You must turn in your code as well as output files. Please generate a report that contains the code and ouput in a single readable format. Getting Started  You may want to download Irfanview image viewing software. It handles pretty much any image type, lets you convert, and provides batch processing.  Download the sample images from the class website. The following question operates on the city.jpg image. (a) Perform image smoothing using a 7×7 averaging filter and a Gaussian filter with σ = 0.5 and 3. Compare the outputs. (b) Perform edge enhancement using the Sobel operator (Matlab's default parameters). Repeat using the Laplacian and Laplacian of Guassian operators. Compare the outputs 2. Frequency Domain Filtering The following question operates on the city.jpg image. (a) Find the Fourier transform of the image. Be sure to center the frequencies. (b) Perform image smoothing in the frequency domain using the filters defined in the previous problem. Compare the output images from the two methods (spatial and frequency) and the time for operation. (c) Perform edge enhancement using the filters defined in the previous problem. (d) Define a lowpass filter in the frequency domain with radius of 1/4 the height. Show the result. Repeat with a similar sized Guassian and compare the results. Give the σ parameter you used and show the output transform image. (e) Repeat with a rectangular filter with the same dimension as the ideal lowpass. Compare the results between the ideal filter and the rectangular approximation. 3. Canny Edge Detection (a) Give the convolution kernels for determining the gradient. You may examine the function gradient.m to help with the explanation. (It may be easiest to apply the gradient to an impulse and inspect the results. (b) Implement the simplified version of the Canny edge detector (single scale). The syntax of the function should be where E contains the detected edges, M the smoothed gradient magnitude, A contains the gradient angle, I is the input image, sig is the σ parameter for the smoothing filter, and tau= [τ h , τ l ] is the two element vector containing the hysteresis thresholds. See Algorithm 6.4 for non-maximal suppression and Algorithm 6.5 for hysteresis thresh-olding. (It may be more efficient to implement the hysteresis as edge tracking)

