Kulyk A.Y., Kryvogubchenko S.G., Svyetlov A.V.

Vinnitsia national technical university

SALE FILTERING FOR DIGITAL SIGNAL PROCESSING

The process of digital signal processing plays an important role in the systems for information management, image processing, radio and sonar, information-measurement and others. An integral part of digital signal processing is filtering.

Synthesis of digital filters consists of two phases: determination of characteristics and their further implementation. On the first stage it must be found in the discrete plane the complex variables of transfer function, which should meet certain requirements. Transfer function as a mathematical model ignores many features of the hardware implementation. First of all – this is restriction of the bits transfer function coefficients, which follows by the formation of errors in calculating of the signal at the output of digital filter due to rounding or rejection of the results of arithmetic operations. Choice of bits of the coefficients transfer function is caused mainly by two factors: the margin of stability and frequency distortion characteristics of digital filters. For approximation of the characteristics of digital filters we widely use methods based on finding of the transfer function of analog prototype in p-plane, and then its discrete model in the z-plane [1, 2, 3]. It greatly complicates the optimization of pole distance of transfer function due to the complexity of its control during the transition to the z-plane. The most effective are the direct methods for synthesis of digital filters. The problem of optimization of digital filters is reduced to determining of the distance pole transfer function.

During the design of digital filters is the problem of determining the characteristics and their implementation is quite complicated. Thus, it is necessary to define the features of the process and design techniques to build their account.

To meet the requirements of the amplitude-frequency and phase-frequency characteristics of the digital filter and its physical implementation it is necessary to performe the following conditions:

, j = 1, 2,…, 2d_x+d_k; ; (1)

(2)

; (3)

where ;

– relative deviation of the amplitude-frequency response;

– absolute deviation of the phase-frequency characteristics;

– relative deviation (due to quantization) of coefficient a_j of transfer function level H_j;

- function of the relative sensitivity to changes a_i.

From point of stability this purpose relates to the problem of nonlinear programming, solution methods which are presented in the literature [4]. With a high order of filter the optimization procedure is quite complicated. In connection with this method it can offer to maximize pole distance for digital filter transfer function, described by expression (1). The poles distance from the unit circle increases with increase of value of N. Thus, in terms of increased resistance potential digital filter without increasing its order value of N is better to accept equal .

In terms of frequency characteristic distortion the solution (2), (3) involves minimizing the real (imaginary) parts of the sensuality functions . To resolve this problem, you must find expressions for the possible before. In the case of the use of such links is sufficient to analyze only once.

Admission to the deviation of frequency characteristics of digital filters due to quantization of coefficients of transfer functions can be divided into stages. As the most sensitive to changes bits level are the links of high merit [7], so the admission should be allocated in proportion to merit poles.

The basis of further calculation are the expressions (2) and (3), in which the left part of the equations are tolerances to frequency characteristics units. As for the solution of right-hand parts of equations (2) and (3), there are two possible approaches. The first one is to minimize the approximation . In this case the radical solution is increasing bits to describe the microprocessor digital filter coefficients, but it significantly increases the hardware cost of implementation. In some cases is more rational decision when the approximation is performed with maximum absolute value of coefficients. Since as the microprocessor implementation of digital filters are typically used fractional fixed-point arithmetic, so it is desirable that the transfer function coefficients would be close to one (but not more than it).

Task of minimization under given bits of coefficients representation and specific number of units n can be formulated as follows. Find:

; (4)

; (5)

, (6)

where і=1, 2, ..., n;

α_j, β_j, γ_j take value of 0 or 1.

The task can be solved by means of vector optimization, for which as objective functions are the

(7)

or

. (8)

Take into consideration the complexity to solve the problem of vector optimization with nonlinear constraints [5], it is advisable to consider other methods of minimization .

Sensitivity to changes of coefficients can be reduced by increasing the number of links. At the same time more stringent requirements can be revealed to the frequency characteristics of the designing filter, which in their turn would have reduced the impact of the quantization coefficient by reducing. However, increasing the number of links may be undesirable because of possible increase of hardware costs and reducing performance.

It is known [6] that the sensitivity of frequency characteristics of digital filters to change the coefficients of transfer function depends on remoteness of the poles from the unit circle. Thus, the minimization problemcan be examined like the problem of minimizing of pole distance, which was considered earlier.

In order to improve performance it should be used the approximation with coefficients, which is represented in binary code containing a small number of units. It is used for uncritical zoom coefficients and allows to reduce the multiplication to some simple shift operations with high performance. The task of units reducing in binary record of the coefficients should be looked when its used for the digital filter with transfer function

, (9)

which coefficients in a binary code are represented as

; (10)

; (11)

, (12)

that , are calculated in accordance with the following expression for

the calculation of the poles

(13)

that l=1, 2, …, n/2.

α_іj, β_іj, γ_іj take values of zero or one.

In this case solution of the problem to improve performance is to find

(14)

if

On the one hand increasing of n and N increases the stability of digital filters, on the other - it is necessary to reduce them to increase the sensitivity to change of transfer functions coefficients, reducing hardware costs and increase performance.

In terms to improve the performance of digital filters it is necessary to appropriate approximation with the coefficients of transfer function, which representation in binary code contains a small number of units. This approach allows to reduce the multiplication to some simple shift operations with high performance. The application opens the prospects to expand the frequency range in which signals are used to process digital filters on microprocessors of Atmel and Texas Instruments.

Take into consideration that the task of designing of digital filters is quite complex, its features were found and proposed measures for their microprocessor implementation.

REFERENCES

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