Forums

Modelling of a RC series connection in time and frequency domain

Started by Eugen Artus September 7, 2009
Hi!

Suppose there is resistor R and a capacitor C connected in series. A voltage 
U_0 is applied at time t=0. The current in the time domain is then

I(t) = U_0/R * e^(-t/RC)                (1)

Taking a Laplace transform and substituting s by j*2*\pi*f, I get the 
frequency domain current:

I(f) = U_0*C/(j*2*\pi*f*R*C+1)  	(2)

Now the problem: I defined values for U_0, R and C, calculated the time 
domain signal by Eq. 1 and transformed it by DFT (using the Python function 
numpy.fft.rfft). Then, I normalized the result by multiplying with 2/N, 
except the first and the last sample, which were multiplied with 1/N. 
N is the number of samples of the time signal. 

A comparison with the signal predicted by Eq. 2 shows a significant 
difference. What am I doing wrong?

Many thanks!

Eugen

On 7 Sep, 13:38, Eugen Artus <eugen.ar...@arcor.de> wrote:
> Hi! > > Suppose there is resistor R and a capacitor C connected in series. A volt=
age
> U_0 is applied at time t=3D0. The current in the time domain is then > > I(t) =3D U_0/R * e^(-t/RC) =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0(1) > > Taking a Laplace transform and substituting s by j*2*\pi*f, I get the > frequency domain current: > > I(f) =3D U_0*C/(j*2*\pi*f*R*C+1) =A0 =A0 =A0 =A0 =A0(2) > > Now the problem: I defined values for U_0, R and C, calculated the time > domain signal by Eq. 1 and transformed it by DFT (using the Python functi=
on
> numpy.fft.rfft). Then, I normalized the result by multiplying with 2/N, > except the first and the last sample, which were multiplied with 1/N. > N is the number of samples of the time signal. > > A comparison with the signal predicted by Eq. 2 shows a significant > difference. What am I doing wrong?
The error you make is that you think that the DFT and the Fourier Transform are the seame operation. In fact, they are fundamentally different: - The FT works with infinitely long, continuous-time signals. - The DFT works with dicrete-time data of finite length. These differences have a couple of consequences, like - The DFT produces a discrete, finite-length spectrum - The FT produces a continuous, infinitely wide spectrum The FT and the DFT can be linked, but then you have to apply the sampling theorem, map the frequency response of the RC cirquit to Z-transform domain by using a BiLinear Transform (BLT), and so on. Rune
On Sep 7, 8:00=A0am, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 7 Sep, 13:38, Eugen Artus <eugen.ar...@arcor.de> wrote: > > > > > Hi! > > > Suppose there is resistor R and a capacitor C connected in series. A vo=
ltage
> > U_0 is applied at time t=3D0. The current in the time domain is then > > > I(t) =3D U_0/R * e^(-t/RC) =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0(1) > > > Taking a Laplace transform and substituting s by j*2*\pi*f, I get the > > frequency domain current: > > > I(f) =3D U_0*C/(j*2*\pi*f*R*C+1) =A0 =A0 =A0 =A0 =A0(2) > > > Now the problem: I defined values for U_0, R and C, calculated the time > > domain signal by Eq. 1 and transformed it by DFT (using the Python func=
tion
> > numpy.fft.rfft). Then, I normalized the result by multiplying with 2/N, > > except the first and the last sample, which were multiplied with 1/N. > > N is the number of samples of the time signal. > > > A comparison with the signal predicted by Eq. 2 shows a significant > > difference. What am I doing wrong? > > The error you make is that you think that the DFT and the > Fourier Transform are the seame operation. In fact, they are > fundamentally different: > > - The FT works with infinitely long, continuous-time signals. > - The DFT works with dicrete-time data of finite length. > > These differences have a couple of consequences, like > > - The DFT produces a discrete, finite-length spectrum > - The FT produces a continuous, infinitely wide spectrum > > The FT and the DFT can be linked, but then you have to > apply the sampling theorem, map the frequency response > of the RC cirquit to Z-transform domain by using a > BiLinear Transform (BLT), and so on. > > Rune
To add to Rune's comment, the DFT operates on discrete, finite, PERIODIC time series and produces a discrete, finite, PERIODIC frequency series. One way to get around this is to use zero padding judiciously. Julius
On 7 Sep, 15:13, julius <juli...@gmail.com> wrote:
