How to find whether signal x(n)=sin(3pi/4)+sin(pi/3) is periodic signal or not?

Discussion in 'Homework Help' started by avan93, Oct 29, 2013.

1. avan93 Thread Starter New Member

Oct 29, 2013
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0
How to find whether signal x(n)=sin(3pi/4*n)+sin(pi/3*n) is periodic signal or not?

my solution:
x(n+N)=sin(3pi/4*n+N)+sin(pi/3*n+N)
=sin(3pi/4*n)cos(N)+ cos(3pi/4*n)sin(N)+sin(pi/3*n)cos(N)+cos(pi/3*n)sin(N)
=sin(N){cos(3pi/4*n)+cos(pi/3*n)}+cos(N){sin(3pi/4*n)+sin(pi/3)}
at N=2npi
sin(2npi)=0
cos(2npi)=1
= sin(3pi/4*n)+sin(pi/3*n)
=x(n)
hence periodic signal
period=2npi

Last edited: Oct 29, 2013
2. anhnha Active Member

Apr 19, 2012
776
48
Then:

$x(n) = \frac{ \sqrt{2} }{2} + \frac{ \sqrt{3} }{2}$

I think, maybe, this is not what you meant.
What are you missing here?

3. WBahn Moderator

Mar 31, 2012
18,085
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x(n)=sin(3pi/4)+sin(pi/3)

sin(3pi/4) = 1 / √2
sin(pi/3) = √3 / 2

Thus

x(n) = (√2 + √3)/2 ≈ 1.573

x(n+N) = x(n) for N=1, hence it's periodic.

Interestingly, √2+√3 = ∏ within about 0.1%. No significance, just interesting.

Oct 29, 2013
5
0

Oct 29, 2013
5
0

Apr 19, 2012
776
48

7. avan93 Thread Starter New Member

Oct 29, 2013
5
0
how to solve it??

8. Danm1 Member

Jul 19, 2010
55
7
isn't there some therom that says any sum of periodic signals is also periodic?
Something needed for the Fourier Transform to work I think.

9. WBahn Moderator

Mar 31, 2012
18,085
4,917
Engineering and mathematics require pretty careful attention to detail. What you have here is:

x(n+N)=sin(3pi/4*n+N)+sin(pi/3*n+N)

Which is actually:

$
x[n+N] \, = \, \sin $$\, \( \frac{3 \pi}{4}n$$ \, + \, N \) \, + \, \sin $$\, \( \frac{\pi}{3}n$$ \, + \, N \)
$

Is this really what you meant?

Don't be sloppy and rely on others to correctly read your mind and infer what you meant to say instead of them being able to rely on you correctly saying what you mean.

10. WBahn Moderator

Mar 31, 2012
18,085
4,917
So would something that had a period of √2 added to something that had a period of √3 be periodic?

In order for the sum to be periodic, there must be a period that they have in common -- which means that when one signal goes through K1 periods, the other signal goes through K2 periods where K1 and K2 are integers.

Hint -- right there is the key you need to focus on.

11. Danm1 Member

Jul 19, 2010
55
7
yes any periodic waveform can be composed by a sum of sinusoidal warforms.

So take a square wave as an extreme example, this can be reduced to a sum of sine waves. Many of the DSP analysis techniques are based on this, e.g the powerful FFT operations.

For a quick discussion of this concept, see the entry for "Square Wave" on Wikipedia.

Last edited: Oct 29, 2013
12. WBahn Moderator

Mar 31, 2012
18,085
4,917
There is a fundamental difference between the following statements:

Any periodic waveform may be composed of a sum of sinusoids.

and

Any waveform composed of a sum of sinusoids is periodic.

You're making the same logic error that you would if you took the statement, "Any battery is a DC voltage supply," and concluded from that that "Any DC voltage supply is a battery."

13. Danm1 Member

Jul 19, 2010
55
7
I'm not trying to offer up my comments as a formal proof, but I would venture to say that I believe the sum of those two periodic sine wave formulas would produce a periodic wave form.

Although I have not spent the time to look it up, the math related to the proof might be through representing the sine waves as a Taylor series, and proceeding from there. I confess that I havn't done this level of math for a few years, but only suggesting that if a proof is needed that this might be a good direction to go.

14. WBahn Moderator

Mar 31, 2012
18,085
4,917
Consider what I said a while back:

In order for the sum to be periodic, there must be a period that they have in common -- which means that when one signal goes through K1 periods, the other signal goes through K2 periods where K1 and K2 are integers.

Hint -- right there is the key you need to focus on.

This means that you need K1*√2 = K2*√3 where K1 and K2 are integers.

This means that you need

K1 = K2*√(3/2)

Is this possible? If not, then the sum of these two periodic waveforms is not periodic. Conversely, if the sum of these two periodic waveforms is periodic, the you must be able to find two integers, K1 and K2, that satisfy this relationship.