# Power spectrum

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Thread starter 6 years ago
#1
Hello there

I wonder if anyone may be able to offer me some feedback regarding my answers to the following problem?

i) Frequency 2 and frequency 10? Is Frequency 2 the most important (first frequency)?

ii) Individual waves? (Scaling not precise).

New signal? (Scaling not precise)

I don't think I understand this concept. Please might anyone be able to advise me?

Thank you very much
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6 years ago
#2
From what I can see in the question (it's not easy to read the values off the axes) you have a waveform which is made up of a main wave of frequency 1 Hz. (Maybe 2Hz I can't really tell. Use whatever value you think it is.) This wave has power 40 units.
On top of that you have a wave of much smaller power (and amplitude) of frequency 10Hz. It looks like the power is 2 units. (Again, difficult to read the value from the axis. You decide what it is yourself.)

Remember that power depends on the square of the amplitude. So if you put the larger wave now as amplitude =1 the smaller wave will have an amplitude equal to the square root of the fraction it is of the power.

You now have to add the waves together.
Here is a diagram showing what happens when you take a main wave of frequency 1 unit (blue wave) and amplitude 1 unit and combine it with a wave of frequency 5 times this amount (brown wave) but with an amplitude of 0.1
The resultant is the black wave. (Click for enlarged view.)

You need to do this but using the values from the question.
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Thread starter 6 years ago
#3
(Original post by Stonebridge)
From what I can see in the question (it's not easy to read the values off the axes) you have a waveform which is made up of a main wave of frequency 1 Hz. (Maybe 2Hz I can't really tell. Use whatever value you think it is.) This wave has power 40 units.
On top of that you have a wave of much smaller power (and amplitude) of frequency 10Hz. It looks like the power is 2 units. (Again, difficult to read the value from the axis. You decide what it is yourself.)

Remember that power depends on the square of the amplitude. So if you put the larger wave now as amplitude =1 the smaller wave will have an amplitude equal to the square root of the fraction it is of the power.

You now have to add the waves together.
Here is a diagram showing what happens when you take a main wave of frequency 1 unit (blue wave) and amplitude 1 unit and combine it with a wave of frequency 5 times this amount (brown wave) but with an amplitude of 0.1
The resultant is the black wave. (Click for enlarged view.)

You need to do this but using the values from the question.
Thank you very much for your excellent explanation, it is really appreciated.

If the smaller wave has a power that is the square root of the fraction it is of the big wave's amplitude, does that mean that for the smaller wave in your example (amplitude 0.1), I should do: 0.1/1 = 0.1 x 100 = 10sqrt = 3.16 power of smaller wave?

I'm sorry if I've confused myself!
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6 years ago
#4
(Original post by Ggdf)
Thank you very much for your excellent explanation, it is really appreciated.

If the smaller wave has a power that is the square root of the fraction it is of the big wave's amplitude, does that mean that for the smaller wave in your example (amplitude 0.1), I should do: 0.1/1 = 0.1 x 100 = 10sqrt = 3.16 power of smaller wave?

I'm sorry if I've confused myself!
AS power depends on the square of the amplitude, if you had a wave of amplitude 1 unit and a wave of amplitude 2 units, then the power of the waves would be such that if wave 1 had a power of 1 unit, wave two would have a power of 4 units (22)

So the ratio of the powers of two waves is equal to the ratio of the squares of the amplitudes.

Conversely, the amplitudes of two waves are in the ratio of the square root of the powers.

In my case if the powers are in the ratio 10 to 1 the amplitudes are in the ratio √10 to √1
If the amplitudes are in the ratio 10 to 1 the powers are in the ratio 100 to 1
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Thread starter 6 years ago
#5
(Original post by Stonebridge)
AS power depends on the square of the amplitude, if you had a wave of amplitude 1 unit and a wave of amplitude 2 units, then the power of the waves would be such that if wave 1 had a power of 1 unit, wave two would have a power of 4 units (22)

So the ratio of the powers of two waves is equal to the ratio of the squares of the amplitudes.

Conversely, the amplitudes of two waves are in the ratio of the square root of the powers.

In my case if the powers are in the ratio 10 to 1 the amplitudes are in the ratio √10 to √1
If the amplitudes are in the ratio 10 to 1 the powers are in the ratio 100 to 1
Thank you

If I set the amplitude of the big wave to 1, would the ratio of the amplitudes of the big and small wave be 10 to 2 and the powers 100 to 4?

If I set the big wave to amplitude 1 in my drawing, should I leave the small wave as amplitude 2? (It would therefore now be the bigger wave than the big wave.)

Alternatively, should I draw the big wave at power 100 and the small wave at power 4?
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6 years ago
#6
For example
If the amplitudes are 10 and 2 and you set the big one to 1 then the small one would be 0.2
The big one is always 5 times bigger than the small one.
The small one is 1 fifth of the big one.

The question is asking you to draw amplitude not power.
Amplitude would be square root of power, as I explained before.
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Thread starter 6 years ago
#7
(Original post by Stonebridge)
For example
If the amplitudes are 10 and 2 and you set the big one to 1 then the small one would be 0.2
The big one is always 5 times bigger than the small one.
The small one is 1 fifth of the big one.

