Design of Optimum Weighting Functions for LFM Signals

Linear frequency-modulated (LFM) pulse signals are probably the most common type of pulse compression waveforms for various radar systems (Barton, 2005; CookB Curlander & McDonough, 1991; Levanon & Mozeson, 2004; Richards, 2005; Skolnik, 2008). LFM signals are also often the waveform of choice for wideband systems, where the required bandwidth may be hundreds of megahertz. The ambiguity function of the LFM signal suffers from significant sidelobes, both in delay (range) and in Doppler. It is known, for example, that the first range sidelobe is approximately 13 dB below the main peak of the ambiguity function. Such sidelobes may be unacceptable in many applications due to system performance degradation caused by high sidelobes (Cook & Bernfeld, 1967; Levanon & Mozeson, 2004; Richards, 2005). To suppress the sidelobes some form of weighting can be applied to the matched filter response. The main drawbacks associated with conventional weighting functions (e.g., Hamming, Kaiser windows) are the broadening of the main lobe of the ambiguity function cut along the time axis and an inevitable attenuation in the peak response which decreases the signal-to-noise ratio. The chapter provides theoretical justification for a new approach, which is being applied to the design of discrete weighting function, or in other words, digital mismatched receiving filters. This approach considers the design of weighting functions as a problem of finding such a digital mismatched filter that will maximize the proportion of the total response power that is concentrated in the specified time-frequency region. Two applications of the proposed approach are theoretically addressed in sections 2 and 3. First, in section 2, we apply it to the problem of the optimum Doppler-tolerant pair signalfilter design when a given signal is specified to be an LFM signal. Section 3 addresses the specification of weighting functions in interferometric synthetic aperture radars with the purpose of improving the height measurement accuracy. Both of these sections are supplimented with numerical results, which demonstrate benefits that one can derive from using the proposed optimum weigthing functions as compared to conventional weighting functions. Conclusions are given in section 4.


Introduction
Linear frequency-modulated (LFM) pulse signals are probably the most common type of pulse compression waveforms for various radar systems (Barton, 2005;Cook&Bernfeld, 1967;Curlander & McDonough, 1991;Levanon & Mozeson, 2004;Richards, 2005;Skolnik, 2008). LFM signals are also often the waveform of choice for wideband systems, where the required bandwidth may be hundreds of megahertz. The ambiguity function of the LFM signal suffers from significant sidelobes, both in delay (range) and in Doppler. It is known, for example, that the first range sidelobe is approximately 13 dB below the main peak of the ambiguity function. Such sidelobes may be unacceptable in many applications due to system performance degradation caused by high sidelobes (Cook & Bernfeld, 1967;Levanon & Mozeson, 2004;Richards, 2005). To suppress the sidelobes some form of weighting can be applied to the matched filter response. The main drawbacks associated with conventional weighting functions (e.g., Hamming, Kaiser windows) are the broadening of the main lobe of the ambiguity function cut along the time axis and an inevitable attenuation in the peak response which decreases the signal-to-noise ratio. The chapter provides theoretical justification for a new approach, which is being applied to the design of discrete weighting function, or in other words, digital mismatched receiving filters. This approach considers the design of weighting functions as a problem of finding such a digital mismatched filter that will maximize the proportion of the total response power that is concentrated in the specified time-frequency region. Two applications of the proposed approach are theoretically addressed in sections 2 and 3. First, in section 2, we apply it to the problem of the optimum Doppler-tolerant pair signalfilter design when a given signal is specified to be an LFM signal. Section 3 addresses the specification of weighting functions in interferometric synthetic aperture radars with the purpose of improving the height measurement accuracy. Both of these sections are supplimented with numerical results, which demonstrate benefits that one can derive from using the proposed optimum weigthing functions as compared to conventional weighting functions. Conclusions are given in section 4.

Design of Doppler-tolerant weighting functions
In this section we develop a method for designing weighting functions or, in other words, mismatched filters for LFM signals in order to minimize the sidelobe level at the filter output with respect to that of the LFM ambiguity function and preserve its Doppler-tolerant behaviour as much as possible. First, we review the Doppler tolerance of LFM signals in terms of their analog and digital ambiguity functions, then we briefly discuss the concept of digital cross-ambiguity function and then proceed to the design of the optimum Dopplertolerant weighting functions and numerical examples.

