Image Processing Reference
In-Depth Information
accurately determine the results of the linear operations (additions, subtractions, constant
multiplications and delays), and the purpose of the filters is to perform a given linear
transformation of the input signal. Consequently, the features offered by AA match
perfectly with the requirements of the interval-based simulations of the unquantised digital
filter structures.
When the quantization operations are included in this type of analysis, the affine forms
must be adjusted to include all the possible values of the results. Since AA keeps track of the
effects of the uncertainty sources (the noise terms can be seen as the first-order relationship
between each uncertainty source and the signals), the affine forms are easily modified to
simulate the effects of the quantization operations in the structures containing feedback
loops.
In summary, one of the most important problems of IA to perform accurate interval-based
simulations of the DSP realizations is the cancellation problem. The use of AA, in
combination with the modification of the affine forms in the quantization operations, solves
this problem and allows performing accurate analysis of the linear structures, even when
they contain feedback loops.
3.2 Computation of the fourier transform of deterministic interval-based signals
The analysis of deterministic signals in DSP systems is of great importance, since most
systems use or modify their properties in the frequency domain to send the information. In
this sense, the decomposition of the signals using the Fourier transform as finite or infinite
sums of sinusoids allows to evaluate these properties. Conversely, it is also widely known
that a sufficient condition to characterize the linear systems is to determine the variations of
the properties of the sinusoids of the different frequencies.
The following experiment shows the variations of the properties of deterministic signals
when intervals of a given width are included in one or all of their samples. These widths
represent the possible uncertainties in these signals and their effect on their associated
signals in the transformed domain.
First, we evaluate the effects of including uncertainties of the same width in all the samples
of the sequence. The steps required to perform this example are as follows:
1. Generate the Fast Fourier Transform (FFT) program file, specifying the number of
stages.
2. Generate the sampled sinusoidal signals to be used as inputs.
3. Include the uncertainty specifications in the input signals.
4. Compute the Fourier Transform (run the interval-based simulation).
5. Repeat the steps 1-4 modifying the widths of the intervals of step 3.
6. Repeat the previous steps modifying the periods of the sinusoids of step 2.
Steps 1 to 4 generate the FFT of the interval-based sinusoidal signals. Step 5 has been
included to investigate the effects of incorporating uncertainties of a given width to all input
samples of the FFT. By superposition, this should be equal to the numerical FFT of the mean
values of the original signal, plus another FFT in which all the input intervals are centered in
zero and they all have the same width. Finally, step 6 allows us to investigate the variations
of the computed results according to the periods of the sinusoids.
Figure 3 shows two examples of cosine signals with equal-width intervals in all the
samples and their respective computed FFTs. Figure 3.a corresponds to a cosine signal of
amplitude 1, length 1024, period 32, and width 1/8 in all the samples, and Figure 3.c
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