After an image has been segmented into regions ; the resulting pixels is usually is represented and described in suitable form for further computer processing.
After an image has been segmented into regions ; the resulting pixels is usually is represented and described in suitable form for further computer processing.
Improving The Performance of Viterbi Decoder using Window System IJECEIAES
An efficient Viterbi decoder is introduced in this paper; it is called Viterbi decoder with window system. The simulation results, over Gaussian channels, are performed from rate 1/2, 1/3 and 2/3 joined to TCM encoder with memory in order of 2, 3. These results show that the proposed scheme outperforms the classical Viterbi by a gain of 1 dB. On the other hand, we propose a function called RSCPOLY2TRELLIS, for recursive systematic convolutional (RSC) encoder which creates the trellis structure of a recursive systematic convolutional encoder from the matrix “H”. Moreover, we present a comparison between the decoding algorithms of the TCM encoder like Viterbi soft and hard, and the variants of the MAP decoder known as BCJR or forward-backward algorithm which is very performant in decoding TCM, but depends on the size of the code, the memory, and the CPU requirements of the application.
MDCT audio coding with pulse vector quantizersEricsson
This paper describes a novel audio coding algorithm that is a building block in the recently standardized 3GPP EVS codec. The presented scheme operates in the Modified Discrete Cosine Transform (MDCT) domain and deploys a Split-PVQ pulse coding quantizer, a noise-fill, and a gain control optimized for the quantizer’s properties. A complexity analysis in terms of WMOPS is presented to illustrate that the proposed Split-PVQ concept and dynamic range optimized MPVQ-indexing are suitable for real-time audio coding.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Design and Performance Analysis of Convolutional Encoder and Viterbi Decoder ...IJERA Editor
In digital communication forward error correction methods have a great practical importance when channel is
noisy. Convolutional error correction code can correct both type of errors random and burst. Convolution
encoding has been used in digital communication systems including deep space communication and wireless
communication. The error correction capability of convolutional code depends on code rate and constraint
length. The low code rate and high constraint length has more error correction capabilities but that also
introduce large overhead. This paper introduces convolutional encoders for various constraint lengths. By
increasing the constraint length the error correction capability can be increased. The performance and error
correction also depends on the selection of generator polynomial. This paper also introduces a good generator
polynomial which has high performance and error correction capabilities.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
A Density Control Based Adaptive Hexahedral Mesh Generation Algorithmijeei-iaes
A density control based adaptive hexahedral mesh generation algorithm for three dimensional models is presented in this paper. The first step of this algorithm is to identify the characteristic boundary of the solid model which needs to be meshed. Secondly, the refinement fields are constructed and modified according to the conformal refinement templates, and used as a metric to generate an initial grid structure. Thirdly, a jagged core mesh is generated by removing all the elements in the exterior of the solid model. Fourthly, all of the surface nodes of the jagged core mesh are matching to the surfaces of the model through a node projection process. Finally, the mesh quality such as topology and shape is improved by using corresponding optimization techniques.
The part is axisymmetrically modeled in solidworks(2D) before importing to ansys workbench where the boundary zones are identified and appropriate mesh settings is applied. The model is then imported in Fluent for analysis . Significant setting changes are Density based solver , Enhanced Eddy viscosity model with near wall treatment , solution steering , FMG initialization etc.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Manifold Blurring Mean Shift algorithms for manifold denoising, report, 2012Florent Renucci
(General) To retrieve a clean dataset by deleting outliers.
(Computer Vision) the recovery of a digital image that has been contaminated by additive white Gaussian noise.
Improving The Performance of Viterbi Decoder using Window System IJECEIAES
An efficient Viterbi decoder is introduced in this paper; it is called Viterbi decoder with window system. The simulation results, over Gaussian channels, are performed from rate 1/2, 1/3 and 2/3 joined to TCM encoder with memory in order of 2, 3. These results show that the proposed scheme outperforms the classical Viterbi by a gain of 1 dB. On the other hand, we propose a function called RSCPOLY2TRELLIS, for recursive systematic convolutional (RSC) encoder which creates the trellis structure of a recursive systematic convolutional encoder from the matrix “H”. Moreover, we present a comparison between the decoding algorithms of the TCM encoder like Viterbi soft and hard, and the variants of the MAP decoder known as BCJR or forward-backward algorithm which is very performant in decoding TCM, but depends on the size of the code, the memory, and the CPU requirements of the application.
