Simulations

We will start by using the FFT method and then apply MUSIC in the same conditions. The latter algorithm has been more extensively tested for two main reasons: firstly because the FFT is less flexible and doesn't provide so many alternatives to be considered and secondly the superresolution technique must be further explored to reveal more information required to its full conq>rehension.

The simulations presented here are based on planewaves and circular waves models incident on a uniform linear array configuration, as discussed in the previous sections.

These arrays have been specified with the same test parameters used in the measuring sections, allowing straight conq)arisons to be made between this results and the ones to be obtained in the immediate section. These simulations, however, can lighten with their results some details that may escape firom the actual dataset analysis when eventually awkward conditions are met. They also provide the opportunity to compare the two algorithms in focus.

It is noticeable that these array response simulations and data collected by the probe in the experimental tests can be treated indistmctly in this analysis. This is totally accepted because the coupling o f the array elements is normally ignored in theory and the probe measurements phase and amphtude are relative quantities independent o f time, except for the noise parcel assumed at very low level.

There are several controls that can be managed to shape the overall pattern o f an array, namely:

a) the geometrical configuration o f the array;

b) the relative displacement between elements;

c) the gain of each element;

d) the phase shifl; o f each element; and

la this case a linear uniformly spaced array o f identical elements is being considered so these parameters are reduced by the symmetry o f the problem leaving us fewer choices to shape the array pattern. The geometrical configuration reduces to a sim$)le matter o f total number o f elements. The relative displacement can be broken down into the inter element spacing and the wavelength. The gain and phase shift o f each element are removed by the assumption o f identical elements and the element relative pattern can be discarded by assuming it omnidirectional.

All these assumptions and snq)lifications are also logical to the measuring method adopted during the experiments, so the simulations will still reflect the actual experimental conditions. The only thinkable exception is on the sin^lification done to the element pattern by considering it uniform and this may be contested in practice. However, if this detail is not considered at all, it is only when the analysis is made along the vertical axis that the results can be altered by the non uniformity o f the vertical dipole pattern. Furthermore, this will only attenuate the magnitude o f some reflections, but won't change their angle o f arrival.

The initial variables to be used in simulations will be chosen to produce favourable conditions for an algorithm smooth performance. For carrying out the measurement routine, one needs basically to settle the wavelength, the inter element spacing and the total number o f sangles (or the array length), as it has been explained in section 3.1.

To optimise the FFT it is likely that the number o f elements is given a power o f 2 value, rendering the fast radix- 2 method to be used. Also, the inter element spacing is bounded to be shorter than the half wavelength, and the array length is limited by the probe positioner at about 0.35 metres, which altogether can outline a test simulation.

Hence, let's consider a planewave arriving at 0 degrees to the linear array formed by 64 elements displaced from each other by ^ 6 and assume a frequency o f lOGHz. Both algorithms have been run for this signal and array configurations and the results are presented in figure 3.4 (a) and (b).

When the planewave angle of arrival is gently offset from the array broadside the angular estimation responses become as it is displayed in the remaining plots o f figure 3.4, where (c) and (d) resulted from a 1 degree offset and (e) and (Q emerged from a 2 degree displacement respectively.

20 - 4 0 0 _ 10 log ( Pmu _“200 - “400

(a)

Figure 3.4 FFT and MUSIC planewave angular spectrum estimation comparison.

By observing this results, it is significant the revelation that the MUSIC algorithm maintains its sharp angular resolution independently of the angle o f arrival changes while the FFT doesn't. The spectral impulse in the former algorithm will eventually reduce when the angle o f arrival gets closer to the array fire end but this is caused by the array factor decay. The phenomenon seen in the FFT's plots in figure 3.4 is much stronger and relies on the intrinsic characteristics of a discrete transform. As mentioned in the preceding chapter,

this aliasing is consequence o f the real signal harmonics being suppressed and substituted by the FFT's own misplaced harmonics.

The logarithmic scales are multiplied by different constants because while the FFT is related to the signal level, the MUSIC estimator considers the covariance matrix, which stands for the power spectrum

Another fact to be aware o f is that the graphics' independent variable is not the same for the two algorithms, as displayed in figure 3.4. To make it easier for conq)arison, equation (3.11) can be used to generate a FFT plot con^atible variable. Alternatively the FFT domain can be changed by

where X and D are respectively the wavelength and the array length, and n is the index of the FFT's output term If the number of elements is not large enough, the factor

( # - l ) / # must be multiplied to the inverse sine argument.

To obtain a better view on the magnitude o f their performance differences, in the succeeding paragraphs we will look at the resolution o f these two algorithms, by simulating multiple planewaves, travelling in close bearings, and arriving at the same 64- element array.

We consider the situation o f two planewaves with an angular displacement o f 6 degrees arriving at this array. The angular spectrum is corcçuted by FFT and MUSIC with spatial smoothing and the results are shown in figure 3.5.

This time the independent variable o f the FFT has been changed by using equation (3.12), conforming the two obtained graphics to a unique representation domain.

From figure 3.5 (a) it is observed that the FFT has again displaced the wavefi*ont location to one o f its harmonics. The level of the peak to the background is about 10 dB below the level found for the worst case o f a single planewave (remember that the worst case is when the signal spatial-spectrum is located in the middle o f two consecutive harmonics). As expected, this is a result poorer than the one obtained for a single arriving planewave.

2 0 log ( FFT - 4 0 - 9 0 - 8 0 - 7 0 - 6 0 - 5 0 - 4 0 “ 30 ^ 0 - 1 0 0 10 20 30 40 50 60 70 80 90

(a)

Figure 3.5 Spatial-spectrum generated by a) FFT and b) MUSIC fo r two interfering

planewaves arriving at -3 and +3 degrees.

In a further analysis it can be seen that the behaviour of the FFT has an awkward pattern throughout the angular spectrum when the angle of arrival is varied along the semi-plane.

The MUSIC angular pattern in frame 3.5 (b) is showing an accurate angular positioning and a large peak size, whose value is almost the same even when compared to the one in figure 3.4, where only one signal is present.

If we let a couple of incoming planewaves travel in closer directions the two algorithms will end up failing to resolve them, but MUSIC does it starting from smaller angles.

In figure 3.6 a similar test for two wavefronts is shown, however this time the separation between sources is only 2 degrees.