how to slove this question?

If two periodic signals x(t) and y(t) are to be combined in different ways (mentioned below), then how can the Fourier series of resultant signal z(t) can be expressed in terms of Fourier series of x(t) and y(t)? The periodic signals are to be combined as follows:
a. z(t)= x(y)+y(t)
b. z(t)= x(y).y(t)
c. z(t)= x(y)*y(t)

 

You need to define the operations indicated by the dot and the star. Is the dot multiplication and star convolution?

With linear operators the answer to a) should be straightforward.

 

how to slove this question?

If two periodic signals x(t) and y(t) are to be combined in different ways (mentioned below), then how can the Fourier series of resultant signal z(t) can be expressed in terms of Fourier series of x(t) and y(t)? The periodic signals are to be combined as follows:
a. z(t)= x(y)+y(t)
b. z(t)= x(y).y(t)
c. z(t)= x(y)*y(t)

What do you know about the relationship between multiplication and convolution with respect to a time-domain signal and it's frequency domain counterpart?

 

dot is the product and * is the convolution

 

I don't think you have the definitions of the symbols correct. The dot product applies to vectors, not functions of time. I believe it refers to point-wise multiplication of two functions of time.

There are a couple of key phrases that you should be familiar with:

1. Multiplication in the time domain is equivalent to (??) in the frequency domain
2. Convolution in the time domain is equivalent to (??) in the frequency domain

If you don't already know the answers, Id suggest Google for starters.
If that doesn't work for you try "Convolution Theorem"

 

I don't think you have the definitions of the symbols correct. The dot product applies to vectors, not functions of time. I believe it refers to point-wise multiplication of two functions of time.

Easy to miss, but the OP didn't say that the period was the dot product, he said that the dot is the product, aka, the period indicated multiplication.

 

Yeah he got me with that one. Too bad English is my first language instead [of] my second or third. I guess maybe I should switch to a Russian forum.

 

I know if I switched to a non-English forum I would NEVER make a similiar mistake! Of course, every response I made would be in the one word that I would make a point of learning in that language -- "Huh?"