# How to do convolution in Fourier Series?

Two signals are given to me :

$$x(t)=\cos(4\pi t)$$ $$y(t)=\sin(4\pi t)$$

I have founded their coefficients as follows:

$$a_k = a_1=a_{-1}=\frac{1}{2}$$

$$b_k = b_1=b^*_{-1}=\frac{1}{2j}$$

Now I am stuck at convolution. In the formula of the convolution:

$$\sum_{l=-\infty}^{\infty}a_lb_{k-l}$$

What are these $$l$$ and $$k$$ and from the coefficients I found what is their respective value?

• Welcome to DSP.SE! Is the Fourier series coefficients of the signal $z(t)=x(t)y(t)$ what you are looking for? – Tendero Nov 9 '18 at 14:17
• @Tendero yes that's what I am trying to find. – Muhammad Ahmad Nov 9 '18 at 14:24

Following your notation, if we define the coefficients of the Fourier series of $$z(t)=x(t)y(t)$$ as $$c_k$$:

$$c_k=\sum_{l=-\infty}^{\infty}a_lb_{k-l}$$

Notice that the $$k$$ in which you evaluate the left side of the equality has to be the same as the $$k$$ in the summation. Therefore, for example, if you want to find $$c_0$$, you should calculate:

$$c_0=\sum_{l=-\infty}^{\infty}a_lb_{-l}$$

Then you have to calculate that summation. Given that $$a_l$$ and $$b_{-l}$$ are non-zero only if $$l$$ equals $$1$$ or $$-1$$:

\begin{align} c_0 &=\sum_{l=-\infty}^{\infty}a_lb_{-l} \\ &=a_{-1}b_1 + a_1b_{-1} \\ &=\frac12 \frac{1}{2j} - \frac12 \frac{1}{2j}=0 \end{align}

I think this might have clarified the notation for you. You can go on and calculate the rest of the coefficients $$c_k$$ for all values of $$k$$.