## 1 Introduction  
Maximally decimated, uniform, cosine-modulated filter banks play a major role in applications where signals must be processed in sub-bands [1; 2; 3; 4; 5]. There are two main methods obtaining a cosine-modulated filter bank. (1) Design analysis and synthesis filter banks respectively under the certain conditions; (2) Design one or two prototype filters and get the analysis and synthesis filter banks by cosine modulating. In this paper, we choose the later one because the design is straightforward, e.g., this method focus on designing a single prototype filter.  
There are several methods to design the prototype filter of cosine-modulated filter banks [6; 7; 8]. The weighted constrained least square (WCLS) algorithm proposed in reference [6] casts the design problem as a linear minimization of prototype filter coefficients, which makes the value at \(\omega=\pi/2M\) is 0.707. However, the order of the prototype filter is very high. It is generally true that group delay increases with the order of the filter. The quantum-behaved particle swarm optimization (QBPSO) algorithm proposed in reference [7] appropriately relaxes the limitation of perfect-reconstruction conditions. This method designs the prototype filter by making an unconstrained optimization to adjust the pass band cutoff frequency of objective function, which minimizes the cost function to meet the reconstruction conditions. Although the amplitude distortion is guaranteed, the aliasing distortion cannot meet the perfect-reconstruction condition of filter banks, and the attenuation of prototype filter is restricted. The design method based on gradient information (GI) in reference [8] uses the gradient vector of objective function to design prototype filter. However, the order of prototype coefficient is high, the aliasing distortion and the amplitude distortion of filter banks are not controlled well.  
The order of NPR cosine-modulated filter bank designed by these methods is high and not sparse, which makes an increasing number of the multipliers and adders in practical application, and results in a high hardware cost and power consumption.  
In this paper, we present a fast design paradigm for sparse NPR cosine-modulated filter banks. A sparse FIR filter bank contains a large number of zero coefficients, such that the multipliers and adders corresponding to these zero coefficients are not required, which results in a lower hardware cost and power consumption with allowed accuracy of distortion. The design procedure can be divided into two stages. In the first stage, an approximation function \(f(\omega)\) is introduced to reduce the non-convex quadratically constrained optimization problem to a linearly constrained optimization problem. The desired sparse linear phase FIR prototype filter can be derived through the orthogonal matching pursuit [9] performed under the weighted \(l_{2}\) norm in the second stage. The simulation results demonstrate that the proposed scheme is an effective paradigm to design sparse NPR cosine-modulated filter banks.  
The paper organization is as follows: In Section 2, we briefly summarize the design framework of perfect-reconstruction cosine-modulated filter banks. The design of sparse linear phase FIR prototype filter of NPR cosine-modulated filter banks is showed in Section 3. The representative experimental results and comparison data are discussed in Section 4. The conclusion of proposed scheme is showed in Section 5.