> On Sep 7, 8:00=A0am, Rune Allnor <all...@tele.ntnu.no> wrote: > > > > > > > On 7 Sep, 13:38, Eugen Artus <eugen.ar...@arcor.de> wrote: > > > > Hi! > > > > Suppose there is resistor R and a capacitor C connected in series. A =
voltage
> > > U_0 is applied at time t=3D0. The current in the time domain is then > > > > I(t) =3D U_0/R * e^(-t/RC) =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0(1) > > > > Taking a Laplace transform and substituting s by j*2*\pi*f, I get the > > > frequency domain current: > > > > I(f) =3D U_0*C/(j*2*\pi*f*R*C+1) =A0 =A0 =A0 =A0 =A0(2) > > > > Now the problem: I defined values for U_0, R and C, calculated the ti=
me
> > > domain signal by Eq. 1 and transformed it by DFT (using the Python fu=
nction
> > > numpy.fft.rfft). Then, I normalized the result by multiplying with 2/=
N,
> > > except the first and the last sample, which were multiplied with 1/N. > > > N is the number of samples of the time signal. > > > > A comparison with the signal predicted by Eq. 2 shows a significant > > > difference. What am I doing wrong? > > > The error you make is that you think that the DFT and the > > Fourier Transform are the seame operation. In fact, they are > > fundamentally different: > > > - The FT works with infinitely long, continuous-time signals. > > - The DFT works with dicrete-time data of finite length. > > > These differences have a couple of consequences, like > > > - The DFT produces a discrete, finite-length spectrum > > - The FT produces a continuous, infinitely wide spectrum > > > The FT and the DFT can be linked, but then you have to > > apply the sampling theorem, map the frequency response > > of the RC cirquit to Z-transform domain by using a > > BiLinear Transform (BLT), and so on. > > > Rune > > To add to Rune's comment, the DFT operates on discrete, > finite, PERIODIC time series and produces a discrete, finite, > PERIODIC frequency series. > > One way to get around this is to use zero padding judiciously.
Ah. Here are some data I have lying around: x =3D [1,2,3,4,5,6,7,8,9]; Those numbers are all there is to this data set. No more. No less. Remind me, where is that period? Rune
Rune Allnor wrote:
> On 7 Sep, 15:13, julius <juli...@gmail.com> wrote: >> On Sep 7, 8:00 am, Rune Allnor <all...@tele.ntnu.no> wrote: >> >> >> >> >> >>> On 7 Sep, 13:38, Eugen Artus <eugen.ar...@arcor.de> wrote: >>>> Hi! >>>> Suppose there is resistor R and a capacitor C connected in series. A voltage >>>> U_0 is applied at time t=0. The current in the time domain is then >>>> I(t) = U_0/R * e^(-t/RC) (1) >>>> Taking a Laplace transform and substituting s by j*2*\pi*f, I get the >>>> frequency domain current: >>>> I(f) = U_0*C/(j*2*\pi*f*R*C+1) (2) >>>> Now the problem: I defined values for U_0, R and C, calculated the time >>>> domain signal by Eq. 1 and transformed it by DFT (using the Python function >>>> numpy.fft.rfft). Then, I normalized the result by multiplying with 2/N, >>>> except the first and the last sample, which were multiplied with 1/N. >>>> N is the number of samples of the time signal. >>>> A comparison with the signal predicted by Eq. 2 shows a significant >>>> difference. What am I doing wrong? >>> The error you make is that you think that the DFT and the >>> Fourier Transform are the seame operation. In fact, they are >>> fundamentally different: >>> - The FT works with infinitely long, continuous-time signals. >>> - The DFT works with dicrete-time data of finite length. >>> These differences have a couple of consequences, like >>> - The DFT produces a discrete, finite-length spectrum >>> - The FT produces a continuous, infinitely wide spectrum >>> The FT and the DFT can be linked, but then you have to >>> apply the sampling theorem, map the frequency response >>> of the RC cirquit to Z-transform domain by using a >>> BiLinear Transform (BLT), and so on. >>> Rune >> To add to Rune's comment, the DFT operates on discrete, >> finite, PERIODIC time series and produces a discrete, finite, >> PERIODIC frequency series. >> >> One way to get around this is to use zero padding judiciously. > > Ah. Here are some data I have lying around: > > x = [1,2,3,4,5,6,7,8,9]; > > Those numbers are all there is to this data set. > No more. No less. > > Remind me, where is that period?
The period is not associated with the numbers, but rather with what the FFT does. The result of performing an FFT on your data is exactly the same as performing one on ...1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,.... Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
On Mon, 07 Sep 2009 08:30:10 -0700, Rune Allnor wrote:

> On 7 Sep, 15:13, julius <juli...@gmail.com> wrote: >> On Sep 7, 8:00&nbsp;am, Rune Allnor <all...@tele.ntnu.no> wrote: >> >> >> >> >> >> > On 7 Sep, 13:38, Eugen Artus <eugen.ar...@arcor.de> wrote: >> >> > > Hi! >> >> > > Suppose there is resistor R and a capacitor C connected in series. >> > > A voltage U_0 is applied at time t=0. The current in the time >> > > domain is then >> >> > > I(t) = U_0/R * e^(-t/RC) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(1) >> >> > > Taking a Laplace transform and substituting s by j*2*\pi*f, I get >> > > the frequency domain current: >> >> > > I(f) = U_0*C/(j*2*\pi*f*R*C+1) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(2) >> >> > > Now the problem: I defined values for U_0, R and C, calculated the >> > > time domain signal by Eq. 1 and transformed it by DFT (using the >> > > Python function numpy.fft.rfft). Then, I normalized the result by >> > > multiplying with 2/N, except the first and the last sample, which >> > > were multiplied with 1/N. N is the number of samples of the time >> > > signal. >> >> > > A comparison with the signal predicted by Eq. 2 shows a significant >> > > difference. What am I doing wrong? >> >> > The error you make is that you think that the DFT and the Fourier >> > Transform are the seame operation. In fact, they are fundamentally >> > different: >> >> > - The FT works with infinitely long, continuous-time signals. - The >> > DFT works with dicrete-time data of finite length. >> >> > These differences have a couple of consequences, like >> >> > - The DFT produces a discrete, finite-length spectrum - The FT >> > produces a continuous, infinitely wide spectrum >> >> > The FT and the DFT can be linked, but then you have to apply the >> > sampling theorem, map the frequency response of the RC cirquit to >> > Z-transform domain by using a BiLinear Transform (BLT), and so on. >> >> > Rune >> >> To add to Rune's comment, the DFT operates on discrete, finite, >> PERIODIC time series and produces a discrete, finite, PERIODIC >> frequency series. >> >> One way to get around this is to use zero padding judiciously. > > Ah. Here are some data I have lying around: > > x = [1,2,3,4,5,6,7,8,9]; > > Those numbers are all there is to this data set. No more. No less. > > Remind me, where is that period?
No where. Which is why you should use the DFT judiciously on this sequence, because the DFT implicitly assumes a periodic, evenly sampled sequence and your sequence is not periodic. -- www.wescottdesign.com
On 7 Sep, 17:36, Tim Wescott <t...@seemywebsite.com> wrote:
> On Mon, 07 Sep 2009 08:30:10 -0700, Rune Allnor wrote: > > On 7 Sep, 15:13, julius <juli...@gmail.com> wrote: > >> On Sep 7, 8:00=A0am, Rune Allnor <all...@tele.ntnu.no> wrote: > > >> > On 7 Sep, 13:38, Eugen Artus <eugen.ar...@arcor.de> wrote: > > >> > > Hi! > > >> > > Suppose there is resistor R and a capacitor C connected in series. > >> > > A voltage U_0 is applied at time t=3D0. The current in the time > >> > > domain is then > > >> > > I(t) =3D U_0/R * e^(-t/RC) =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0(1) > > >> > > Taking a Laplace transform and substituting s by j*2*\pi*f, I get > >> > > the frequency domain current: > > >> > > I(f) =3D U_0*C/(j*2*\pi*f*R*C+1) =A0 =A0 =A0 =A0 =A0(2) > > >> > > Now the problem: I defined values for U_0, R and C, calculated the > >> > > time domain signal by Eq. 1 and transformed it by DFT (using the > >> > > Python function numpy.fft.rfft). Then, I normalized the result by > >> > > multiplying with 2/N, except the first and the last sample, which > >> > > were multiplied with 1/N. N is the number of samples of the time > >> > > signal. > > >> > > A comparison with the signal predicted by Eq. 2 shows a significan=
t
> >> > > difference. What am I doing wrong? > > >> > The error you make is that you think that the DFT and the Fourier > >> > Transform are the seame operation. In fact, they are fundamentally > >> > different: > > >> > - The FT works with infinitely long, continuous-time signals. - The > >> > DFT works with dicrete-time data of finite length. > > >> > These differences have a couple of consequences, like > > >> > - The DFT produces a discrete, finite-length spectrum - The FT > >> > produces a continuous, infinitely wide spectrum > > >> > The FT and the DFT can be linked, but then you have to apply the > >> > sampling theorem, map the frequency response of the RC cirquit to > >> > Z-transform domain by using a BiLinear Transform (BLT), and so on. > > >> > Rune > > >> To add to Rune's comment, the DFT operates on discrete, finite, > >> PERIODIC time series and produces a discrete, finite, PERIODIC > >> frequency series. > > >> One way to get around this is to use zero padding judiciously. > > > Ah. Here are some data I have lying around: > > > x =3D [1,2,3,4,5,6,7,8,9]; > > > Those numbers are all there is to this data set. No more. No less. > > > Remind me, where is that period? > > No where. =A0Which is why you should use the DFT judiciously on this > sequence, because the DFT implicitly assumes a periodic, evenly sampled > sequence and your sequence is not periodic.
... so you seem to mean that the DFT can not be used on this sequence? If so, what does the numbers X =3D fft(x) =3D [ 45.0000 -4.5000 +12.3636i -4.5000 + 5.3629i -4.5000 + 2.5981i -4.5000 + 0.7935i -4.5000 - 0.7935i -4.5000 - 2.5981i -4.5000 - 5.3629i -4.5000 -12.3636i] signify? Rune
On Mon, 07 Sep 2009 11:35:23 -0400, Jerry Avins wrote:

