I can't make sense of the amplitude spectrum of a signal. Help please I have a sinusoidal signal namely:
signal = sin(2*pi*0.08*t) + sin(2*pi*0.2*t) + sin(2*pi*0.32*t) + sin(2*pi*0.4*t);

so the segment of the code becomes:
freqs = [0.08, 0.2, 0.32, 0.4];
periods = 1./ freqs;
% Let the max value of t be 4 times the largest period
t_max = 4 * periods(1);
% Let us have 500 samples over the interval [0, t_max].
t = linspace(0, t_max, 500);
% The synthesized signal 
signal = sin(2*pi*0.08*t) + sin(2*pi*0.2*t) + sin(2*pi*0.32*t) + sin(2*pi*0.4*t);

Now I wanna view the amplitude spectrum for the first 50 samples.
So I did this:
signal_first_50_samples= signal (1:50);
% Display its magnitude spectrum. 
N = 64;
signal_spect = abs (fft(signal_first_50_samples,N));
signal_spect = fftshift(signal_spect);
F = [-N/2:N/2-1]/N;
plot (F, signal_spect) 
set(gca, 'XTick',[-0.5:0.1:1])
grid on;

I was expecting spikes at each of the frequencies, however, I got this:

I re-did the sampling part as follows, to include more samples:
%% Synthesizing a sampled signal that consists of 4 sine waves 
% We are given 4 different frequencies 
freqs = [0.08, 0.2, 0.32, 0.4];
periods = 1./ freqs;
% Let the max value of t be 4 times the largest period
t_max = 4 * periods(1);
% Let us have 500 samples over the interval [0, t_max].
t = linspace(0, t_max, 500);
% The synthesized signal 
signal = sin(2*pi*0.08*t) + sin(2*pi*0.2*t) + sin(2*pi*0.32*t) + sin(2*pi*0.4*t);
% Visualising the synthesized signal 
plot(t, signal, 'b-', 'LineWidth', 1.5);
grid on;
title('The Synthesized Signal', 'FontSize', fontSize);
xlabel('t', 'FontSize', fontSize);
ylabel('Y', 'FontSize', fontSize);
% Display its magnitude spectrum. 
% i was expecting spikes at each freq 
N = 64;
signal_spect = abs (fft(signal,N));
signal_spect = fftshift(signal_spect);
F = [-N/2:N/2-1]/N;
plot (F, signal_spect) 
%set(gca, 'XTick',[-0.5:0.05:1])
grid on;

but i still got approx the same result, not what i expected

another edit:
i did something else i had only 50 samples for the singal and took all of them
%% 50 samples taking 50
clc;    
clear;  
% We are given 4 different frequencies 
freqs = [0.08, 0.2, 0.32, 0.4];
periods = 1./ freqs;
% Let the max value of t be 4 times the largest period
t_max = 4 * periods(1);
% Let us have 50 samples over the interval [0, t_max].
t = linspace(0, t_max, 50);
% The synthesized signal 
signal = sin(2*pi*0.08*t) + sin(2*pi*0.2*t) + sin(2*pi*0.32*t) + sin(2*pi*0.4*t);
% Determine the first 50 samples of this signal 
% signal_first_50_samples= signal (1:50);
% Display its magnitude spectrum. 
% ??????????????????????????????????????????????????????????????????????
% I was expecting spikes at each freq 
N = 64;
signal_spect = abs (fft(signal,N));
signal_spect = fftshift(signal_spect);
F = [-N/2:N/2-1]/N;
figure
plot (F, signal_spect) 
%set(gca, 'XTick',[-0.5:0.05:1])
grid on;

I got this

which looks about right
but I wonder why it didn't work when i had 500 samples and i took all 500 of them
any ideas?
A few suggestions and comments:


You calculate the frequency vector with F = [-N/2:N/2-1]/N;. This is wrong. You need to take the sampling frequency into account.
The way you organized your code, you don't have direct control over the sampling frequency. I recommend specifying both the time length and the sampling frequency explicitly in your code.
You need to set up your analysis so that you get high resolution in the frequency domain (so that you can resolve peaks that are very near to each other). The only way to achieve this is to sample for a long time. The more you oversample the signal (that is, the higher the Nyquist frequency is), the longer time you need to observe the signal. This is the reason you want to have explicit control over sampling rate and signal duration.
Do not limit the FFT size by choosing an explicit N, since that may lower your frequency resolution. If you want to set the FFT to a power of 2, then use nextpow2 or similar.