co-relation between frequency and rate of change

Sirs,

I would like to get a clarification with respect to the frequency component
and rate of change. 

A system I am trying to analyze is characterized by the rate of change of
its output. But I am not sure if rate of change has any direct co-relation
with frequency and whether DSP can help in this analysis.

With this background, I have the following question for the following sine
signals. The sample points are same for both the signals but you can notice
the amplitude varies

sine_1 = [0, 2, 3, 4, 2, 0, 2, 3, 4, 2, 0]
sine_2 = [0, 3, 4, 5, 3, 0, 3, 4, 5, 3, 0]

As far as the frequency is concerned, they are the same but the amplitude
at various sample points vary. Question is, as far as frequency analysis is
concerned (e.g. FFT), would both of them show as having same freqency
component?

As I have mentioned above, for the problem I am working on, the rate of
change matters. So, the second signal has more rate of change.

Can someone help me understand.

THanks, manish

	 

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>Sirs,
>
>I would like to get a clarification with respect to the frequency
component
>and rate of change. 
>
>A system I am trying to analyze is characterized by the rate of change of
>its output. But I am not sure if rate of change has any direct
co-relation
>with frequency and whether DSP can help in this analysis.
>
>With this background, I have the following question for the following
sine
>signals. The sample points are same for both the signals but you can
notice
>the amplitude varies
>
>sine_1 = [0, 2, 3, 4, 2, 0, 2, 3, 4, 2, 0]
>sine_2 = [0, 3, 4, 5, 3, 0, 3, 4, 5, 3, 0]
>
>As far as the frequency is concerned, they are the same but the amplitude
>at various sample points vary. Question is, as far as frequency analysis
is
>concerned (e.g. FFT), would both of them show as having same freqency
>component?
>
>As I have mentioned above, for the problem I am working on, the rate of
>change matters. So, the second signal has more rate of change.
>
>Can someone help me understand.
>
>THanks, manish
>
>	 
>
>_____________________________		
>Posted through www.DSPRelated.com
>

Hmmm, ignoring that neither of those signals is a sine wave...  What is the
derivative of a sine wave and how does that relate to frequency, phase and
amplitude?	 

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On Tue, 25 Jun 2013 07:46:18 -0500, manishp wrote:

> Sirs,
> 
> I would like to get a clarification with respect to the frequency
> component and rate of change.
> 
> A system I am trying to analyze is characterized by the rate of change
> of its output. But I am not sure if rate of change has any direct
> co-relation with frequency and whether DSP can help in this analysis.
> 
> With this background, I have the following question for the following
> sine signals. The sample points are same for both the signals but you
> can notice the amplitude varies
> 
> sine_1 = [0, 2, 3, 4, 2, 0, 2, 3, 4, 2, 0]
> sine_2 = [0, 3, 4, 5, 3, 0, 3, 4, 5, 3, 0]
> 
> As far as the frequency is concerned, they are the same but the
> amplitude at various sample points vary. Question is, as far as
> frequency analysis is concerned (e.g. FFT), would both of them show as
> having same freqency component?
> 
> As I have mentioned above, for the problem I am working on, the rate of
> change matters. So, the second signal has more rate of change.
> 
> Can someone help me understand.

First, if those really were sine waves of different amplitudes, then an 
FFT would show a frequency response of identical shape and different 
amplitudes, from which you could infer rate of change.

Second, "DSP" and "FFT" are not synonyms.  There's a lot that you can do 
with digital signal processing without using the FFT, and much of it is 
stuff that you cannot reasonably do _with_ the FFT.  So if your approach 
to DSP problems (or signal processing problems in general) is "I'll get 
this into the frequency domain, now what do I do?" then your approach is, 
in general, wrong.

Third, does the word "differentiator" mean anything to you?

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Tim Wescott <tim@seemywebsite.really> wrote:
> On Tue, 25 Jun 2013 07:46:18 -0500, manishp wrote:

(snip)
>> I would like to get a clarification with respect to the frequency
>> component and rate of change.
 
(snip)
>> Can someone help me understand.
 
> First, if those really were sine waves of different amplitudes, then an 
> FFT would show a frequency response of identical shape and different 
> amplitudes, from which you could infer rate of change.
 
> Second, "DSP" and "FFT" are not synonyms.  There's a lot that you can do 
> with digital signal processing without using the FFT, and much of it is 
> stuff that you cannot reasonably do _with_ the FFT.  So if your approach 
> to DSP problems (or signal processing problems in general) is "I'll get 
> this into the frequency domain, now what do I do?" then your approach is, 
> in general, wrong.

I agree if you mean the FFT algorithm, or the DFT for that matter.
But DSP pretty much requires understanding time domain, frequency
domain, and the connection between them. ASP you might be able to
do entirely in the frequency domain, computing the R, L, and C
you need to build RLC filters.

