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In this brief, efficient multiplierless design of lattice quadrature mirror filter bank is presented. Previous work by the authors has shown that splitting each lattice stage into cascade of subrotations results in larger stopband attenuation of filter than the conventional direct quantization. This brief extends the work further by exploiting the subrotations which yield more flexible sum of signed powers-of-two quantization. This enables us to find more possible discrete representations, and hence to reduce the quantization error. Also, an algorithm for the efficient gathering of candidate discrete coefficients is developed, based on the trellis-based searching approach. It substantially alleviates the overheads of optimization program, especially when the wordlengths and the number of nonzero digits are large. Several design examples are provided to show that the proposed structure with the candidate gathering algorithm provides improved frequency response. Index Terms--Lattice, multiplierless, quadrature mirror filter (QMF), signed power-of-two (SPT), trellis-based searching (TBS).

Lattice structures have several advantages over the tapped delay line form, especially for the hardware implementation of general digital filters. It is also efficient for the implementation of quadrature mirror filter (QMF), because the perfect reconstruction is preserved under the coefficient quantization. Moreover, if lattice coefficients are implemented in signed powers-of-two (SPT), the hardware complexity can also be reduced. But the discrete coefficient space with the SPT representation is sparse when the number of nonzero bits is small. This paper proposes a structure of orthogonal QMF lattice with SPT coefficients, which has much denser discrete coefficient space than the conventional structure. While the conventional approaches directly quantize the lattice coefficients into SPT form, the proposed algorithm considers the quantization in the SPT angle space. For this, each lattice stage is implemented by the cascade of several variants of COordinate Rotation Digital Computer. The resulting angle space and corresponding discrete coefficient space is much denser than the one generated by the conventional direct quantization approach. An efficient coefficient search algorithm for this structure is also proposed. Since the proposed architecture provides denser coefficient space, it shows less coefficient quantization error than the conventional QMF lattice. Index Terms--COordinate Rotation Digital Computer (CORDIC), lattice filter, perfect reconstruction (PR), quadrature mirror filter (QMF), signed powers-of-two (SPT).

This paper presents a fixed-point mean-square error (MSE) analysis of coordinate rotation digital computer (CORDIC) processors based on the variance propagation method, whereas the conventional approaches provide only the error bound which results in large discrepancy between the analysis and actual implementation. The MSE analysis is aimed at obtaining a more accurate analysis of digital signal processing systems with CORDIC processor, especially when the design specification is given by the signal-to-noise ratio or MSE. For the MSE analysis, the error source and models are first defined and the output error is derived in terms of MSE in the rotation mode of the conventional CORDIC processor. It is shown that the proposed analysis can also be applied to the modified CORDIC algorithms. As an example of practical application, a fast Fourier transform processor using the CORDIC processor is presented in this paper, and its output error variance is analyzed with respect to the wordlength of CORDIC. The results show a close match between the analysis and simulation. Index Terms--Coordinate rotation digital computer, covariance matrices, error analysis, fixed-point error analysis, fast Fourier transform (FFT), mean-square error (MSE),.

This paper proposes an adaptive quantization algorithm for video coding using the information obtained from the previously encoded image. Before quantizing the discrete cosine transform coefficients, the properties of reconstruction error of each macro block (MB) are estimated from the previous frame. For the estimation of the error of current MB, a block with the size of MB in the previous frame is chosen. Since the original and reconstructed images of the previous frame are available in the encoder, we can evaluate the tendency of reconstruction error of this block in advance. Then, this error is considered as the expected error of the current MB if it is quantized with the same step size and bit rate. Comparing the error of the MB with the average of overall MB's, if it is larger than the average, a small step size is given for this MB, and vice versa. As a result, the error distribution of the MB is more concentrated to the average, yielding low variance and improved image quality. Especially for low bit application, the proposed algorithm yields much smaller error variance and higher peak signal-to-noise ratio compared to the conventional TM5. We also propose a modified algorithm for efficient hardware implementation.

An algorithm to design optimal one and two-dimensional FIR filters over a discrete coefficient space is presented. It is based on the close relationship of the equiripple frequencies of a subproblem (SP) in the brand and bound algorithm, to those of adjacent SPs. The algorithm reduces the number of constraints needed to solve each SP, therefore overall computational load for FIR filter design is alleviated.

In this paper, tracking properties of the adaptive lattice notch filter (ALNF) for input sinusoids with linearly and randomly varying frequencies are investigated. Expressions for the asymptotic tracking error and the mean square error (MSE) of the frequency estimate are derived. The expressions show that the forgetting factor, or the step-size constant of the algorithm, plays an important role in controlling the tracking error, the MSE of the frequency estimate, and the output SNR. It is shown that the tracking error decreases as the forgetting factor increases. However, the MSE of the frequency estimate becomes large either for large and small forgetting factor. The output SNR also becomes small either for too large and small forgetting factors. Hence, the optimal forgetting factors, which minimize the MSE and maximize the output SNR for given frequency variation models, are also derived from the expressions for MSE. Intensive computer simulations are performed to augment the theoretical analysis, and the results are discussed.

Recently, a new fast algorithm for 2-D N X N DCT, where N = 2 to the m power, has been proposed. It requires only half number of multiplications compared to the conventional row-column approach. However, the signal flow graph for the post-addition stage seems very complicated and the order of the output index is seemingly irregular, because the post-addition stage was not based on the mathematical expressions. Consequently, derivation of the signal flow graph becomes complicated as the transform size increases. Hence, In this paper, we present a systematic expressions for the post-addition stage in the 2-D DCT algorithm, which enable us to implement any N X N DCT in a straightforward manner. The results show that the signal flow graph from input to output has the recursive structure in which the structure for smaller N reappears for larger N. However, the number of additions increases in the new signal flow graph at the expense of improving the regularity in the structure.

A four by four discrete cosine transform (DCT) algorithm is presented. It consists of four, decomposed, four-point one dimensional DCTs which, in conjunction with the appropriate amount of additions and multiplications, may be used for the speedy computation of a recursive, two-dimensional 2m by 2m DCT. The proposed algorithm, which employs only 16 multiplications, is evaluated in light of the performance of other polynomial transform based algorithms.

This paper introduces the concept of warped discrete cosine transform (WDCT) and an image-compression algorithm based on the WDCT. The proposed WDCT is a cascade connection of a conventional DCT and all-pass filters whose parameters can be adjusted to provide frequency warping. Because only the first-order all-pass filters are considered in this paper, the WDCT can be implemented by Laguerre network connected with the DCT. For the more efficient software implementation, we propose truncated and approximated FIR filter banks which can be used instead of the Laguerre network. As a result, the input--output relationship of the WDCT can be represented by a single matrix-vector multiplication, like the DCT. In the proposed image-compression scheme, the frequency response of the all-pass filter is controlled by a fixed set of parameters from which a specified warping parameter is used for a specified frequency range. Also, for each parameter, the corresponding WDCT matrices are computed a priori. For each image block, the best parameter is chosen from the set and the index is sent to the decoder as side information along with the result of corresponding WDCT matrix computation. At the decoder, an inverse WDCT is performed to reconstruct the image. The WDCT based compression outperforms the DCT based compression, for high bit rate applications and for images with high-frequency components. It results in 1.1-3.1-dB PSNR gain over conventional DCT at 1.5 bpp for natural images, and provides more gain for compound images with texts. Index Terms--All-pass filter, DCT, image compression, warped DCT.