to noise ratio (SNR) rather than to the exact waveform of the signal received. Hence we are more interested in maximizing the SNR than in preserving the shape of the signal. A specific matched filter is a linear filter whose impulse response is determined by a specific signal in a way what will result in the maximum attainable SNR at the output of the filter when that particular signal and white noise are passed through the filter.

Matched filter^[3] can be derived for baseband as well as for bandpass real signals. For the latter case it will usually suffice to implement a filter matched to the complex of signal. Hence, we need to be able to design matched filters for complex signals as well.

The impulse response of the matched filter is

*

( ) ( 0 )

h t = Ks t −t (2-8)

This says that the impulse response is a delayed mirror image of the conjugate of the signal.

It can easily be derived using convolution between the signal and the matched filter impulse response,

*

( ) ( ) * ( ) ( ) ( ) ( ) [ 0 ( )]

s to s t h t ^∞ s τ h t τ τd ^∞ s τ Ks t t τ τd

−∞ −∞

= =

∫

− =

∫

− − ^(2-9)

If we assume that t₀ =0 _and K =1, then

( ) ( ) (* ) s to ^∞ s τ s τ t dτ

=

∫

−∞ − (2-10) the right-hand side of Eq. (2-10) is recognized as the autocorrelation function of

( ) s t .

We can now summarize the main results concerning the matched filter: The impulse response is linearly related to the time-inverted complex-conjugate signal; when the input to the matched filter is the correct signal plus white noise, the peak output response is linearly relate to the signal’s energy. At that instance, the SNR is the highest attainable, which is 2 /E N0; elsewhere, the response is described by the autocorrelation function of the signal.

2.3 LFM Pulse Compression

Linear frequency modulation pulse compression is accomplished by adding frequency modulation to a long pulse at transmission, and by using a matched filter receiver in order to compress the received signal.

The matched filter impulse response of s t( ) _is

( ) *( )

h t = s −t (2-11)

Substituting Eq. (2-4) into Eq. (2-11) yields

2 2

2 2

Combining Eq. (2-13) with Eq. (2-14) yields

2

Eq. (2-15) is the output of LFM signal passing through the matched filter. It is a signal of fixed carrier frequency f_c.

When t ≤T_p, the envelope is similar to a sinc function

t t

( ) ( )Rect( ) ( )Rect( )

2T 2T

s to =TSa πkTt =TSa πBt (2-16)

As illustrated in Fig. 2-7, 1

t = ± B is the first zero point when πBt = ±π ;

When

Bt π2

π = ± _, 1 t 2

= ± B .It is usually to define the presently pulsewidth as the compressed pulsewidth

1 1

2 x2

B B

τ = = (2-17)

As a result, the matched filter output is compressed by a factor ξ = BTp_,

where τ ' is the pulsewidth and B is the bandwidth. Thus, by using long pulses and wideband LFM modulation large compression ratio can be achieved.

The time axis is normalized in Fig. 2-8, (t/(1/ )B = ×t B). The result simulated in Fig. 2-8 is as good as the theory. The relative amplitude is -13.4 dB when the first zero point appears at ±1 ( 1

±B ). As illustrated in Fig. 2-9, it is equal to the analysis in theory and the relative amplitude is -4dB when the compressed pulsewidth is similar to 1

B ⁽ 1

±2B _).

2.3.1 Correlation Processor

Radar operations area usually carried out over a specified range window, referred to as the receive window and defined by the difference between the radar maximum and minimum range. Returns from all targets within the receive window area collected and passed through matched filter circuitry to perform pulse compression. One implementation of such analog processors is the Surface Acoustic Wave (SAW) devices. Because of the recent advances in digital computer development, the correlation processor^[4] is often performed digitally using FFT. This digital implementation is called Fast Convolution Processing (FCP) and can be implemented at base-band. The fast convolution process is illustrated in Fig. 2-9.

