**2. Problem and approach**

2 Will-be-set-by-IN-TECH

constituents such as proteins and carbohydrates. Any that are dependent on other factors are eliminated in the regression analysis having no significant correlation with the material properties. Rather than using measurements in the frequency domain, of course it is possible to transform time domain measurements to the frequency domain using Fourier or other forms of transformation. For time varying data acquired by TDR the inverse Fourier transform

> +∞ −∞

where h(t) is a time dependent function the Fourier transform of which is a frequency dependent spectrum g(f). Examination of equation (1) shows that at any instant t every component part of the spectrum contributes to the value of h. Because the subsequent multivariate analysis can be thought of as dealing with shape and is concerned only with the variations in g then transformation of h(t) to the frequency domain is not required, since related variations are present in h(t) and the shape of the time domain function is equally useful. A further justification for eliminating the transformation step concerns the difficulties with which it is fraught. Firstly, the truncation of the pulse after a finite measurement interval can introduce undesirable distortions in the integral from convolution of the pulse with the rectangular time window (windowing). Secondly, the act of sampling the pulse at regular intervals means that frequencies present with periods shorter than the sampling interval are incorporated as lower frequency information (aliasing), and thirdly, to accurately reconstruct the reflection coefficients in the frequency domain exact time referencing of the pulse is required, else phase errors cause large inaccuracies at the high frequency end of the spectrum. By carrying out the analysis on the raw, sampled TDR pulse, all the errors above can be avoided. This was tried and demonstrably gave the same results as the spectral data, with less computer effort and fewer error generating problems. The first problem to which this was applied was something less tangible than water content: it was in fact the quality of various seafoods as defined by more subjective methods [8]. The success of this approach naturally led to attempts to measure dielectric objects in a non-contacting fashion, using firstly transmitted UWB quasi-Gaussian pulses of 400ps width [9]. The transmitting and receiving antennas in this initial work were double ridged horns and the sample was arranged in a wide layer of uniform thickness. In such a situation, additional interfering variables can be the position of the antennae, polarization of the transmitted wave, multiple path effects and a host of others, all of which may be eliminated by multivariate analysis. Effects due to dielectric properties can be separated from those due to geometry and other exterior factors. Thereafter, further work followed with samples of increasingly complex shapes and different orientations, beginning with simple rectangles and progressing to other shapes such as triangles and circles [10, 11], albeit still of constant thickness. At the same time, various forms of UWB antennas were investigated but currently the choice is an array of simple dipoles for the receiving antennas with a horn antenna transmitting the pulse. The multivariate data analysis still uses PCA as a first step to reduce the data and provide shape descriptors, but because PCR is not entirely suitable for non-linear processes (being a linear regression of variables) the PCs are used as input to a non-linear ANN. A great deal of work has now been done, gradually broadening the application parameters until now it is possible to measure UWB dielectric properties of objects with any shape, thickness, orientation and without contact [12–20]. This has been the subject of the project 'ISOPerm' (irregularly shaped

*g*(*f*)*e*

<sup>2</sup>*πift*d*f* , (1)

in its most general form can be written as in equation 1.

*<sup>h</sup>*(*t*) = <sup>1</sup> 2*π*

objects-permittivity) the methods and results of which will now be described.

There is no appropriate method for the determination of the dielectric or related properties of irregular shaped objects in free-space. The investigated objects are considered to be small compared to the range of wavelengths and the footprints of the antennas used. A visualization of the problem is depicted in Figure 1. An electromagnetic wave illuminates an irregularly shaped object with its frequency and temperature dependent dielectric properties. It is surrounded by air. Portions of the scattered field are collected by multiple field probes, i.e. a line array of receiving antennas. The scattered signals contain information about geometry as well as the dielectric properties. Because of the complexity of the problem it is assumed that the development of a physical model (as used in conventional free-space methods having a plane parallel plate) would be too complex. Furthermore, an on-line method suitable for characterizing many objects in a short time is anticipated. Therefore, multivariate calibration methods are applied in order to separate dielectric from geometric influences. The objects are considered to be homogeneous and non-magnetic (*μ<sup>r</sup>* = 1). However, it is assumed that it would also be possible to predict both average permittivity and permeability. In research and industrial applications other properties, for example the water content, moisture, freshness or quality of foodstuffs, are of great interest. They are often strongly correlated to dielectric properties and can be determined directly without knowing the permittivity.
