C_0 coefficient of Complex Fourier transforms

[Wminus]

Mod note: Moved from technical math section, so no template was used.
Hey! So the complex Fourier transform of the square wave

$$
f(x) = \begin{cases}
2 & x \in [0,2] \\
-1 & x \in [2,3] \\
\end{cases}, \space \space f(x+3) = f(x)$$

is $C_k = \frac{3j}{2 \pi k}( e^{-j \frac{4 \pi k}{3}} -1)$, correct? However, setting $k = 0$ and using L' Hopital rule I get that $C_0 = 2$, while it should be equal to 1 since that's the average of the function.

Where is the error? My expression for $C_k$ is correct, right?

 

[Ray Vickson]

Mod note: Moved from technical math section, so no template was used.
Hey! So the complex Fourier transform of the square wave

$$
f(x) = \begin{cases}
2 & x \in [0,2] \\
-1 & x \in [2,3] \\
\end{cases}, \space \space f(x+3) = f(x)$$

is $C_k = \frac{3j}{2 \pi k}( e^{-j \frac{4 \pi k}{3}} -1)$, correct? However, setting $k = 0$ and using L' Hopital rule I get that $C_0 = 2$, while it should be equal to 1 since that's the average of the function.

Where is the error? My expression for $C_k$ is correct, right?

Are actually doing the Fourier transform, or do you mean the Fourier series? These are very different things.

 

[Wminus]

Are actually doing the Fourier transform, or do you mean the Fourier series? These are very different things.

I'm sorry, I mixed them up. I meant I was finding the the complex Fourier series of $f(x)$ , with the $C_k$s being its coefficients.

 

[Ray Vickson]

I'm sorry, I mixed them up. I meant I was finding the the complex Fourier series of $f(x)$ , with the $C_k$s being its coefficients.

OK, so to clarify: are you writing the Fourier series as
$$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inax}$$
for an appropriate value of the parameter $a$? If so, what is your $a$, and what is the formula you are using to compute the $c_n$?

 

[Wminus]

OK, so to clarify: are you writing the Fourier series as
$$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inax}$$
for an appropriate value of the parameter $a$? If so, what is your $a$, and what is the formula you are using to compute the $c_n$?

Sorry, it appears my OP was really dang. Yeah, I am using that form and I use equation 3 from here http://www.thefouriertransform.com/series/complexcoefficients.php to calculate the Fourier coefficients. What I don't get is why my C_0 coefficient is twice as large as it's supposed to be..

 

[Ray Vickson]

Sorry, it appears my OP was really ****. Yeah, I am using that form and I use equation 3 from here http://www.thefouriertransform.com/series/complexcoefficients.php to calculate the Fourier coefficients. What I don't get is why my C_0 coefficient is twice as large as it's supposed to be..

That's odd: when I do the integral, I get $c_0 = \frac{1}{3} \int_0^3 f(t) \, dt = 1$, not the value 2 that you claim. I have not checked your other values for $n \neq 0$.

 

[Wminus]

Yes, I get that too. That is the correct value. My problem is that I don't get this result with the expression of the $C_k$ from the OP.

I'll try to explain my problem as best I can:

(1) I got my $C_k=\frac{3j}{2 \pi k}( e^{-j \frac{4 \pi k}{3}} -1)$ from the definition of the complex-fourier series, and that definition states: $C_k = \frac{1}{T} \int_0^T f(t) e^{-j k \omega t} dt, \space \omega = 2 \pi /T$. This is defined for all $k \geq 0$, right?

(2) Further, by setting $k=0$, we get that $C_0 = \frac{1}{T} \int_0^T f(t) e^{-j k \omega t} dt$, which is $C_0 =1$ in our case.

However, from my earlier expression (1), I have that setting $k=0$ gives $lim_{k\to0} C_k = lim_{k\to0} \frac{3j}{2 \pi k}( e^{-j \frac{4 \pi k}{3}} -1) = 2 \neq 1$. This isn't what I get in (2), and I fail to understand this discrepancy!

 

[Ray Vickson]

Yes, I get that too. That is the correct value. My problem is that I don't get this result with the expression of the $C_k$ from the OP.

I'll try to explain my problem as best I can:

(1) I got my $C_k=\frac{3j}{2 \pi k}( e^{-j \frac{4 \pi k}{3}} -1)$ from the definition of the complex-fourier series, and that definition states: $C_k = \frac{1}{T} \int_0^T f(t) e^{-j k \omega t} dt, \space \omega = 2 \pi /T$. This is defined for all $k \geq 0$, right?

(2) Further, by setting $k=0$, we get that $C_0 = \frac{1}{T} \int_0^T f(t) e^{-j k \omega t} dt$, which is $C_0 =1$ in our case.