> Rune Allnor wrote: >> On 7 Sep, 15:13, julius <juli...@gmail.com> wrote: >>> On Sep 7, 8:00 am, Rune Allnor <all...@tele.ntnu.no> wrote: >>> >>> >>> >>> >>> >>>> On 7 Sep, 13:38, Eugen Artus <eugen.ar...@arcor.de> wrote: >>>>> Hi! >>>>> Suppose there is resistor R and a capacitor C connected in series. A >>>>> voltage U_0 is applied at time t=0. The current in the time domain >>>>> is then I(t) = U_0/R * e^(-t/RC) (1) Taking a Laplace >>>>> transform and substituting s by j*2*\pi*f, I get the frequency >>>>> domain current: >>>>> I(f) = U_0*C/(j*2*\pi*f*R*C+1) (2) Now the problem: I >>>>> defined values for U_0, R and C, calculated the time domain signal >>>>> by Eq. 1 and transformed it by DFT (using the Python function >>>>> numpy.fft.rfft). Then, I normalized the result by multiplying with >>>>> 2/N, except the first and the last sample, which were multiplied >>>>> with 1/N. N is the number of samples of the time signal. A >>>>> comparison with the signal predicted by Eq. 2 shows a significant >>>>> difference. What am I doing wrong? >>>> The error you make is that you think that the DFT and the Fourier >>>> Transform are the seame operation. In fact, they are fundamentally >>>> different: >>>> - The FT works with infinitely long, continuous-time signals. - The >>>> DFT works with dicrete-time data of finite length. These differences >>>> have a couple of consequences, like - The DFT produces a discrete, >>>> finite-length spectrum - The FT produces a continuous, infinitely >>>> wide spectrum The FT and the DFT can be linked, but then you have to >>>> apply the sampling theorem, map the frequency response of the RC >>>> cirquit to Z-transform domain by using a BiLinear Transform (BLT), >>>> and so on. >>>> Rune >>> To add to Rune's comment, the DFT operates on discrete, finite, >>> PERIODIC time series and produces a discrete, finite, PERIODIC >>> frequency series. >>> >>> One way to get around this is to use zero padding judiciously. >> >> Ah. Here are some data I have lying around: >> >> x = [1,2,3,4,5,6,7,8,9]; >> >> Those numbers are all there is to this data set. No more. No less. >> >> Remind me, where is that period? > > The period is not associated with the numbers, but rather with what the > FFT does. The result of performing an FFT on your data is exactly the > same as performing one on ...1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,....
After I wrote my response, I realized that one doesn't have to insist on periodicity -- one can insist, instead, on data that lives on a 1-sphere (i.e. a circle). Then the periodicity becomes an illusion generated by our insistence on data existing on a line. The two are mathematically equivalent, of course; only how one thinks about one or the other is different between the two cases. The _important_ thing is that -- to the DFT -- if you have a sequence [x0, x1, ... , x{n-1}] then x{n-1} is exactly as adjacent to x0 x1 is. Thus if you use the DFT to analyze a sequence of numbers that are all there is in the data set, no more, then you need to either make sure that the data set is circular to match the DFT, or you have to do whatever is necessary to insure that the approximation to the DFT input is valid. -- www.wescottdesign.com
On 7 Sep, 18:55, Tim Wescott <t...@seemywebsite.com> wrote:
> On Mon, 07 Sep 2009 11:35:23 -0400, Jerry Avins wrote: > > Rune Allnor wrote: > >> On 7 Sep, 15:13, julius <juli...@gmail.com> wrote: > >>> On Sep 7, 8:00 am, Rune Allnor <all...@tele.ntnu.no> wrote: > > >>>> On 7 Sep, 13:38, Eugen Artus <eugen.ar...@arcor.de> wrote: > >>>>> Hi! > >>>>> Suppose there is resistor R and a capacitor C connected in series. =
A
> >>>>> voltage U_0 is applied at time t=3D0. The current in the time domai=
n
> >>>>> is then I(t) =3D U_0/R * e^(-t/RC) =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0(=
1) Taking a Laplace
> >>>>> transform and substituting s by j*2*\pi*f, I get the frequency > >>>>> domain current: > >>>>> I(f) =3D U_0*C/(j*2*\pi*f*R*C+1) =A0 =A0 =A0 =A0 =A0(2) Now the pro=
blem: I
> >>>>> defined values for U_0, R and C, calculated the time domain signal > >>>>> by Eq. 1 and transformed it by DFT (using the Python function > >>>>> numpy.fft.rfft). Then, I normalized the result by multiplying with > >>>>> 2/N, except the first and the last sample, which were multiplied > >>>>> with 1/N. N is the number of samples of the time signal. A > >>>>> comparison with the signal predicted by Eq. 2 shows a significant > >>>>> difference. What am I doing wrong? > >>>> The error you make is that you think that the DFT and the Fourier > >>>> Transform are the seame operation. In fact, they are fundamentally > >>>> different: > >>>> - The FT works with infinitely long, continuous-time signals. - The > >>>> DFT works with dicrete-time data of finite length. These differences > >>>> have a couple of consequences, like - The DFT produces a discrete, > >>>> finite-length spectrum - The FT produces a continuous, infinitely > >>>> wide spectrum The FT and the DFT can be linked, but then you have to > >>>> apply the sampling theorem, map the frequency response of the RC > >>>> cirquit to Z-transform domain by using a BiLinear Transform (BLT), > >>>> and so on. > >>>> Rune > >>> To add to Rune's comment, the DFT operates on discrete, finite, > >>> PERIODIC time series and produces a discrete, finite, PERIODIC > >>> frequency series. > > >>> One way to get around this is to use zero padding judiciously. > > >> Ah. Here are some data I have lying around: > > >> x =3D [1,2,3,4,5,6,7,8,9]; > > >> Those numbers are all there is to this data set. No more. No less. > > >> Remind me, where is that period? > > > The period is not associated with the numbers, but rather with what the > > FFT does. The result of performing an FFT on your data is exactly the > > same as performing one on ...1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,.... > > After I wrote my response, I realized that one doesn't have to insist on > periodicity -- one can insist, instead, on data that lives on a 1-sphere > (i.e. a circle). =A0Then the periodicity becomes an illusion generated by > our insistence on data existing on a line.
Very good. Not perfect, but a lot better than most of the sheep who make their opinions known here. Of course, once you think a little bit more about this, you realize that your argument is the usual relation between the DFT and ZT, just applied in reverse. Usually one starts out with the ZT (which uses the rectangular form of complex numbers) and argue that it is related to the DFT, which represents the unit circle on polar form. In that case the 'periodicity' of the DFT becomes an illusion caused by the ambiguity of the polar representation of complex numbers. If you think a little bit more about this, you will no doubt discover that even the complex plane is irrelevant, as the DFT only maps an N-length complex-valued vector to a different set of basis vectors. Rune
On Mon, 07 Sep 2009 09:48:56 -0700, Rune Allnor wrote:

