time domain and frequency domain

Sirs,

In one of the books, I am reading, as a part of DFT duality properties, it
is mentioned that:
  - a point in frequency domain is equivalent to siusoid in time domain
  - a point in time domain is equivalent to sinusoid in frequency domain

While I am able to appreciate the first point that a point in frequency
domain is equivalent to sinusoid in time domain (all the information in
time domain is concentrated in a single frequency).

I am unable to clearly understand point 2 as to how a point in time domain
can result in multitude of frequencies in the frequency domain. How a
single point can be decomposed into many frequencies.

Thanks ...

On Tue, 17 Jul 2012 10:56:38 -0500
"manishp" <58525@dsprelated> wrote:

> Sirs,
> 
> In one of the books, I am reading, as a part of DFT duality properties, it
> is mentioned that:
>   - a point in frequency domain is equivalent to siusoid in time domain
>   - a point in time domain is equivalent to sinusoid in frequency domain
> 
> While I am able to appreciate the first point that a point in frequency
> domain is equivalent to sinusoid in time domain (all the information in
> time domain is concentrated in a single frequency).
> 
> I am unable to clearly understand point 2 as to how a point in time domain
> can result in multitude of frequencies in the frequency domain. How a
> single point can be decomposed into many frequencies.
> 
> Thanks ...
> 
> 

Because that single point doesn't exist in isolation, it's a single
point of non zero amplitude in all of time, extending infinitely in
both directions, the rest of which has no amplitude.

The basis function in frequency space is the sinewave (well, complex
exponential, but...).  No sinewave looks like that; a single isolated
point in all the void.  So instead, what you've got there at t=0 is the
result of constructive interference between all possible frequencies,
and which sum to zero everywhere else.

-- 
Rob Gaddi, Highland Technology -- www.highlandtechnology.com
Email address domain is currently out of order.  See above to fix.

On Jul 17, 10:56=A0am, "manishp" <58525@dsprelated> wrote:

> In one of the books, I am reading, as a part of DFT duality properties, i=
t
> is mentioned that:
> =A0 - a point in frequency domain is equivalent to siusoid in time domain
> =A0 - a point in time domain is equivalent to sinusoid in frequency domai=
n

Are you sure that it is the DFT =3D Discrete Fourier Transform that
you are talking about and not the continuous-time Fourier Transform?
There is a difference between the two concepts, and your questions
seem to relate more to the continuous-time Fourier Transform than
the DFT.  So, read your book very very carefully and tell us again
which duality properties you are talking about. A DFT transforms
one (periodic) sequence of numbers into another (periodic)
sequence of numbers and there is nothing sinusoidal about these
sequences, in time domain or frequency domain.

Dilip Sarwate

dvsarwate <dvsarwate@yahoo.com> wrote:

(snip)
> Are you sure that it is the DFT = Discrete Fourier Transform that
> you are talking about and not the continuous-time Fourier Transform?
> There is a difference between the two concepts, and your questions
> seem to relate more to the continuous-time Fourier Transform than
> the DFT.  

Well, you can do the continuous Fourier transform on a series of
equally spaced delta functions, and get the DFT result. (Especially
if the series is periodic with an appropriate period.)

One can also consider the continuous-time transform as the limit
of the DFT as the sample period goes to zero. 

You then get the Fourier transform pair:

  discrete <--> periodic

Applied twice, you get:

  discrete periodic <--> discrete periodic

-- glen

"manishp" <58525@dsprelated> writes:

> Sirs,
>
> In one of the books, I am reading, as a part of DFT duality properties, it
> is mentioned that:
>   - a point in frequency domain is equivalent to siusoid in time domain
>   - a point in time domain is equivalent to sinusoid in frequency domain
>
> While I am able to appreciate the first point that a point in frequency
> domain is equivalent to sinusoid in time domain (all the information in
> time domain is concentrated in a single frequency).
>
> I am unable to clearly understand point 2 as to how a point in time domain
> can result in multitude of frequencies in the frequency domain. How a
> single point can be decomposed into many frequencies.
>
> Thanks ...

You're talking about the Fourier transform (continuous-time). A point
in the time domain has zero energy and no FT. A Dirac delta function
in time has an infinite frequency spectrum. To see why, study the
sifting property of the Dirac delta function, especially as seen
in the (forward) Fourier transform integral. 
-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

On 7/17/2012 8:56 AM, manishp wrote:
> Sirs,
>
> In one of the books, I am reading, as a part of DFT duality properties, it
> is mentioned that:
>    - a point in frequency domain is equivalent to siusoid in time domain
>    - a point in time domain is equivalent to sinusoid in frequency domain
>
> While I am able to appreciate the first point that a point in frequency
> domain is equivalent to sinusoid in time domain (all the information in
> time domain is concentrated in a single frequency).
>
> I am unable to clearly understand point 2 as to how a point in time domain
> can result in multitude of frequencies in the frequency domain. How a
> single point can be decomposed into many frequencies.
>
> Thanks ...
>
>

I agree that you must be talking about the Fourier Transform 
(continuous) instead of the DFT (discrete).