MDCT audio coding with pulse vector quantizersEricsson
This paper describes a novel audio coding algorithm that is a building block in the recently standardized 3GPP EVS codec. The presented scheme operates in the Modified Discrete Cosine Transform (MDCT) domain and deploys a Split-PVQ pulse coding quantizer, a noise-fill, and a gain control optimized for the quantizer’s properties. A complexity analysis in terms of WMOPS is presented to illustrate that the proposed Split-PVQ concept and dynamic range optimized MPVQ-indexing are suitable for real-time audio coding.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Design and Performance Analysis of Convolutional Encoder and Viterbi Decoder ...IJERA Editor
In digital communication forward error correction methods have a great practical importance when channel is
noisy. Convolutional error correction code can correct both type of errors random and burst. Convolution
encoding has been used in digital communication systems including deep space communication and wireless
communication. The error correction capability of convolutional code depends on code rate and constraint
length. The low code rate and high constraint length has more error correction capabilities but that also
introduce large overhead. This paper introduces convolutional encoders for various constraint lengths. By
increasing the constraint length the error correction capability can be increased. The performance and error
correction also depends on the selection of generator polynomial. This paper also introduces a good generator
polynomial which has high performance and error correction capabilities.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
A Density Control Based Adaptive Hexahedral Mesh Generation Algorithmijeei-iaes
A density control based adaptive hexahedral mesh generation algorithm for three dimensional models is presented in this paper. The first step of this algorithm is to identify the characteristic boundary of the solid model which needs to be meshed. Secondly, the refinement fields are constructed and modified according to the conformal refinement templates, and used as a metric to generate an initial grid structure. Thirdly, a jagged core mesh is generated by removing all the elements in the exterior of the solid model. Fourthly, all of the surface nodes of the jagged core mesh are matching to the surfaces of the model through a node projection process. Finally, the mesh quality such as topology and shape is improved by using corresponding optimization techniques.
The part is axisymmetrically modeled in solidworks(2D) before importing to ansys workbench where the boundary zones are identified and appropriate mesh settings is applied. The model is then imported in Fluent for analysis . Significant setting changes are Density based solver , Enhanced Eddy viscosity model with near wall treatment , solution steering , FMG initialization etc.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Manifold Blurring Mean Shift algorithms for manifold denoising, report, 2012Florent Renucci
(General) To retrieve a clean dataset by deleting outliers.
(Computer Vision) the recovery of a digital image that has been contaminated by additive white Gaussian noise.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Online aptitude test management system project report.pdfKamal Acharya
The purpose of on-line aptitude test system is to take online test in an efficient manner and no time wasting for checking the paper. The main objective of on-line aptitude test system is to efficiently evaluate the candidate thoroughly through a fully automated system that not only saves lot of time but also gives fast results. For students they give papers according to their convenience and time and there is no need of using extra thing like paper, pen etc. This can be used in educational institutions as well as in corporate world. Can be used anywhere any time as it is a web based application (user Location doesn’t matter). No restriction that examiner has to be present when the candidate takes the test.
Every time when lecturers/professors need to conduct examinations they have to sit down think about the questions and then create a whole new set of questions for each and every exam. In some cases the professor may want to give an open book online exam that is the student can take the exam any time anywhere, but the student might have to answer the questions in a limited time period. The professor may want to change the sequence of questions for every student. The problem that a student has is whenever a date for the exam is declared the student has to take it and there is no way he can take it at some other time. This project will create an interface for the examiner to create and store questions in a repository. It will also create an interface for the student to take examinations at his convenience and the questions and/or exams may be timed. Thereby creating an application which can be used by examiners and examinee’s simultaneously.
Examination System is very useful for Teachers/Professors. As in the teaching profession, you are responsible for writing question papers. In the conventional method, you write the question paper on paper, keep question papers separate from answers and all this information you have to keep in a locker to avoid unauthorized access. Using the Examination System you can create a question paper and everything will be written to a single exam file in encrypted format. You can set the General and Administrator password to avoid unauthorized access to your question paper. Every time you start the examination, the program shuffles all the questions and selects them randomly from the database, which reduces the chances of memorizing the questions.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
2. After an image has been segmented into regions ; the resulting
pixels is usually is represented and described in suitable form for
further computer processing.