Doppler tolerance of LFM signals
The complex envelope u(t) of a linear frequency modulated pulse (LFM pulse or signal) of duration T , bandwidth B and unity energy is given by where α is the frequency slope. The time -bandwidth product of the LFM signal is simply defined as BT. Hence, for the LFM pulse to be qualified as a pulse compression waveform, this product has to meet the condition BT >> 1. The ambiguity function χ( , ) of an LFM pulse can be analytically presented as (2-2) where τ is the shift in time and ν is the Doppler shift of the received signal relative to the nominal values expected by the matched filter. From (2-2), the peak of this sinc-like function occurs when + α =0, i.e., at points lying on the straight line = ∓ T/B. All the peaks occur along this line represent a so-called skewed "ridge" of the ambiguity function of LFM signals. Hence, when = s ≠ 0 , which means that there is a Doppler mismatch between an LFM signal received from a moving point target and corresponding matched filter, the peak of (2-2) will not occur at = 0 as it takes place when there is no Doppler mismatch ( = 0 ). Instead, the peak will be shifted with respect to the point = 0 by an amount s that is proportional to the Doppler shift s / TB This phenomenon is termed "range-Doppler" coupling. An estimate of the target range is based on the position of this peak in time. Hence, any non-zero Doppler shift s of an LFM signal results in a range measurement error proportional to the shift in time specified by equation (2-3). As also follows from (2-2), taking into account (2-3), the amplitude of the peak will be reduced by the factor (2-4) Let the reduction factor be bounded below by a value of b; it is clear that 0≤ b ≤ ≤1. Using (2-4) yields the corresponding inequality for the acceptable Doppler shift (2)(3)(4)(5) and, from equation | s |=2|V t |/ , for the radial target velocity V t www.intechopen.com where λ is the wavelength of the transmitted signal and b is specified in decibels. Assuming B = 5 MHz, λ = 0.03m, and b = −0.1dB (this corresponds to b=0.9886≤ ≤ 1) and using (2-5) and (2-6) yields |ν s | / B ≤ 0.0114 and |V t | ≤ 3090 km/h, respectively. Hence, even for such a wide range of radial target velocity the reduction in amplitude at the output of the LFM matched filter does not exceed 0.1 dB (relative to the output peak for a zero-Doppler shift). Thus, one can conclude that Doppler tolerance is an intrinsic feature of the LFM signal, which reveals itself through the ridge of the LFM ambiguity function. Athough the skewed ridge associated with the LFM ambiguity function causes errors in range and/or Doppler measurements it should be noted that the Doppler-tolerance of LFM signals can be an advantage in some applications. An example is in a search application since a preffered waveform to be used for target search should be Doppler-tolerant, so large Doppler shifts of targets whose velocity is not known does not prevent their detection due to a weak peak of the response at the matched filter output. This is also beneficial in terms of simplifying signal processor and detection hardware. If one were to use a biphase-coded signal in the same application one would need to have a bank of signal processors matched to a range of expected target Doppler shifts. This is because of the biphase-coded signal "thumbtack" ambiguity function, which features a single narrow central peak concentrated at the point = = 0 , with the remaining energy spread uniformly all over the ( , ) plane (Levanon & Mozeson, 2004;Richards, 2005). Such features are useful for systems intended for high-resolution measurements in range and Doppler, or radar imaging. Fig.2-2 compares the contours for the LFM ambiguity function and the ambiguity function of a biphase-coded signal. These contours correspond to a 3.92 dB level (Barton, 2005;Peebles, 1998) below maximum at ( = 0, = 0). Both of the signals have equal pulse duration of T=MT 0 and bandwidth of B ≈ 1/T 0 , where M is the number of subpulses for the biphasecoded signal and T 0 is the duration of each subpulse. A disadvantage of the LFM ambiguity function is that it suffers from significant sidelobes, both in delay (range) and in Doppler. Consider, for example, a zero-Doppler cut of (2-2) χ( , 0), which is just the LFM matched filter output when there is no Doppler mismatch.
Analyzing χ( , 0) under the condition BT >> 1 , yelds that it exhibits the distinctly sinc-like main lobe and first few sidelobes (in magnitude). Due to this the first (and highest) range sidelobe in χ( , 0) is approximately equal to -13 dB and this value does not depend on the time-bandwidth product BT . Such a high sidelobe level is unacceptable in many applications since it may cause appreciable system performance degradation (Cook & Bernfeld, 1967;Levanon & Mozeson, 2004;Richards, 2005;Rosen et al., 2000).

