Doubt regarding DFT and sinc function

Hello,

I am kind of new to DSP and presently going through the book "
Understnading DSP" by Richard lyons.

On the topic on DFT it says that the max amplitude/gain of the main lobe
in the magnitude response of DFT increases with N ( N-point DFT) and the
bandwidth reduces. Makes sense.

The it goeson derive the DFT of a rectangular function whose magnitude
rsponse is simalar to that of a sinc function. Now it sates there that

" main lobe width is 2N/K " on page 97. Though the derivation on how it
reached 2N/K makes perfect sense to me but it now leads to me to conclude
that width  of main lobe is increasing with N (N point DFT).

Both seems contarxting to each other. I ma sure there is something I am
missing out on. Would be great if somebody can throw a light on it.

Vivek

On Feb 24, 5:03 am, "mudraraksha" <jindal.vi...@gmail.com> wrote:
> I am kind of new to DSP and presently going through the book "
> Understnading DSP" by Richard lyons.
>
> On the topic on DFT it says that the max amplitude/gain of the main lobe
> in the magnitude response of DFT increases with N ( N-point DFT) and the
> bandwidth reduces. Makes sense.
>
> The it goeson derive the DFT of a rectangular function whose magnitude
> rsponse is simalar to that of a sinc function. Now it sates there that
>
> " main lobe width is 2N/K " on page 97. Though the derivation on how it
> reached 2N/K makes perfect sense to me but it now leads to me to conclude
> that width  of main lobe is increasing with N (N point DFT).
>
> Both seems contarxting to each other. I ma sure there is something I am
> missing out on. Would be great if somebody can throw a light on it.

There are 2 ways to increase N.  One is to sample for a longer
interval of time, the other is to sample faster during the same
interval of time.  Think about how the DFT results might differ
for those two ways, given a constant f(t) time domain function.


IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M

On 24 Feb, 14:03, "mudraraksha" <jindal.vi...@gmail.com> wrote:
> Hello,
>
> I am kind of new to DSP and presently going through the book "
> Understnading DSP" by Richard lyons.
>
> On the topic on DFT it says that the max amplitude/gain of the main lobe
> in the magnitude response of DFT increases with N ( N-point DFT) and the
> bandwidth reduces. Makes sense.
>
> The it goeson derive the DFT of a rectangular function whose magnitude
> rsponse is simalar to that of a sinc function. Now it sates there that
>
> " main lobe width is 2N/K " on page 97. Though the derivation on how it
> reached 2N/K makes perfect sense to me but it now leads to me to conclude
> that width  of main lobe is increasing with N (N point DFT).
>
> Both seems contarxting to each other. I ma sure there is something I am
> missing out on. Would be great if somebody can throw a light on it.

You need to keep track of what N and K represent: N is the total
number of
elements in the DFT, K is the number of non-zero (unit) elements in
the DFT.

Keep N fixed and increase K, and the lobe width becomes smaller. Keep
K fixed and increase N, and the lobe width increases.

Rune

On Feb 26, 1:19 pm, "Rune Allnor" <all...@tele.ntnu.no> wrote:
> On 24 Feb, 14:03, "mudraraksha" <jindal.vi...@gmail.com> wrote:
>
>
>
> > Hello,
>
> > I am kind of new to DSP and presently going through the book "
> > Understnading DSP" by Richard lyons.
>
> > On the topic on DFT it says that the max amplitude/gain of the main lobe
> > in the magnitude response of DFT increases with N ( N-point DFT) and the
> > bandwidth reduces. Makes sense.
>
> > The it goeson derive the DFT of a rectangular function whose magnitude
> > rsponse is simalar to that of a sinc function. Now it sates there that
>
> > " main lobe width is 2N/K " on page 97. Though the derivation on how it
> > reached 2N/K makes perfect sense to me but it now leads to me to conclude
> > that width  of main lobe is increasing with N (N point DFT).
>
> > Both seems contarxting to each other. I ma sure there is something I am
> > missing out on. Would be great if somebody can throw a light on it.
>
> You need to keep track of what N and K represent: N is the total
> number of
> elements in the DFT, K is the number of non-zero (unit) elements in
> the DFT.
>
> Keep N fixed and increase K, and the lobe width becomes smaller. Keep
> K fixed and increase N, and the lobe width increases.
>
> Rune


K (with the sampling frequency)  determines the width of the main
lobe. N determines the number of samples in the main lobe. That is,
there are more samples at a smaller seperation as N increases. Fixed K
(and sampling frequency) and increasing N is zero extention: no change
in size, just interpolation.

Dale B. Dalrymple

On Feb 26, 1:41 pm, "dbd" <d...@ieee.org> wrote:
> On Feb 26, 1:19 pm, "Rune Allnor" <all...@tele.ntnu.no> wrote:
>
>
>
> > On 24 Feb, 14:03, "mudraraksha" <jindal.vi...@gmail.com> wrote:
>
> > > Hello,
>
> > > I am kind of new to DSP and presently going through the book "
> > > Understnading DSP" by Richard lyons.
>
> > > On the topic on DFT it says that the max amplitude/gain of the main lobe
> > > in the magnitude response of DFT increases with N ( N-point DFT) and the
> > > bandwidth reduces. Makes sense.
>
> > > The it goeson derive the DFT of a rectangular function whose magnitude
> > > rsponse is simalar to that of a sinc function. Now it sates there that
>
> > > " main lobe width is 2N/K " on page 97. Though the derivation on how it
> > > reached 2N/K makes perfect sense to me but it now leads to me to conclude
> > > that width  of main lobe is increasing with N (N point DFT).
>
> > > Both seems contarxting to each other. I ma sure there is something I am
> > > missing out on. Would be great if somebody can throw a light on it.
>
> > You need to keep track of what N and K represent: N is the total
> > number of
> > elements in the DFT, K is the number of non-zero (unit) elements in
> > the DFT.
>
> > Keep N fixed and increase K, and the lobe width becomes smaller. Keep
> > K fixed and increase N, and the lobe width increases.
>
> > Rune
>
> K (with the sampling frequency)  determines the width of the main
> lobe. N determines the number of samples in the main lobe. That is,
> there are more samples at a smaller seperation as N increases. Fixed K
> (and sampling frequency) and increasing N is zero extention: no change
> in size, just interpolation.

The word "size" above is ambiguous.  Zero padding doesn't increase
the lobe's size in absolute frequency, but does increase the size in
terms of the number of frequency samples (e.g. by all the added
interpolated samples).


IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M