2.6 Mathematical Methods of Linear Model Based Image Processing Procedures Processing Procedures
2.6.7 Theory and basic laws of sampling
Those mathematical tools and procedures have been written in the previous chapters – discussion of linear system properties – will be applied for describing and characterizing of the shift invariant quantized i.e. discretized system in the followings. Operation of quantization is not a linear transformation. There are mapping from the continuous value set to the discrete parameter space i.e. to the set of discrete numbers. The basic laws and consequences of the discretization process by one dimension approach are discussed in the chapter. Parameter value denoted by is the time in the current description, i.e. the quantization process will be discussed by the time dependent events.
Relations between the analogue and digital information evaluation process
It is possible to see by Figure 33., the data processing unit determines the various typical parameters of the process taking place in a system. Characteristic parameters of the system process can be evaluated by analogue way too detouring the quantization operations /as was written and mentioned in the previous chapters/. signal processing can be executed by digital way only, after the quantization of the parameter domain as well as the discretization of its corresponded value set i.e. both set are converted into digital values.
Figure 33.
Consequently, the digitization of happen by the subsequent two steps:
a) Quantization of , i.e. quantization of the parameter domain
b) Quantization of , i.e. discretization of the corresponded value set (codomain).
Definitions:
a) Quantize of the domain value, i.e. time values - as the independent variables - is called Sampling, or more precisely Sampling & Hold (S&H).
b) Discretization unit executes the quantization of the amplitude i.e. the corresponded value set (codomain ) is called Analogue to Digital Conversion denoting ADC or A/D.
Sampling-unit:
Sampling converts the continuous signal (both in the domain and in the codomain) into continuous signal series only in codomain: , where is equivalent with the corresponded value at moment of the process (see Figure 34.). is called sampling time (or “aperture” in abstract way).
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Figure 34.
Quantizing Unit:
Quantizing unit converts the continuous signal to discrete set (discrete number set). The quantization is a non-linear operation having the following description:
, for , where N is the number of sampling, is the
output, while is the input of the quantizing unit (see Figure 35.)
Figure 35.
Sampling law
Let‟s investigate the relation between the particular represented sample and its Fourier transformation (see Figure 36. and 37.)
125 Figure 36.
Figure 37.
Fourier transformation of the signal is as follow:
Let‟s repeat the signal periodically in the domain.
series of this periodic function exists and can be written as follow:
, where the coefficients of Fourier series can be determined:
, i.e. set of will determine injectively the periodic function.
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. It is possible to conclude: which means, if Fourier coefficients are known, then values of Fourier spectrum of
function at the points can be derived. These values consequently will determine the values of function at every moment.
First law of sampling:
Fourier transformation of signals being finite in time i.e. function are determined by the values at every points, where .
Let‟s consider the following band limited spectrum (see Figure 38.), where the bandwidth is in the Fourier spectrum.
Figure 38.
It is possible to tell, that the dependent signal, function can be derived by the above shown band limited spectrum:
Let‟s repeat the function by periodicity in the whole frequency domain by similar way as was described above in time domain case.
Thus, .
127 Fourier series of the periodic signal in frequency domain:
, where the coefficients are
Consequently,
, where it is obtained: i.e. the set of will determine the periodic function. Furthermore, it is possible to see, that coefficients determine the value of function at each moment. Since all the existing coefficients will determine function, then they will determine also function by similar way as was used previously (i.e.
coefficients will determine function everywhere, consequently is also determined in every points).
Second law of sampling:
The time domain (or the generalized parameter domain) questions and problems will be under consideration by the second law of sampling. The main interpretation of the second law of sampling is the signal (function) determining by a band limited spectrum with the values at points.
Let‟s see, how is possible to get from the sampled values the dependent function.
Thus:
Let‟s determine the integral:
i.e.
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Let‟s execute the necessary substitution:
Mathematical interpretation of the second law of sampling is the following:
Any of the finite band limited and Fourier transformable signal (or function) can be expandable into series on the function set and the coefficient of element of the function will be the value of point.
Mathematical interpretation of the first law of sampling:
Similar steps and methods have to apply as above. Result of the expanding into series in the frequency domain of the signal (or function) with time width (or parameter width) will be the following:
, where the coefficient of the the element of the series is the value of at the point.
Physical interpretation of the sampling laws:
Main goal of the physical interpretation is to clarify the relation between the and functions, where is the input function, while is the sampled process (signal or function) of S&H. Let‟s consider Figure 39., 40., 41 and 42.
129 Figure 39. Input function of sampling-unit undergoing to sampling
Figure 40. Figure 41.
Simple switching diagram of the ideal sampling-unit , where
Figure 42. Sampled process depending on the time
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The sampled process can be expressed in the domain as follow:
Let‟s execute the Fourier transformation on both sides:
, where , is a
convolution in the frequency domain. Since switching function is a periodic function, the Fourier function of can be derived on the following way:
Let‟s apply the main properties of Dirac-delta:
If an function is periodic, then the Fourier series can be written as follow:
Consequently, the Fourier transformation of an periodic function is:
Thus the Fourier transformation of the switching function can be written with help of its Fourier series - where coefficient is need from the Fourier series-.
Consequently, Fourier series of switching function:
131 Then the Fourier transformation of the switching function:
Let‟s write down the sample process in the frequency domain (by the Fourier transformation) by substituting and formula and applying the convolution rule:
Let‟s see, what conclusions are possible to do by the obtained expression:
1.) Fourier transformation of the sampled signal may be obtained by the weighted periodic recurrence of the spectrum of input signal among the frequency axis (see Figure 43.).
Figure 43.
2.) If the model of sampling is an “Ideal electric gate”, then the energy content of signal will be changed by ratio. In case of ideal sampling is possible to prescribe the equivalence of the energy content between the single element of series and . It is need to provide the increasing of the transferred energy with the decreasing . If the amplitude of the switching function will change by way, then condition is satisfied.
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3.) If condition is satisfied in case of ideal sampling, then the periodic recurrence spectra won‟t overlap each other, consequently the input signal can be restored without distortion by an ideal bandwidth low-pass filter.
4.) If the bandwidth and the sampling time (more generally the sampling aperture) i.e. satisfy the sampling law, then the signal can be restored without distortion independently of the mid-frequency point position. The sampling time (or aperture) is determined by the band-width of the signal not by the high pass frequency limit.