CONTRASTACION DE LA HIPOTESIS

• Introduction

• Computer simulations of the correlator performance • Ferroelectric liquid crystal spatial light modulator • Correlation system results without SLM

• System Experiment with the spatial light modulator

8.1 Introduction

This chapter describes the final choice of design parameters for our incoherent correlator system. First, it starts with the computer simulation of the system under real constraints (Section 8.2). These simulation results not only model the performance of the system in practice, they also helped us to choose the test patterns for the actual experiment. In section 8.3, the effect of the SLM in the system performance is investigated. In section 8.3.2, we analyse the reason for the lack of contrast. Simple correlation experiments without the use of the SLM are then demonstrated and compared with the simulation results (section 8.4). In section 8.5, the final experiment with the SLM in place is described.

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8.2 Computer simulations of the correlator performance

8.2.1 Introduction

In this section, the correlator system is simulated using a computer. Since the shared microlens correlator behaves in a very similar way to the shadow casting correlator, the simulation results will apply to both systems. The different techniques for the simulation are first described and results are shown for correlation of binary patterns with 10:1 contrast ratio similar to that of our FLC-SLM. It will be followed by how the resulting signal-to-noise ratio can be improved using bipolar and randomised patterns. These results also helped us to generate masks for the input and filter planes to give more easily observable results.

8.2.2 Simulation techniques

In chapter 4 we showed that correlation can be achieved in two domains: - real-space and Fourier-space. Similarly, computer simulation of the system can also be achieved digitally in both domains. In the Fourier-space domain. Digital Fourier Transforms are involved and an algorithm called the Fast Fourier Transform (FFT) is commonly used. A number of steps is required to prevent the aliasing of the data due to sampling. For a symmetric signal f[p) in 1-D which is represented by N pixels, the signal is repeated every N pixels in real space due to the sampling in the Fourier space (fig. 8.1a). Usually, only the positive part of the signal is taken to avoid negative indexing in a computer. Since the pattern repeats,/(N - p) =f[-p) and the negative part can be taken into account. Then this rearranged signal undergoes a FFT which gives the frequency sprectrum in a similar manner, i.e., the negative half is shifted by N pixels. After shifting back the negative half of the signal, the correlation signal in Fourier domain G(co), is given by:

G(co) = F(co)//*(co) = F(cû)H(cû) (8.1)

since the filter /i(p) is a real and symmetric function. Then the pattern is split once again in a similar manner before the inverse FFT is operated and the two halves are recombined afterwards.

For a 2-D pattern, the splitting and shifting operation is done similarly and is described in fig. 8.1(b).

In the real-space simulation, the digital correlation is achieved by the discrete form of the correlation equation, which involves the shifting and multiplication operations. For a 2-D pattern, the intensity at the output pixel 0(p,q) is given by:

N N

0 ( p , ^) = ^ 7(m, n ) T{ m + p , n + q) (8 .2)

where I(m,n) is the intensity at the input plane and T(m,n) is the transmittance at the filter plane. Since the shared microlens correlator operates in real-space, we use the latter simulation although both give the same results.

J

f(p )

f(N-p)=f(-p)

Split the negative h alf & shift by N pixels f(p )

1

i

Discrete Fourier T ransform

Real space FFT F(-Cù)=F(M-cù) Fourier space shift by M pixels in opp. direction

( a )

Fig. 8.1 The split and shift operation before and after the FFT in (a) one dimension and (b) two dimensions

Correlator system : O ptim u m system design, analysis, sim ulations and experim en ts 188

8.2.3 Simulation results of binary patterns

The correlator system is simulated by using Matlab^ programming on a Sun SPARC Unix workstation (Appendix IV). Since the FLC spatial light modulator has binary amplitude, the input and mask (filter) patterns can take values of either 1 or 0. When the contrast ratio of the SLM under LED illumination (10:1) is taken into account (described in chapter 6), the values of a pixel will be either 10 (if turned on) or 1 (if it is off). The effect is that the resultant correlation pattern will be scaled by 10 times. A pattern which shows an inverted letter ‘A’ is used as the filter (fig. 8.2a). The inversion is due to the software representation of matrices. The input pattern depicts two displaced letter ‘A’s and a letter V (fig. 8.2b). The patterns are assumed to be displayed on a SLM. If the SLM has infinite contrast, the correlation pattern of these is given in fig. 8.2c In reality, the contrast ratio is only 10:1 and the resulting correlation is shown in fig. 8.2e in which it is harder to distinguish the correlation peaks from the background.

Each correlation pattern between the letters is symmetrical as expected. The auto­ correlation patterns (the correlation of the letter ‘A’ with itself) feature a cross with a horizontal stroke through the centre bright spot. The diagonal of the cross signifies the correlations of the diagonals of the letter ‘A ’ and the horizontal line is the result of the correlation of the corresponding horizontal stroke in the letter. These are the sidelobes which reduces the signal-to-noise ratio. The cross correlation between the letter ‘A ’ (inverted) and ‘V ’ just gives a cross with a lower correlation peak.

One of the useful parameters in a correlation is the signal-to-largest sidelobe ratio (SLSR) in 2-D and it is found to be 3.2 in the infinite contrast case (from fig. 8.2d). For the correlation from patterns with CR of 10:1, the SSLR is reduced to 1.6 (fig. 8.2f). If a linescan is run across the correlation peaks as shown in fig. 8.3a, the signal-to-sidelobe ratio (SSR) along that scan is 15:1 for infinite contrast and 2.2:1 for 10:1 CR. This line does not necessary include the largest sidelobe but will serve as a convenient point for comparison.

In the case of the patterns with CR of 10 : 1, the low signal-to-noise ratio observed in the correlation is due to in the leakage of a tenth of the intensity of the bright pixels (those turned on) through the supposedly dark pixels (those which are off).