8 Optimization Potentials
8.4 Pattern Tapering on Transmit
in the spatial sampling. In contrast to single-platform systems, in sparse arrays the spatial posi- tion of adjacent samples is not bound to the same transmitted pulse. Equation (96) reveals that variations in the PRF result in variations of the position Δxj,i amplified by a factor (p-q) and
hence only small changes in the PRF might already have a large impact on the spatial distribu- tion of the samples. As shown in Fig. 72, this results in two clearly different sample distributions for two similar PRF values. This offers wide flexibility to adapt the sampling even if only a small variation of the PRF is possible and thus allows for compensating for the high sensitivity of the systems’ SNR and AASRN regarding PRF variations.
8.4 Pattern Tapering on Transmit
As derived in Section 5.5, all spectral energy outside the band [-N·PRF/2, N·PRF/2] is not properly cancelled by the multi-channel reconstruction and causes aliasing in the reconstructed signal, finally resulting in ambiguities in the SAR image. This can be mitigated by confining the Doppler bandwidth of the signal to N·PRF by an appropriate joint antenna pattern. In a very sim- ple approach, one could just enlarge the dimension of the transmit antenna resulting in a nar- rower pattern, but being at the expense of resolution. A bigger antenna in combination with an adapted tapering is required to provide an improved suppression of the ambiguous frequency bands without degrading the resolution. Furthermore, the better the pattern is limited to the rele- vant Doppler frequencies, the better the emitted power is used as less power is lost by illuminat- ing unwanted areas. The basic idea of concentrating the transmitted energy in the mainlobe ac- cording to the processed Doppler bandwidth BD while reducing the emission to directions corre- sponding to ambiguous returns is visualized in Fig. 73: A minimum gain level of the joint an- tenna pattern within the interval BD ensures sufficient signal energy, while a maximum allowable gain outside the system band restricts the ambiguous energy.
BD
Joint Antenna Pattern
f Maximum Ambiguity Level Minimum Signal Level Minimize Aliasing Minimize Aliasing Optimize δaz& SNR N·PRF/2 -N·PRF/2
Fig. 73. Joint Tx-Rx antenna pattern in azimuth (dashed red) illustrating the basic principle of pattern tapering on transmit. Suppression of aliased energy by imposing a maximum allowable
ambiguity level in combination with a minimum signal level to optimize the signal energy. It should be noted that the presented tapers can be either realized by a separate transmit an- tenna, or by using an active array based on transmit-receive (T/R) technology. While the first
requires less sophisticated hardware, the latter offers the flexibility to use parts of the receiving antenna for transmit and hence allows for benefiting from its length.
To demonstrate the potential of pattern tapering on transmit, the system presented in Chapter 7 is investigated for different combinations of transmit antenna dimensions and amplitude excita- tions.26 The transmit antenna of the example system of length 3 m with a uniform taper entailing a sin(fx)/fx pattern characteristic is compared to three different pattern tapering concepts. The
respective transmit antenna lengths daz,tx, the applied excitation, and the resulting far field pattern are summarized in Table 8, where the notation sinc(x) is used to denote the sin(x)/x function.
daz,tx Excitation (Azimuth) Far Field (Azimuth) δaz Represented by
3 m uniform sinc(fx) < 1 m “Sinc”
4 m cos(x) sinc(fx-π/2) + sinc(fx+π/2) < 1 m “Cos”
4.6 m triangular sinc2(f
x) < 1 m “Sinc2”
11.2 m sinc(x) rect(fx/ BD) < 1 m “Rect”
Table 8. Pattern tapering on transmit: Excitations and resulting pattern characteristics. Further, it is well-known that excitation and far field are related by Fourier transform with the variables x and fx, respectively. The variable x represents azimuth dimension while fx de-
scribes the spatial frequency given by (97). Recalling the relation between azimuth angle Θ and Doppler frequency f (cf. (98)), one recognizes that fx and f are linked by the sensor velocity.
( )
( )
s sin 2 x f f v Θ Θ λ = = ⋅ (97)( )
2 vs sin( )
f Θ Θ λ ⋅ = ⋅ (98)Fig. 74 shows the resulting patterns with normalized amplitude, which is given in depend- ency on the Doppler frequency f(Θ). This allows for highlighting the potential of the different tapers to confine the transmitted energy to the desired azimuth region, i.e. the processed Doppler band, which is marked in gray. Consequently, Fig. 74 does not take into account a possible gain loss due to the amplitude taper, which is investigated in the second part of this section. Finally, to give a better overview, two approaches are combined in each of the plots.
26Although phase coefficients avoid a tapering loss by attenuation, they cannot be determined
straightforwardly like the amplitude coefficients. An exhaustive analysis of both methods is beyond the scope of this work and thus only amplitude tapering is investigated to demonstrate the potential this technique.
