Chapter 10. Discrete Fourier Transform
Sample problems
10-S1
Let us consider the discrete version of an ideal low pass filter, (Figure 10.13). Derive a discrete form of the low pass filter.
Solution to 10-S1
The response of the ideal low pass filter in the frequency space is given by,
The response of the filter in the coordinate space is,
Figure P10.1.Discrete band pass filter in the coordinate space.
Figure P10.1 is the discrete version of the low pass filter (Figure 10.13) in the frequency space. It represents the weights in the x space of the filter that will produce a low-pass filter with cut-off frequency in the frequency space. It is possible to see the symmetry of the weights with respect to the coordinate space. One should remember that linear filtering is an operation that modifies the input pixel values through the application of an algorithm producing an output that is a linear combination of the input values. The discrete aspect of the transform is due to the fact one operates with the pixel values (discrete values) of the involved variables. The plot of Figure P10.1 can be done by extracting the discrete values from a table of the sinc function.
10-S2
The filtering operation can be applied to the amplitude of the signal or to the phase of the signal. The transfer function of a filter can be represented by,
The input can be represented by and the filtered output by
With this notation one can write,
, hence and
A filter can affect the phase and the amplitude of a signal. If the filter is such that , then the filter is an amplitude filter. If , the filter is a phase filter. Is the low-pass filter of sample problem 10-S1 a phase filter, or an amplitude filter?
Solution to 10-S2
For an amplitude filter,
, the amplitude of the spectrum is modified by the factor , but the phase is unchanged.
With the type of filters being utilized the amplitude of the signal in certain frequencieswill be changing, that is .
The low-pass filter is an amplitude filter since it changes only the amplitude of the signal.
10-S3
From sample problem 10-S2amplitude filters are ideal to remove undesired components in the input without altering the phase of the signal, the phase of the signal contains the information of interest in fringe pattern analysis and hence should not be changed. These filters are also called finite impulse response filters (FIR). Figure P10.1 is an example, in this figure only the central portion of the filter has been represented, and then only one of the infinite number of additional regions called lateral lobes. How can the infinite filter of P10.1 be transformed into a finite filter?
Solution to 10-S3
This problem can be formulated in the following way,
,
the filter is represented as a function of the exponential function and ω the angular frequency.. The basic idea is to obtain a finite length impulse function by truncating the infinite response filter or replacing the infinite series by a finite series where the h(n) have the role of a Fourier series coefficients. The coefficients can be represented by the product , where w(n) is a finite response window. If N is the total number of discrete samples, the desired result is
then. The meaning of this result is that the transfer function of the filter is the product of the convolution of the desired transfer function with the FT of the window.
.
Figure P10.2.Convolution of the desired transfer function of a filter, the rect function with the FT of a rect function that generates the finite response low-pass filter.
The desired transfer function is a rect function, the windowing function is the FT of a rect function, the sinc function. It is interesting to observe that the sudden transition at the end of the cut-off frequency of the rectfunction is replaced by a smoother transition that reduces the Gibbs’ phenomenon (See Figure 10.16). As a result of the introduction of a transition error bands are created as shown in Figure P10.2. The effective design of the filter consists of minimizing these three types of errors.
10-S4
The filtering operation has the purpose of extracting the information of interest from the overall signal that was experimentally obtained. All the components of the obtained signal that are not of interest are called noise. This statement is very general because it not only refers to the random noise which is always present in any experimental determination but to all other components that are not of interest. Let us consider the case of additive noise, that is the experimentally obtained function called f(x,y), this signal is the addition of the signal of interest s(x,y), plus all other signals ni(x,y), with i=0,1,2, ...n; 0 corresponds to random noise, the rest are all other signals. Is there an optimum filter?
Solution to 10-S4
It starts by writing,
To remove the noise a linear filter that will be characterize by the unit impulse response h(x,y) is introduced such that
g(x,y) indicates the best signal that can be extracted from the experimental signal f(x,y). The best possible will be g(x,y)=f(x,y). With the above substitution and taking the FT of the convolution gives,
.
From this equation we get,
dividing numerator and denominator by the FT of the noise,
It is possible to recognize that the numerator of the preceding fraction is the signal to noise ratio. If the ratio of signal to noise is a very large quantity the limit of the above expression is 1. If the signal to noise ratio is 1, then . Although the above expression does not provide us a complete answer it tell us that knowledge of the actual signal or of the composition of the noise will help to achieve the best possible filter.
Problems to solve
10.1
The result of the analysis presented in sample problem 10-S4indicates that some knowledge of the signal we are trying to recover is a basic element in the process of elimination of unwanted information in the experimental signal.What are the essential elements of the model of the optical signal presented in section 10.3.
10.2
What is the importance of the result obtained in equation (10.38)?
10.3
What additional information is added in section 10.3.2 that contributes to give an answer to sample problem 10-S4.
10.4
Point out the additional important facts added by section 10.3.3 to the solution of the noise problem.
10.5
Why you think that is important to formulate the answer to the problem of signal recovery in terms of the phase concept?
10.6
What important drawback does the signals-in-quadrature method present?
10.7
Why do you think that the phase-stepping technique is the preferred method for phase retrieval?
10.8
Under what circumstances can the two alternatives presented in problems 6and 7 be considered equivalent?
10.9
In problems of Continuum Mechanics is it possible to observe fringe dislocation type of singularities?
10.10
What remedies in phase unwrapping can be utilized to remove singularities?
10.11
What alternative solution can be utilized to solve the problem of phase unwrapping in problems where the source of fringe dislocations is due to noise?