In the following, the first example for a cascaded beamforming network is given to demon- strate the potential of the technique. In the present section, an analog pre-processing technique of the multi-channel RF signals is chosen as a representation of such a cascaded structure. The method is referred to as “pre-beamshaping on receive” and it requires a more complex Rx an- tenna architecture, as multiple independent receiving elements are needed to enable the pre- processing. Thus, in the following, K=168 receiving elements are assumed, each allowing for a tapering in amplitude and phase. This means that in the reference case – given by the “conven- tional” system according to Section 8.5 – K0 =24 receiving elements are grouped to a single vir- tual channel to obtain N0 =7 cannels without mutual overlap. In the cascaded case, the pre- processing network is configured such that mutually overlapping sub-apertures of 2.625 m length each are formed; i.e. each channel consists of K1 =36 single elements. This yields a decreased phase center spacing entailing a new optimum PRF of 1346 Hz according to (111). In order to keep the beamwidth of the receiving pattern constant and to suppress its sidelobes, a cosine taper is applied to each of the sub-apertures, leading to different impacts on signal and noise power as will be discussed in the following. In detail, the chosen weighting functions are as follows:
( )
az,el 1 1'
1 1
cos π cos π ; where ,
2 2 ij ij i d i K K w f w i d K ⋅ ⎛ ⎞ ⎛ ⎞ ⎡ ⎤ = = ⎜ ⎟= ⎜ ⎟ ∈ −⎢ ⎥ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ (137)
Concerning the signal power in the pre-beamshaping case, the new antenna dimensions and the pre-processing network entail for the considered bandwidth of BD =7.6 kHz an increased azimuth loss Laz of 3.05 dB compared to Laz =2.9 dB in the conventional case (cf. (61)). This loss of 0.15 dB is increased by a new maximum gain on receive in the pre-beamshaping case that is worsened by ~0.55 dB compared to the maximum gain in the conventional case. It results from the applied tapering and takes into account the increased sub-aperture length. This maximum gain is calculated accordingly to (125), where K0 =24, K1 =36, and the tapering coefficients as
given by (137) are considered. Note that the loss of 0.55 dB not only depends on the applied ta- pering but also on the number of elements. In the limit given by continuous amplitude tapering of the antenna, i.e. K1→∞ and del→0, the loss reduces to 0.4 dB. Regarding the noise power scaling, the amplitude tapering of the pre-processing network attenuates the noise power in a similar way as the signal power. Due to the large number of elements that are combined and due to the applied amplitude taper, the ratio of the noise powers for the respective uniform PRF val- ues of the two investigated systems cannot be derived illustratively as the signal power above. Nevertheless, the simulation of the SNR scaling factor for the pre-beamshaping scenario and its estimation according to (135) and (136), respectively, show very good coincidence and prove the validity of the derived equations (cf. Fig. 92).
Fig. 92. Simulated SNR scaling factor Φbf of the pre-beamshaping scenario normalized to the uni-
form value of the conventional case. Results before (dashed green) and after focusing with
BD =7.6 kHz (solid orange). Selected samples of the respective analytic predictions according to
Section 8.7.6 are overlaid with the diamond symbols.
According to the decreased phase center spacing, the optimum PRF and consequently the optimum SNR scaling factor moved towards higher PRF values and provides good results for the higher PRF region. The shown SNR scaling factor of the cascaded network is normalized to Φbf obtained for uniform sampling of the conventional network.27 It already incorporates the impact of the sub-aperture dimension and taper coefficients on noise and signal power and allows hence for a direct comparison to the conventional beamforming approach. This comparison of the two characteristics regarding the SNR scaling in the focused image is given in Fig. 93. Again, both curves are normalized to the uniform value of the conventional case.
Fig. 93 shows that an improved SNR scaling factor is obtained for the pre-beamshaping sce- nario for PRF values above 1240 Hz (solid orange line) while the conventionally operated sys- tem is favorable for PRF values below 1240 Hz (dashed blue line). Further, the combination of both operational modes guarantees a sufficiently low SNR scaling over the complete range of
27It should be noted that in Fig. 92 the normalization of the SNR scaling is not (and in the following
will not be) done to the single-element SNR, but to the SNR of a single virtual aperture in the conven-
8.7 Cascaded Beamforming Networks 129
PRF. Compared to the example system of Chapter 7, a better SNR scaling is obtained for all op- erated PRF values, thus ensuring a NESZ below -19.2 dB.
In a next step, an analysis of the geometric resolution in azimuth yields a value of 1.03 m for both conventional and pre-beamshaping operation.
Finally, the ambiguous energy suppression of the modified system is investigated. As men- tioned above, the transmit antenna was slightly increased to 3.15 m to guarantee a minimum am- biguity suppression of -21 dB in conventional operation of the system (cf. Fig. 94, dashed blue line). When the pre-processing network is applied, the suppression becomes better for PRF val- ues above ~1210 Hz and is clearly improved for higher PRF values due to reduced sidelobes of the receive pattern and minimized amplification of the ambiguous energy caused by the adapted phase centers (cf. Fig. 94, solid orange line).
Fig. 94. AASRN vs. PRF with pre-beamshaping network (solid orange) compared to conventional
digital beamforming (dashed blue) (BD =7.6 kHz). The peak in the conventional approach occurs at
the “singular” PRF=1440 Hz (cf. Section 4.3.3).
Fig. 93. Simulated SNR scaling factor Φbf of the conventional digital beamforming network (dashed
blue) compared to the case where a pre-beamshaping network is added (solid orange) (BD =7.6 kHz).
In conclusion, it can be stated that the application of an appropriate analog pre-processing network clearly extends the full performance range with respect to the operating PRF. In combi- nation, the results for AASRN, resolution, and the SNR scaling factor Φbf show that it is favorable to use the conventional configuration below a PRF of 1210 Hz and apply the pre-beamshaping for PRF values higher than 1240 Hz. In between, the focus can be either turned to the NESZ by choosing the conventional approach or the pre-processing is applied to concentrate on the opti- mization of the AASRN.