Digital ambiguity function
The discussion in section 2.1 deals with analog LFM ambiguity function, i.e. it assumes that the LFM signal in radar receiver is processed by an ideal analog matched filter. Modern radar systems employ digital technologies so that it is instructive to discuss the features of a digital LFM ambiguity function that characterizes the response of a radar receiver when matched filtering is implemented by a digital signal processor. First, we introduce a general relationship between the analog ambiguity function (analog matched filtering) and digital ambiguity function (digital implementation of the matched filter). In case of a point target the complex envelope of the received signal is given (ignoring where subscript a stands for "analog". The analog ambiguity function is defined as the magnitude of (2-8) (2-9) In digital receivers, the sampled complex envelope of the received signal can be represented as u(nT s − ) exp[ −j2π (nT s − )], where n = ...,− 2,−1, 0, 1, 2, ..., and T s is the sampling period that can be presented as T s = 1/(gB), where g ≥ 1 is the oversampling factor. Since the unit sample response of the digital version of the matched filter is given by u*( −nT s ) the complex digital ambiguity function can be easily shown to be (2-10) The definition for the digital ambiguity function is similar to (2-9) (2-11) As has been shown (Blankenship & Hofstetter, 1975), there exists a simple relationship between the analog and digital ambiguity functions, which is given by where F s = 1/T s is the sampling frequency. Thus, the complex digital ambiguity function is a sum of the replicas of the complex analog ambiguity function displaced to all frequencies nF s . Inspecting (2-12) yields that it is similar to the well known relationship between the Fourier transforms of an analog signal and its sampled version. It is also clear that equation (2-12) allows to quickly draw contour plots (ambiguity diagrams) for the digital LFM ambiguity functions at different sampling frequencies by using contour plots for the analog LFM anbuguity finction. The contour plots provide insight into the gross behavior of the LFM digital ambiguity function. Thus, for the purpose of detailed analysis an exact mathematical expresson for this function is necessary. As has been shown (Blankenship & Hofstetter, 1975), the exact formula for the digital LFM ambiguity function is given by Where N=BT, y= T, and M =T/T s is the integer number of the LFM signal samples taken over the total signal duration T. Formula (2-13) assumes =(k+x)T s , where k is an integer and 0≤ x <1.

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Blankenship and Hofstetter studied the digital LFM ambiguity function by using (2-13) and contour plots. In particular, they have shown that in the digital case even with Nyquist rate sampling, i.e. with T s = 1/B, the sidelobes do not fall off uniformly with increasing | |, but increase as | |nears the ends of the response interval (±T). The reason for this is that the aliased replicas of the analog ambiguity function produce increased sidelobe levels for large values of | |. Hence, increasing the sampling rate should reduce this effect. Indeed, it has been shown that for sampling at twice the Nyquist rate, i.e., with T s = 1/(2B), the behavior of the digital LFM ambiguity functions is essentially indistinguishable from that of analog one.
Here we supplement that study by illustrating the ridge (peak signal amplitude) for the digital LFM ambiguity function ( Fig. 2-3) computed for T s = 1/B and T s = 1/(2B). Fig. 2-3 clearly demonstrates appreciable improvement in the Doppler tolerance for sampling at twice the Nyquist rate.

Digital cross-ambiguity function
This section introduces the concept of digital cross-ambiguity function that is directly related to digital signal processing based on mismatched filtering, which is more general structure than that based on the matched filter. By analogy with (2-10) we have for the complex digital cross-ambiguity function where w*(nT s ) corresponds to the unit sample response of the digital mismatched filter. By letting =(k+x)T s , where k is an integer and 0 ≤ x <1, one can represent the samples of the received signal where . The integer k is omitted since in computing the output of a digital filter the time shift kT s is a shift of the train of N received samples [vector www.intechopen.com u x ( )] by k samples [to the left if k < 0 or to the right if k ≥ 0 when we look at the filtering operation as a complex correlator] with respect to N samples of the reference sequence specified by the vector u 0 (0) . Next, define a receiving filter vector w of length M ≥ N (2-16) Since the filter length M can be equal to or greater than N we will use the extended signal vector , which is zero-padded vector u x ( ) , to match the length M of the vector w . We define this extended signal vector as (2-17) Then the digital cross-ambiguity function (DCAF) is given by the cross-correlation function between the vectors s x ( ) and w as (2-18) where − (M−1) ≤ k ≤ M−1, and must be identically zero for n−k < 1 and n−k > M. Next, consider the DCAF for integer delay k. As follows from (2-18), it is given by where the row vector represents the DCAF (2-19) according to the relation (2-22) and the symbol "~" denotes a hermitian or conjugate transpose. From (2-19) and (2-22), one can observe that r M (0) corresponds to the peak value of the DCAF at the point =0, =0.

