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11.4. FILTERS BASED ON FOURIER TRANSFORMATION

Lecture



The FFT convolution algorithm, discussed in the previous section, is often used in computer simulation of linear analog filters. In this section, the errors inherent in this method of modeling are analyzed, and methods are described for determining the discrete frequency response from a given continuous frequency response. In order to simplify the presentation, only one-dimensional signals are described here.

Consider a long one-dimensional continuous signal.   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION whose spectrum   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION equals zero if   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION more cutoff frequency   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . It is necessary to find the convolution of the signal with a continuous impulse response.   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION whose frequency response is   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION also limited by frequency   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . As mentioned in ch. 1, convolution can be performed either in the spatial domain in accordance with the ratio

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION , (11.4.1а)

either in the frequency domain

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . (11.4.1b)

In ch. 9 describes the discretization method of the convolution integral (11.4.1). Continuous impulse response   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION must be truncated by multiplying it by the weight function   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . The result is a weighted impulse response.

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION , (11.4.2)

Where   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION at   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . The weight function reduces the effects associated with truncation. The convolution integral is approximated by an expression.

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . (11.4.3)

Then in   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION point take samples of the output signal with an interval   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION , and the continuous integral is replaced by the sum with the same step   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . The result is a discrete representation.

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION , (11.4.4)

Where   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION - integer closest to the fraction value   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION .

To calculate the sum (11.4.4) using the discrete Fourier transform, you can apply the algorithm described in sect. 11.3. In the first stage as the first   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION items   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION -element sequences are taken weighted impulse response responses, and as follow   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION elements are zeros. In this way,

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION , (11.4.5)

Where   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . Sequence elements   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION can be obtained from continuous impulse response   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION and weight function   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION by discretizing the product of these functions, i.e.

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . (11.4.6)

In the next step, the discrete Fourier transform is calculated.   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION at   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION points

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION , (11.4.7)

Where   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . After substitution of the expression for   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION in the formula (11.4.7) and transformations it turns out that the discrete frequency response of the filter is associated with the analog frequency response   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION and Fourier Spectrum   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION weight function by

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION , (11.4.8а)

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION , (11.4.8b)

Where   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION , but   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION .

Equalities (11.4.8) define the desired connection between discrete and continuous frequency characteristics. If continuous frequency response   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION and Fourier spectrum   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION the weight function is known and specified in an analytical form, then in principle the samples of the discrete frequency response can be obtained by analytically performing a convolution (11.4.8b) and finding the numerical values ​​of the function obtained at the same time in points   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION for each parameter value   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . In practice, it is often not possible to calculate the convolution analytically, especially for two-dimensional signals, and it turns out to be easier to perform the inverse Fourier transform of the frequency response.   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION to get the analytical expression of the impulse response and then take the samples   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION in accordance with the formula (11.4.6). You can use another approach: according to equality (11.4.8), take samples of the discrete inverse Fourier transform   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION multiply the extended sequence of samples of the impulse response to the weighting function, and then, by performing a discrete Fourier transform, obtain   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION .

Multiplication by a weight function, which can be performed according to equality (11.4.6) or implicitly in the spectral domain using the relation (11.4.8), is absolutely necessary if it is necessary to suppress those described in ch. 9 cyclical errors. When filtering images, a typical error is usually made when a sequence of samples of a continuous pulse response is taken as a discrete impulse response. Then, in general, all   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION elements of the corresponding extended discrete impulse response will be nonzero, i.e. the length   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION discrete impulse response, "immersed" in the extended vector (11.4.5), will be implicitly taken to be   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . Therefore, all filtered samples will be distorted due to cyclical error.

It is known [10-12] several types of weight functions suitable for use in discrete linear filtering. Some of them, most often found in practice, are given in table. 11.4.1, and their graphics - in Fig. 11.4.1. In fig. 11.4.2 shows the corresponding spectra consisting of the main lobe and a set of side lobes, the amplitudes of which usually decrease with increasing frequency. Analysis of the expression (11.4.8) shows how the shape of the weight function and its spectrum affect the output signal. The shape of the main lobe of the spectrum of the weight function determines the signal distortion in the range from 0 to   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION . Side lobes cause spectral overlap because the weighted impulse response   11.4.  FILTERS BASED ON FOURIER TRANSFORMATION has a range of unlimited width. Smooth weight functions have weak side lobes, which reduces the overlapping errors of the spectra, but the main lobe in these cases turns out to be wider, which leads to a smoothing of the signal spectrum. When designing filters, it is necessary to find a compromise between these two types of errors. Both errors can be reduced by increasing the length of the weighted impulse response, but this will either reduce the length of the processed signal or increase the amount of computation.

Table 11.4.1. Weight functions

Function

Definition

Rectangular

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION

Bartlett (triangular)

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION

Hannah

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION

Hamming

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION

Blackman

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION

Kaiser

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION ,

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION - modified Bessel function of zero order of the first kind

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION

Fig. 11.4.1. One-dimensional weight functions [10].

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION

  11.4.  FILTERS BASED ON FOURIER TRANSFORMATION

Fig. 11.4.2. Spectra of one-dimensional weight functions [10].

a - rectangular; b - triangular (Bartlett); in - Hanna; g - Hamming; d - Blackman.

created: 2016-09-09
updated: 2021-03-13
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Digital image processing

Terms: Digital image processing