Representing regions incolves in to two choices: in terms of it’s
1) external characteristics ( boundary)
2) internal characteristics ( pixels)
Above task is making data useful to computer
Next task is to describe the region on bases of representation
3. External representation is chosen when primary focus on
shape characteristics.
Internal representation is chosen when primary focus on
regionl properties like color and texture characteristics.
Sometime it is possible to choose both characteristics.
Features selected as descriptors should be insensitive to
changes in size, tanslation & rotation.
4. It deals with compaction of segmented data into
representations that facilitate the computation of
descriptors.
Technique of Representation :-
a. Chain codes
b. PolygonalApproximations
- Minimum perimeter polygones
- Merging technique
- Splitting technique
c. Signatures
d. BoundarySegments
e. Skeletons
5. 0
3
2
They are used to represent a boundary by a connected
sequence of straight line segments of specified length &
direction.
Typically this representation is based on 4- or 8- connectivity
of segments.
The direction of each segments is coded by using a
numbering scheme.
2
1 1
0
3
4
5
7
4-connectivity
6
8-connectivity
6. A boundary code formrd as a sequence of such directional
number is referred as a Freeman chain code.
Digital images are acquired & processed in a grid format with
equal spacing in x and y directions.
So a chain code can be generated by following a boundary
(say clockwise direction) and assigning a direction to the
segments connecting every pair of pixels.
Unacceptable method: (because)
1) Resulting chain tends to be quite long
2) Any small disturbances along the boundary due to noise
or imperfect segmentation can cause changes in code.
7. A solution to this problem is to resample the boundary by
selecting a larger grid spacing.
Then, as the boundary is traversed, a boundary point is
assigned to each node of the large grid, depending upon the
proximity of original boundary to that node.
The re-sampled boundary can now be represented by a 4- or
8-code.
The accuracy of the resulting code representation depends
on the spacing of the sampling grid.
Example:-
Start ChainCode:
0, 0, 0, 3, 0, 0, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1
8. The chain code of a boundary depends upon the starting point.
However the code can be normalized:
–Assume the chain is a circular sequence
(given a chain of 1 to N codes ; N+1 = 1)
– Redefine the starting point such that we generate an integer of
smallest magnitude
Chain Code 1 Chain Code 2
0, 0, 0, 3, 0, 0, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 3, 0, 0, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 1,1, 1, 1, 1, 1, 0, 0, 0
NormalizedCode 0, 0, 0, 3, 0, 0, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1
9. 0
Chain code depend on orientation
– a rotation results in a different chain codeOne solution
– Use the “first difference” of the chain code instead of the
code itself
The difference is obtained by simply counting (counter-
clockwise) the number of directions that separate two
adjacent elements
1
2
3
Difference:Count the number of separating
directions inan anti-clockwise fashion
2 1 0 1 0 3 3 2 2 2 ChainCode
3 3 1 3 3 0 3 0 DifferenceCode
Ex. First difference of 4-direction chain code
10103322 is 3133030.
Considering the circular sequence first
element is calculated by subtracting last and
first component. Ex. 33133030
10.
11. An approach for generating an algorithm to compute MPPs
is to enclose a boundary by a set of concatenated cells.
Allow the boundary to shrink as a rubber band.
This shrinking produces the shape of a polygon of min.
perimeter.
Size of cells determine the accuracy of the polygonal
approximation.
In the limit if size of each cell corresponds to a pixel in the
boundary , the error in each cell between the boundary & the
MPP approx. at most would be √2d, where d-min possible
pixel distance.
The objective is to use the largest possible cell size
acceptable in a given application.
Thus, producing MPPs with fewest no. of vertices.
12. The cellular approach reduces the shape of the object
enclosed by the original boundary.
Min. Perimeter
Polygon
13. Techniques based on average error or other criteria have
been applied to the problem of polygonal approximation.
One approach is to merge points along a boundary until the
least square error line fit of the points merged so far exceeds
a preset threshold.
When this condition occurs, parameters of line are stored,
the error is set to 0, the procedure is repeated, merging new
points along the boundary until the error again exceeds the
threshold.
Merging technique problem:
– No guarantee for corner detection
Solution:
– SplittingTechniques
14. to subdivide a segment successively into two parts until a
given criterion is satisfied.
–Objective: seeking prominent inflection points
Step:-
1. Start with an initial guess, e.g., based on majority axes
2. Calculate the orthogonal distance from lines to all points
3. If maximum distance > threshold, create new vertex there
4. Repeat until no points exceed criterion
16. It is a 1D functional representation of a boundary & may be
generated in various ways.