Optimum weighting functions
Following (Harris, 1978;Nuttal, 1981;Van Trees, 2002) one can choose among several different approaches to the design of optimum weighting functions. In view of our intention www.intechopen.com to minimize sidelobe level and preserve as much as possible the ridge-like behaviour of the mismatched filter response we shall follow an approach based on the physical meaning of the optimal discrete prolate sequences. For instance, in (Donald & James, 1970;Greene, 2007) the optimal discrete prolate sequence is interpreted as such a sequence that maximizes the proportion of its total power in a specified frequency interval or, referring to (Van Trees, 2002), as a weighting function that maximizes the percentage of the total power (radiated by an antenna) that is concentrated in a given angular region. By analogy, the problem in question can be formulated as a problem of finding such a weighting function (mismatched filter) that will maximize the percentage of the total power that is concentrated in the specified mainlobe region of the DCAF. This percentage can be written as (2-23) The numerator in (2-23) represents the power for a specified mainlobe region ( Fig. 2-4), which is simply a sum of the squared samples of the DCAF , which belong to the set R k that contains the points associated with the ridge for k th Doppler cut specified by the frequency , where is the step in Doppler frequency and max is the maximum Doppler shift of interest. The denominator represents the full power, which is the total sum of the squared samples of the DCAF taken over all points in range (n = 1, 2,..., 2M−1 ) for each Doppler cut. The integer parameter d k represents the shift of the peak value of the DCAF for k th Doppler cut with respect to the point = 0, = 0 (this is the point n = M, L+1 = 0 in Fig. 2-4). To preserve the skew of the ridge it is reasonable to select the values of d k to be equal to those of corresponding Doppler cuts from the digital LFM ambiguity function. The parameter m is a positive integer value that determines the width of the mainlobe region. For example, setting m=g (if the oversampling factor g is a positive integer) means that the mainlobe extent is equal to that of the standard sinc-function (at the first zero points = ±1/B) that correspons to a signal having rectangular spectrum of width B or to that of a zero-Doppler cut of the LFM ambiguity function χ( , 0) as can be seen from (2-2) provided BT >> 1. In general, the value of m can be varied depending on the Doppler cut. To formally specify the mainlobe region for kth Doppler cut of the DCAF we shall use a weighting vector of the length 2M-1. As can be seen from Fig. 2-5, for a zero-Doppler cut this vector can be written as (2-24) For arbitrary k th Doppler cut this vector is given by The total power in the mainlobe region is where the M -by-M matrix B ML is given by The total power in the k th Doppler cut is The matrix Q TL in (2-31) is a (2M −1)-by-(2M −1) unity matrix since (2-31) will present the total power only when the diagonal of Q TL is equal to the weighting vector V TL with all elements that are unity (2-32) Using (2-21) and (2-31) yields the total power for all Doppler cuts  where the M -by-M matrix B TL is given by Thus, the ratio (2-23), which is to be maximized with respect to w, can be written as The maximization of the ratio (2-35) is equivalent to the following minimization problem (2-36) The solution to (2-36) can be found by minimizing the function where is the Langrange multiplier. Taking the gradient of (2-37) with respect to w and equating it to zero, we find that the solution to (2-36) is given by the following generalized eigenvalue problem  where can be interpreted as a corresponding generalized eigenvalue. It should be noted that all generalized eigenvalue in (2-38) are positive real numbers. Indeed, multiplying both sides of (2-38) by As follows from (2-26)-(2-29) and (2-31)-(2-33) P TL >0 and P ML >0 because either of the sums in (2-29) and (2-33) contains at least one real positive term | r M (0)| 2 , hence, the matrices B TL and B ML are positive definite. This proves that is always real positive value. Therefore, the solution to (2-37) is the generalized eigenvector corresponding to the minimum generalized eigenvalue of (2-38). Multiplying (2-38) by yields (2-39) Since is always real and positive the minimum generalized eigenvalue min in (2-38) corresponds to the maximum eigenvalue ζ max =1/ min in (2-39). By using this fact, the optimum weighting function (weighting coefficients of the optimum mismatched filter) can be represented as (2-40) where is the operator, which returns the principal eigenvector of a matrix A, that is, the eigenvector corresponding to its maximum eigenvalue. As follows from (2-36), the optimum vector w opt has to be normalized to satisfy the constraint in (2-36). Since any eigenvector can be normalized arbitrarily, is is clear that multiplying w opt by an appropriate non-zero constant gives a vector , which meets the requirement . On the other hand, it is clear from (2-35) that multiplying w opt by any non-zero constant does not affect the ratio (2-35). Therefore, such normalization of w opt is unimportant. In this chapter we use the vector of signal u x (s x ) and the weighting vectors w (both for the optimum and conventional weighting functions) that are normalized to be of unit norm vectors, that is, . The condition m = g means that the mainlobe width of the DCAF is specified to be equal to that of a zero-Doppler cut of the LFM ambiguity function (2-2) measured at the first zeros. is selected to provide the same mainlobe width as that for the optimum weighting function. The maximum sidelobe level, the loss in the signal-to-noise ratio (SNR) and 3 dB mainlobe broadening (all these parameters are measured in a zero-Doppler cut) for the optimum filter and Kaiser window are also included in Table 2-1. One can easily show that the SNR loss due to weigthing is given by (2-41) The 3dB mainlobe broadening is defined as the ratio b ML =W WF /W MF where W WF and W MF are the mainlobe width at -3 dB level for a zero-Doppler cut of the DCAF in case of mismatched filtering (for conventional or optimum weighting functions) and digital LFM ambiguity function (matched filtering), respectively.  