Frequency response of Sinc function

Hi everyone, 
I am trying to understand more the practical aspects of FFT. 
I was reading some articles, like the one posted below:
http://zone.ni.com/devzone/cda/ph/p/id/200

And, in this article the duality relation for sinc function is illustrated
in figure 3.14
Which the definition is similar to definitions that I see in my text books.


However, in the same article, figure 3.19 shows a different relationship !

In fact, there is another article, which has the same relationship as the
one in figure 3.19

http://www.ni.com/white-paper/3876/en
  
Now, I am confused ! Which one is the right one?
I would personally think that the second one (figure 3.19) makes more
sense, because the Sinc's main-lobe width corresponds to the inverse of the
frequency response width, which this makes more sense to me. 
Now which one is the correct drawing? figure 3.14, or figure 3.19?

Can someone explain please!

Thanks, 
--Rudy




	 

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On Tue, 23 Jul 2013 17:51:39 -0500, "rudykeram" <51467@dsprelated>
wrote:

>Hi everyone, 
>I am trying to understand more the practical aspects of FFT. 
>I was reading some articles, like the one posted below:
>http://zone.ni.com/devzone/cda/ph/p/id/200
>
>And, in this article the duality relation for sinc function is illustrated
>in figure 3.14
>Which the definition is similar to definitions that I see in my text books.
>
>
>However, in the same article, figure 3.19 shows a different relationship !
>
>In fact, there is another article, which has the same relationship as the
>one in figure 3.19
>
>http://www.ni.com/white-paper/3876/en
>  
>Now, I am confused ! Which one is the right one?
>I would personally think that the second one (figure 3.19) makes more
>sense, because the Sinc's main-lobe width corresponds to the inverse of the
>frequency response width, which this makes more sense to me. 
>Now which one is the correct drawing? figure 3.14, or figure 3.19?
>
>Can someone explain please!
>
>Thanks, 
>--Rudy

They all show the same thing.    The only difference is that Fig.
3.14, which I'll call the Pi figure (for obvious reasons), shows the
scaling in terms of the bit rate for a communications pulse.   In
other words, 1/rb = T/2.

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com

>On Tue, 23 Jul 2013 17:51:39 -0500, "rudykeram" <51467@dsprelated>
>wrote:
>
>>Hi everyone, 
>>I am trying to understand more the practical aspects of FFT. 
>>I was reading some articles, like the one posted below:
>>http://zone.ni.com/devzone/cda/ph/p/id/200
>>
>>And, in this article the duality relation for sinc function is
illustrated
>>in figure 3.14
>>Which the definition is similar to definitions that I see in my text
books.
>>
>>
>>However, in the same article, figure 3.19 shows a different relationship
!
>>
>>In fact, there is another article, which has the same relationship as
the
>>one in figure 3.19
>>
>>http://www.ni.com/white-paper/3876/en
>>  
>>Now, I am confused ! Which one is the right one?
>>I would personally think that the second one (figure 3.19) makes more
>>sense, because the Sinc's main-lobe width corresponds to the inverse of
the
>>frequency response width, which this makes more sense to me. 
>>Now which one is the correct drawing? figure 3.14, or figure 3.19?
>>
>>Can someone explain please!
>>
>>Thanks, 
>>--Rudy
>
>They all show the same thing.    The only difference is that Fig.
>3.14, which I'll call the Pi figure (for obvious reasons), shows the
>scaling in terms of the bit rate for a communications pulse.   In
>other words, 1/rb = T/2.
>
>Eric Jacobsen
>Anchor Hill Communications
>http://www.anchorhill.com
>

Eric, 
Thanks for the reply. I think I am still having problem. 
Let's look at the two figures completely separate, as if they are not
related. 
If I just look at figure 3.19, I can see that in the time domain, the width
of the main-lobe is (T).  Similarly, if I look at the frequency response, I
will see that the entire width of the response is (1/2T)+(1/2T) = (1/T)

So, my interpretation of this figure is that the width of the frequency
response is exactly the inverse of the width of the main-lobe.


Now, if I do the same on figure 3.14, 
I can see that the main-lobe width is (2/rb), but the width of its
frequency response is (rb/2)+(rb/2)=rb

So, by looking at this figure, it tells me that the width of the frequency
response is TWICE the inverse of the width of the main-lobe !!

I hope I was able to address my confusion properly. 

Thanks, 
--Rudy

	 

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On Tue, 23 Jul 2013 18:51:52 -0500, "rudykeram" <51467@dsprelated>
wrote:

>>On Tue, 23 Jul 2013 17:51:39 -0500, "rudykeram" <51467@dsprelated>
>>wrote:
>>
>>>Hi everyone, 
>>>I am trying to understand more the practical aspects of FFT. 
>>>I was reading some articles, like the one posted below:
>>>http://zone.ni.com/devzone/cda/ph/p/id/200
>>>
>>>And, in this article the duality relation for sinc function is
>illustrated
>>>in figure 3.14
>>>Which the definition is similar to definitions that I see in my text
>books.
>>>
>>>
>>>However, in the same article, figure 3.19 shows a different relationship
>!
>>>
>>>In fact, there is another article, which has the same relationship as
>the
>>>one in figure 3.19
>>>
>>>http://www.ni.com/white-paper/3876/en
>>>  
>>>Now, I am confused ! Which one is the right one?
>>>I would personally think that the second one (figure 3.19) makes more
>>>sense, because the Sinc's main-lobe width corresponds to the inverse of
>the
>>>frequency response width, which this makes more sense to me. 
>>>Now which one is the correct drawing? figure 3.14, or figure 3.19?
>>>
>>>Can someone explain please!
>>>
>>>Thanks, 
>>>--Rudy
>>
>>They all show the same thing.    The only difference is that Fig.
>>3.14, which I'll call the Pi figure (for obvious reasons), shows the
>>scaling in terms of the bit rate for a communications pulse.   In
>>other words, 1/rb = T/2.
>>
>>Eric Jacobsen
>>Anchor Hill Communications
>>http://www.anchorhill.com
>>
>
>Eric, 
>Thanks for the reply. I think I am still having problem. 
>Let's look at the two figures completely separate, as if they are not
>related. 
>If I just look at figure 3.19, I can see that in the time domain, the width
>of the main-lobe is (T).  Similarly, if I look at the frequency response, I
>will see that the entire width of the response is (1/2T)+(1/2T) = (1/T)
>
>So, my interpretation of this figure is that the width of the frequency
>response is exactly the inverse of the width of the main-lobe.
>
>
>Now, if I do the same on figure 3.14, 
>I can see that the main-lobe width is (2/rb), but the width of its
>frequency response is (rb/2)+(rb/2)=rb
>
>So, by looking at this figure, it tells me that the width of the frequency
>response is TWICE the inverse of the width of the main-lobe !!
>
>I hope I was able to address my confusion properly. 
>
>Thanks, 
>--Rudy

Good catch.   I'd missed that.   Fig. 3.19 has a disparity of a factor
of two.    Fig. 3.14 - 3.16 are correct.

You can search on terms like Fourier Transform Pairs and then look for
rectangular function or sinc function, and you should get consistent
info.

Here's a quick one:

http://www.thefouriertransform.com/pairs/box.php


Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com

On Tuesday, July 23, 2013 9:20:02 PM UTC-5, Eric Jacobsen wrote:

> 
> You can search on terms like Fourier Transform Pairs and then look for
> 
> rectangular function or sinc function, and you should get consistent
> 
> info.

One thing to be careful about when consulting tables
of Fourier Transform pairs is to remember that there
are two different definitions of the sinc function.
Typically (though not always), those who define Fourier
Transforms in terms of Hertzian frequency f use

sinc(t) = [sin(pi t)]/(pi t)

while those who use radian frequency omega use

sinc(t) = [sin(t)]/t.

Sometimes those who insist on one definition
consider those who use the other definition
to be in a state of mortal sin (or sinc, heh heh)
while more pragmatic folks just live and let
live, and don't trust tables completely till they
have checked what sinc means.

Dilip Sarwate

On Wed, 24 Jul 2013 05:57:14 -0700 (PDT), dvsarwate
<dvsarwate@yahoo.com> wrote:

>On Tuesday, July 23, 2013 9:20:02 PM UTC-5, Eric Jacobsen wrote:
>
>> 
>> You can search on terms like Fourier Transform Pairs and then look for
>> 
>> rectangular function or sinc function, and you should get consistent
>> 
>> info.
>
>One thing to be careful about when consulting tables
>of Fourier Transform pairs is to remember that there
>are two different definitions of the sinc function.
>Typically (though not always), those who define Fourier
>Transforms in terms of Hertzian frequency f use
>
>sinc(t) = [sin(pi t)]/(pi t)
>
>while those who use radian frequency omega use
>
>sinc(t) = [sin(t)]/t.
>
>Sometimes those who insist on one definition
>consider those who use the other definition
>to be in a state of mortal sin (or sinc, heh heh)
>while more pragmatic folks just live and let
>live, and don't trust tables completely till they
>have checked what sinc means.
>
>Dilip Sarwate
>

That's also important when calling sinc() functions that you didn't
write yourself.   Found that out recently, that Matlab/Octave uses the
"normalized sinc" which includes the pi factor in the argument.   Took
me a little while to sort that out, after having done it the other way
for a long time.


Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com

>On Wed, 24 Jul 2013 05:57:14 -0700 (PDT), dvsarwate
><dvsarwate@yahoo.com> wrote:
>
>>On Tuesday, July 23, 2013 9:20:02 PM UTC-5, Eric Jacobsen wrote:
>>
>>> 
>>> You can search on terms like Fourier Transform Pairs and then look for
>>> 
>>> rectangular function or sinc function, and you should get consistent
>>> 
>>> info.
>>
>>One thing to be careful about when consulting tables
>>of Fourier Transform pairs is to remember that there
>>are two different definitions of the sinc function.
>>Typically (though not always), those who define Fourier
>>Transforms in terms of Hertzian frequency f use
>>
>>sinc(t) = [sin(pi t)]/(pi t)
>>
>>while those who use radian frequency omega use
>>
>>sinc(t) = [sin(t)]/t.
>>
>>Sometimes those who insist on one definition
>>consider those who use the other definition
>>to be in a state of mortal sin (or sinc, heh heh)
>>while more pragmatic folks just live and let
>>live, and don't trust tables completely till they
>>have checked what sinc means.
>>
>>Dilip Sarwate
>>
>
>That's also important when calling sinc() functions that you didn't
>write yourself.   Found that out recently, that Matlab/Octave uses the
>"normalized sinc" which includes the pi factor in the argument.   Took
>me a little while to sort that out, after having done it the other way
>for a long time.
>
>
>Eric Jacobsen
>Anchor Hill Communications
>http://www.anchorhill.com
>

Thanks for the explanation. 
I just wanted to follow up just a little bit on this concept. Whatever you
explained so far makes sense.

Now let's refer to the figure that you mentioned:
http://www.thefouriertransform.com/pairs/box.php
 
I know if you look at this mathematically it makes sense. But let's just
look at this from a pure practical angle, as if there is no math involved.

My thinking has always been that frequency domain (1/T) is the inverse of
time domain (T). So, for example, if I have a function in time, that only 
stretches over ONE second in time domain, this should correspond to be 1 Hz
in frequency.

Now, having this vision, I would've thought that the figures in
http://www.ni.com/white-paper/3876/en
and figure 3.19 in  
http://zone.ni.com/devzone/cda/ph/p/id/200
would've made more sense. 
But, we know that these are off by a factor of two (mathematically)!!! 

And the mathematically correct ones are the figures in:
 http://www.thefouriertransform.com/pairs/box.php
and figure 3.15 in
http://zone.ni.com/devzone/cda/ph/p/id/200

I just want to see if there is a visual/practical explanation for this?

Thanks,
--Rudy	 

_____________________________		
Posted through www.DSPRelated.com

On Wed, 24 Jul 2013 10:52:07 -0500, "rudykeram" <51467@dsprelated>
wrote:

>>On Wed, 24 Jul 2013 05:57:14 -0700 (PDT), dvsarwate
>><dvsarwate@yahoo.com> wrote:
>>
>>>On Tuesday, July 23, 2013 9:20:02 PM UTC-5, Eric Jacobsen wrote:
>>>
>>>> 
>>>> You can search on terms like Fourier Transform Pairs and then look for
>>>> 
>>>> rectangular function or sinc function, and you should get consistent
>>>> 
>>>> info.
>>>
>>>One thing to be careful about when consulting tables
>>>of Fourier Transform pairs is to remember that there
>>>are two different definitions of the sinc function.
>>>Typically (though not always), those who define Fourier
>>>Transforms in terms of Hertzian frequency f use
>>>
>>>sinc(t) = [sin(pi t)]/(pi t)
>>>
>>>while those who use radian frequency omega use
>>>
>>>sinc(t) = [sin(t)]/t.
>>>
>>>Sometimes those who insist on one definition
>>>consider those who use the other definition
>>>to be in a state of mortal sin (or sinc, heh heh)
>>>while more pragmatic folks just live and let
>>>live, and don't trust tables completely till they
>>>have checked what sinc means.
>>>
>>>Dilip Sarwate
>>>
>>
>>That's also important when calling sinc() functions that you didn't
>>write yourself.   Found that out recently, that Matlab/Octave uses the
>>"normalized sinc" which includes the pi factor in the argument.   Took
>>me a little while to sort that out, after having done it the other way
>>for a long time.
>>
>>
>>Eric Jacobsen
>>Anchor Hill Communications
>>http://www.anchorhill.com
>>
>
>Thanks for the explanation. 
>I just wanted to follow up just a little bit on this concept. Whatever you
>explained so far makes sense.
>
>Now let's refer to the figure that you mentioned:
>http://www.thefouriertransform.com/pairs/box.php
> 
>I know if you look at this mathematically it makes sense. But let's just
>look at this from a pure practical angle, as if there is no math involved.
>
>My thinking has always been that frequency domain (1/T) is the inverse of
>time domain (T). So, for example, if I have a function in time, that only 
>stretches over ONE second in time domain, this should correspond to be 1 Hz
>in frequency.

A sine wave with a *period* of one second corresponds to a frequency
of 1Hz, not a *duration* of one second. 

Several things should be noted:

For a signal that is not a sine wave, even if it has a period of one
second, it may contain frequencies much, much higher than 1 Hz.

For time-limited signals, generally the shorter they get the broader
the bandwidth they occupy.   An impulse with an infinitely small
duration has an infinite bandwidth.   As the duration increases, the
bandwidth decreases.

The sinc/rectangle transform pair have this behavior.   As the width
of the sinc mainlobe decreases, the bandwidth increases, that is, the
width of the rectangle function increases.


>Now, having this vision, I would've thought that the figures in
>http://www.ni.com/white-paper/3876/en
>and figure 3.19 in  
>http://zone.ni.com/devzone/cda/ph/p/id/200
>would've made more sense. 
>But, we know that these are off by a factor of two (mathematically)!!! 
>
>And the mathematically correct ones are the figures in:
> http://www.thefouriertransform.com/pairs/box.php
>and figure 3.15 in
>http://zone.ni.com/devzone/cda/ph/p/id/200
>
>I just want to see if there is a visual/practical explanation for this?
>
>Thanks,
>--Rudy	 
>
>_____________________________		
>Posted through www.DSPRelated.com

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com

Eric Jacobsen <eric.jacobsen@ieee.org> wrote:

(snip)

> Good catch.   I'd missed that.   Fig. 3.19 has a disparity of a factor
> of two.    Fig. 3.14 - 3.16 are correct.
 
> You can search on terms like Fourier Transform Pairs and then look for
> rectangular function or sinc function, and you should get consistent
> info.

Having not looked at a lot of transform pair tables recently, I
wouldn't be surprised to see some in terms of f, and some in terms
of omega (2 pi f). 

It does seem, though, that sinc(x) is sometimes sin(x)/x and
sometimes sin(pi x)/(pi x). That could confuse some reading of
transform pair tables. (According to one reference, the latter is
usual in DSP.)
 
> Here's a quick one:
 
> http://www.thefouriertransform.com/pairs/box.php

-- glen

On 7/24/13 1:36 PM, glen herrmannsfeldt wrote:
>
> Having not looked at a lot of transform pair tables recently, I
> wouldn't be surprised to see some in terms of f, and some in terms
> of omega (2 pi f).
>
> It does seem, though, that sinc(x) is sometimes sin(x)/x and
> sometimes sin(pi x)/(pi x). That could confuse some reading of
> transform pair tables. (According to one reference, the latter is
> usual in DSP.)
>

from my POV, i became totally convinced of the non-angular frequency 
version of the continuous Fourier transform because of combination of 
scaling (what a *unit* rect() looks like) and the symmetry of the f vs. 
t roles for applying duality.  and then, in that context, a the unit 
rect() and sinc() functions both have a height and area of 1.  it's also 
why the unit gaussian pulse is e^(-pi*t^2).

i would be in favor of an historical convention to forever change the 
definition of sinc(x) to sin(pi*x)/(pi*x) despite the original 
definition in mathematics history.

it's one of these "right things to do", similar to indiscriminately 
using positive or non-positive indices in arrays.  dunno what the 
problem is.


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."