One of the simplest is to plot the distance from the centroid
to the boundary as a function of angle.
Signatures generated by this method are invariant to
translation but they depend on rotation and scaling.
Normalization with respect to rotation can be achieved by
finding a way to select the same starting point to generate
the signature , according to shape.
One way to normalize this is to scale all functions so that
they always span the same range of values. e.g [0,1]
17. Advantage :
- it’s simplicity
disadvantage :
- scaling of the entire function depends upon only two
values: maximum & minimum.
18. The boundary can be decomposed into segments.
–Useful to extract information from concave parts of the
objects.
A good way to achieve this is to calculate the convex Hull of
the region enclosed by the boundary Hull.
Can be a bit noise sensitive
1. Smooth prior toConvex hull calculation
2. CalculateConvex Hull on polygon approximation
19. Convex Hull, H, of an arbitrary set, S, is the smallest set
containingS
H minusS (H-S)
– Is called the convex deficiency
We can use the convex deficiency to mark features
Follow the contour ofS and mark points that transitions
into or out of the convex deficiency
20. UsingConvex Hull and its Deficiency
This data can be used to describe the region
– Number of pixels in convex deficiency
– Number of components in convex deficiency
– Ratio lengths of the transition points
– so on
21. An important approach to represent the structural shape of a
plane region is to reduce it to a graph.
The reduction may be accomplished by obtaining the
skeleton of the region via thinning (skeletonizing) algorithm.
Skeleton of a region may be defined as the medial axis
transformation (MAT).
MAT of a region R with border B is as follows:
- For every point p in R, we find its closest neighbor in B. If
p has more than such neighbor, it is said to belong to the
medial axis (skeleton).
‘prairie fire concept’
22. Implementation involves calculating the distance from every
interior point to every boundary point on region.
Thinning algorithm deals with deleting the boundary points
of a region subject to condition that deleting these points:
1) Does not remove end points.
2) Does not break connectivity &
3) Does not cause excessive erosion of the region.
23. Thinning algorithm:
Region points are assumed to have value 1 & background
points are assumed to have value 0.
p9 p2 p3
p8 p1 p4
p7 p6 p5
Step 1
flags a contour point for deletion if the followingconditions
are satisfied:
a) 2 ≤ N(p1) ≤ 6
b)T(p1) = 1
c) p2.p4.p6 = 0
d) p4.p6.p8 = 0
24. p2,
where N(p1) is the number of nonzero neighbors of p1;
i.e. N(p1) = p2 + p3 + …… + p8 + p9
where pi is either 0 or 1.
T(p1) is number of 0-1 transitions in the ordered sequence
p3, ….., p8, p9., p2
InStep 2:
conditions a & b remain the same, but conditions c & d are
changed to
c’) p2.p4.p8 = 0
d’) p2.p6.p8 = 0
0 0 1
1 p1 0
1 0 1
Step 1 is applied to every border pixel in binary region under
Consideration If 1 or more of conditions a to d are violated, the
value of point in question is not changed.
25. If all conditions are satisfied, the point is flagged
for deletion.
But not deleted until all border points have been
processed.
This delay prevents the change of structure of
data during execution of algorithm.
After step 1 been applied to all border points,
those who are flagged are deleted (changed to
0).
Step 2 is applied to resulting data in exactly
same manner as step1.
26. Some simple discriptors:-
Length of the contour
– Simply count the number of pixels along the border
–You may consider diagonally connected pixels to count as √2
Diameter of the boundary B
– Diam(B) = max[D(pi,pj)]
– this is the major axis of the region
Curvature
– Rate of change of the slope
Bounding Box
–Smallest rectangle (aligned with the image axis) that can bound
the region
27. Shape number
– compute the chain code difference
– re-order this to create the minimum integer
– this is called the shape number
Example:-
28. Fourier Descriptors
Consider an N-point digital boundary in the xy plane
This forms a coordinate pairs (xo, yo), (x1, y2), . . . . ., (xn-1, yn-1)
We can consider this as two vectors
– x(k) = xk
– y(k) = yk
Furthermore
– We could consider this a complex number
– s(k) = x(k) + jy(k) where j=sqrt(-1)
29. Using the vector s(k)
Compute the 1-D Discrete FourierTransform