Design of NPR DFT-Modulated Filter Banks via Iterative Updating Algorithm

Abstract

In this paper, an efficient algorithm is proposed to design nearly-perfect-reconstruction (NPR) DFT-modulated filter banks. First, the perfect-reconstruction (PR) condition of the oversampled DFT-modulated filter banks in the frequency domain is transformed into a set of quadratic equations with respect to the prototype filter (PF) in the time domain. Second, the design problem is formulated as an unconstrained optimization problem that involves PR condition and stopband energy of the PF. With the gradient vector of the objective function, an efficient iterative algorithm is presented to design the PF, which is updated with linear matrix equations at each iteration. The algorithm is identified as a modified Newton’s method, and its convergence is proved. Numerical examples and comparison with many other existing methods are included to demonstrate the effectiveness of the proposed method.

Appendices

Appendix A: Proof of Property 3.1

Let G(h)=A(h)h. Then the objective function can be written as a function of G(h) and h:

(27)

Similar to that in [15], the derivative of G ^T(h) with respect to h is

$$ \frac{\partial \mathbf{G}^{T}(\mathbf{h})}{\partial \mathbf{h}} = \frac{\partial \mathbf{h}^{T}}{\partial \mathbf{h}}\mathbf{A}^{T}( \mathbf{h}) + \bigl(\mathbf{I}_{N} \otimes\mathbf{h}^{T} \bigr)\frac{\partial \mathbf{A}^{T}(\mathbf{h})}{\partial \mathbf{h}} = 2\mathbf{A}^{T}(\mathbf{h}), $$

(28)

where I _N is the N×N identity matrix, and ⊗ denotes the Kronecker product. As a result, the gradient vector can be derived by

(29)

The proof is completed. □

Appendix B: Efficient calculation of A ^T(h ₀)

From the definition of matrices , the matrix A ^T(h ₀) can be calculated without using any matrix-vector multiple. Given k,l,r _i , we obtain

(30)

In (30), $\mathbf{h}_{0}^{T}\mathbf{U}_{l_{0}}\mathbf{D}_{n}\mathbf{U}_{l_{1}}^{T}$ can be fast calculated with the following three steps:

• (1) In terms of (10), $\mathbf{d}_{l_{0}} = \mathbf{h}_{0}^{T}\mathbf{U}_{l_{0}}$ is calculated by decimating $[\mathbf{h}_{0}^{T},\mathbf{0}^{T}]$ with offset factor l ₀, where 0 is the (P−2)×1 zero vector, and the zero padding is performed to make $\mathbf{d}_{l_{0}}$ have the same size for different l ₀.

• (2) The matrix D _n is a highly sparse matrix, with the coordinates of its nonzero entries, $\mathbf{d}_{l_{0},n} = \mathbf{h}_{0}^{T}\mathbf{U}_{l_{0}}\mathbf{D}_{n}$ can be quickly formed without any multiple.

• (3) Interpolating $\mathbf{d}_{l_{0},n}$ with offset factor l ₁ to form a 1×((m+1)P−1) vector g, we have $\mathbf{h}_{0}^{T}\mathbf{U}_{l_{0}}\mathbf{D}_{n}\mathbf{U}_{l_{1}}^{T} = [g(1), \ldots,g(mP + 1)]$, which is sparsely represented in numerical simulation.

As a result, the row vector can be efficiently calculated only with several additions. Then A ^T(h ₀) is formed by orderly stacking with different k,l,r _i . □