What is the meaning of $Ta_k$ of fourier series or transform?

What is the meaning of $Ta_k$ of fourier series or transform? I am taking a course on signal and systems.

In 286 page of my textbook, it says that as T becomes arbitrarily large the original periodic square wave approaches a rectangular pulse. Also it says that all that remains in the time domain is an aperiodic signal corresponding to one period of the square wave. (textbook: siganls and systems sencond edition, author: oppenheim)

I have a difficulty understanding this.. I can't connect this idea with fourier transform.

• Slide 4-6: T is the length (in time) of the period over which you are calculating the Fourier transform. $a_k$ is the $k$th Fourier coefficient (the "amount" of frequency $k$.) In the graph at the bottom they are varying $k$ on the $x$-axis and showing the value of $a_k$ on the $y$-axis. They are multiplying the values of $a_k$ by $T$ because as $T$ increases all the Fourier coefficients get smaller. (They are $1/T$ times an integral.) And then in slide 7 they take $lim_{T\rightarrow \infty} T a_k$, which would be 0 if we didn't multiply back by $T$. – Wandering Logic May 8 '13 at 11:56
• I don't know what is on page 286 of (what book are you using: author, title, year?) But in the linked slides a square wave is not approaching a rectangular pulse. Rather there is a rectangular pulse of width $S$ that is being repeated at interval $T$. As $T$ lengthens to $\infty$ there is just a single pulse (still of width $S$). – Wandering Logic May 8 '13 at 12:03
• Sorry,, I edited my original post. – jakeoung May 8 '13 at 12:05
• $Ta_k \to X(jw)$. Is this fact an accidental result? or is there a meaning? – jakeoung May 8 '13 at 12:22
• I view it as the definition of $X(j\omega)$. As $T\rightarrow\infty$ the number of frequencies goes from countably infinite to uncountably infinite (i.e., continuous), and rather than a function repeating at intervals $T$ you have a single function that extends (non repeating) to $\infty$. – Wandering Logic May 8 '13 at 12:29

The idea is that a Fourier series is only defined for periodic signals. In the discussion in the linked slides, the author is considering a rectangular pulse train with period $T$. That is, a pulse of width $2S$ repeats periodically with a spacing of $T$ between them. The pulses are therefore centered at:
$$[\ldots, -2T, T, 0, T, 2T, \ldots ]$$
Now, consider what happens as $T \to \infty$: in the limit, the only pulse that remains is the one centered at zero; the others are infinitely far away. When the author makes the claim that:
$$\lim_{T\to \infty} T a_k = E(\omega)$$
The multiplication by $T$ in the limit is to account for the differences in definition between the Fourier series and Fourier transform: the series representation typically has a factor of $\frac{1}{T}$, while the transform does not. I don't know that there is a lot of insight to be gained via this analysis, but it shows that the series and transform representations are intimately related.