I am trying to convert a signal into frequency domain using fft then construct a time domain signal (with any length) from the frequency response. With my matlab code, it works fine when there is no noise but there is a weird behaviour when noise is introduced in to the signal.

So let's first look at the case when there is no noise.

<pre><code>clear
Fs = 100;          % Frequency (Hz)
dt = 1/Fs;         % Sampling Time (s)
Tsim_original = 3; % 3 seconds of original data
t = (0:dt:Tsim_original-dt)';     % Time array
signal = sin(2*pi*2*t) + 2 * sin(2*pi*3*t);    % Original signal

% Compute FFT - I'm using Matlab's example on their FFT page
Y = fft(signal);
L = length(signal);
P2 = abs(Y/L);
P1 = P2(1:floor(L/2)+1);
P1(2:end-1) = 2*P1(2:end-1);
f = (Fs*(0:(L/2))/L)';

figure(1)
clf
stem(f, P1, 'Marker', 'none')
hold on
grid on
xlabel('Frequency (Hz)')
ylabel('|P1(f)|')


Tsim = 10;    % Time of constructed signal - I'm using longer time here, will be important later
signal2 = zeros(ceil(Tsim/dt), 1);          % Predefine constructed signal array
t2 = linspace(0, Tsim, length(signal2))';   % Time array

% Construct signal from FFT
% Loop through all the frequencies and add them to gether
for kk = 1 : length(f)
    if f(kk) > 0   % Ignore 0 Hz 
        temp = P1(kk) * sin(2*pi*f(kk)*t2);
        signal2 = signal2 + temp;
    end
end

figure(2)
clf
plot(t, signal)
hold on
grid on
plot(t2, signal2, '--')
legend('Original Signal', 'Constructed Signal')
xlabel('Time (s)')
</pre></code>

So this works fine and the constructed signal pretty much looks identical to the original signal. The FFT also looks good, with correct frequency and amplitude.
 
[![enter image description here][1]][1]


Here's where the problem is. If I add some noise into the signal

<pre><code>signal = sin(2*pi*2*t) + 2 * sin(2*pi*3*t) + randn(length(t), 1);    % Original signal
</pre></code>

The resulting signal has weird peaks happening every 3 seconds (and at the start), which is equivalent to the length of the original signal. If I change the length of the original signal to a different number, the frequency of the peaks also change accordingly. For example, if I change the the original signal to be 5 seconds long, the constructed signal has a peak every 5 seconds also.

I need the constructed signal to look like the figure below but without the peaks every X seconds. Note that I do not want to filter the noise, but rather I want to keep the noise.

So my questions are:
 - What is the cause of these peaks? Am I doing something wrong?
 - Is there a better way to reconstruct the time domain signal (with any length)?
 - I understand matlab has the `ifft` function available already, but I am not sure how to get it to work with the absolute power `P1` of the frequency response. I kind of have to keep this FFT format as it's already in other parts of the code.

[![enter image description here][2]][2]

When zoomed in, the peak looks like this
[![enter image description here][3]][3]

Thank you so much for your help!!


  [1]: https://i.sstatic.net/0Tw8a.png
  [2]: https://i.sstatic.net/1b4jB.png
  [3]: https://i.sstatic.net/5EUXX.png