As can be seen from Table 2-1, the percentage of the mainlobe power ( ) for the optimum weigthing exceeds those for the matched filter and Kaiser window by about 8.5% and 2.5%, respectively, for a wide range of the Doppler interval of interest (| max |/B ≤ 0.4). Despite these relatively small gains in the optimum weigthing provides appreciable reduction in sidelobes. As one can see, the maximum sidelobe level measured in a zero-Doppler cut is about -29 dB; this is by 15 dB below that for the matched filter and by about 9 dB below that for the KW (under the conditions the data in Table 2-1 are computed). The price paid for the sidelobe suppression is quite low. Indeed, the SNR loss for the OF is about -0.8 dB with respect to that of the MF and -(0.3 … 0.5) dB as compared to that for the KW. Second, the mainlobe broadening against the MF is also relatively small, it ranges from 1.21 to 1.16 for | max |/B = 0, 0.05, 0.1, 0.2 and 0.4, respectively. Fig. 2-6 illustrates the sidelobe suppression for the optimum weighting by comparing a zero-Doppler cut of the digital cross-ambiguity function for the optimum filter (designed for max /B = 0.1) with corresponding zero-Doppler cuts for the Kaiser window and matched filter. To get more pictorial view for comparison all these cuts are normalized so that the magnitudes of their peak values are equal to unity (zero on the dB-scale). The Doppler tolerance of the optimum filter is illustrated in Fig. 2-7 that compares the ridge (peak signal amplitude) of the DCAF for the optimum filter ( max /B = 0.1) with those for the www.intechopen.com   Fig. 2-7, for | |/B ≤ 0.1 the ridge of the DCAF for the optimal filter goes slightly below that for the Kaiser window (due to the SNR loss of -0.36 dB). This means that the optimum filter maximizes the ratio (2-35) chiefly through the sidelobe suppression. In the second example we illustrate a new feature of the proposed optimum weighting that convetional weightings do not possess in principle. This feature can be realized when the parameter m, which determines the width of the mainlobe region in time delay, is less than www.intechopen.com the oversampling factor g. The condition m<g means that the mainlobe width of the DCAF is specified to be less than that for a zero-Doppler cut of (2-2) measured at the first zeros. Since the optimum filter is designed under this condition the maximum percentage of the mainlobe power is concentrated within a narrower "strip" than for the matched filter. Hence, one should expect that there will be no mainlobe broadening and no range resolution degradation with respect to the matched filter. Table 2-2 presents the percentage of the mainlobe power, maximum sidelobe level, the SNR loss and 3dB mainlobe broadening (in zero-Doppler cut) for the OF of length M=1.2N=48 designed under the condition m = g − 1 = 1. Since there is no simple formula that directly relates the SNR loss to m and M for the OF, it was numerically found for N ≤ M ≤ 1.5N that the SNR loss does not exceed 3 dB and the maximum sidelobe level is approximately minimized at M= 1.2N= 48 for all max /B given in Table 2-2. As can be seen from Table 2-2, for the optimum weighting with M = 48, m = 1 the 3dB mainlobe broadening b ML <1, as expected. Therefore, there is no degradation in the range resolution with respect to the matched filter. The price paid for this is relatively large SNR loss (from -1.5dB to -2.8dB depending on the Doppler extent max /B) and an increase in the maximum sidelobes (about 8 dB) with respect to the OF designed under the condition m=g (see Table 2-1). At the same time the maximum sidelobe level is somewhat lower (by about 1.5 dB) than that for the Kaiser window from  Fig. 2-8 compares a zero-Doppler cut of the DCAF for the OF with that for the matched filter. The Doppler tolerance of this filter is illustrated in Fig. 2-9. The digital cross-ambigutiy functions (3-D presentations) and corresponding contour plots (at two levels of -3.92dB and -13.32dB) for the optimum filters with M = 40, m = 2 and M = 48, m = 1 (both are designed for max /B=0.1) are shown in Fig. 2-10 (a) and (b), respectively. Analyzing the plots in Fig. 2-8 and Fig. 2-9 and the percentage of the mainlobe power for the OF and MF (Table 2-2) we again arrive at the conclusion that the optimum filter (2-40) chiefly maximizes the ratio (2-35) by supressing the sidelobes. Comparing Fig 2-10 (a) and (b) reveals increased sidelobes: additional contour lines at a level of -13.32 dB in the latter. This increase in sidelobe level is due to narrower mainlobe region for the OF with M =48, m =1 against that for the OF with M =40, m =2. The fundamental property of the crossambiguty function says that it is impossible to remove energy from one portion of the crossambiguity surface without placing it somewhere else on the ( , ) plane.