http://ieeexplore.ieee.org/iel5/5/31355/01457626.pdf

Multidimensional digital signal processing

This new textbook by R. E. Blahut, which deals with the theory and design of efficient algorithms for digital signal processing, contains perhaps the most comprehensive coverage of fast algorithms todate.Alargecollectionofalgorithmsistreated,withanemphasis on implementing the two canonical signal processing operations of convolution and discrete Fourier transformation. In recent years, there has been much work done on fast algorithms,andBlahutdoesafinejobofblendingmaterialfromdiverse sources to form a coherent and self-contained approach to his subject.The mathematical level of this book is high, reflecting the rather abstract nature of the theoretical underpinnings of fast computational techniques. Although electrical engineers are forthe most part mathematically sophisticated, they tend to lack training in abstract algebra and number theory, both of which are essential to any thorough discussion of fast algorithms. Thus this audience should find the tutorial chapters which the text provides on these topics to be quite helpful. An additional feature of the text, which the nonspecialist should find useful, is that each new algorithm is described through three different formats: a simple example, a flowchart, and a set of matrix equations. This use of repetition assists the reader in grasping subject matter which for the most part is nonintuitive. Operation counts (as measured by the number of multiplications and the number of additions) for each algorithm are tabulated for avarietyof blocklengths (i.e., lengths of data segments), making performance comparisons easy. As the author points out, run-time comparisons may be quite different. Each chapter concludes with a set of problems of varying difficulty. These problems are well integrated with the text and serve to supplement the many examples worked out in the text. The book is devoted to how one rapidly computes various mathematical operators such as transform and convolutions. For a deeper understanding of the meaning of theseoperators,one must consult other sources in which their use is discussed. The text emphasizes algorithms which employ a reduced, or minimum, numberof muItiplications,althoughadditioncountsarealsotaken into consideration. However, an algorithm which is the “fastest” as measured in arithmetic operation counts may not be the fastest in execution time, particularly if dedicated hardware is employed. Indeed in practiceother considerationsfrequentlypropel oneaway from the computationally ”optimal” algorithm. Much work has been done on the theory and application of signal processing algorithms which are “efficient” in terms other than rnultiply/add counts, such as roundoff noise, limit cycles, coefficient quantization, memory access, hardware costs, etc. It is clearly necessary to limit the scope of any treatise, and the exclusion of differing performance measures is certainly appropriate. A description of the contents of the book will now be given, followed by some concluding remarks of a more general nature. Chapter 2 i s a tutorial on abstract algebra. It i s quite readable and is liberally laced with examples. In addition to the standard modern algebra fare (groups, rings, fields, vector spaces, matrices), the ubiquitous Chinese remainder theorem is discussed in detail. Chapters 3 and 4, and their extensions in Chapters 7and 8, form the core of the text. The third chapter addresses fast algorithms for short convolutions. The Cook-Toom convolution algorithm is discussed, followed by the Winograd convolution algorithm. A proof of the optimality of the Winograd algorithm, with respect to multiplications, for performing cyclic convolutions, is presented at the close of the chapter. The fourth chapter addresses fast algorithms for computing the discrete Fourier transform. The CooleyTukey algorithm is considered first. The approach taken is to view this algorithm as a means of mapping a onedimensional Fourier transform into a multidimensional transform. Variations of the algorithm are discussed, including the Rader-Brenner transform. Next, the Good-Thomas algorithm is discussed. This algorithm is again presented as a means of mapping a onedimensional transform into a higher dimensional transform, this time based on the Chinese remainder theorem. Rader’s algorithm for computing primelength Fouriertransforms by useofconvolution ispresented next. Extensions of the algorithm to blocklengths which are the power of an odd prime are considered. The chapter closes with the Winograd-Fourier transform which builds upon the Rader prime algorithm. Certain short blocklengthsareconsidered in detail,and the corresponding algorithms are compiled into an Appendix. Chapter 5 i s a mathematical interlude, tutorially covering items from number theory and algebraic field theory which are needed in later chapters. Topics include the totient function, Euler’s theorem,  Fermat’s theorem, minimal polynomials, and cyclotomic polynomials. Chapter 6 is devoted to number theoretic transforms. These transforms proceed by representing the data values themselves in the field of integers modulo a prime. Convolution in integer fields is also covered. Chapters 7and 8 extend the convolution and transform methods of Chapters 3 and 4 to higher dimensions. Multidimensional transforms (convolutions) are used both to efficiently compute onedimensional transforms (convolutions) and to process data which are inherently higher dimensional. Both applications are treated in these chapters. Topics include the Agarwal-Cooley convolution algorithm, polynomial transforms, the family of Johnson-Burrus transforms and the Nussbaumer-Quandalle FFT. Chapter 9 discusses architectures for transforms and digital filters and includes treatmentsof FFT butterfly networks and overlapadd convolution. The remaining three chapters are mostly independent from the rest of the book. Chapter 10 covers fast algorithms based on doubling strategies. Computational tasks for which such fast algorithms are derived include sorting, matrix transposition, matrix multiplication, polynomial division, computation of trigonometric functions, and coordinate rotation. Many of theseoperations arise as steps in the solution of oneor more signal processing problems. Fast algorithms for solving Toeplitz systems is the theme of Chapter 11. There is a variety of fast algorithms discussed, the proper choice of which depends on the specific structure of the Toeplitz system at hand (such as whether or not the system is symmetric and whether or not the right-hand vector is arbitrary). The final chapter addresses fast algorithms for Trellis and tree search and includes the Viterbi, Stack, and Fano algorithms. These

Fast algorithms for digital signal processing

http://ieeexplore.ieee.org/iel5/5/31294/01456141.pdf

Number theory in digital signal processing

http://ieeexplore.ieee.org/iel5/5/31261/01455125.pdf

Orthogonal transforms for digital signal processing

This brief proposes an efficient radix-2 single-path delay commutator (SDC) pipelined architecture to implement the fast Walsh–Hadamard–Fourier transform (FWFT) algorithm. The proposed architecture includes $(\log_{2}N-1)$ SDC stages, which are implemented by merged half-butterfly. The merged half-butterfly is proposed to achieve 100% hardware utilization and minimum buffer usage by sharing common merged half-butterflies in the time-multiplexed approach. Compared with the conventional pipelined radix-2 FFT+Walsh–Hadamard Transform (WHT) designs, the proposed architecture reduces the number of buffers by 50% and of adders by 25%. The required number of complex multipliers is decreased to $0.5\log_{2}N-0.5$ , which is roughly the minimum number. Moreover, the proposed architecture can be applied to FFT/WHT/sequence-ordered complex Hadamard transform (SCHT).

