In signal processing, the filter designs are the system that performs the mathematical operations for the discrete-time signal samples, and it is also reduced the aspects these signals. In the systems of discrete-time , the filter designs are the implement through the transfer function and also has linear constant difference equations by the Z- transforms of transfer functions, which also has two polynomials rations. The designs of the filter are very complex topics. Thus the filter is calculated the practical challenges for implementation and design of advanced researches. Whereas in digital filter there are two categories;

· Recursive filter

· Non-recursive filter

In this assignment, the recursive filter is discussed, which is referred the infinite impulse response (IIR).

Aims of Filter Design and Signal Analysis by using the MATLAB software

The aim of this assignment to solved these three questions (Filter Design and Signal Analysis ; and image processing) by using the MATLAB software , then provide the scripts of Matlab code with proper comments and write the report of these questions.

Analysis of Filter Design and Signal Analysis by using the MATLAB software

The first problem is about the filter designs and the signal analysis, where the filter has the input-output difference equations, where the by the IIR filter designs different equations explained. Whereas the filters included both feed-forward as well as feedback, and the system of function should be rationales, which also has poles and zeros. The general forms of difference equation are given below;

Problem 1 (1) of Filter Design and Signal Analysis by using the MATLAB software

Difference equations are the formula for computing the output samples at the time n, which is based on past output samples and presented input samples in a time domain [3];

The difference equations may be written as;

Now apply filter function to the above difference equations in MATLAB;

And the MATLAB Scripts/Code of difference equation is given below in appendix 1;

Now in the below appendix 1, the filter function or the difference equations is shown in the 3^rd line of the code, then initialize the random generators, and put the right hand of difference equations as shown in the below code, and apply function of the filter on this code with the proper length vector, and plot on the “filter function to difference equation'

“and the legend('Input Data’, ‘Filtered Data')as shown in the below graph.

Graph:

Graph: Apply filter function to a difference equation

Problem 1 (2) of Filter Design and Signal Analysis by using the MATLAB software

In MATLAB the recursive filter is “yulewalk” and its syntax is;

Yulewalk is the designs of the recursive IIR digital filter by using the least square fit to the specified frequency response.

The output of filter coefficients is;

According to the part , the difference equation has order n= 8 ; so now we design low pass filter which has 8 order , with the normalized cutoff frequency ; and plot of frequency response and overlay response is shown below , in ideal filter ; the MATLAB code/Script is given in below appendix 2 ;

According to the requirements implement the recursive filter in MATLAB, then for recursive filter that is applied on the difference equations; and its function is “ yulewalk” and the cutoff frequency 8 and apply this function then plot the magnitude and frequency as who in the below graph 2,

Problem 1 (3):

Solutio of Filter Design and Signal Analysis by using the MATLAB software

Whereas in the above equation is the input signal, and is the output signals, and a and b are coefficients; [4]

Problem 2

Solution of Filter Design and Signal Analysis by using the MATLAB software

The DFT is Discrete Fourier transform, which is the primary tools and in the digital signal processing, and the method of computing DFT by the reduced exceptions times.

For input sequence x as well as its transformed version X (the discrete-time Fourier transform at equally spaced frequencies around the unit circle), two functions implement relationships

And;

In the above equation the series subscripts begin with 1 instead of 0 because of the MATLAB vector indexing scheme, and

Performed on MATLAB and the MATLAB code is given below in appendix 3;

In this code, Spectral analysis is the study of the frequency where the spectrum contained in discretely sampled data. Whereas the Fourier transforms reveals the components of frequency for the time-space. Now in the below appendices 3, the Matlab code for DFT of the spectral analysis signals , in first line of the code, the time of the spectral analysis is set; then the given requirements are fixed into code. In line 4 of MTAALB code, the FFT function is used to compute DFT of spectral analysis signals. Then power the spectrum like the frequency function, where the noise diagnoses the signal frequency by the complements, which are based on the time-space. Then plot magnitude and phase, as shown in the below graphs 3

Part II (Image Processing) of Filter Design and Signal Analysis by using the MATLAB software

The problem is about written Matlab function where the input is Grayscale image with the n integers. The read function is supported general syntax. A= read (filename, jpg) reads the grayscale image, which is specified form the filename as shown in the below code appendix 4. First of all store the image in your existed folder, like in this assignment. First of all, the image of Apple is store, and then the image read the grayscale code.