8.4 Pattern Tapering on Transmit 105
Regarding the suppression of azimuth ambiguities, Fig. 75 shows the AASRN that is achieved
for the different antenna patterns, respectively. It is stated that the suppression is already clearly improved for higher PRF values by applying the 4 m antenna with a cos(x)-like excitation (dashed blue). This results from the reduced sidelobes of the modified antenna pattern as can be seen in the left plot of Fig. 74. The suppression becomes even better with a 4.6 m antenna with a triangular tapering which entails a sinc(fx)2 characteristic (dotted green). However, to provide an
improvement in suppression also for lower PRF values, a larger transmit antenna is necessary. An unconventional realization is given by the quasi-optimum – as it approximates a rectangular pattern – sinc(x) excitation in combination with a transmit antenna of 11.2 m (dotted dashed vio- let). This corresponds to the antenna on receive and represents consequently the maximum avail- Fig. 74. Normalized amplitude of azimuth transmit pattern vs. Doppler frequency f(Θ) for excita-
tions and antenna dimensions according to Table 8. BD is marked by the gray-shaded area. Left:
3 m antenna with uniform excitation (“Sinc”, solid red) and 4 m antenna with cosine excitation (“Cos”, dashed blue). Right: 4.6 m antenna with triangular excitation (“Sinc2”, dotted green) and
11.2 m antenna with sin(x)/x excitation (“Rect”, dotted dashed violet).
Fig. 75. Azimuth ambiguity suppression vs. PRF for different transmit antenna sizes and patterns according to Table 8: 3 m antenna with sinc(fx) characteristic (solid red) 4 m antenna with cos(x)
excitation (dashed blue), 4.6 m antenna with sinc2(f
x) characteristic (dotted green) and approxi-
able length if T/R modules are used. In the present simulation, it is realized by 35 elements of 0.32 m length each. Especially this sinc(x)-excitation demonstrates the potential of tapering to effectively cancel the spurious spectral components while preserving the resolution. A full ex- ploitation of the benefits of tapering requires a fine adjustment of antenna dimensions, PRF and BD, taking into account the trade-off between resolution and ambiguity suppression.
Further, an analysis of the respective point target impulse response shows, that all scenarios yield an azimuth resolution of better than 1 m for a processed bandwidth BD =7.6 kHz.
To conclude this section, the impact of the tapering on the NESZ is investigated. As men- tioned above, the enlarged antenna size in combination with an appropriate tapering improves the distribution of the transmitted energy. This can be quantified by the joint Tx/Rx antenna azi- muth loss Laz, which accounts for the spectral shape of the pattern (cf. (60) and (61)). Further, the overall transmitted energy changes with antenna length and amplitude taper. Hence, additionally to Laz, the maximum power gain Gtx of the antenna characteristic has to be considered. Note that the calculation of the modified gain takes into account the new antenna length and is based on the assumption of a constant transmit power per antenna area, i.e. an equal feeding volt- age/power of each individual transmit antenna element. The respective values with respect to the original system defined by Gtx,0, Laz,0 and NESZ0 are summarized in the following Table 9. It gives the resulting improvement (“-”) or degradation (“+”) of the new NESZ that can be directly applied to Fig. 70 to obtain the NESZ corresponding to the modified system. Equivalently, Table 9 allows for deducing how the peak transmit power per antenna area can be reduced (“-”) or has to be increased (“+”) to keep the NESZ constant. Finally, the efficiency of the taper itself is measured by relating the different peak power gains if the antenna lengths were all the same. The results are listed in Table 9 in the very right column.
Excitation Maximum Transmit Power Gain Gtx/Gtx,0
Azimuth Loss
Laz/Laz,0 NESZ/NESZ0 Efficiency
uniform 0 dB 0 dB 0 dB 0 dB
cos(x) +1.42 dB -0.03 dB +1.39 dB +3.92 dB
triangular +2.30 dB +0.18 dB +2.48 dB +6.02 dB
sinc(x) +9.30 dB -0.97 dB +8.33 dB +20.74 dB
Table 9. Pattern tapering on transmit: Maximum antenna gain Gtx, azimuth loss Laz and resulting
NESZ relative to the original system defined by Gtx,0, Laz,0 and NESZ0. Improvement is indi-
cated by “-” while “+” means a degradation.
As one can see from the change in the NESZ, particularly the cosine taper seems to be a good choice if a clearly improved azimuth ambiguity suppression is needed while a moderate degrada- tion of the NESZ can be afforded. Especially for PRF values above 1300 Hz, an AASRN im-
proved by ~10 dB is achieved at the cost of a SNR decreased by only ~1.4 dB. The triangular taper yields an even faster drop of the residual ambiguities, but entails a degraded NESZ by al- ready ~2.5 dB. Finally, the approximated rectangular pattern was only evaluated to demonstrate its potential regarding the suppression of azimuth ambiguous energy, as the transmitted signal power decreases too strongly with respect to the original system.