FFT vs DFT Run Time Comparison (Complexity Analysis) in MATLAB
Abhinav Jain, Welcome to DSP Community.

I build for you a proper testing of the run time comparison.

Few tips about timing in MATLAB:


Never time in a script. Always call a function to do the heavy lifting. When you run something from script it runs in the global scope which means MATLAB can't optimize it as well as it could if it was in a function.
When you time a function, make few iterations of the measurement. I like taking the median of this iterations. Others like the minimum. Usually the mean isn't good as it is sensitive to outliers.
MATLAB is a JIT based Script language. It means the first run of anything takes more time. Hence (2) is crucial.
Pay attetion that the result of the FFT / DFT is complex. Hence when you allocate memory for a complex array you should use - vArrayName = complex(zeros(arrayLength, 1));.
When you need the imaginary number $ i = \sqrt{-1} $ in MATLAB you should use 1i or 1j (No need for multiplication). Of course you can use 5i as well. Yet no need to multiply as it assists the JIT engine to understand you mean the imaginary number and not a variable.


Here are the results:



As you can see, indeed the DFT Implementation has running time which behaves as O(n^2).

The code goes like:

%% Simulation Parameters

vNumSamples     = 2:2:1024;
numIterations   = 6;


%% Generate Data

mDftTime = zeros(numIterations, length(vNumSamples));
mFftTime = zeros(numIterations, length(vNumSamples));

for jj = 1:length(vNumSamples)
    numSamples = vNumSamples(jj);
    vX = randn(numSamples, 1);

    for ii = 1:numIterations
        hDftTimer           = tic();
        vXDft               = ApplyDft(vX, numSamples);
        mDftTime(ii, jj)    = toc(hDftTimer);

        hFftTimer           = tic();
        vXFft               = fft(vX);
        mFftTime(ii, jj)    = toc(hFftTimer);
    end

end


%% Run Time Analysis

vDftMedian = median(mDftTime).';
vFftMedian = median(mFftTime).';

vDftMean = mean(mDftTime).';
vFftMean = mean(mFftTime).';

vDftMax = max(mDftTime).';
vFftMax = max(mFftTime).';

vDftMin = min(mDftTime).';
vFftMin = min(mFftTime).';


%% Display Results

figureIdx = figureIdx + 1;

hFigure = figure('Position', figPosLarge);
hAxes   = subplot(4, 1, 1);
% set(hAxes, 'NextPlot', 'add');
hLineSeries = plot(vNumSamples, [vDftMedian, vFftMedian]);
set(hLineSeries, 'LineWidth', lineWidthNormal);
% set(hLineSeries(2), 'LineStyle', ':');
set(get(hAxes, 'Title'), 'String', {['DFT vs. FFT Run Time - Median']}, ...
    'FontSize', fontSizeTitle);
set(get(hAxes, 'XLabel'), 'String', {['Input Size']}, ...
    'FontSize', fontSizeAxis);
set(get(hAxes, 'YLabel'), 'String', {['Run Time [Sec]']}, ...
    'FontSize', fontSizeAxis);
hLegend = ClickableLegend({['DFT'], ['FFT']});

hAxes   = subplot(4, 1, 2);
% set(hAxes, 'NextPlot', 'add');
hLineSeries = plot(vNumSamples, [vDftMean, vFftMean]);
set(hLineSeries, 'LineWidth', lineWidthNormal);
% set(hLineSeries(2), 'LineStyle', ':');
set(get(hAxes, 'Title'), 'String', {['DFT vs. FFT Run Time - Mean']}, ...
    'FontSize', fontSizeTitle);
set(get(hAxes, 'XLabel'), 'String', {['Input Size']}, ...
    'FontSize', fontSizeAxis);
set(get(hAxes, 'YLabel'), 'String', {['Run Time [Sec]']}, ...
    'FontSize', fontSizeAxis);
hLegend = ClickableLegend({['DFT'], ['FFT']});

hAxes   = subplot(4, 1, 3);
% set(hAxes, 'NextPlot', 'add');
hLineSeries = plot(vNumSamples, [vDftMax, vFftMax]);
set(hLineSeries, 'LineWidth', lineWidthNormal);
% set(hLineSeries(2), 'LineStyle', ':');
set(get(hAxes, 'Title'), 'String', {['DFT vs. FFT Run Time - Max']}, ...
    'FontSize', fontSizeTitle);
set(get(hAxes, 'XLabel'), 'String', {['Input Size']}, ...
    'FontSize', fontSizeAxis);
set(get(hAxes, 'YLabel'), 'String', {['Run Time [Sec]']}, ...
    'FontSize', fontSizeAxis);
hLegend = ClickableLegend({['DFT'], ['FFT']});

hAxes   = subplot(4, 1, 4);
% set(hAxes, 'NextPlot', 'add');
hLineSeries = plot(vNumSamples, [vDftMin, vFftMin]);
set(hLineSeries, 'LineWidth', lineWidthNormal);
% set(hLineSeries(2), 'LineStyle', ':');
set(get(hAxes, 'Title'), 'String', {['DFT vs. FFT Run Time - Min']}, ...
    'FontSize', fontSizeTitle);
set(get(hAxes, 'XLabel'), 'String', {['Input Size']}, ...
    'FontSize', fontSizeAxis);
set(get(hAxes, 'YLabel'), 'String', {['Run Time [Sec]']}, ...
    'FontSize', fontSizeAxis);
hLegend = ClickableLegend({['DFT'], ['FFT']});

if(generateFigures == ON)
    saveas(hFigure,['Figure', num2str(figureIdx, figureCounterSpec), '.png']);
end


As you can see in the code above I run numIterations = 6 each algorithm for each number of samples.
Then, on the time array mDftTime / mFftTime I can analyze the Median / Mean / Min / Max run time.
This is the proper way to analyze function run time in MATLAB.

The full code is available on my StackExchange Signal Processing Q51516 GitHub Repository.

Remark
You're doing your first steps in DSP world and big part of being modern engineer is about programming code.
You should invest time in learning to properly write code and style in a way it is easy to under stand and manage for you and your partners.
Try learning from other people examples.