Pipelined Architecture for a Radix-2 Fast Walsh–Hadamard–Fourier Transform Algorithm

An efficient fast Walsh-Hadamard-Fourier transform algorithm which combines the calculation of the Walsh-Hadamard transform (WHT) and the discrete Fourier transform (DFT) is introduced. This can be used in Walsh-Hadamard precoded orthogonal frequency division multiplexing systems (WHT-OFDM) to increase speed and reduce the implementation cost. The algorithm is developed through the sparse matrices factorization method using the Kronecker product technique, and implemented in an integrated butterfly structure. The proposed algorithm has significantly lower arithmetic complexity, shorter delays and simpler indexing schemes than existing algorithms based on the concatenation of the WHT and FFT, and saves about 70%-36% in computer run-time for transform lengths of 16-4096.

Fast Walsh–Hadamard–Fourier Transform Algorithm

https://eprints.ncl.ac.uk/file_store/production/197541/DD7BC629-F2AA-4E17-938E-44293A4706C1.pdf

Efficient algorithms for computing the new Mersenne number transform

Two new number theoretic transforms named as odd and odd-squared new Mersenne number transforms are introduced for incorporation into a generalized new Mersenne number transforms (GNMNTs) suite, which are defined in finite fields modulo Mersenne primes where arithmetic operations and residue reductions are simple to implement. This suite is categorized by type, with detailed instructions regarding their derivations. An example is given which shows their suitability for the calculation of different types of convolutions, along with an analysis of their arithmetic complexities for radix-2 and split radix algorithms. This in turn shows that these new transforms are suitable for fast error free calculation of convolutions/correlations for signal processing and other applications.

Generalized New Mersenne Number Transforms

In this paper, a new fast Hartley transform (FHT) algorithm-radix-22 suitable for pipeline implementation of the discrete Hartley transform (DHT) is presented. The proposed algorithm is developed by integrating two stages of the twiddle factor decomposition together into single butterfly, and applying the multidimensional index mapping technique. Radix-22 algorithm achieves at the same time both a simple and regular butterfly structure as a radix-2 algorithm and a reduced number of twiddle factor multiplication provided by a radix-4 algorithm and, unlike radix-4, can be applied to any transform length that is power-of-two with simple bit reversing for ordering the output sequence. The algorithm performance is analyzed and the number of multiplications and additions are calculated. Furthermore, a method for reducing the number of multiplications and additions is proposed, making it possible to noticeably improve the arithmetic complexity as compared with the existing FHT algorithms.

New radix-based FHT algorithm for computing the discrete Hartley transform

Error-free convolutions and correlations can be efficiently computed using number theoretic transforms. One particular transform, which is known as the new Mersenne number transform (NMNT), can be used for the calculation of long-length convolutions/correlations. In this brief, a new decimation-in-time algorithm for the fast calculation of the NMNT based on the Rader-Brenner approach is proposed. The structure of the NMNT can be further improved by means of the developed algorithm, which involves multiplication by constants that are simpler than vector rotation. In terms of arithmetic operation, the proposed algorithm achieves the lowest number of multiplication among all known butterfly-style algorithms. An example of large integer multiplication is given to prove the validity of the developed algorithm.

Mounir T. Hamood

Papers

Fast Walsh–Hadamard–Fourier Transform Algorithm

Efficient algorithms for computing the new Mersenne number transform

Generalized New Mersenne Number Transforms

New radix-based FHT algorithm for computing the discrete Hartley transform

Rader–Brenner Algorithm for Computing New Mersenne Number Transform