Direct methods for computing discrete sinusoidal transforms

In this letter, we first propose new nearly tridiagonal commuting matrices of discrete Fourier transform (DFT) matrix and generalized DFT (GDFT) matrix. Then, using the block diagonalizations technique of circular-centrosymmetric and centrosymmetric matrices, we derive novel tridiagonal commuting matrices for each discrete cosine transform (DCT) and discrete sine transform (DST) matrices of the types I, IV, V, and VIII from the commuting matrices of the DFT and GDFT based on the relationships in matrix forms among DFT, GDFT, and various types of DCT and DST. Moreover, the novel tridiagonal commuting matrices of various types of DCT and DST do not have multiple eigenvalues. Last, with these novel commuting matrices, we can easily determine an orthonormal set of Hermite-like eigenvectors for each of their corresponding DCT or DST matrix.

In this paper, we propose Speaker Identification using the frequency distribution of various transforms like DFT (Discrete Fourier Transform), DCT (Discrete Cosine Transform), DST (Discrete Sine Transform), Hartley, Walsh, Haar and Kekre transforms. The speech signal spoken by a particular speaker is converted into frequency domain by applying the different transform techniques. The distribution in the transform domain is utilized to extract the feature vectors in the training and the matching phases. The results obtained by using all the seven transform techniques have been analyzed and compared. It can be seen that DFT, DCT, DST and Hartley transform give comparatively similar results (Above 96%). The results obtained by using Haar and Kekre transform are very poor. The best results are obtained by using DFT (97.19% for a feature vector of size 40).

This paper proposes a high performance and low cost inverse discrete cosine transform (IDCT) processor for high definition Television (HDTV) applications by using cyclic convolution and hardwired multipliers. By properly arranging the input sequence, we formulate the one-dimensional (1-D) IDCT into cyclic convolution that is regular and suitable for VLSI implementation. The hardwired multiplier that implements multiplication with IDCT coefficients are first scaled and optimized by using the common sub-expression techniques. Based on these techniques, the data-path in the proposed two-dimensional (2-D) IDCT design costs 7504 gates plus 1024 bits of memory with 100 M pixels/sec throughput according to the cost estimation based on the cell library of COMPASS 0.6 μm SPDM CMOS technology. Also, we have verified that the precision analysis of the proposed 2-D 8 × 8 IDCT meets the demands of IEEE Std. 1180-1990. Due to the good performance in the computing speed as well as the hardware cost, the proposed design is compact and suitable for HDTV applications. This design methodology can be applied to forward DCT as well as other transforms like discrete sine transform (DST), discrete Fourier transform (DFT), and discrete Hartley transform (DHT).

This paper presents a new property called the Duality Theorem for the Discrete Sine Transform (DST). Discrete Sine Transform is a finite — length discrete transform which is related to the renowned Discrete Fourier Transform (DFT) and is quite popular in signal processing arena, but has remained in the oblivion from pure mathematicians. Many of the properties of the Discrete Sine Transform are akin to those of the Discrete Fourier Transform and Discrete Cosine Transform, subject to minor differences. A formal derivation of the Duality Theorem corresponding to the Discrete Sine Transform is given which was hitherto not mentioned or derived in the literature. The usage of Duality Theorem helps in finding the discrete time — domain function from the DST frequency domain and vice versa thereby reducing considerable labour involved in the evaluation of the summation and hence, saves computation time and cost of implementation to a considerable extent. DST finds applications in Image Processing, Signal Processing applications for Communication Systems, and in the numerical solutions of differential equations as well as partial differential equations of mathematics.

