Find fourier series of my signal

Question

T = 3*(1/25);
fs = 1000;
t = 0:1/fs:T-1/fs;
x = sawtooth(2*pi*25*t,1/2);
x = (x + 1)*0.5;
subplot(3,1,1)
plot(t,x)
grid on
xlabel('Time t (sec)')
ylabel('Amplitude')

I want to take DFT(Discrete Fourier Transform) of my signal using FFT and FFTShift and find frequency spectrum. However,the graphics I found did not seem right to me. Is there anyone who can help? If there are errors in what I do, I want to find and fix it.

Following code for second graph:

y = fft(x);
n = length(x);        
f = (0:n-1)*(fs/n);     
power = abs(y).^2/n;   
subplot(3,1,2)
plot(f,power)
xlabel('Frequency')
ylabel('Power')

Answer 1

First, your period is $T = 1/25s=0.04s$ which means you do not need to take integral of 3 periods for getting fourier series. A plot of the DFT is given below when I took only first $T=0.04s$ is given by

As you can see, all even multiples of $25\text{Hz}$ is zero, while odd multiples of $25\text{Hz}$ is decreasing with increasing frequency. From the plot $|X[5]|^2 = 0.002914$, while $|X[7]|^2=0.0008386$. Even multiples of $25\text{Hz}$ have very low values ($\lt 10^{-12}$). Taking DFT is not the correct method, so read on.

The above values are expected because fourier series of your even symmetric signal with non-zero DC has all $b_k =0$. For computing $a_k$, your signal $x(t)=2t/T$ for $0 \le t \le T/2$ and $x(t)=-2t/T-2$ for $T/2 \le t \le T$. I am not going into details of how to compute fourier series coefficients but I will give hints $$ a_0=\frac{2}{T}\int_0^T x(t)dt\\ a_k=\frac{2}{T}\int_0^T x(t)\cos(2\pi kt/T)dt \\ =\frac{2}{T}\int_0^{T/2} 2t/T\cos(2\pi kt/T)dt + \frac{2}{T}\int_{T/2}^T (-2t/T-2)\cos(2\pi kt/T)dt\\ =\frac{2((-1)^k-1)}{\pi^2k^2}\\ a_k=\frac{-4}{\pi^2k^2}\text{ for k odd}\\ a_k=0\text{ for k even} $$ So as you can see, the even harmonics are all zeros, while odd harmonics decrease by a factor of $1/k^2$. So $$a_5=\frac{-4}{\pi^2 25}\\ a_7=\frac{-4}{\pi^2 49} $$