There might be some people who write FFT when they actually
mean the continuous Fourier (integral) transform. Less characters
to type, for one.
 
> Third, does the word "differentiator" mean anything to you?

-- glen

manishp wrote:
> Sirs,
>
> I would like to get a clarification with respect to the frequency component
> and rate of change.
>
> A system I am trying to analyze is characterized by the rate of change of
> its output. But I am not sure if rate of change has any direct co-relation
> with frequency and whether DSP can help in this analysis.

Paraphrasing some instructors I had as an undergrad E.E. student, 
"When confused, go back to first principles."

>
> With this background, I have the following question for the following sine
> signals. The sample points are same for both the signals but you can notice
> the amplitude varies
>
> sine_1 = [0, 2, 3, 4, 2, 0, 2, 3, 4, 2, 0]
> sine_2 = [0, 3, 4, 5, 3, 0, 3, 4, 5, 3, 0]
> [snip]



You are confusing yourself labeling those as "sine...". Instead 
call them Y1 and Y2.

Plot Y1 and Y2 on the same set of axes.
Plot a sine-wave of matching period whose amplitude equals 4 on 
the same plot.

On a separate plot:
Plot Y1 vs Y2
Plot (4*(sin(q))) vs (5*(sin(q)))

What is a "basis set"/"basis vector"?

Suggested reading
https://en.wikipedia.org/wiki/Fourier_series
https://en.wikipedia.org/wiki/Fourier_transform
https://en.wikipedia.org/wiki/Discrete_Fourier_transform
https://en.wikipedia.org/wiki/Fourier_analysis

I have not proof read those pages. BUT reading them and 
referenced pages will, I believe, help you to ask yourself 
illuminating questions.

HTH

On Tue, 25 Jun 2013 17:49:51 +0000, glen herrmannsfeldt wrote:

> Tim Wescott <tim@seemywebsite.really> wrote:
>> On Tue, 25 Jun 2013 07:46:18 -0500, manishp wrote:
> 
> (snip)
>>> I would like to get a clarification with respect to the frequency
>>> component and rate of change.
>  
> (snip)
>>> Can someone help me understand.
>  
>> First, if those really were sine waves of different amplitudes, then an
>> FFT would show a frequency response of identical shape and different
>> amplitudes, from which you could infer rate of change.
>  
>> Second, "DSP" and "FFT" are not synonyms.  There's a lot that you can
>> do with digital signal processing without using the FFT, and much of it
>> is stuff that you cannot reasonably do _with_ the FFT.  So if your
>> approach to DSP problems (or signal processing problems in general) is
>> "I'll get this into the frequency domain, now what do I do?" then your
>> approach is,
>> in general, wrong.
> 
> I agree if you mean the FFT algorithm, or the DFT for that matter.
> But DSP pretty much requires understanding time domain, frequency
> domain, and the connection between them. ASP you might be able to do
> entirely in the frequency domain, computing the R, L, and C you need to
> build RLC filters.
> 
> There might be some people who write FFT when they actually mean the
> continuous Fourier (integral) transform. Less characters to type, for
> one.

I'm not saying that you can get very far doing digital signal processing 
without being facile with computations in both the time and the frequency 
domain.

But the OP seems to think that the way to do his analysis goes roughly as:

1: collect data
2: perform a FFT
3: what now?

I think his error comes not in not knowing the step to perform at 3, but 
in specifying an entirely wrong step to perform at 2.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Tim Wescott <tim@seemywebsite.really> wrote:

(snip)

>>> Second, "DSP" and "FFT" are not synonyms.  There's a lot that you can
>>> do with digital signal processing without using the FFT, and much of it
>>> is stuff that you cannot reasonably do _with_ the FFT.  

(snip, I wrote)
>> I agree if you mean the FFT algorithm, or the DFT for that matter.
>> But DSP pretty much requires understanding time domain, frequency
>> domain, and the connection between them. ASP you might be able to do
>> entirely in the frequency domain, computing the R, L, and C you need to
>> build RLC filters.
 
>> There might be some people who write FFT when they actually mean the
>> continuous Fourier (integral) transform. Less characters to type, for
>> one.
 
> I'm not saying that you can get very far doing digital signal 
> processing without being facile with computations in both 
> the time and the frequency domain.
 
> But the OP seems to think that the way to do his analysis 
> goes roughly as:
 
> 1: collect data
> 2: perform a FFT
> 3: what now?
 
> I think his error comes not in not knowing the step to perform 
> at 3, but in specifying an entirely wrong step to perform at 2.

I hadn't thought that when I read it first, but now rereading it
I can see that it could be that way. Partly I read the words but
didn't look at the numbers.

I was thinking about the not so unusual "my signal has sharp
edges, why does that require high frequencies" question.

But the data doesn't have sharp edges, and isn't so bad an
approximation to sines. Still, it shouldn't be hard to see 
the difference (by subtracting) between the two.

As far as the perform FFT, what now, that might not be so unusual
for people with a physics background. Fourier transforms and Fourier
series are pretty important in physics, but FIR and IIR filters
are pretty much ignored. Also, the FFT is, more than in DSP, seen
as an approximation to the continuous transform. 

Even more, in physics frequency/time or spatial frequency/distance
are used much less often than energy/time or momentum/distance,
where energy is angular frequency times hbar, and momentum is
wave number (k) times hbar. Quantum mechanics doesn't use FIR
and IIR filters, though.

-- glen
 

On Tue, 25 Jun 2013 07:46:18 -0500, "manishp" <58525@dsprelated>
wrote:

>Sirs,

Hi manishp,
   Wow.  You ask strange questions.  And you ask 
them in a way that causes people to struggle to 
understand your questions.

>I would like to get a clarification with respect to the frequency component
>and rate of change. 
>
>A system I am trying to analyze is characterized by the rate of change of
>its output. 

Is your system an analog system or a DSP system?
What is the nature of the input to your system?
What is the nature of the output to your system?
What is the exact function of your system (what 
are you trying to do)?

>But I am not sure if rate of change has any direct co-relation
>with frequency and whether DSP can help in this analysis.

Yes, of course!!   Higher-frequency sinusoids have 
a greater "time rate of change" than lower-frequency 
sinusoids.  

Tim pointed you in the right direction when 
he wrote, "differentiators."

Rate of change is a derivative.  The derivative of 
a sinusoid is another sinusoid whose amplitude is 
proportional to the first sinusoid's frequency.  Right?

>
>With this background, I have the following question for the following sine
>signals. The sample points are same for both the signals but you can notice
>the amplitude varies
>
>sine_1 = [0, 2, 3, 4, 2, 0, 2, 3, 4, 2, 0]
>sine_2 = [0, 3, 4, 5, 3, 0, 3, 4, 5, 3, 0]

Those aren't "sine signals."  Also, if two sequences 
are not identical, as yours are not, then you CANNOT 
say "the sample points are the same."  The sample 
points are NOT the same.

You don't mean that "the amplitude varies".  You mean 
that the "amplitudes are different."  Do you see 
why people have trouble understanding your questions?

>
>As far as the frequency is concerned, they are the same but the amplitude
>at various sample points vary. 

What does the word "they" mean?  What do the words "the same" 
mean?  The same in what way?   What "sample points" are 
you referring to, time samples or frequency samples?  

>Question is, as far as frequency analysis is
>concerned (e.g. FFT), would both of them show as having same freqency
>component?

If two time-domain sequences are different, then their 
discrete Fourier transforms (DFTs) will be different.  When 
you write, "... having same frequency component" do you 
mean "same spectrum?"

>As I have mentioned above, for the problem I am working on, the rate of
>change matters. 

Again, what is the problem you are trying to solve?

>So, the second signal has more rate of change.

Yes, the sine_2 sequence has a greater "time rate of 
change" than the sine_1 sequence.

[-Rick-]

>Hi manishp,
>   Wow.  You ask strange questions.  And you ask 
>them in a way that causes people to struggle to 
>understand your questions.
>
>>I would like to get a clarification with respect to the frequency
component
>>and rate of change. 
>>
>>A system I am trying to analyze is characterized by the rate of change
of
>>its output. 
>
>Is your system an analog system or a DSP system?
>What is the nature of the input to your system?
>What is the nature of the output to your system?
>What is the exact function of your system (what 
>are you trying to do)?

the input to the system are a set of parameters. The parameters are bound
within an interval. Also, the parmeters have relationship between each
other which are defined as inputs to the system.

The system also generates output parameters. As a general rule, taking just
one parameter at a time, it is generally assumed to be a good performance
by the system, is the system generates values which vary a lot wrt to the
previous values. Basically, temporal variance is what defines the system.

The point to be noted here is that sometimes when system restricts its
outputs, its not always due to its internal algorithm but mostly due to
incorrect parameters passed to it.

So, in a way, we are measuring the quality of input to the system by
looking at the quality of the output.

Rate of change is one such parameter through which the system and inputs
are graded.