However, from my earlier expression (1), I have that setting $k=0$ gives $lim_{k\to0} C_k = lim_{k\to0} \frac{3j}{2 \pi k}( e^{-j \frac{4 \pi k}{3}} -1) = 2 \neq 1$. This isn't what I get in (2), and I fail to understand this discrepancy!

The problem seems to be that for $k \neq 0$ there are different ways of writing $C_k$ that are all the same as your expression when $k$ is an integer. For example, when I compute $C_k$ for general $k > 0$ using Maple, I get
$$C_k = \frac{i}{2 \pi k} \left( -2 + 3e^{-4 \pi i k/3} - e^{-2 \pi i k} \right)$$
This expression has limit $\lim_{k \to 0} C_k = 1$, as it should (from continuity of $e^{- i k 2 \pi/3}$). When $k$ is an integer, it reduces to your previous expression for $C_k$, but that expression does not apply when $k$ is fractional.

 

[Wminus]

The problem seems to be that for $k \neq 0$ there are different ways of writing $C_k$ that are all the same as your expression when $k$ is an integer. For example, when I compute $C_k$ for general $k > 0$ using Maple, I get
$$C_k = \frac{i}{2 \pi k} \left( -2 + 3e^{-4 \pi i k/3} - e^{-2 \pi i k} \right)$$
This expression has limit $\lim_{k \to 0} C_k = 1$, as it should (from continuity of $e^{- i k 2 \pi/3}$). When $k$ is an integer, it reduces to your previous expression for $C_k$, but that expression does not apply when $k$ is fractional.

Kind of interesting how all this weird stuff starts happening when you start dividing by zero... When I derived $C_k$ I made no assumptions what-so-ever, so both mine and your $C_k$ should be mathematically equivalent to $C_k = \frac{1}{T} \int_0^T f(t) e^{-j k \omega t} dt, \space \omega = 2 \pi /T$. Yet despite this, they produce different answers. Strange.

But at the end of the day L'Hopital's rule is used to evaluate these limits and I've only ever applied it when the variable of differentiation is continuous, while here $k$ is discrete.

 

[Ray Vickson]

Kind of interesting how all this weird stuff starts happening when you start dividing by zero... When I derived $C_k$ I made no assumptions what-so-ever, so both mine and your $C_k$ should be mathematically equivalent to $C_k = \frac{1}{T} \int_0^T f(t) e^{-j k \omega t} dt, \space \omega = 2 \pi /T$. Yet despite this, they produce different answers. Strange.

But at the end of the day L'Hopital's rule is used to evaluate these limits and I've only ever applied it when the variable of differentiation is continuous, while here $k$ is discrete.

Well, my $C_k$ above IS $\frac{1}{T} \int_0^T f(t) e^{-j \omega k t}\, dt$ without any assumptions about $k$. However, the moment you make $k$ an integer, you can (and often will) make $e^{-2 \pi i k} = 1$ automatically; for example, most (or, at least, many) computer algebra systems will make that simplification right away, and that will be enough to change the limit.

Basically, it amounts to saying that $C_k = N(k)/k$, and that we can have two different numerator functions $N_1(k)$ and $N_2(k)$ such that $N_1(k) = N_2(k)$ for all integers $k$, but with different derivatives $N_i'(k)$ at the integer values of $k$. That will make the limits of $N_1(k)/k$ and $N_2(k)/k$ come out different as $k \to 0$.

 

[Wminus]

Well, my $C_k$ above IS $\frac{1}{T} \int_0^T f(t) e^{-j \omega k t}\, dt$ without any assumptions about $k$. However, the moment you make $k$ an integer, you can (and often will) make $e^{-2 \pi i k} = 1$ automatically; for example, most (or, at least, many) computer algebra systems will make that simplification right away, and that will be enough to change the limit.

Basically, it amounts to saying that $C_k = N(k)/k$, and that we can have two different numerator functions $N_1(k)$ and $N_2(k)$ such that $N_1(k) = N_2(k)$ for all integers $k$, but with different derivatives $N_i'(k)$ at the integer values of $k$. That will make the limits of $N_1(k)/k$ and $N_2(k)/k$ come out different as $k \to 0$.

Ah yes, you're right. I think I also set $e^{-2 \pi i k} = 1$ in my calculations, unwittingly making the mathematical assumption that the $k$s are integers, and thus changed the limit that comes from L'Hopital. I didn't realize the assumption I made because I thought it was a given that the $k$s are integers, but just looking at the definition of $C_k$ I see it wasn't.

Thanks man, I really appreciate that you cleared this up for me. :)