Fast Split-Radix and Radix-4 Discrete Cosine Transform Algorithms

The Discrete Fourier Transform (DFT) has a plethora of applications in applied mathematics and electrical engineering. Discrete Cosine Transform (DCT) is a real-arithmetic analogue of DFT. DCTs with orthogonal trigonometric transforms have been especially popular in recent decades due to their applications in digital video technology and high efficiency video coding. One can say that DCT is the key transform in image processing, signal processing, finger print enhancement, quick response code (QR code), multi-mode interface, etc. In this talk, we first introduce sparse and scaled orthogonal factorization for the DCT and inverse DCT. Afterwards, we present fast split-radix and radix-4 DCT and inverse DCT algorithms. We show that the proposed algorithms attain the lowest theoretical multiplication complexity and arithmetic complexity for 8-point DCT II/III matrices. We perform execution time of the proposed algorithms while verifying the connection to the order of the arithmetic complexity. Finally, the language of signal flow graph representation of digital structures is used to describe potential for real-world circuit implementation

Low Power Implementation of Non Power-of-Two FFTs on Coarse-Grain Reconfigurable Architectures

The DRM standard for digital radio broadcast in the AM band requires integrated devices for radio receivers at very low power. A System on Chip (SoC) call DiMITRI was developed based on a dual ARM9 RISC core architecture. Analyses showed that most computation power is used in the Coded Orthogonal Frequency Division Multiplexing (COFDM) demodulation to compute Fast Fourier Transforms (FFT) and inverse transforms (IFFT) on complex samples. These FFTs have to be computed on non power-of-two numbers of samples, which is very uncommon in the signal processing world. The results obtained with this chip, lead to the objective to decrease the power dissipated by the COFDM demodulation part using a coarse-grain reconfigurable structure as a coprocessor. This paper introduces two different coarse-grain architectures: PACT XPP technology and the Montium, developed by the University of Twente, and presents the implementation of a\ud Fast Fourier Transform on 1920 complex samples. The implementation result on the Montium shows a saving of a factor 35 in terms of processing time, and 14 in terms of power consumption compared to the RISC implementation, and a\ud smaller area. Then, as a conclusion, the paper presents the next steps of the development and some development issues

Signal compression and enhancement using a new orthogonal-polynomial-based discrete transform

Discrete orthogonal functions are important tools in digital signal processing. These functions received considerable attention in the last few decades. This study proposes a new set of orthogonal functions called discrete Krawtchouk-Tchebichef transform (DKTT). Two traditional orthogonal polynomials, namely, Krawtchouk and Tchebichef, are combined to form DKTT. The theoretical and mathematical frameworks of the proposed transform are provided. DKTT was tested using speech and image signals from a well-known database under clean and noisy environments. DKTT was applied in a speech enhancement algorithm to evaluate the efficient removal of noise from speech signal. The performance of DKTT was compared with that of standard transforms. Different types of distance (similarity index) and objective measures in terms of image quality, speech quality, and speech intelligibility assessments were used for comparison. Experimental tests show that DKTT exhibited remarkable achievements and excellent results in signal compression and speech enhancement. Therefore, DKTT can be considered as a new set of orthogonal functions for futuristic applications of signal processing

Signal Flow Graph Approach to Efficient DST I-IV Algorithms

In this paper, fast and efficient discrete sine transformation (DST) algorithms are presented based on the factorization of sparse, scaled orthogonal, rotation, rotation-reflection, and butterfly matrices. These algorithms are completely recursive and solely based on DST I-IV. The presented algorithms have low arithmetic cost compared to the known fast DST algorithms. Furthermore, the language of signal flow graph representation of digital structures is used to describe these efficient and recursive DST algorithms having $(n-1)$ points signal flow graph for DST-I and $n$ points signal flow graphs for DST II-IV

The Orthogonal Band Decomposition of the Finite Dirichlet Matrix and its Applications

In my work I establish and extend the theory of finite D-matrices for the purposes of signal processing applications in the finite, digital setting. Finite D-matrices are obtained by truncating infinite D-matrices to upper-left corners. I show that finite D-matrices are furnished with a number-theoretical structure that is not present in their infinite counterparts. In particular, I show that the columns of every finite D-matrix of size $N\times N$ admits a natural, non-trivial, Orthogonal Band Decomposition, induced by the Floor Band Decomposition on the finite set $\{1,2,\dots,N\}$. When the D-matrix is invertible, its Orthogonal Band Decomposition induces a non-trivial resolution of the identity. Furthermore, for every finite D-matrix $A$, I show that the sum $P$ of the orthogonal projections corresponding to each band of $A$ admits the following sparse representation $P=A\Lambda^{-1} A^*$, where $\Lambda$ is a special diagonal matrix and $A^\star$ is the Hermitian adjoint of $A$. I also show that the matrix $P$ and its inverse induce another non-trivial resolution of the identity. Being a sum of projection matrices, I call the matrix $P$ the associated P-matrix of $A$. Both the finite D-matrices and their associated P-matrices can be applied in the processing of digital signals. For example, given a D-matrix $A$, its associated P-matrix allows us to pass from a signal representation in the Fourier basis to a representation, as a sum of projections, in the basis induced by the Orthogonal Band Decomposition of $A$. Preliminary experiments suggest that the error of approximating signals with partial sums of projections might offer a more suitable metric to choose D-matrix representations in specific applications. Significantly, computations with finite D-matrices and P-matrices can be carried out via fast algorithms, which makes these transforms computationally competitive

Non-parametric linear time-invariant system identification by discrete wavelet transforms

We describe the use of the discrete wavelet transform (DWT) for non-parametric linear time-invariant system identification. Identification is achieved by using a test excitation to the system under test (SUT) that also acts as the analyzing function for the DWT of the SUT's output, so as to recover the impulse response. The method uses as excitation any signal that gives an orthogonal inner product in the DWT at some step size (that cannot be 1). We favor wavelet scaling coefficients as excitations, with a step size of 2. However, the system impulse or frequency response can then only be estimated at half the available number of points of the sampled output sequence, introducing a multirate problem that means we have to 'oversample' the SUT output. The method has several advantages over existing techniques, e.g., it uses a simple, easy to generate excitation, and avoids the singularity problems and the (unbounded) accumulation of round-off errors that can occur with standard techniques. In extensive simulations, identification of a variety of finite and infinite impulse response systems is shown to be considerably better than with conventional system identification methods.Department of Computin