Fourier series representation Conceptually when we want to represent a peroidic series, for example a pulse train, we find the Fourier coefficients and get a representation in the time domain. 

However what is conceptually wrong with using say an infinite sum of time shifted rect functions to represent it? 



Sorry I had a formatting problem, so i am putting this in the bottom post...

My method is as such:

Assuming we have a periodic pulse $x(t)$ such that $x(t) = 1$ for $0 &lt; t&lt; T$ and $0$ for $T &lt; t &lt; T_p$; thus $x(t)$ has a period of $T_p$.

Finding the fourier coefficients Ck via:

$$Ck = \frac{1}{T_p} \int_{-\infty}^{\infty} x(t) * e^\tfrac{-j2\pi kt}{T_p}$$

over 1 period, and thus we can represent x(t) as:

$$x(t) = \sum_k C_k e^\tfrac{j2\pi kt}{T_p}$$

and doing the Fourier transform we will get this form:

$$X(f) = \sum_k C_k \delta(f-kf_p)$$

which is discrete.

However if we consider $x(t)$ to be of this form:

$$x(t) = \sum_n \text{rect} \left(\frac{t-nT_p}{T} \right) $$

Applying fourier transform of $x(t)$ to get (in the form):

$$X(f) = \sum \text{sinc} * e^{-j*W}$$ -- (2)

where sinc() is due to the FT of rect and the e^(-j*W) comes out due to the time shifting property of FT.

Comparing X(f) in (1) and (2), we see that 1 is discrete and the other continuous.

However they come from the same x(t), so isn't this a contradiction?

Sorry for the long post.
However what is conceptually wrong with using say an infinite sum of time shifted rect functions to represent it? 


There is nothing conceptually wrong with it.  Fourier transforms decompose a signal into a sum of complex sinusoids, but you can also decompose a signal into many other things, which may be more useful in certain applications.  The Haar wavelet transform, for instance, breaks up a signal a sum of rectangular pulses:

 source

We use sinusoids in lots of applications because it makes the most sense in those applications.  For instance, why do we almost always decompose audio signals into sinusoids?  Because our cochleas do the same thing: