Calculating the Fourier Transform of a Digital Signal

[andrey21]

Ive just noticed I may have made an error. For the Fourier transform mentioned in post 18.
I concluded that:

8i sin2x + 4i sinx

could be written as:

4i (sin2x+sinx)

However I have missed a 2 out from sin2x.

So should it be:

4i (2sin2x + sinx)

marcusl or jbunni does this look correct to u?

 

[jbunniii]

Ive just noticed I may have made an error. For the Fourier transform mentioned in post 18.
I concluded that:

8i sin2x + 4i sinx

could be written as:

4i (sin2x+sinx)

However I have missed a 2 out from sin2x.

So should it be:

4i (2sin2x + sinx)

marcusl or jbunni does this look correct to u?

It looks right to me. I hadn't noticed the earlier error you pointed out.

 

[andrey21]

Thanks Jbunnnii I only just noticed it myself. For the first Fourier transform i sketched the magnitude of the function and it oscillates between 9 and 6.708. From -3pi to 3pi. This seem correct to u??

 

[andrey21]

Also would the magnitude of:
4i (2sin2x + sinx)
be

SQRT-16 .((2sin2x)^2 + (sinx)^2)

which would give me imaginary numbers so cannot be plotted.

 

[jbunniii]

Also would the magnitude of:
4i (2sin2x + sinx)
be

SQRT-16 .((2sin2x)^2 + (sinx)^2)

which would give me imaginary numbers so cannot be plotted.

No, that's not right.

What is the magnitude of 4i? It's 4, not -4.

 

[andrey21]

Ah i see so the magnitude would be:

4.SQRT((2sin2x)^2 + (sinx)^2)

 

[jbunniii]

Ah i see so the magnitude would be:

4.SQRT((2sin2x)^2 + (sinx)^2)

No, you left out a term.

The magnitude is

$$4 \sqrt{(2 \sin 2x + \sin x)^2}$$

When you expand the square, you get three terms, not two.

Or, you can simply express it as

$$4 |2 \sin 2x + \sin x|$$

which is probably easier to work with if all you want to do is plot it.

 

[andrey21]

Ok I thought magnitude was fo example:

SQRT((a)^2 +(b)^2) not SQRT (a+b)^2

SO to plot magnitude i must expand the brackets to give me three terms. Could u expand the brackets I am little confused.

 

[jbunniii]

Ok I thought magnitude was fo example:

SQRT((a)^2 +(b)^2) not SQRT (a+b)^2

The first form is not true in general, but only if a and b are the lengths of two perpendicular vectors and you want the magnitude of the sum of the vectors. (Think Pythagorean theorem.) For example, if z was a complex number, with real component a and imaginary component b, then

$$|z| = \sqrt{|z|^2} = \sqrt{|a + bi|^2} = \sqrt{(a + bi)(a - bi)} = \sqrt{a^2 + abi - abi + b^2} = \sqrt{a^2 + b^2}$$

Notice that the middle two terms, $abi$ and $-abi$, canceled each other. This happens precisely because $a$ and $bi$ are perpendicular to each other.

But you have a different situation. You have a real number,

$$2 \sin 2x + \sin x$$,

and you want its magnitude (absolute value).

In general, if $r$ is a real number, then its absolute value is

$$|r| = \sqrt{|r|^2} = \sqrt{r^2}$$

So in your case

$$|2 \sin 2x + \sin x| = \sqrt{(2 \sin 2x + \sin x)^2}$$

Now expand the square:

$$(2 \sin 2x + \sin x)^2 = 4\sin^2 2x + 4 \sin 2x \sin x + \sin^2 x$$

The middle term is the one you are missing.

 

[andrey21]

Rite ok I understand now how to calculate magnitude. All I have to do now is sub values form -3pi t0 3pi in order to gain the plot required.

Should I adopt the same technique for:

$$3\cdot|1 + 2\cos \omega + 2 \cos 2\omega |$$

 

[jbunniii]

Rite ok I understand now how to calculate magnitude. All I have to do now is sub values form -3pi t0 3pi in order to gain the plot required.

Should I adopt the same technique for:

$$3\cdot|1 + 2\cos \omega + 2 \cos 2\omega |$$

Yes, although as I mentioned before, it might be easier if you use the equivalent form

$$3 \cdot \left| \frac{\sin(2.5 \omega)}{\sin(0.5 \omega)} \right|$$

 

[andrey21]

Ye I will adopt ur equvalent form to plot with. With the magnitude obtained for the second Fourier transform where u expanded the square. Would it be easier to rewrite:

sin^(2) x using the chain rule to obtain 2sin(x)cos(x)

Just as I am using Excel to plot the magnitudes. And would that give:

4sin^(2) 2x = 8sin2xcos2x