Summary

This assignment involves writing a computer program in the $C$ programming language. It will read in a file of the values of a signal and then calculate its Discrete Fourier Transform $($DFT$).$ The program will then attempt to 'smooth out' the signal using a moving average and then recalculate the transform.

Throughout the following questions, the 'length of the record' is denoted by $N_2.$ The data you have generated from the spreadsheet 'Assignmentdata' will consist of $5$ columns of numbers. The first is the time value and the next four are four distinct signals. You should copy and paste this data into a plain text file to use as your input.

There are three parts to the assignment, be sure and answer the questions from all three parts.

Part $1$

Two values will be used for your calculations here; $N_2=N_1$ and $N_2=2N_1.$ Your program should carry out the following steps.

Read in the list of values in your input file, noting the number $N_1,$ the number of values in each signal $($that is$,$ each column of numbers$)$ in your data. Let $x_1$ to $x_4$ be the four signals in your data. The following steps will be carried out for each of the four signals you have.

For the first signal in your data, $x_1[n],$ calculate the DFT of the signal $x_1[n]$ for $N_2=N_1.$ The result of your calculation should be a list of values in a file. Explain in your program comments or elsewhere in your submission how your program handles the complex values of the transform widehat$(x_1)[k].$

Repeat this process for length of record $N_2=2N_1.$

Part $2$

A second signal $y_1$ is constructed as the $5-$point moving average of the signal $x_1.$ Using $N_1$ as the length of the signal, this is defined as follows:

$y_1[0]=x_1[0]$

$y_1[1]=\frac{1}{2}(x_1[0]+x_1[1]),$

$y_1[2]=\frac{1}{3}(x_1[0]+x_1[1]+x_1[2]),$

$y_1[2]=\frac{1}{3}(x_1[0]+x_1[1]+x_1[2]),$

$y_1[3]=\frac{1}{4}(x_1[0]+x_1[1]+x_1[2]+x_1[3]),$

For all $n>3$ and $($:$x_1[n+1]+x_1[n+2]\}($:$x_1[n+1]+x_1[n+2]\}$

$y_1[N_1-4]=\frac{1}{4}(x_1[N_1-4]+x_1[N_1-3]+x_1[N_1-2]+x_1[N_1-1])$

$y_1[N_1-3]=\frac{1}{3}(x_1[N_1-3]+x_1[N_1-2]+x_1[N_1-1])$

$y_1[N_1-2]=\frac{1}{2}(x_1[N_1-2]+x_1[N_1-1])$

$y_1[N_1-1]=x_1[N_1-1].$

Your program should now carry out the following steps.

Generate the signal $y_1$ from signal $x_1.$

Calculate the DFT of the signal $y_1[n]$ for both values of the length of the record $N_2.$

Part $3$

You should now have data files with the signal $x_1,$ the smoothed signal $y_1$ and the transforms of both signals for lengths of record $N_1$ and $2N_1.$

Draw a plot of the real and complex components of transform widehat$(x_1)[k].$

Draw plots of $|(widehat(x_1))[k]|$ and arg$((widehat(x_1))[k]),$ where $|z|$ represents the modulus of the complex number $z$ and arg$(z)$ is the argument of $z.$ So if $r=|z|$ and $A=$arg$(z)$ then $z=r{e}^{jA}.$

Explain the difference in your output when $N_2=2N_1.$

Draw a plot of the real and complex components of transform widehat$(y_1)[k].$

Draw a plot of $|(widehat(y_1))[k]|$ and arg$((widehat(y_1))[k]).$

Explain what the transform you have calculated is telling you about the signal $x_1.$ You should consider the meaning of the Fourier

Note: I have extensive Excel spreadsheet including signal data in the C programming language. However, due to its size, I am unable to upload it here. May I send it to you through an alternative way?