Fourier Transform: given X(f), find x(t)

1. Jan 16, 2009

hastings

Hi everyone!
I'm not sure if I'm posting this question in the right section. Please don't be mad at me if I'm mistaken.

Calculate the value of the signal x(t), given its spectrum (see figure in attachment), at the time t=2/W.

Attempted solution:
I don't know how to write the module |X(f)| and phase Φ(f) of X(f). I suppose X(f) should be written in polar form:

$$X(f)=|X(f)|e^{j\Phi}$$

Maybe the module could be written like this:

$$|X(f)|=\frac{1}{16}f \mbox{ rect}_{4W}(f+4W) -\frac{1}{16}f \mbox{ rect}_{4W}(f-4W)$$

The 4W appearing in subscript of the signal rect, indicates the duration of rect.

The $$\pm 4W$$ inside the round brackets, tells us that rect is centered in $$\pm 4W$$

As to the PHASE, I don't have any idea on how to write it. Does it have anything to do with arctg (...)?

Attached Files:

• moduloFase.bmp
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2. Jan 17, 2009

hastings

Did I get the module correct?

3. Jan 19, 2009

hastings

Can someone try and help me? Please :(

4. Jan 19, 2009

Redbelly98

Staff Emeritus
I would write the amplitude as 2 separate expressions, depending on whether f is between -6W & -2W, or between 2W and 6W:

|X(f)| = ***** , -6W < f < -2W
|X(f)| = ***** , 2W < f < 6W

And similarly for the phase angle. Note that phase is a simple linear function of f.

That's right. Once you have that expression, do the integrals that give you the inverse transform and then you'll have x(t).

5. Jan 19, 2009

hastings

That's rather generic. Can you give me another hint for the amplitude?

I suppose it assumes the form of a slope: $$\phi(f)= m\cdot f$$

6. Jan 19, 2009

Redbelly98

Staff Emeritus
From the graph, the amplitude is a straight line within each region. You need to use the information in the graph to get the equation of the line in each region.

I.e.,

|X(f)| = (slope · f) + intercept

and you find the values of "slope" and "intercept" from the graph.

Yes, and by looking at the numbers given in the graph, you can figure out what the slope "m" is.

7. Jan 20, 2009

hastings

So you're saying the Amplitude should be something like:

$$\frac{1}{16}f +\frac{3}{8}W \mbox{, for -6W \leq f \leq -2W}$$

$$-\frac{1}{16}f +\frac{3}{8}W \mbox{, for 6W \leq f \leq 2W} \right.$$

As for the PHASE, it should be:
$$\phi(f)=-\frac{\pi}{72W}f$$

Which by the way means that at f=-2W
$$\phi(-2W)=\frac{\pi}{36}$$

If I'm right about the phase, how do I express the fact that the phase slope is sort of interrupted between -2W and 2W?

8. Jan 20, 2009

Redbelly98

Staff Emeritus

Since the amplitude is zero in that region, the phase is not relevant.

9. Jan 23, 2009

hastings

Sorry I'm late in replying, I was little busy.

So X(f) should be:

$$X(f)= \left\{ \begin{array}{ll} (\frac{1}{16}f +\frac{3}{8}W)e^{-j(-\frac{2\pi}{144W}f)} &, \text{ for -6W \leq f \leq -2W} \\ (-\frac{1}{16}f +\frac{3}{8}W)e^{-j(-\frac{2\pi}{144W}f)} &,\text{ for 2W \leq f \leq 6W} \end{array} \right.$$

I wrote the exponentials in the form $$e^{-j2\pi t_0 f}$$ since an exponential in the frequency domain indicates a time shift, might be useful later on.

Ok, I guess I'm ready to antitransform the equations of the system above.
Let's see the first equation:
$$\mathcal{F}^{-1}\biggl[ \biggl( \frac{1}{16}f +\frac{3}{8}W \biggr) e^{j(\frac{2\pi}{144W}f)} \bigg|_{-6W}^{-2W} \quad \biggr]= \int_{-6w}^{-2W} \biggl[ \biggl(\frac{1}{16}f +\frac{3}{8}W \biggr) e^{j(\frac{2\pi}{144W}f)} \biggr] e^{j2\pi f t}df$$

Is the integral right? Can I go on?

10. Jan 23, 2009

Redbelly98

Staff Emeritus
Looks right so far. I forget how the +/- signs work in the exponential terms, but trust that you have looked up the formulas and done that part correctly.

11. Jan 24, 2009

hastings

I thought of breaking the equation in 2 pieces:
$$\underbrace{\mathcal{F}^{-1}[ \frac{1}{16}f e^{j2\pi \frac{1}{144W}f }]}_{\text{1st addend}}+ \underbrace{\mathcal{F}^{-1}[\frac{3}{8}We^{j2\pi\frac{1}{144W}f } ]}_{\text{2nd addend}}$$

I know that $$\mathcal{F}[\delta(t-t_0) ]=e^{j2\pi t_0 f}$$ therefore the 2nd addend should be

$$\mathcal{F}^{-1}[\frac{3}{8}We^{j2\pi\frac{1}{144W}f } ] =\frac{3W}{8}\delta(t+\frac{1}{144W})$$
However I don't know how to evaluate this last expression for $$-6W \leq f \leq -2W$$

As to the 1st addend $$\mathcal{F}^{-1}[ \frac{1}{16}f e^{j2\pi \frac{1}{144W}f }] = \frac{1}{16}\int_{-6W}^{-2W} f e^{j2\pi \frac{1}{144W}f } e^{j2\pi t f} df= \frac{1}{16}\int_{-6W}^{-2W} f e^{j2\pi (\frac{1}{144W}+t) f }df$$

I looked up the integration by part formula:
$$\int f(x)g'(x) dx= f(x)g(x) -\int f'(x) g(x) dx$$

To simplify notations I called $$a=j2\pi(\frac{1}{144W}+t)$$ since I'm integrating in f.

Here's the work out:
$$\int f \cdot e^{af}df= f \frac{e^{af}}{a} -\int \frac{e^{af}}{a}df = f \frac{e^{af}}{a} - \frac{e^{af}}{a^2}$$

where $$f(x)=f \text{, } g'(x)= e^{af} \text{ and } g(x)= \frac{e^{af}}{a}$$

Now, introducing the extremes of integration, the solution should be:
$$\frac{1}{16}\bigg[ f \frac{e^{af}}{a} - \frac{e^{af}}{a^2} \bigg]_{-6W}^{-2W} = \frac{1}{16} \bigg\{ e^{-6Wa} \bigg[\frac{6W}{a}+\frac{1}{a^2}\bigg] -e^{-2Wa}\bigg[\frac{2W}{a} + \frac{1}{a^2}\bigg] \bigg\}$$

I'm sure there's some mistake somewhere, please correct me if I'm wrong.

12. Jan 25, 2009

Redbelly98

Staff Emeritus
... for -∞ < f < ∞.

But that is not what we have, since your expression holds only for
-6W < f < -2W​
and is zero outside that range. So it is not a δ-function in the time domain.

Can you do the integral for the inverse transform? Your expressions are of the forms
A ∫ ecx dx

and
A ∫ x ecx dx

These are standard integrals, which you can look up here:
http://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions

13. Jan 25, 2009

hastings

Regarding the 1st addend, is it correct?

where

Meanwhile I'll work out the 2nd addend.

Last edited: Jan 25, 2009
14. Jan 25, 2009

Redbelly98

Staff Emeritus
Looks good so far.