Optimization of InSAR accuracy performance
This section addresses the improvement of the interferometric synthetic aperture radar (InSAR) performance, namely, minimization of the standard deviation of the height estimate through the optimization of the weighting coefficients of a digital filter to be used in the InSAR receiving channels for pulse compression in range and in azimuth dimension. For simplicity we consider only the pulse compression in range. The matched filter is an optimum filter under the criterion of maximum signal-to-noise ratio (SNR) in the presence of white noise. This section deals with such a filter that approximates to an optimum filter under a different criterion: the minimum of the height-standard deviation h . As is well known, (e.g., Rosen et al., 2000) applying conventional weighting in the InSAR receiving channels allows decreasing h against matched filtering although the SNR in the event of weighting is inevitably less than that in the event of matched filtering. Thus, strictly speaking, the optimum filter in terms of the minimum of the height-standard deviation h belongs to a class of mismatched filters.

Height estimation accuracy
The magnitude of the full correlation function between InSAR channels is given by = G N Z T (Rosen et al., 2000), where G is the geometric (baseline) correlation, N is the correlation due to thermal noise in the interferometric channels, Z is the correlation due to volume scattering, and T represents the temporal correlation (in repeat-pass systems). From (Rodriguez & Martin, 1992;Rosen et al., 2000) it follows that increasing results in decreasing the standard deviation of inferred interferometric phase estimates and correspondingly derived height values. As has been pointed out in (Baskakov&Ка, 2000;Rosen et al., 2000), modifying the system point target response (PTR) in range by applying different weighting functions in the InSAR receiving channels, one can change the shape of the geometric correlation G to reduce the phase noise. It should be noted, that as has been shown (Gatteli et al., 1994) one can obtain G = 1 (at least theoretically) by using a kind of band-pass filtering (Rosen et al., 2000) of signals from both channels so that they have slightly different center frequencies. The relationship between these frequencies depends on the look angle and surface slope, so that an adaptive iterative procedure is needed to implement this approach (Rosen et al., 2000). In this paper we will try deriving benefits from practically much simpler approach, which is based on using weighting.

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The major implication of weighting in an InSAR is that it allows increasing the total correlation due to increasing the geometric correlation G , and, consequently, reducing height estimation errors, other conditions being equal. Examples given in (Rosen, et al., 2000) for conventional weighting functions demonstrate improved shapes of geometric correlation with weightings against that for a standard sinc response with no weighting. However, at least two significant problems associated with weighting have not been discussed in (Rosen et al., 2000). First, it is well-known, that applying weighting inevitably leads to the signal-to-noise (SNR) degradation in the interferometric channels, which, in turn, results in decreasing the correlation due to thermal noise N . If this decrease in N prevails over an increase in G , then the full correlation will degrade that finally results in deterioration of the height estimation accuracy. Another problem associated with weighting is degradation of the InSAR resolution due to mainlobe broadening of the PTR in range. The effect of conventional weighting on the InSAR performance was analyzed in (Baskakov & Ка, 2000) for Hamming and Gauss weightings. It has been claimed that conventional weighting gives an improvement in the height accuracy only for small relative baselines [the ratios of B(see Fig. 3-1) to the wavelength ], while for large relative baselines weighting leads to significant deterioration in performance. The maximum-likelihood (ML) estimator presented in (Rodriguez & Martin, 1992) can be used as an accurate estimator of the interferometric phase from homogeneous distributed targets. That ML estimator is unbiased modulo 2π, and its standard deviation can be easily obtained by using Monte-Carlo simulations. The Cramér-Rao lower bound (CRLB) on the standard deviation Φ σ in unbiased estimation of the interferometric phase is given by  where N L is the number of looks. It has been shown (Rodriguez & Martin, 1992) that the phase-standard deviation of the ML estimator approaches the limit given by (3-1) very rapidly with number of looks for the first four looks, especially if the correlation is high. Hence, formula (3-1) gives a reasonable approximation for the phase-standard deviation of the ML estimator when the inequality N L ≥ 4 holds true. Using (3-1) one can relate the full correlation to height estimation errors as (Rodriguez & Martin, 1992;Ка & Kononov 2007) ( 3-2) where h is the standard deviation of height errors, H is the height of a spacecraft, B n = B cos (α −θ) is the projection of the baseline B (in section 2 B denoted the LFM signal bandwidth) onto the direction perpendicular to the look direction, and α and θ are the baseline tilt and look angles, respectively (Fig. 3-1). As was shown (Rodriguez & Martin, 1992) , Φ σ is minimized, hence, h is also minimized, by choosing α =θ.
In what follows we assume that the system operates in standard mode of data collection (Rosen et al., 2000) and its two interferometric channels have identical PTRs of the separable form χ r (ρ)· χ a (s), where χ r (ρ) and χ a (s) are the system PTR in range (ρ) and azimuth (s), respectively. For simplicity, we confine our analysis to the range dimension although the proposed technique can be also used in the azimuth dimension. To reveal the potential of www.intechopen.com  (Rosen et al., 2000) reduces to  where is the interferometric fringe wavenumber in range ( is the wavenumber) and r o = H/cosθ is the slant range from the system to the middle point of a distributed resolution cell at the centre of the swath. When the PTR in range is given by a standard sinc-function , where is the intrinsic range resolution (c is the speed of light and F m is the system bandwidth), the geometric correlation was shown (Rodriguez & Martin, 1992;Ка & Kononov, 2007) to be given by (3)(4) The second term N is the correlation due to thermal noise alone. As was shown in (Rodriguez & Martin, 1992), assuming isotropic backscatter coefficient within the SAR resolution element, N can be represented as (3)(4)(5) where q is the system signal-to-noise ratio (SNR), which is assumed to be equal in both channels. The correlation term Z is given by (Rodriguez & Martin, 1992;Ка & Kononov, 2007) (3-6) www.intechopen.com where s is the height standard deviation of specular points with respect to the mean topography. The term T is specific to repeat-pass systems (Rosen et al., 2000;Zebker & Villasenor, 1992) and represents a measure of decorrelation between two images (forming an interferogram) due to change in the surface itself over the temporal interval between the times when the images are acquired. In what follows we assume T = 1.

Optimization of PTR
Examining (3-3) yields that its right-hand side has the form of a Fourier transform. From the Fourier transform properties, it follows that the narrower the PTR χ r (ρ) in time (range) domain, the wider the baseline correlation function G (3-3) and the better InSAR height estimation performance, other conditions being equal. For a given transmit signal, one of the possible ways to concentrate the PTR in range domain is designing such a receiving filter, which maximizes the percentage of the total power within a specified time interval around the PTR peak value point. To control the range resolution the length of this interval (mainlobe region) can be adjusted. Noting that the module of PTR in range is a zero-Doppler cut (in case of ideal frequency stability of radar transmitter) of the cross-ambiguity function allows concluding that the method of section 2 can be directly used to optimize the PTR for the purpose of improving the accuracy of InSAR height measurements. Then, as follows from equation (2-21), the digital PTR can be written as (3-7) where the row vector represents the digital , i.e., a zero-Doppler cut of the DCAF, according to the relation (3-8) Next, as can be seen from (2-23), for the problem in question the ratio to be maximized is (3)(4)(5)(6)(7)(8)(9) where the set represents the specified mainlobe region around the PTR peak point. Based on the derivation in section 2.4 one can write the solution for the optimum weighting function that maximally concentrates the PTR in range in terms of the ratio (3-9) as (3-10) The matrices B TL and B ML in (3-10) are given by (3)(4)(5)(6)(7)(8)(9)(10)(11) respectively, where S=S(0) [see formula (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)] and the matrix Q ML is (2M −1)-by-(2M −1) diagonal matrix: its diagonal is equal to the weighting vector [see (2-24)] and nondiagonal elements are zeros.

