TECHNIQUE FOR CONSTRUCTING THE STRUCTURE OF A RECURSIVE FILTER WITH A FINITE IMPULSE RESPONSE IN THE FORM OF A FUNCTION APPROXIMATING THE HANN WINDOW

Abstract

Filters with an impulse response (IR) in the form of a weighting (smoothing) function are used in
completely different areas of digital signal processing, such as spectral analysis - in order to reduce the
Gibbs effect, in the formation of an amplitude distribution – to reduce the level of side lobes, including for
radio engineering systems with synthesized aperture and others. The article considers the structure of a
recursive FIR filter (RFIR-filter) with IR in the form of an approximated Hann window with a limited fixed
number of multiplication and summation operations for any window duration. Such a structure has significantly
lower computational complexity compared to the classical structure of the FIR-filter, and it can be
used in embedded systems with limited computing resources. The function approximating the Hann window
is a polynomial of the third degree, the coefficients of which are calculated using a specific integration
of the quasi-sine function. An analytical formula is obtained for the coefficients of the non-recursive
part of the filter by calculating the inverse finite difference of the fourth degree from the approximating
function of the Hann window. The coefficients of the non-recursive part are integers, the values of which
depend on the number of samples (length) of the half-cycle of the quasi-sine function, which simplifies the
implementation of such an RFIR-filter based on a programmable logic integrated circuit (FPGA).
The average absolute approximation error is calculated with an increase in the length of the window.
When the number of samples is less than 600, the error does not exceed 4.5%, which is an indicator of the
high accuracy of matching the approximating function to the Hann window. The authors propose a further
perspective for the development of the structure of the RFIR-filter with IR in the form of an approximating
function of the Hann window. This structure makes it possible to implement a RFIR-filter with a change in
the length of the Hann window in the time domain while maintaining stability by accurately performing
calculation operations using the coefficients of the non-recursive part, which are fixed-point numbers, and
their linear dependence on the half-period length of the quasi-cosine function.