FFT(Fast Fourier Transform) 알고리즘에는 DIT(Decimation-In-Time)와 DIF(Decimation-In-Frequency)가 있다. DIF 알고리즘은 Radix-2/4/8 등의 다양한 종류와 그 구현 방법이 개발되어 사용되고 잇으나, DIT 알고리즘은 순차적인 출력을 낼 수 있는 장점이 있음에도 불구하고 다양한 알고리즘이 연구되지 못하였다. 이 논문에서는 새로운 DIT Radix-4 FFT의 나비연산기(butterfly) 구조를 제안하고 검증하였다. 제안 구조를 사용하여 64-point FFT 구조를 설계하고 Verilog로 코딩하여 구현함으로써 제안 구조의 효용성을 입증하였다. 48개의 곱셈기를 사용하여 합성하였으며 678만 게이트 수를 나타내었다. 따라서 제안된 DIT Radix-4 FFT 구조는 순차적인 FFT 출력을 필요로 하는 OFDM 통신용 SoC(System on a Chip)에 사용될 수 있을 것이다. Two basic FFT(Fast Fourier Transform) algorithms are the DIT(Decimation-In-Time) and the DIF (Decimation-In-Frequency). In spite of the advantage of the DIT algorithm is to generate a sequential output, various structures have not been made. In this paper, a new DIT Radix-4 FFT butterfly structure are proposed and implemented using Verilog coding. Through synthesis, it is shown that the 64-point FFT is implemented by 6.78 million gates. Since the proposed FFT structure has the advantage of a sequential output, it can be used in OFDM communication SoC(System on a Chip) which need a high speed FFT output.

Block frames called <i>directional analytic discrete cosine frames</i> (DADCFs) are proposed for sparse image representation. In contrast to conventional overlapped frames, the proposed DADCFs require a reduced amount of 1) computational complexity, 2) memory usage, and 3) global memory access. These characteristics are strongly required for current high-resolution image processing. Specifically, we propose two DADCFs based on discrete cosine transform (DCT) and discrete sine transform (DST). The first DADCF is constructed from parallel separable transforms of DCT and DST, where the DST is permuted by row. The second DADCF is also designed based on DCT and DST, while the DST is customized to have no DC leakage property which is a desirable property for image processing. Both DADCFs have rich directional selectivity with slightly different characteristics each other and they can be implemented as non-overlapping block-based transforms, especially they form Parseval block frames with low transform redundancy. We perform experiments to evaluate our DADCFs and compare them with conventional directional block transforms in image recovery.

Chapter 1 - Discrete Cosine and Sine Transforms

A discrete cosine transform (DCT) is described and a technique to compute it using fast Fourier transform (FFT) is developed. In this work, DCT of a finite length sequence is obtained by incorporating CORDIC methodology in radix-2 FFT algorithm. The proposed methodology is simple to comprehend and maintains a regular structure, thereby reducing computational complexity. DCTs are used extensively in the area of digital processing for the purpose of pattern recognition. So the efficient computation of DCT maintaining a transparent design flow is highly solicited.

A joint Hyperbolic Tangent Companding (HTC) technique and Discrete Cosine Transform (DCT) has been proposed to decrease of the PAPR ratio effects of the OFDMA System. The proposed schema utilizes DCT instead of the traditional discrete Fourier transform (DFT) as a requisite function to perform DCT OFDMA. HTC is an actual process to minimize the high PAPR of DCT OFDMA. However, the companded DCT OFDMA suffers from the bit-error-rate (BER) performance degradation by companding noise introduced. So, an effective scheme of estimation and noise cancellation for the compressed DCT OFDMA is introduced. By expressing the compressed signals as a combination of the main signals adding with the component of compressed noise, and the estimated compressed noise deduct from the received signals, leading to enhance the BER performance of the proposed system. The new proposed scheme minimizes PAPR by about 6.3 dB and 7.3 dB compare respectively, with the DFT OFDMA and DCT OFDMA without commanding. Also, it is noted that the proposed scheme of estimation and companding noise cancellation for the companded DCT OFDMA provides a great lowest BER degradation that can be neglected comparing with the premier DCT SC-FDMA system without PAPR reduction.

The most vivacious area of the image processing is image enhancement. Newer techniques and ideas are being routinely incorporated to this field for the proper enhancement to make the original image suitable for some specific purposes. However, the techniques that are suitable for X-ray image enhancement may not necessarily be suitable for enhancement of the Satellite Telescope Transmitted (STT) images. Many times, Medical images especially X-ray images are required to be sharper than its original version for proper information extraction that is necessary for diagnostic purposes. Homomorphic filters that actually belong to the group of the sharpening filters can be employed for finding fine details in the X-ray images. The Homomorphic filters use the Discrete Fourier Transforms (DFT) as the core transform. Presently, more efficient tools for transformations are being used; such as the Discrete Cosine Transform (DCT), Discrete Wavelet Transform (DWT), Fast Fourier Transform (FFT) and Fast Wavelet Transform (FWT) in digital image processing. In this paper, along with the conventional DFT technique, DCT, DWT, FFT and FWT are incorporated in the image transformation process that consequently modifies the core process of Homomorphic filters. The performance of the resulting modified Homomorphic filters are compared with that of the original filter. The performance enrichment has become evident from the numerical and the visual comparisons.

In this paper we present the Peak-to-Average Power Ratio decrease method targeted at energy efficiency improvement of the Radio-over-fiber telecommunication systems based on the Orthogonal Frequency Division Multiplexing. The scheme for the proposed method that consists in the signal precoding with discrete function is discussed in details. The simulation results obtained for five different precoding functions: Discrete Hartley Transform, Discrete Cosine Transform, Discrete Sine Transform, Fast Walsh-Hadamard Transform, and Discrete Fourier Transform, show the peak-to-average power ratio decrease of the transmitted signal up to 7 dB without degradation in noise immunity. Such a gain in transmitted power reduction corresponds to energy efficiency improvement by 5.15 dB in the case of 16-position quadrature amplitude modulation. The experimental study confirms the simulation results, providing the 6.5 dB peak-to-average power ratio decrease for Discrete Fourier Transform function.

A spectral 1st level FPGA trigger for detection of very inclined showers based on a 16-point discrete cosine transform for the Pierre Auger Observatory

The processing efficiency of the widely used time-space (t-x) domain vertical seismic profiling (VSP) wavefield separation methods (such as median and singular value decomposition filtering) via one-dimensional discrete Fourier transform (DFT) depends on the wavefield separation method (or algorithm with processing parameter) and the total number of samples in the input VSP data. Once the wavefield separation method is determined, its processing efficiency is set and cannot be optimized. Although the popular frequency-wavenumber (f-k) domain VSP wavefield separation method via two-dimensional DFT has higher processing efficiency than the t-x domain VSP wavefield separation methods, its processing precision is susceptible to the spatial alias and Gibbs effects. For efficiency and precision improvements, we introduced the discrete cosine transform (DCT) operation into VSP wavefield separation for the first time, and proposed a high-efficiency and precision frequency-space (f-x) domain VSP wavefield separation method via DFT and DCT, in which the wavefield separation efficiency and precision can be optimized using the effective bandwidth cutoff frequency of the input VSP data. Based on the relationship between DFT and DCT, we combined their operations in the proposed method into one step (referred to as DCT–DFT) for further efficiency enhancement, thereby designing a high-efficiency and precision f-x domain VSP wavefield separation method via DCT–DFT. Theory analysis and synthetic and field VSP data examples show that the proposed method is highly efficient and precise, and can be widely used for three-dimensional (3D)-VSP data wavefield separation, especially for large distributed acoustic sensing (DAS)-VSP data.

The Cooley-Tukey FFT algorithm decomposes a discrete Fourier transform (DFT) of size n = km into smaller DFT of size k and m. In this paper we present a theorem that decomposes a polynomial transform into smaller polynomial transforms, and show that the FFT is obtained as a special case. Then we use this theorem to derive a new class of recursive algorithms for the discrete cosine transforms (DCT) of type II and type III. In contrast to other approaches, we manipulate polynomial algebras instead of transform matrix entries, which makes the derivation transparent, concise, and gives insight into the algorithms' structure. The derived algorithms have a regular structure and, for 2-power size, minimal arithmetic cost (among known DCT algorithms).

Interpretation of eight-point discrete cosine and sine transforms as 3×3 orthogonal edge masks in terms of the Frei–Chen masks