The question is asking you to draw amplitude not power.
Amplitude would be square root of power, as I explained before.
Ah, I see! Thank you very much for breaking it down so well for me .

I'll have a go at drawing the waves.
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Thread starter 6 years ago
#8
(Original post by Ggdf)
Ah, I see! Thank you very much for breaking it down so well for me .

I'll have a go at drawing the waves.
I had a go at drawing the individual waves:

Do you think this is OK? I realised after doing it that the smaller wave should really look the same as the bigger wave, but just of a smaller scale.

I struggled to draw the whole signal using a laptop mouse pad - it looks terrible! Have I however got the right idea?

Thank you, I'm really grateful for your help!
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Thread starter 6 years ago
#9
(Original post by Ggdf)
I had a go at drawing the individual waves:

Do you think this is OK? I realised after doing it that the smaller wave should really look the same as the bigger wave, but just of a smaller scale.

I struggled to draw the whole signal using a laptop mouse pad - it looks terrible! Have I however got the right idea?

Thank you, I'm really grateful for your help!
My apologies, I accidentally drew 11 waves within the signal instead of 10.
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6 years ago
#10
I think you might have intended to have 10 of the small waves corresponding to 1 of the large but your diagram has just 9 and a half of the small waves.

1 and 0.2 are correct for the amplitudes I gave but the question has power in the ratio 40 to 2 (it looks like 2, anyway) which is 20 to 1.
I only gave those values as an example.
This means the amplitudes in the question are in the ratio √20 to 1
So if you take the big wave as 1 the small one has amplitude 1/√20

Yes it's almost impossible to draw with a mouse.
It probably works better by hand on paper.
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Thread starter 6 years ago
#11
(Original post by Stonebridge)
I think you might have intended to have 10 of the small waves corresponding to 1 of the large but your diagram has just 9 and a half of the small waves.

1 and 0.2 are correct for the amplitudes I gave but the question has power in the ratio 40 to 2 (it looks like 2, anyway) which is 20 to 1.
I only gave those values as an example.
This means the amplitudes in the question are in the ratio √20 to 1
So if you take the big wave as 1 the small one has amplitude 1/√20

Yes it's almost impossible to draw with a mouse.
It probably works better by hand on paper.
Thank you. So the large wave would be set at amplitude 1, and the small wave at 0.22?
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Thread starter 6 years ago
#12
(Original post by Stonebridge)
I think you might have intended to have 10 of the small waves corresponding to 1 of the large but your diagram has just 9 and a half of the small waves.

1 and 0.2 are correct for the amplitudes I gave but the question has power in the ratio 40 to 2 (it looks like 2, anyway) which is 20 to 1.
I only gave those values as an example.
This means the amplitudes in the question are in the ratio √20 to 1
So if you take the big wave as 1 the small one has amplitude 1/√20

Yes it's almost impossible to draw with a mouse.
It probably works better by hand on paper.
Or would the small wave be 1/20? = amplitude 0.05?

Thank you
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6 years ago
#13
It's square root of power for the amplitude, so the small wave would be as I said in post #10 and you agreed in post #11.

1/√20 which is 0.22
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Thread starter 6 years ago
#14
(Original post by Stonebridge)
It's square root of power for the amplitude, so the small wave would be as I said in post #10 and you agreed in post #11.

1/√20 which is 0.22
Thank you very much, Stonebridge. I'll have a go at drawing it on paper and get back to you.

Best wishes
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Thread starter 5 years ago
#15
(Original post by Stonebridge)
It's square root of power for the amplitude, so the small wave would be as I said in post #10 and you agreed in post #11.

1/√20 which is 0.22
Hi Stonebridge

I'm sorry to bother you. I wonder if you may kindly be able to offer me some feedback on my answers to the following exercise (on a related topic to that of this thread)?

If you may be able to advise me at all, I would be enormously appreciative. I'm new to physics (I only started studying it 2 weeks ago) and am struggling a lot.

i) Is the dotted wave more selective in terms of frequency?

Q factor of dotted wave = resonant frequency/frequency width

Resonant frequency = 1.34 (roughly)
Frequency width = 1.45 - 1.19 (values roughly taken by method in below image)

Q factor of dotted wave therefore = 1.34/(1.45-1.19) = 5.15

Q factor of solid wave = resonant frequency/frequency width

Resonant frequency = 1.1 (roughly)
Frequency width = 1.16 - 1.03 (values roughly taken by method in below image)

Q factor of solid wave = 1.1/(1.16-1.03) = 8.46

Solid wave therefore has the higher Q factor?

ii) Distinguishes 1.1 to 1.7? From the peak of the solid wave to the trough of the dotted wave?

iii) One signal can be decomposed to its constituent frequencies like in Fourier decomposition. Unlike Fourier decomposition, we cannot see the power (how much of that frequency is in the signal/how important each frequency is to the signal), only the amplitude. We can however use the amplitude to find the power of the constituent frequencies.

Thank you very much.

Best wishes
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