Efficient frequency-sampling design of one-and two-dimensional FIR filters using structural subband decomposition

<title>Nonseparable 2D FIR filter design using nonuniform frequency sampling</title>

“…An example of this cross-section is shown in Figure 2. Given the 2-D filter specifications such as the support size and band-edges, we first represent the passband and stopband of this cross-section by separate analytic functions, H(w) and H3(w), as done earlier for 1-D lowpass filter design [7]. Then we place samples along the wi axis, at the extrema of these functions [6].…”

“…If we take a cross-section of H(wi , w2) along the diagonal line wi = W2, it looks like a 1-D half-band lowpass response. We approximate the passband of this response by a 1-D function H(w), as used for 1-D half-band lowpass filter design in [7]. The order p of the corresponding Chebyshev polynomial T(x) is (N -1)/2.…”

“…In the following sections, we show how these parameters are chosen for square-and diamond-shaped filters. A common approach used is that a particular cross-seciion of the desired 2-D frequency response is approximated by 1-D analytic functions based on Chebyshev polynomials, similar to those used for 1-D FIR filter design in [6,7]. The samples are then placed on contours that pass through the extrema of this cross-section.…”

System functions for 2D linear phase frequency sampling filters

“…Using an appropriate design method [2,3,5,6,[9][10][11][12], the frequency samples can be chosen to minimize interpolation errors and meet passband and stopband specifications. Because a frequency sampling filter uses frequency samples instead of impulse response values as coefficients in its implementation, it can potentially implement narrowband FIR filters more efficiently than a direct convolution implementation.…”

“…Frequency sampling filters interpolate a frequency response through a set of frequency samples, which are specific values from the filter's frequency response, and use these frequency samples as coefficients in their implementation [1][2][3][4][5][6][7]. In [8], a 2D frequency sampling system function, which this paper denotes as a Type 1-1 frequency sampling filter, is developed that selects its frequency samples as the discrete Fourier transform (DFT) coefficients, H k ðk 1 ; k 2 Þ where ðk 1 ; k 2 ÞAfðk 1 ; k 2 Þ: 0pk 1 pN 1 À 1; 0pk 2 pN 2 À 1g k 1 ; k 2 ; N 1 ; N 2 AI:…”