The pulse waveforms are suitable for monostatic radar configurations. However, together with a pulse code, techniques of pulse compression are needed. In fact these allow to obtain a good compromise between range resolution and radar performance..
In fact, a resolution δ R = cTi
2 of 5km would require a pulse duration Ti equal to 33µs, which
would mean transmitting a peak power (A2/2) too high to get a good detection to receiver which is strength correlated with the signal energy (E =A22Ti).
Through the pulse compression, the range resolution is not proportional to Ti, but depends
on the band (B) of received signal. For this reasons, it is possible to separate the needs relative to detection from those relative to range resolution.
The resolution obtained with pulse compression, is the same that of a no-modulated pulse of duration Ti
D where D is the compression ratio (equation 5.5):
In the plausible hypothesis that the pulse duration is of 4ms, it is necessary to size the system in order to have a compression ratio of 120 (equation 5.6) which allows to obtain a bandwidth B= 30kHz (equation 5.7). It is worth underlining that this bandwidth value is smaller than the maximum extension of frequency (around 100kHz) on which are negligible the non-linearities of ionospheric propagation channel.
δ R = c 2 Ti D ⇒ D= c 2 Ti δ R = 120 (5.6) B= D Ti = 30kHz (5.7)
Of course, in terms of radar applications, together the autocorrelation function, also the response of the matched filter depends on signal encoding. Specifically, a suitable signal encoding, can make the response of the matched filter less sensitive to distortions (e.g., power loss of the peak, symmetry loss of the main lobe as well as its reduction and augmentation of side lobes) caused by Doppler effect.
These effects, and in particular the power loss of the peak, are significant for their impact on the primary function of the radar: the targets detection.
Therefore, it is essential to evaluate the waveform tolerance to the Doppler effect: the instrument typically used for this purpose is the ambiguity function.
The ambiguity function (χ), described in the equation 5.8, is a two-dimensional function of time delay (τ) and Doppler frequency ( fd).
χ (τ , fd) =
Z +∞
−∞
e
S∗(t)eS(t + τ) exp(− j2π fd) dt (5.8)
with eS(t) complex envelope (ℜ{eS(t) exp( j2π fnt)}, fn is the carrier frequency) of transmitted
signal S(t).
Based on the analysis performed, the waveform most appropriate for the primary purposes of the radar, and alternatively characterized by lower contraindications for other applications, is the multi-pulse linear chirp IM. The reasons for this conclusion are that:
1. Lower losses for Doppler mismatch than other encodings analog or digital.
For this purposes in Figure 5.4-Figure 5.7, are reported the ambiguity functions in the case of Linear Frequency Modulation (L-FM or chirp) and Non Linear Modulation (e.g., C-FM, V-FM, QUAD-FM) for the analogue encoding.
Specifically the non linear modulation are characterized by an instantaneous phase and frequency provided by the following empirical equations 5.9:
φ (t) = π βTL i t 2−π βcTi 2 q 1 −4tT2 i fi(t) =φ (t)2π = Tt i[βL+ βc 1 r 1−4t2 Ti ] (5.9)
Figure 5.5: Normalized Ambiguity Function of C-FM pulse with D = 15
Figure 5.7: Normalized Ambiguity Function of QUAD-FM pulse with D = 15
It is worth noting that L-FM (Figure 5.4) and C-FM (Figure 5.5) stress a deviation of the main lobe of the ambiguity function on both the axes.This phenomenon, called range-Doppler coupling, implies a systematic error in range due to the asymmetry of the peak of the matched filter output caused by the Doppler mismatch. The same is not presents in the other analogue encodings, V-FM (Figure 5.6) and QUAD-FM (Figure 5.7) but in these cases there are a decoupling of the main lobe and an its lowering.
With Respect to analog waveforms, digital encodings (Figure 5.8-Figure 5.9) present an irregular variation of the secondary lobes in addition to being subject to the phe- nomenon of range-Doppler coupling.
Figure 5.8: Normalized Ambiguity Function of Frank code of length 16
Figure 5.9: Normalized Ambiguity Function of P4 code of length 16
3. Rapid decay of the side lobes of the autocorrelation function, although not optimal in the presence of multiple targets or clutter characterized by strong spatial decorre- lation;
4. Systematic error in distance estimation due to Doppler mismatch, nevertheless neg- ligible compared to the distances reached by the radar;
5. Good robustness against linear distortions of amplitude and phase, at least for rms phase errors (∆Φrms) less than 20◦− 40◦and for percentages of amplitude distortion
below 10%.
Figure 5.10 shows the combined effect of linear distortions of amplitude and phase as a function of quadratic (∆Φ( f ) = 2γsgn( f )(Bf)2) and cubic (∆Φ( f ) = 2γ(Bf)3) phase deviation. With regard to the effect of amplitude distortion, it was considered the following trend of the transfer function A( f0) = 1/(1 +1−ββ (2 f
0
B ) 2);
Figure 5.10: Mismatch Losses due to linear distortion of amplitude and phase, with β = 0.1 as a function of ∆Φrms for quadratic (blu lines) and cubic (red lines) ∆Φ( f ), for L-FM, C-FM, QUAD-
FM e V-FM modulations
6. Congruity with the waveforms used in existing systems.
• The simultaneous transmission of multiple frequencies (multi-IM), compared to other possible choices multi-carrier, presents the following features:
2. The illumination time for a given azimuth angle in transmission is reduced only to the integration time necessary for detection;
3. The scan time is reduced;
4. It can use all the skip time for stretch the duration of transmitted pulse and thus reduce the peak power.
Although from the theoretical point of view the instantaneous multi-frequency transmission was found to be better than the sequential mode, one must add that from the implementability point of view, have been identified the following issues:
• Variability of the amplitude of input signal to amplifier which rises to increase of the beams number;
• Formation of spurious in band and intermodulation products due to the amplifier nonlin- earity . For the reduction of which would be necessary the introducing of the adaptable tuning filters, difficult to obtain because of the bandwidth and the power involved;
• Gain reduction compared to single beam by a factor equal to the logarithm of the number of simultaneous beams to form;
• Degradation of side lobes;
• Reduction of transmitted energy because of a redistribution of the same on multiple beams, and then reduction of detection performance.
Therefore the choose of sequential multifrequency is most appropriate.
Without excluding any possibility, the numerical results on waveform parameters setting with regard to both sequential and istantaneous multifrequency and a comparison between they, are reported in the following chapter.