> On 7 Sep, 17:36, Tim Wescott <t...@seemywebsite.com> wrote: >> On Mon, 07 Sep 2009 08:30:10 -0700, Rune Allnor wrote: >> > On 7 Sep, 15:13, julius <juli...@gmail.com> wrote: >> >> On Sep 7, 8:00&nbsp;am, Rune Allnor <all...@tele.ntnu.no> wrote: >> >> >> > On 7 Sep, 13:38, Eugen Artus <eugen.ar...@arcor.de> wrote: >> >> >> > > Hi! >> >> >> > > Suppose there is resistor R and a capacitor C connected in >> >> > > series. A voltage U_0 is applied at time t=0. The current in the >> >> > > time domain is then >> >> >> > > I(t) = U_0/R * e^(-t/RC) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(1) >> >> >> > > Taking a Laplace transform and substituting s by j*2*\pi*f, I >> >> > > get the frequency domain current: >> >> >> > > I(f) = U_0*C/(j*2*\pi*f*R*C+1) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(2) >> >> >> > > Now the problem: I defined values for U_0, R and C, calculated >> >> > > the time domain signal by Eq. 1 and transformed it by DFT (using >> >> > > the Python function numpy.fft.rfft). Then, I normalized the >> >> > > result by multiplying with 2/N, except the first and the last >> >> > > sample, which were multiplied with 1/N. N is the number of >> >> > > samples of the time signal. >> >> >> > > A comparison with the signal predicted by Eq. 2 shows a >> >> > > significant difference. What am I doing wrong? >> >> >> > The error you make is that you think that the DFT and the Fourier >> >> > Transform are the seame operation. In fact, they are fundamentally >> >> > different: >> >> >> > - The FT works with infinitely long, continuous-time signals. - >> >> > The DFT works with dicrete-time data of finite length. >> >> >> > These differences have a couple of consequences, like >> >> >> > - The DFT produces a discrete, finite-length spectrum - The FT >> >> > produces a continuous, infinitely wide spectrum >> >> >> > The FT and the DFT can be linked, but then you have to apply the >> >> > sampling theorem, map the frequency response of the RC cirquit to >> >> > Z-transform domain by using a BiLinear Transform (BLT), and so on. >> >> >> > Rune >> >> >> To add to Rune's comment, the DFT operates on discrete, finite, >> >> PERIODIC time series and produces a discrete, finite, PERIODIC >> >> frequency series. >> >> >> One way to get around this is to use zero padding judiciously. >> >> > Ah. Here are some data I have lying around: >> >> > x = [1,2,3,4,5,6,7,8,9]; >> >> > Those numbers are all there is to this data set. No more. No less. >> >> > Remind me, where is that period? >> >> No where. &nbsp;Which is why you should use the DFT judiciously on this >> sequence, because the DFT implicitly assumes a periodic, evenly sampled >> sequence and your sequence is not periodic. > > ... so you seem to mean that the DFT can not be used on this sequence? > If so, what does the numbers > > X = fft(x) = > [ 45.0000 > -4.5000 +12.3636i > -4.5000 + 5.3629i > -4.5000 + 2.5981i > -4.5000 + 0.7935i > -4.5000 - 0.7935i > -4.5000 - 2.5981i > -4.5000 - 5.3629i > -4.5000 -12.3636i] > > signify?
Well, if the sequence is taken from a program written in C, and if the variable in question is an enumeration of fruit, and if the mapping between meaning and numbers goes 1: apples 2: oranges 3: grapes 4: kumquats 5: peaches 6: banannas 7: grapefruit 8: rose hips 9: mangoes, then the sequence you present means Absolutely Nothing. That doesn't mean you can't take the DFT of it, just that you have chosen a mathematical operation that means nothing to the data at hand. So I will repeat and expand my original assertion, in hopes that by repetition I can make up for my lack of brilliance: the DFT implicitly assumes a certain temporal organization of the sequence (not to mention the assumption that the numbers stand for real quantities), and if those assumptions are totally invalid for the input, then the output stands more as an unfortunate comment on the abilities and sensibility of the practitioner than they stand for any meaningful interpretation of the data. Without saying what that sequence means, one may as well pick up a copy of the I Ching, randomly invent a way to translate ten integers into the six lines of a hexagram, and use the sequence to predict what you're going to eat for breakfast tomorrow. Either way the interpretation will be as accurate and meaningful, and the I Ching is always more fun than the output of an FFT. -- www.wescottdesign.com