I would read the transform equations.  They are pretty similar.  So 
similar in fact that you can see how they are almost "the same".  They 
probably are the same when you substitute in simple functions such as 
those you describe.

Can you see the inverse cases:
"A sinusoid becomes a Dirac (your "point")"?

Actually you would be better off with this:
Instead of:
"- a point in time domain is equivalent to sinusoid in frequency domain"
Consider using:
"- a pair of equal, real Diracs in the time domain, equally spaced away 
from zero time, is equivalent to a real sinusoid in the frequency domain"

Once you have that down, you can consider what it means to have but one 
of those Diracs.
- what if the Diracs have the same sign? (you get a cosine)
- what if the same Diracs have the opposite sign? (you get +/- a sine)
- what if you add the two above together to get one Dirac?
(superposition applies so you add their transforms together in frequency).

Fred

dvsarwate <dvsarwate@yahoo.com> wrote:
> On Jul 17, 10:56&#4294967295;am, "manishp" <58525@dsprelated> wrote:
> 
>> In one of the books, I am reading, as a part of DFT duality properties, it
>> is mentioned that:
>> &#4294967295; - a point in frequency domain is equivalent to siusoid in time domain
>> &#4294967295; - a point in time domain is equivalent to sinusoid in frequency domain
> 
> Are you sure that it is the DFT = Discrete Fourier Transform that
> you are talking about and not the continuous-time Fourier Transform?
> There is a difference between the two concepts, and your questions
> seem to relate more to the continuous-time Fourier Transform than
> the DFT.  So, read your book very very carefully and tell us again
> which duality properties you are talking about. A DFT transforms
> one (periodic) sequence of numbers into another (periodic)
> sequence of numbers and there is nothing sinusoidal about these
> sequences, in time domain or frequency domain.


How to interprete results of octave code given below ;)

for i=1:4;l=100;
  x=zeros(1,l);
  x(i)=1;
  subplot(4,1,i);
  y=fft(x);
  plot([1,length(y)],[-1,1],'.',real(y),'-',imag(y));
end

>dvsarwate <dvsarwate@yahoo.com> wrote:
>> On Jul 17, 10:56&#65533;am, "manishp" <58525@dsprelated> wrote:
>> 
>>> In one of the books, I am reading, as a part of DFT duality properties,
it
>>> is mentioned that:
>>> &#65533; - a point in frequency domain is equivalent to siusoid in time
domain
>>> &#65533; - a point in time domain is equivalent to sinusoid in frequency
domain
>> 
>> Are you sure that it is the DFT = Discrete Fourier Transform that
>> you are talking about and not the continuous-time Fourier Transform?
>> There is a difference between the two concepts, and your questions
>> seem to relate more to the continuous-time Fourier Transform than
>> the DFT.  So, read your book very very carefully and tell us again
>> which duality properties you are talking about. A DFT transforms
>> one (periodic) sequence of numbers into another (periodic)
>> sequence of numbers and there is nothing sinusoidal about these
>> sequences, in time domain or frequency domain.
>
>
>How to interprete results of octave code given below ;)
>
>for i=1:4;l=100;
>  x=zeros(1,l);
>  x(i)=1;
>  subplot(4,1,i);
>  y=fft(x);
>  plot([1,length(y)],[-1,1],'.',real(y),'-',imag(y));
>end
>

you are generating x = [1 0 0 0...] 
then x [0 1 0 0 ...] 
then x = [0 0 1 0 ..] 
then x = [0 0 0 1 0...]

then looking at frequency domain.
in all cases abs(y) = 1 over all frequencies
but phase (angle(y) changes depending on location of impulse.
y is real only when x = 1 else it becomes complex corresponding to change
of phase.

kadhiem
 

On Jul 17, 3:24=A0pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:

> Well, you can do the continuous Fourier transform on a series of
> equally spaced delta functions, and get the DFT result. (Especially
> if the series is periodic with an appropriate period.)

You really like to muddy the waters without contributing anything
relevant to the discussion, don't you?

Dilip Sarwate

>On Jul 17, 3:24=A0pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
>
>> Well, you can do the continuous Fourier transform on a series of
>> equally spaced delta functions, and get the DFT result. (Especially
>> if the series is periodic with an appropriate period.)
>
>You really like to muddy the waters without contributing anything
>relevant to the discussion, don't you?
>
>Dilip Sarwate

Below text is from the same book and is mentioned in the context of DFT.

"This symmetry between the time and frequency domains is called duality,
and gives rise to many interesting properties. For example, a single point
in the frequency domain corresponds to a sinusoid in the time domain. By
duality, the inverse is also true, a single point in the time domain
corresponds to a sinusoid in the frequency domain"

Also, thanks for your earlier replies.