Numerical examples
Numerical examples illustrate the InSAR height accuracy and range resolution that can be achieved by using the optimum weighting function (3-10) for an LFM signal with T = 3 μs and B = 20 MHz. The oversampling factor g = 2 was chosen. Hence, the complex envelope of the LFM signal is given by a vector of length N = 120. To compute the baseline correlation function for digital interferometric channels the following digital counterpart of equation (3-3) was used (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) where [, ] [, ,0] To quantify the change in the PTR mainlobe width due to weighting we use the 3dB mainlobe broadening parameter b ML defined in subsection 2.5. It can be shown, for a single SAR resolution element with isotropic backscatter coefficient, that in the event of mismatched filtering the SNR can be represented as (3-13) where q MF is the SNR for the matched filter and is the loss in SNR (due to weighting), which is given by equation (2-41). Since there are no simple closed-form expressions that directly relate the coefficient of loss to m and M for the optimum weighting filter given by (3-10), it was numerically found for m=g−1, g, g+1, i.e., for m = 1, 2, and 3 that the SNR loss is approximately minimized for 1.1N ≤ M ≤ 1.5N. The parameters and b ML for the optimum filter of length M =132 computed from (3-10) for m = 1, 2, 3 are summarized in Table 3-1. To estimate the mainlobe width, the digital PTRs were computed by using (2-18) at integer points −(M−1) ≤ k ≤ M−1 and between them for x = 0.01, 0.02,..., 0.99 . For comparison, the SNR loss and parameter β for the Kaiser window of length M = 132 are also given. The parameter β was selected to approximately provide the same value of b ML , except for m = 1 (it is impossible for the Kaiser window to find such a parameter β , at which b ML < 1). As can be seen from Table 3-1, for the optimum filter of length M = 132 at m = 2 the degradation in range resolution is quite appreciable (mainlobe broadening is 20%) and the SNR loss is relatively small (about -0.7 dB). The optimum filter with M =132, m = 1 gives a quite noticeable enhancement in the range resolution with respect to the matched filter. The price paid for this is relatively large SNR loss (about -3 dB) and an increase of 10% in the filter length. It will be shown that despite the relatively large SNR loss this optimum filter provides appreciable improvement in height accuracy. The range resolution enhancement for the optimum filter with M =132, m = 1 is evident from Fig. 3-2 that shows the magnitude of the digital PTR for the optimum filter in comparison with those for the Kaiser window (β = 3.1) and matched filter. To get more pictorial view, all the PTR's are normalized so that the magnitudes of their peak values are equal to unity (zero on the dB-scale).   Fig. 3-2. For comparison, the baseline correlations for the standard sinc response and for the response of a filter matched to a simple rectangular pulse with pulsewidth 1/ F m are also plotted. A remarkable feature of the PTR for the simple rectangular pulse is that it has zero sidelobes outside of the mainlobe region for m=g. Therefore, the baseline correlation function for the simple rectangular pulse can be used as an example of ideal reference model for comparison. From the inspection of Fig. 3-3, one can conclude that narrowing the mainlobe region in designing an optimum filter allows a distinct improvement in the shape of the baseline correlation function in the sense of the phase (height) noise reduction. As can be seen, the baseline correlation curve for www.intechopen.com the optimal filter goes appreciably higher than those for the Kaiser window and standard sinc function and almost coincides with that corresponding to the simple rectangular pulse. As mentioned above, the SNR degradation due to weighting may result in a significant decrease in the correlation due to thermal noise N and, finally, in performance deterioration. Thus, to draw a final conclusion concerning the effectiveness of the suggested approach one has to evaluate a potential improvement in the InSAR accuracy performances taking into account all the components of the full correlation. In Fig. 3-4 and Fig. 3-5 we plot for the optimal filter with M = 132 and m = 1, the height standard deviation h calculated by using (3-2), (3-5), (3-6), (3-12) and (3-13) with inputs x = 0 , N L = 9, θ =α = 30°, H = 350 km, s = 1 m, and T = 1 versus SNR for B n / = 1500 and versus the relative baseline B n / for SNR=18dB, respectively. The calculations were also done at x = 0.5, which can be interpreted as a subpixel shift. The performance degradation for x = 0.5 (with respect to x = 0) did not exceed 15%. The performances for the matched filter, Kaiser window (β =3.1) and the simple pulse are also plotted (for x = 0). It is clear from Fig. 3-4 that no performance improvement results for SNR<17.79 dB, but the improvement becomes noticeable when the SNR is on the order of 20 dB or larger. As can be seen from Fig. 3-5, despite the relatively large SNR loss this optimum filter provides appreciable improvement in height estimation accuracy against the matched filter and Kaiser window over a wide range of the relative baseline.

Conclusion
This chapter has introduced a new approach for the design of optimum weighting functions. By analogy with optimal discrete prolate sequences this approach considers the design of weighting functions as a problem of finding such a digital mismatched filter that will maximize the proportion of the total response power that is concentrated in the specified time-frequency region. A closed-form matrix equation to numerically design the optimum weigthing functions has been derived. Two applications of the proposed approach have theoretically been addressed in the chapter. First, the problem of the optimum Doppler-tolerant pair signal-filter design when a given signal is specified to be an LFM signal and, second, the specification of weighting functions for interferometric synthetic aperture radars with the purpose of improving the accuracy of height measurements, have been considered. There has been shown, for the first application, that the proposed optimum weighting functions (mismatched filters) provide significant sidelobe suppression with respect to conventional weighting functions at relatively low SNR loss. It has also been shown that they can provide appreciable improvement in the accuracy of height measurements, as compared to conventional weightings, for interferometric synthetic aperture radars. A remarkable feature of the suggested optimum weightings is that improvement in the sidelobe suppression and height accuracy can be achieved without degradation in range resolution that is inevitable in the case of using traditional weighting functions.