Problem 3 (a) Partition the image into 8*8 blocks

In the below code or appendix 5, it presents the partitions each block 8 into 8 pixels; where the image reads then it converts into the grayscale image; whereas in this code line 8 present the size of the block; and at last it presented the figure as shown in below;

% partition of each block 8 into 8 pixel

x=imread('applered.jpg');

I = imread('applered.jpg');

J = rgb2gray(I);

figure, imshow(I), figure, imshow(J);

y=double(x);

[r c]=size(y);

bs=8; %size of each block

nob=(r/bs)*(c/bs);%number of blocks

kk=0;

for i=1:(r/bs)

for j=1:(c/bs)

Block(:,:,kk+j)=y((bs*(i-1)+1:bs*(i-1)+bs),(bs*(j-1)+1:bs*(j-1)+bs));

end

Problem 3 (2)

Computes 2D-DCT of each block

The new fast dimension is 2D 8 for the is the algorism based, which also has characteristics for the basic images of the 2D DCT, and it is shown in these assignments. This algorithm for every 2D –DCT has high number of the frequency components which is more efficient and its existing algorithms [5].

As shown in the below code, appendix 6, it computes the 2D-DCT of each block; in the first line of code just read the image, then convert into the grayscale, and apply the 2D-DCT to the image

Computes inverse 2D-DCT of each block of Filter Design and Signal Analysis by using the MATLAB software

As shown in the first part , just implement the Matlab function to read image , then it converts into 8 by 8 pixel by the partition of the blocks, then the image computes the 2D-DCT for every block . Now in this section apply the inverse 2D-DCT to the grayscale images; as shown in this code and the in the below figure

%Computes inverse of 2D-DCT of each block

J(abs(J)<10) = 0;

K = idct2(J);

a figure, imshow(I)

figure, imshow(K,[0287])

Conclusion of Filter Design and Signal Analysis by using the MATLAB software

Summing up all the discussion, it is concluded that the assignment is about the signal process and the image processing. In this assignment there are three tasks, in which by using the MATLAB software perform this task and determine the accurate solutions. The design of the filter for the difference equation is very complex, which is done in task 1 and provides all the MATLAB code, with the proper figures; in task 1 the second part is designed and implement the recursive filter for this difference question. And the third is image processing for the grayscale image. All objective and the requirements with proper specification is achieved as shown in the above discussion, and the results are also provided in below appendix

Introduction of Filter Design and Signal Analysis by using the MATLAB software

The assignment is about Filter Design and Signal Analysis and image processing by using the MATLAB. Whereas the MATLAB is the high-level language, as well as also has the interactive environment, which also enables to perform computationally intensive tasks by programming language, like C, C++. MATLAB is designed for the calculations of the numerical matrix, and for other different purposes. Now in the MATLAB the image processing tools are the collections of functions that are extended different capabilities for the numeric values of MATLAB by computing environment. Image processing toolbox of the MATLAB is supported great range of the operations of image processing that is including;

References of Filter Design and Signal Analysis by using the MATLAB software

[1]	T. Wang, "MATLAB for Image Processing," 12 February 2010. [Online]. Available: https://www.cs.tau.ac.il/~dcor/Graphics/cg-slides/MATLAB-tutorial.pdf.
[2]	Eas.uccs.edu, "IIR Filters," 2019. [Online]. Available: http://www.eas.uccs.edu/~mwickert/ece2610/lecture_notes/ece2610_chap8.pdf.
[3]	Dsprelated, "DIFFERENCE EQUATION," 2019. [Online]. Available: https://www.dsprelated.com/freebooks/filters/Difference_Equation_I.html.
[4]	S. W. Smith, "The Scientist and Engineer's Guide to," 2019. [Online]. Available: https://www.dspguide.com/ch19/1.htm.
[5]	X. Ji and e. al, "Fast 2-D 8×8 discrete cosine transform algorithm for image coding," Science in China Series F Information Sciences, vol. 52, no. 2, p. 2009, 215-225.