Since the matched filter is a linear time invariant system, its output can be described mathematically by the convolution between its input and its impulse response,

( ) ( ) ( )

y t =s t ∗h t (2-18)

where s t( ) is the input signal, h t( ) is the matched filter impulse response(reference signal), and the ∗ operator symbolically represents convolution. From the Fourier transform properties,

{ ( ) ( )} ( ) ( )

FFT s t ∗h t =S f iH f (2-19)

and when both signals are sampled properly, the compressed signal y t( ) can be computed from

( ) 1{ ( ) ( )}

y t = FFT⁻ S f iH f (2-20)

where FFT⁻¹ is the inverse FFT. When using pulse compression, it is desirable to use modulation schemes that can accomplish a maximum pulse compression ratio, and can significantly reduce the sidelobe levels of the compressed waveform. For the LFM case the first sidelobe is approximately 13.4dB below the main peak, and for most radar applications this may not be sufficient. In practice, high sidelobe levels are not preferable because noise and/or jammers located at the sidelobes my interfere with target returns in the main lobe.

Weighting function (windows) can be used on the compressed pulse spectrum in order to reduce the sidelobe levels. The cost associated with such an approach is a loss on the main lobe resolution, and a reduction in the peak value.

Weighting the time domain transmitted or received signal instead of t he compressed pulse spectrum will theoretically achieve the same goal. However, this approach is rarely used, since amplitude modulating the transmitted waveform introduces extra burdens on the transmitter.

Consider a radar system that utilizes a correlation processor receiver. The receive window in meters is defined by

rec max min

R = R −R (2-21)

where Rmax _and R_min, respectively, define the maximum and minimum range over which the radar performs detection. Typically Rrec is limited to extent of the target complex. The normalized complex transmitted signal has the form

( ) exp 2 ( 2)

c 2

s t = ^⎡⎢⎣j π f t+ kt ^⎤⎥⎦ 0≤ ≤t T_p (2-22)

Tp is the pulsewidth, k = B T/ _p, and B is the bandwidth.

The radar echo signal is similar to the transmitted one with the exception of a time delay and an amplitude change that correspond to the target RCS.

Consider a target at range R₁. The echo received by the radar from this target is

2

1 1 1

( ) exp 2 ( ( ) ( ) )

r c 2

s t =a ^⎡⎢⎣j π f t−τ + k t−τ ^⎤⎥⎦ τ¹ ≤ ≤ +t τ¹ T^p _(2-23)

where a₁ is proportional to target RCS, antenna gain, and range attenuation.

The time delay τ₁ is given by

1 2R c1/

τ = (2-24)

The first step of the processing consists of removing the frequency f_c. This is accomplished by missing s t_r( ) with a reference signal whose phase is 2π f t_c . The phase of the resultant signal, after low pass filtering, is then given by

and the instantaneous frequency is

1

2.3.2 Time Interval of Sampling

The spatial domain shown as Fig. 2-10 support band of the target is

0 0

where T_p is the duration of the pulsed radar signal. The echoed signals that are due to the reflectors that reside between the closest and farthest reflectors fall between the time points Tstart _and T_end.

Thus, to capture the echoed signals from all of the reflectors in the target region x∈[x_C −x x₀, _C +x₀], we have to acquire the time samples of the echoed signal in the following time interval^[5]:

[ _start, _end] t∈ t t

Note that the length of this time interval is

4 0

end start p

t t x T

− = c + (2-29)

2.3.3 Range Sampling Criteria

The number of samples, N, must be chosen so that foldover in the spectrum is avoided. For this purpose, the sampling frequency, f_s based on the Nyquist sampling rate, must be

s 2

f ≥ B (2-30)

and the sampling interval is

1

s 2

T ≤ B (2-31)

Using Eq. (2-26) it can be show that the frequency resolution of the FFT is 1

p

f T

Δ ≤ (2-32)

The minimum required number of samples is

1 p

Equating Eqs. (2-31) and (2-33) yields

2 _p

N ≥ BT (2-34)

Consequently, a total of 2BT_p real samples, or BTp complex samples, is sufficient to completely describe an LFM waveform of duration Tp _and

bandwidth B . For example, an LFM signal of duration T_p =2μs _and bandwidth B=10MHz requires 40 real samples to determine the input signal.