>>But I am not sure if rate of change has any direct co-relation
>>with frequency and whether DSP can help in this analysis.
>
>Yes, of course!!   Higher-frequency sinusoids have 
>a greater "time rate of change" than lower-frequency 
>sinusoids.  
>
>Tim pointed you in the right direction when 
>he wrote, "differentiators."
>
>Rate of change is a derivative.  The derivative of 
>a sinusoid is another sinusoid whose amplitude is 
>proportional to the first sinusoid's frequency.  Right?
>
>>
>>With this background, I have the following question for the following
sine
>>signals. The sample points are same for both the signals but you can
notice
>>the amplitude varies
>>
>>sine_1 = [0, 2, 3, 4, 2, 0, 2, 3, 4, 2, 0]
>>sine_2 = [0, 3, 4, 5, 3, 0, 3, 4, 5, 3, 0]
>
>Those aren't "sine signals."  Also, if two sequences 
>are not identical, as yours are not, then you CANNOT 
>say "the sample points are the same."  The sample 
>points are NOT the same.

sorry to ask. Can you tell me if I have used the term sequence or sample
points incorrectly?
Basically, I want to give an example of sine signals whose frequency is
same but their amplitude varies. Now, take samples of these sine waves and
assume that is what sine_1 and sine_2 represent.

>You don't mean that "the amplitude varies".  You mean 
>that the "amplitudes are different."  Do you see 
>why people have trouble understanding your questions?

Yes, now I understand. Thanks.

>>
>>As far as the frequency is concerned, they are the same but the
amplitude
>>at various sample points vary. 
>
>What does the word "they" mean?  What do the words "the same" 
>mean?  The same in what way?   What "sample points" are 
>you referring to, time samples or frequency samples? 

they = freqyuency of the signals
sample points = the digital samples taken for the sine waves
same = frequency is same
sample = time samples
 
>
>>Question is, as far as frequency analysis is
>>concerned (e.g. FFT), would both of them show as having same freqency
>>component?
>
>If two time-domain sequences are different, then their 
>discrete Fourier transforms (DFTs) will be different.  When 
>you write, "... having same frequency component" do you 
>mean "same spectrum?"

no, I dont mean spectrum. Actually, with my limited understanding of the
subject, we can have two signals with same freuqnecy but with different
spectrum.

>
>>As I have mentioned above, for the problem I am working on, the rate of
>>change matters. 
>
>Again, what is the problem you are trying to solve?

Please let me know if I have explained my problem is. I was thinking if I
can take fft of the output and then look at the power level. If most of the
power is concentrated at higher frequency then it would graded one way and
if power is concentrated at low frequency then it would be graded
differently.

But then when I looked at the system once again, fast rate of change does
not mean high frequency. This is the basis for my question.	 

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manishp <58525@dsprelated> wrote:

(snip)
>>>With this background, I have the following question for 
>>>the following sine signals. The sample points are same for 
>>>both the signals but you can notice
>>>the amplitude varies

>>>sine_1 = [0, 2, 3, 4, 2, 0, 2, 3, 4, 2, 0]
>>>sine_2 = [0, 3, 4, 5, 3, 0, 3, 4, 5, 3, 0]

>>Those aren't "sine signals."  Also, if two sequences 
>>are not identical, as yours are not, then you CANNOT 
>>say "the sample points are the same."  The sample 
>>points are NOT the same.

They might not be so bad as sines with course quantization.

Note that to be different amplitudes of the same signal, 
they should be integer multiples of some common signal, which 
they aren't. So, for example:

   [ 0, 4, 6, 8, 4, 0, 4, 6, 8, 4, 0]
and
   [ 0, 6, 9,12, 6, 0, 6, 9,12, 6, 0]

are two different amplitudes, of the same signal, with
all sample points of the second being 1.5 times the first.

> sorry to ask. Can you tell me if I have used the term 
> sequence or sample points incorrectly?
> Basically, I want to give an example of sine signals whose 
> frequency is same but their amplitude varies. Now, take 
> samples of these sine waves and assume that is what 
> sine_1 and sine_2 represent.

(snip)
> no, I dont mean spectrum. Actually, with my limited understanding of the
> subject, we can have two signals with same freuqnecy but with different
> spectrum.

In the usual DSP (and ASP) sense, a signal can be described either
as amplitude (such as voltage) as a function of time, or as a sum
of (usually) sines and cosines of a specified set (discrete or
continuous) of frequencies. 
 
>>>As I have mentioned above, for the problem I am working on, 
>>>the rate of change matters. 

>>Again, what is the problem you are trying to solve?
 
> Please let me know if I have explained my problem is. I was thinking if I
> can take fft of the output and then look at the power level. If most of the
> power is concentrated at higher frequency then it would graded one way and
> if power is concentrated at low frequency then it would be graded
> differently.

Yes, you can do that.
 
> But then when I looked at the system once again, fast rate of change does
> not mean high frequency. This is the basis for my question.      

Signals with a high rate of change require high frequency
components in the spectrum.

-- glen