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 November 10th, 2013, 02:23 PM #1 Senior Member   Joined: Jan 2009 Posts: 344 Thanks: 3 What are the conditions for the existence of x(t) in freq The Fourier transform of some signal x(t) in continuous time is given by the following equation $X(f)=\int_{-\infty}^{+\infty}x(t)e^{-j2\pi f t}dt$ What are the conditions for the Fourier transform of an arbitrary function or signal x(t) to exist in the frequency domain. Why doesn't $x(t)= cos(2\pi t f_0)$ exist in the frequency domain? Where does the need for delta functions arise? and why are they used to find the fourier transform of sinusoids like sine and cosine? Please provide an example
November 10th, 2013, 10:06 PM   #2
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Re: What are the conditions for the existence of x(t) in fre

Hi sivela,

Quote:
 What are the conditions for the Fourier transform of an arbitrary function or signal x(t) to exist in the frequency domain.
I do not know the answer to this question. Looking in my old textbook, functions that satisfy the Dirichlet conditions are called "energy signals", and as you might expect, there are a number of these types of signals that are calculated as examples and more that are listed in a table. The mathematical operations such as differentiation, integration, delay, modulation, etc. are also shown as examples or in a table.

The (somewhat dated) book is Methods of Signal and System Analysis by Cooper and McGillem (Holt, Rinehart, and Winston, Inc., 1967) and I highly recommend it. I have about 3 books that cover this type of material and this one is by far the best, as evidenced by the fact that it is falling apart due to frequent use over the last 30+ years. I googled this book and it looks like it is available.

After the section on "energy signals", the book covers signals that do not satisfy the Dirichlet conditions ("power signals"), such as a constant, the signum function, a cosine, etc. The book states:

"Many such functions can nevertheless be handled by allowing the Fourier transform to contain impulses, or, in some cases, higher order singularity functions. The procedure can be put on a rigorous mathematical basis by means of the theory of generalized functions ..."

Edit: I added the word 'basis' as I accidentally left it out.

For the preceding statement the following reference was given:

A.H. Zemanian, Distribution Theory and Transform Analysis, McGraw-Hill, 1965.

Let's work one of the examples, finding the inverse Fourier Transform of f(t)=1.

$\mathcal{F}(1)=\lim_{\alpha \to 0} \int_{-\infty}\,^{\infty}e^{-\alpha |t|}e^{-jwt}\ dt$

$\mathcal{F}(1)=\lim_{\alpha \to 0}\left[\int_{-\infty}\,^0 e^{(\alpha-jw)t}\ dt \ + \ \int_0\,^{\infty} e^{-(\alpha+jw)t}\ dt\right]$

$\mathcal{F}(1)=\lim_{\alpha \to 0}\left[\left. \frac{e^{(\alpha-jw)t}}{\alpha - jw}\right|_{-\infty}^0 \ + \ \left. \frac{e^{-(\alpha+jw)t}}{\alpha+jw}\right|_0^{\infty}\right]$

$\mathcal{F}(1)=\lim_{\alpha \to 0}\left[\frac{1}{\alpha -jw}+\frac{1}{\alpha+jw}\right]$

$\mathcal{F}(1)=\lim_{\alpha \to 0}\left[\frac{2 \alpha}{\alpha^2+w^2\right]$

The limit is 0 except when $\ w=0 \$. When $\ w=0 \$ an intermediate form can be evaluated by L'Hopital's rule to be

$\lim_{\alpha \to 0} \left[\frac{2}{2\alpha}\right]=\infty \$ (hence a delta function).

The area under the curve (the weight of the delta function) is:

$\text{Area}=\int_{-\infty}\,^{\infty}\frac{2\alpha}{\alpha^2+w^2} \ dw=\left. 2\arctan \left(\frac{w}{a}\right) \right |_{-\infty}^{\infty}$

$\text{Area}=2\left[\frac{\pi}{2}+\frac{\pi}{2}\right]=2\pi$

So $\ \mathcal{F}(1)=2\pi\delta(w)$

Quote:
 Why doesn't $x(t)= cos(2\pi t f_0)$ exist in the frequency domain?
It does. Using the result given above, and the modulation property:

$\mathcal{F}(e^{jw_0t}\cdot 1)=\delta(w-w_0) \$ Edit: There was an errant jw coefficient in front of the delta function.

Since $\ \cos(wt)= \frac{e^{jw_0t}}{2}+\frac{e^{-jw_0t}}{2}$

$\mathcal{F}(\cos(wt))=\pi\left[\delta(w-w_0)+\delta(w+w_0)\right]$

November 11th, 2013, 12:24 AM   #3
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Re: What are the conditions for the existence of x(t) in fre

Thanks for the response.
Quote:
 Originally Posted by jks It does. Using the result given above, and the modulation property: $\mathcal{F}(e^{jw_0t}\cdot 1)=jw\delta(w-w_0)$ Since $\ \cos(wt)= \frac{e^{jw_0t}}{2}+\frac{e^{-jw_0t}}{2}$ $\mathcal{F}(\cos(wt))=\pi\left[\delta(w-w_0)+\delta(w+w_0)\right]$
Mr.Dilip Sarwate doesn't think so. http://dsp.stackexchange.com/questio...osine-and-sine at least in the usual sense.

 November 11th, 2013, 04:01 PM #4 Senior Member     Joined: Jul 2012 From: DFW Area Posts: 635 Thanks: 96 Math Focus: Electrical Engineering Applications Re: What are the conditions for the existence of x(t) in fre Hmm, I do not see any disagreement between what I stated and what Mr. Sarwate stated. We both agree that a cosine wave does not meet the normal conditions (presumably Dirichlet) but using other means we arrive at the same answer for the FT of a cosine wave. But I cannot give you any absolute rules on what functions do and do not have a FT. I gave a reference that might be what you are looking for but I do not know for sure, or if it is even available. Such a reference might take some effort to find.

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