For better implementation of the FFT N is extended to the next power of two, by zero padding. Thus, the total number of samples, for some positive integer m, is

2^m

NFFT = ≥ N (2-35)

2.4 Simulation for Range Processing

As an example shown as Fig. 2-11, consider the case where the parameters are list in table 2-1.The simulation of range processing can be achieved by using MATLAB as Fig. 2-12, and the reference signal is

(

²

)

( ) exp ( )

ref r

s t = j k tπ −τ τ_r ≤ ≤ +t τ_r T_p (2-36)

The τ_r is given by

2 _mid

r

R

τ = c (2-37)

where R_mid is the center distance of receive window.

Because of the properties of Fourier transform

{ ( )} ( )

F s t = S w (2-38)

{ ( )} ( )

F s −t = − −S w (2-39)

* *

{ ( )} ( )

F s t =S −w ) (2-40)

It can be known that

* *

{ ( )} ( )

F s −t = −S w (2-41)

The simulation can be achieved by another way illustrated in Fig. 2-13.

Note that the compressed pulse range resolution is ten meters. Figs. 2-14 and 2-15 show the transmit signal and reference signal respectively. Fig 2-16 shows the uncompressed echo and the compressed matched filter output.

Chapter 3 Azimuth Processing

SAR synthesize a long antenna by transmitting electromagnetic energy and coherently adding the successively reflected and received pulses to obtain high resolution in azimuth direction. To achieve a coherent integration is called azimuth compression.

3.1 Azimuth Resolution

Azimuth (cross-range) is resolved with antenna beamwidth. The azimuth resolution of radars are

R R

A R R

D θ λ

Δ = = (3-1)

S S 2

syn

A R R

L θ λ

Δ = = (3-2)

where ΔAR is the azimuth resolution using the real antenna, ΔAS _{is the}

azimuth resolution using the synthetic antenna, θR _and θ_S are the 3-dB beamwidth of the real antenna and synthetic antenna respectively,D is the width of the real antenna, Lsyn is the effective length of the synthetic antenna,

λ is the wavelength, and R is the range to the targets being resolved.

Azimuth resolution is enhanced by narrowing the antenna beamwidth. With real antennas, this requires enlarging the physical antenna size or decreasing wavelength, which are often not possible because of physical constraints The same effect can be accomplished with a small real antenna moving to a number of locations to simulate a large antenna, called a synthetic, or synthetic aperture as illustrated in Fig. 3-1.

The azimuth resolution of a sidelooking SAR system is described in terms of the geometry of Fig. 3-2. The effective length of the synthetic antenna (Lsyn₎

is the distance the radar moves while a scatterer remains in the beam. It has the same value as the azimuth resolution of the real antenna. For small real beamwidths, this value is

R syn R

A L R R

D θ λ

Δ = = = (3-3)

The azimuth resolution of the synthetic antenna at range R is, by basic resolution relationships

S S

A θ R

Δ = (3-4)

Substituting Eq. (3-2) into Eq. (3-4) yields

S 2

syn

A R

L

Δ = λ (3-5)

Substituting Eq. (3-3) into Eq. (3-5) yields

S 2 A R D

R λ

Δ = λ (3-6)

Algebra reduces this equation to

S 2 A D

Δ = (3-7)

3.2 Properties of Echo Signal

In order to make it convenient for the mathematic analysis, the geometry of airborne SAR^[6] should be discussed firstly. As illustrated in Figs. 3-1 and 3-2, the aircraft flies along the straight line in x direction with uniform velocity v and altitude h . The antenna of airborne SAR transmits the radio waves broadside with regular slant angle β . Assume that he 3dB beamwidth is θ , the measured swath is W , the maximum synthetic aperture length is L.

