Q: fft: Smith Book ch 12 p 230

Greetings,

Smith book, 1st ed., ch 12 p.230

1.

I don't understand the book statement that a 1 point time domain
signal is the same as a one point freq spectrum.

I know the DFT of a single point in time is Cosine shaped wave form in f.
But a Cosine wave in f has points all across the f spectrum.
And each time the t point is shifted toward the right,
the f Cosine wave is merely increased in freq.

I'm sure that in the book, there is something that states
how a one point transforms to a one point. However, I don't see it.

2. 

Fig 12-4 shows the interlacing of FFT synthesis.  I don't recall
where in the book it discussed of this mirroring.  I'm sure it does,
but I cannot find it !! silly me !!  I need to see this discussed.

thanks.

Walt....

waltech <waltechmail@yahoo.com> wrote:
< Smith book, 1st ed., ch 12 p.230

< I don't understand the book statement that a 1 point time domain
< signal is the same as a one point freq spectrum.
 
< I know the DFT of a single point in time is Cosine shaped 
< wave form in f.

A one point signal in frequency space is cos(0*t).

< But a Cosine wave in f has points all across the f spectrum.

-- glen

for single point  (N=1)

X(0) = x(0)  DFT

x(0) = X(0)  IDFT

I recognize the duality..
a one point in f is a cosine in t, as you say.
This is the reverse of my statement, and shows the duality.
But there has to be something simple I'm misunderstanding.

What I'm reading is that one point in one domain transforms to one point
in the other domain.  Hence my confusion.

thx,

Walt.....




>waltech <waltechmail@yahoo.com> wrote:
>< Smith book, 1st ed., ch 12 p.230
>
>< I don't understand the book statement that a 1 point time domain
>< signal is the same as a one point freq spectrum.
> 
>< I know the DFT of a single point in time is Cosine shaped 
>< wave form in f.
>
>A one point signal in frequency space is cos(0*t).
>
>< But a Cosine wave in f has points all across the f spectrum.
>
>-- glen
>

OK, this is blowing my mind.
It took a few minutes to get: you're saying that
the total signal I'm looking at is one and only one sample ?
(N=1).

I need to think about how that stacks up when the book example
shows 4 adjacent samples. If true, this is, in appearance to me,
a totally different thing than a series of samples..
So I need to consider each as a totally separate sample each with
length of 1...


thanks,

Walt.....



>for single point  (N=1)
>
>X(0) = x(0)  DFT
>
>x(0) = X(0)  IDFT
>
>

On Sep 29, 3:52&#4294967295;pm, "waltech" <waltechm...@yahoo.com> wrote:
> I recognize the duality..
> a one point in f is a cosine in t, as you say.
> This is the reverse of my statement, and shows the duality.
> But there has to be something simple I'm misunderstanding.
>
> What I'm reading is that one point in one domain transforms to one point
> in the other domain. &#4294967295;Hence my confusion.
>
> thx,
>
> Walt.....
>
>
>
> >waltech <waltechm...@yahoo.com> wrote:
> >< Smith book, 1st ed., ch 12 p.230
>
> >< I don't understand the book statement that a 1 point time domain
> >< signal is the same as a one point freq spectrum.
>
> >< I know the DFT of a single point in time is Cosine shaped
> >< wave form in f.
>
> >A one point signal in frequency space is cos(0*t).
>
> >< But a Cosine wave in f has points all across the f spectrum.
>
> >-- glen- Hide quoted text -
>
> - Show quoted text -

Walt,
Look at it this way,

X(k) = 1/N SUM( x(n) * exp(-2*pi*k*n/N) ), k = 0 ...... N-1

for a one-point time-domain signal, n = 0 and you have just x(n) = x
(0).
However, exp(-2*pi*k*n/N) = 1 since n = 0 and exp(0) = 1.
Therefore, X(k) = 1/N SUM( x(n) * exp(-2*pi*k*n/N) ), k = 0 ...... N-1
is just X(k) = x(n)/N
which is x(0)/N

Consequently, a signal with one point in the time domain is a signal
with one point in the frequency domain

Maurice Givens

waltech <waltechmail@yahoo.com> wrote:
(snip)
 
< I need to think about how that stacks up when the book example
< shows 4 adjacent samples. If true, this is, in appearance to me,
< a totally different thing than a series of samples..
< So I need to consider each as a totally separate sample each with
< length of 1...

Without actually looking at the book, the basic idea of
the FFT is that you can convert an N point transform into
two N/2 point transforms.  You then apply that recursively
until you get N one point transforms, where the result it trivial.

The N one point transforms are then combined with the appropriate
phase factor to generate the N point transform.

That is pretty much the only time you are likely to run into
a one point transform, but it is an important point.

-- glen

On Sep 29, 5:05&#4294967295;pm, "waltech" <waltechm...@yahoo.com> wrote:
> OK, this is blowing my mind.
> It took a few minutes to get: you're saying that
> the total signal I'm looking at is one and only one sample ?
> (N=1).

Yes, for say N=64 and one non-zero sample or freq component (I think
that is what you were thinking), you get 64 constant magnitude values
when transformed (either way)

Yes, I was locked into trying to think in terms of
each point being a sample, but amongst a number of samples.
Certainly the author said what he meant !
I didn't quite get the significance because of adjacent samples
as shown below. i.e one non-zero point among zeros  makes
an impulse in t with resulting cosine in f in previous presentation.
 
I appreciate the new viewpoint better.  I followed the math
and arguments as presented.
 

But,
Fig 12-4 shows 4 consecutive points of t, abcd.
Their transform is abcd => ABCD where ABCD is in f. 

Which certainly doesn't look like a transform with a cosine
produced. (It agrees with the responders).  Granted, abcd are 
interlaced samples and hence were not consecutive in the original 
signal.  But then the example continues- take the abcs and dilute
with zeros: a0b0c0d0  This looks to me like 8 samples in a row,
even if interlaced, rather than 8 completely separate samples.


so transform , a0b0c0d0 =>  ABCDABCD
and the other, 0e0f0g0h => EFGH times sinusoid=> EFGHEFGH

doesn't look right in terms of 8 samples of some one signal,
but does look correct in terms of N=1 each. So I still don't get why
positioning on the one hand produces a mirror, and the other needs
some sinusoid.

I hope I can continue to fill in the intuitive blanks.  Then I'll
feel comfortable using it.  But real math appreciated if it helps.

>Yes, for say N=3D64 and one non-zero sample or freq component (I think
>that is what you were thinking), you get 64 constant magnitude values
>when transformed (either way) 
>
>

thx,

Walt............

On Sep 29, 6:38&#4294967295;pm, "waltech" <waltechm...@yahoo.com> wrote:
> Yes, I was locked into trying to think in terms of
> each point being a sample, but amongst a number of samples.
> Certainly the author said what he meant !
> I didn't quite get the significance because of adjacent samples
> as shown below. i.e one non-zero point among zeros &#4294967295;makes
> an impulse in t with resulting cosine in f in previous presentation.
>
> I appreciate the new viewpoint better. &#4294967295;I followed the math
> and arguments as presented.
>
> But,
> Fig 12-4 shows 4 consecutive points of t, abcd.
> Their transform is abcd => ABCD where ABCD is in f.
>
> Which certainly doesn't look like a transform with a cosine
> produced. (It agrees with the responders). &#4294967295;Granted, abcd are
> interlaced samples and hence were not consecutive in the original
> signal. &#4294967295;But then the example continues- take the abcs and dilute
> with zeros: a0b0c0d0 &#4294967295;This looks to me like 8 samples in a row,
> even if interlaced, rather than 8 completely separate samples.
>
> so transform , a0b0c0d0 => &#4294967295;ABCDABCD
> and the other, 0e0f0g0h => EFGH times sinusoid=> EFGHEFGH
>
> doesn't look right in terms of 8 samples of some one signal,
> but does look correct in terms of N=1 each.

It looks right for 8 samples of one signal ("A" here is not the DFT of
the single point "a"), "ABCD" here is the frequency spectrum of the 4
point time signal "abcd" and if you add zeros "ABCDABCD" is the
frequency spectrum of the 8 point time signal "a0b0c0d0".