What Frequencies Emerge in a Sampled Signal of Mixed Sine Waves?

[skrat]

Homework Statement

A continuous signal $$x(t)=15\sin (10t)+5\sin(30t)+3\sin(50t)$$ is sampled with frequency $\Omega =4.0$ Hz. Which frequencies are present in the sampled signal?

Homework Equations

The Attempt at a Solution

No idea really. I seriously doubt it is that easy:

Frequencies 10,30,50 are all greater than 4 Hz, meaning all three frequencies and probably even more? th question is how to find the others?

 

[cpscdave]

Frequencies 10,30,50 are all greater than 4 Hz, meaning all three frequencies and probably even more?

Are you sure about that?? What does the Nyquist therom state?

 

[skrat]

Are you sure about that?? What does the Nyquist therom state?

It basically says that all frequencies above $\Omega /2$ are removed from the sampled signal to prevent aliasing of higher frequencies.

 

[cpscdave]

Unless you pass the signal through some filter before hand the frequencies are not removed by doing the sampling.
So this question is asking you to figure out what the aliased frequencies your going to find are.

 

[skrat]

Unless you pass the signal through some filter before hand the frequencies are not removed by doing the sampling.

Ok, good point.

Than I assume of course all frequencies from the input signal (10 Hz, 30 Hz and 50 Hz) and ..
$$f_0=10+2\pi n$$$$f_1=30+2\pi n$$$$f_2=50+2\pi n$$

Any others?

 

[cpscdave]

I think you've missed something here :)

As I always understood it Nyquist said basically that you can accurate represent frequencies upto HALF the sample rate. Anything that is higher than that will get aliased. (Have you covered frequency domain stuffs? these questions are super easy in the freq domain)
So you are not going to see in your question 10, 30 and 50 in the sampled signal. Instead you are going to see whatever those frequencies alias too.

 

[cpscdave]

take a look at http://www.ni.com/white-paper/3000/en/

 

[skrat]

We have covered frequency domain stuff but we haven't done much examples, therefore I am having some troubles now.

Ok doing Fourier transformation on my $x(t)$ leaves me with three frequencies $f_1=10$ Hz, $f_2=30$ Hz and $f_3=50$ Hz. Since all those frequencies are greater than the half of my sampling rate $\Omega =4$ Hz, my sample will contain only the aliased frequencies - if I can express myself that way.
And according to the link you provided $$f_a=|Rn-\Omega|$$ meaning
$$f_{a1}=|4\cdot 2 -10|Hz =2 Hz$$ $$f_{a2}=|4\cdot 7 -30|Hz =2 Hz$$ $$f_{a3}=|4\cdot 12 -50|Hz =2 Hz$$ which leads to a conclusion that despite the original system consists of three different frequencies, the sampled signal will have only one, which is 2 Hz.

 

[cpscdave]

Are you sure you use the SAMPLE rate for the calc?
*EDIT* Apparently you do :)

 

[skrat]

No I'm not sure, but it's what they do here: https://sem.org/PDF/Effects_of_aliasing.pdf on "Aliasing Example" on page 10/21.

 

[cpscdave]

No I'm not sure, but it's what they do here: https://sem.org/PDF/Effects_of_aliasing.pdf on "Aliasing Example" on page 10/21.

Yeah I should've read what I linked more closely! :)

 

[rude man]

First off, your frequencies are not 10, 20 and 30 Hz. They are 10/2π, 20/2π and 30/2π Hz. So back to square 1.
Then, each of the three frequencies in your signal are being mixed with a set of frequencies 0, 4, 8, 12, etc. Hz. That's because the sampler operating at 4 Hz can be represented in the frequency domain by a set of Dirac delta functions spaced 4 Hz apart, starting at dc (0Hz).

So what happens when you mix (multiply) 0 Hz with 10/2π Hz? 0 Hz with 20/2π Hz? 4 Hz with 10/2π Hz? Etc. etc. etc.
Remember that all signals are sinusoids.

 

[skrat]

Ok i am all confused now. I have absolutely no idea what I am doing or how this works. So, the results in post #8 are wrong?

 

[rude man]

Ok i am all confused now. I have absolutely no idea what I am doing or how this works. So, the results in post #8 are wrong?

Yes.
First thing you need to do is understand that a signal of the form sin wt has RADIAN frequency w, not Hz frequency.
I don't know what they taught you about the math for the sampler - it varies. But by assuming the "comb" spectrum I defined you'll get all the frequencies (of course there are an infinite number of them).

 

[skrat]

Ok, in that case the input signal has frequencies $\omega _1=10/2\pi$, $\omega _2=30/2\pi$ and $\omega _3=50/2\pi$.

Now just to make things clear.. http://www.eas.uccs.edu/~mwickert/ece2610/lecture_notes/ece2610_chap4.pdf equations 4.13 and 4.14 are the ones that really matter in my case right?

 

[rude man]

Ok, in that case the input signal has frequencies $\omega _1=10/2\pi$, $\omega _2=30/2\pi$ and $\omega _3=50/2\pi$.

No. ω is the RADIAN frequency, in rads/sec. f is ω/2π and is in Hz (CYCLES per second).

Now just to make things clear.. http://www.eas.uccs.edu/~mwickert/ece2610/lecture_notes/ece2610_chap4.pdf equations 4.13 and 4.14 are the ones that really matter in my case right?

4.13 which is in the frequency domain.
You're running Matlab so I assume you'll want to run Matlab to get your answer. But since you're only interested in the frequencies themselves rather than those plus the respective harmonic magnitudes, the comb transform I have indicated will yield the same set of frequencies.

 

[skrat]

Nono, I am not running Matlab. :D It's just the only useful link I found because I am shooting blinds here..
I have to do this on a paper. :)

If I understand you correctly: $\omega _0=10$ rad/s and accordingly $f_0=10/2\pi$ Hz.

 

[rude man]

Nono, I am not running Matlab. :D It's just the only useful link I found because I am shooting blinds here..
I have to do this on a paper. :)

If I understand you correctly: $\omega _0=10$ rad/s and accordingly $f_0=10/2\pi$ Hz.

Right.
OK, use 4.13 which gives you your frequencies. Remember l runs from 1 + infinity in integer steps.
So you have one f_s and three separate f₀ and you get all frequencies associated with each f₀.

 

[skrat]

But are all frequencies ok or only up to f_s/2?

 

[rude man]

But are all frequencies ok or only up to f_s/2?

No, all frequencies up to infinity.
If you post-filter the signal after sampling with an ideal low-pass filter with cutoff frequency f_s/2 you will restore the original, unsampled signal perfectly IF the sampling frequency is at least twice the highest frequency contained in the incoming signal (this statement is not rigorous but for you it will do). But this is not what the question asked, it asked what are the frequencies after sampling but without any post-filtering.

Post-filtering, Nyquist etc are different ideas. Example: you may know that a typical sampling frequency for say an iPod is around 44KHz. This is something more than twice the highest frequency audio input. Dig?