The detected target is assumed as a ideal point target p and the slant range between target pand flight path x is R₀. Therefore, the coordinate plane can be composed of flight path x and slant range R₀. The aircraft is in the origin of coordinate when t =0 and the location of aircraft is x=vt in instantaneous time t. The location of point target p in this coordinate is fixed and its coordinate is (x R_p, 0) . In instantaneous time t , the distance R

2 2

The high frequency pulse which is transmitted from the antenna is periodic and coherent equally amplitude. Suppose the frequency is fc, the amplitude is A, the pulse repeatation frequency (PRF) is fr, the repeatation period (PRT)

is 1

r r

T = f , and the pulsewidth is τ . The first step is assuming the signal

transmitted from antenna is a continuous cosine wave for analysis. The actual transmitted periodic pulse is viewed as the samples of continuous wave and its sampling frequency is PRF fr. The amplitude of cosine wave is normalized as 1 and the beginning phase is 0. As a result, It can be expressed by using exponential function with complex number

2 0( ) ^{j f tc}

S t =e ^π (3-10)

The signal after transmitted from antenna is a type of electromagnetic wave.

When arriving the target p, the electromagnetic wave is started to scatter, and a part of backscatter energy is received by antenna, thus, being called echo signal S t_r( ). Assume that the RCS of point target p is σ , S t_r( ) _{can be} expressed by

2 ( 0)

Substituting Eq. (3-9) into Eq. (3-12) yields

2 2

Substituting Eq. (3-13) into Eq. (3-11), the echo signal can be expressed by

2

After simplified and normalized, we get

2

where λ denotes transmitted wave length (

c

c

λ = f ). Choosing the real part and making it to become the form of cosine function

2

Observing the phase of this signal, it is composed of three terms and can be

can be neglected when observing variation of phase.

2

most important phase term and is the key in technique of SAR signal processing.

And x=vt, xp =vtp_{, where} t_p denotes the among of time which the aircraft

It is the quadratic phase term changing with time in the presence of square-law, where v is the velocity of aircraft. Differentiating the phase with respect to time time, then divided by 2π , the instantaneous frequency can be obtained

2

where f_c is the carrier frequency of transmitted signal. The second is the Doppler history caused from the relative motion between antenna and target and it is usually denoted as

2

It changes with time linearly and it is perceived that the echo signal is a linear frequency modulation waveform where the chirp rate is

2

For the convenience of further discussion on the property of echo signal, the variation of phase and frequency of echo signal was compared with that of transmit signal as illustrated in Fig. 3-3.

As shown in Eq. (3-20) and Fig. 3-3, the Doppler shift caused from point target p ranges between negative and positive with a center at t =tp_{. When}

t =tp, the antenna will be located at the slant range between flying path and target p. The quantity of aircraft velocity v along the direction of target p is the radial velocity and is equal to zero. Before t =tp_, f_d is positive and its maximum value occurs at

/ 2 synthetic aperture interval. At this moment, the Doppler shift is

2

At this moment, the Doppler shift is

2

2

The bandwidth of Doppler history of target p can be obtained From either Eqs.

(3-23) and (3-25) or Fig. 3-3, namely, identical to the vertical slant range of the flying path, both being R_{0. However} their azimuth is different. As illustrated in Fig. 3-4, their coordinate in x direction are x1 _and x₂ respectively. According to the previous discussion, the echo signal of two targets are both linear modulation signal and their bandwidth is equal to their Doppler shift that is

2

But It is different to start and end of the Doppler frequency. In frequency domain, the instantaneous frequency of two echo signal is not the same time.

When the location of aircraft is x=vt, the instantaneous frequency of first echo signal is

1 1

and the instantaneous frequency of first echo signal is

2 2

The difference of both Doppler frequency is

2 1 2

If the difference can be recognized, the distance between two targets can be resolved either.

In the technique of radio, the method that can solve this problem is called correlation and the procedure of autocorrelation is equal to the matched filter. If the linear modulation of echo signal is used in matched filter, the output is presented as a sinc function and it is called azimuth compression^[7]. For this

In the technique of radio, the method that can solve this problem is called correlation and the procedure of autocorrelation is equal to the matched filter. If the linear modulation of echo signal is used in matched filter, the output is presented as a sinc function and it is called azimuth compression^[7]. For this

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