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From: Jerry Avins on 11 Aug 2010 22:18
On 8/11/2010 9:54 PM, Steve Pope wrote:
> Fred Marshall<fmarshall_xremove_the_xs(a)xacm.org> wrote:
>
>> Jerry Avins wrote:
>
>>> A pulse shaped like a raised cosine in the time domain has
>>> much less "splatter" -- broadband energy -- than a rectangular pulse of
>>> the same width. What has that to do with windowing the data fed to a DFT
>>> routine?
>
>> Jerry,
>
>> Oh, I'd say "everything"! :-)
>
> I'd tend to agree. And it depends upon why you are applying a DFT.
>
> If you are trying to estimate the relative amplitudes and phases
> of expected components within the passband of a signal, often there is
> no windowing applied first.
>
> If instead you are trying to find out how much unwanted energy there
> is in the stopband, outside of the signal of interest, you pretty much
> have to apply a window.
>
> (The above is of course a very broad generalization, to which there
> are many exceptions, but I think it holds up pretty often.)

The window used in windowed sinc filter design and for applying to data
to be DFTed obey the same underlying math. It isn't very useful to apply
that math to the window used to soften the edges of square pulses --
think keyed CW -- thus controlling adjacent-channel interference.

Jerry
--
Engineering is the art of making what you want from things you can get.
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From: Fred Marshall on 12 Aug 2010 15:43
Jerry Avins wrote:

>
> The window used in windowed sinc filter design and for applying to data
> to be DFTed obey the same underlying math. It isn't very useful to apply
> that math to the window used to soften the edges of square pulses --
> think keyed CW -- thus controlling adjacent-channel interference.
>
> Jerry

Jerry,

I still think it does. To soften the edges of a square pulse, one way
to think of doing that (about as simple a way as possible) is to
convolve the square pulse with another, shorter square pulse - giving
ramped edges.

The Fourier Transform of the shorter square pulse is a broad sinc
relative to the narrower sinc of the original square pulse. It has the
effect of reducing adjacent-cahnnel interference.

Obviously we use better window functions than a square pulse but I hope
this conveys the idea as clearly as possible.

Windowing in filter design is the Fourier Transform dual of the pulse
shaping process:
- A perfect lowpass or bandpass filter is the dual of the original
rectangular pulse and
- removing the "unrealistic" time response or making it FIR through
multiplication of a temporal window is the dual of controlling
adjacent-channel interference.

So, maybe the perspective is in *how* the pulse shaping is done?

Fred
From: Eric Jacobsen on 12 Aug 2010 18:21
On Thu, 12 Aug 2010 12:43:08 -0700, Fred Marshall
<fmarshall_xremove_the_xs(a)xacm.org> wrote:

>Jerry Avins wrote:
>
>>
>> The window used in windowed sinc filter design and for applying to data
>> to be DFTed obey the same underlying math. It isn't very useful to apply
>> that math to the window used to soften the edges of square pulses --
>> think keyed CW -- thus controlling adjacent-channel interference.
>>
>> Jerry
>
>Jerry,
>
>I still think it does. To soften the edges of a square pulse, one way
>to think of doing that (about as simple a way as possible) is to
>convolve the square pulse with another, shorter square pulse - giving
>ramped edges.
>
>The Fourier Transform of the shorter square pulse is a broad sinc
>relative to the narrower sinc of the original square pulse. It has the
>effect of reducing adjacent-cahnnel interference.
>
>Obviously we use better window functions than a square pulse but I hope
>this conveys the idea as clearly as possible.
>
>Windowing in filter design is the Fourier Transform dual of the pulse
>shaping process:
>- A perfect lowpass or bandpass filter is the dual of the original
>rectangular pulse and
>- removing the "unrealistic" time response or making it FIR through
>multiplication of a temporal window is the dual of controlling
>adjacent-channel interference.
>
>So, maybe the perspective is in *how* the pulse shaping is done?
>
>Fred

I'm not sure if this is what you're thinking about, but as I mentioned
to the OP, "Raised Cosine" as applied to FFT and frequency analysis
windowing (or even filter design) is not the same thing as "Raised
Cosine" as applied to pulse-shaping for Nyquist filters (i.e., matched
filters) for communication systems.

Wikipedia is useful to illustrate the difference. For window
functions, the most relevant windows are the Hann and Hamming:

http://en.wikipedia.org/wiki/Window_function

Note that the idea is just to take a cycle of a cosine function and
adjust the amplitude and DC offset. That becomes the "Raised Cosine"
window function.

For a comm system pulse-shaping application the function, and
interpretation, is not the same:

http://en.wikipedia.org/wiki/Raised-cosine_filter

Note that the argument of the cosine function is very different, and
contains beta, the so-called "rolloff factor" or "excess bandwidth"
metric. This beta function takes the center passband of the filter
and stretches it out so that there is a flat portion in the middle.
This most resembles the Tukey window in the first Wiki link on window
functions, but it's still not quite the same thing.

In the pulse-shaping example the "Raised Cosine" function is directly
describing the spectrum of the signal and has nothing to do with
resolution or sidelobe control. So the basic use of the function
also differs from the spectral windowing application.

The idea of the spectral and time-domain dualities still holds for
both cases, and there is some conceptual overlap, but that's
fundamental for pretty much anything.

It's clear that in both applications "Raised Cosine" is not an
inappropriate name for the functions. It is just unfortunate that
the same name being applied for two different things (gosh, that never
happens otherwise in DSP ;) ) causes regular confusion.


Eric Jacobsen
Minister of Algorithms
Abineau Communications
http://www.abineau.com
From: Fred Marshall on 12 Aug 2010 22:17
Eric Jacobsen wrote:
> On Thu, 12 Aug 2010 12:43:08 -0700, Fred Marshall
> <fmarshall_xremove_the_xs(a)xacm.org> wrote:
>
>> Jerry Avins wrote:
>>
>>> The window used in windowed sinc filter design and for applying to data
>>> to be DFTed obey the same underlying math. It isn't very useful to apply
>>> that math to the window used to soften the edges of square pulses --
>>> think keyed CW -- thus controlling adjacent-channel interference.
>>>
>>> Jerry
>> Jerry,
>>
>> I still think it does. To soften the edges of a square pulse, one way
>> to think of doing that (about as simple a way as possible) is to
>> convolve the square pulse with another, shorter square pulse - giving
>> ramped edges.
>>
>> The Fourier Transform of the shorter square pulse is a broad sinc
>> relative to the narrower sinc of the original square pulse. It has the
>> effect of reducing adjacent-cahnnel interference.
>>
>> Obviously we use better window functions than a square pulse but I hope
>> this conveys the idea as clearly as possible.
>>
>> Windowing in filter design is the Fourier Transform dual of the pulse
>> shaping process:
>> - A perfect lowpass or bandpass filter is the dual of the original
>> rectangular pulse and
>> - removing the "unrealistic" time response or making it FIR through
>> multiplication of a temporal window is the dual of controlling
>> adjacent-channel interference.
>>
>> So, maybe the perspective is in *how* the pulse shaping is done?
>>
>> Fred
>
> I'm not sure if this is what you're thinking about, but as I mentioned
> to the OP, "Raised Cosine" as applied to FFT and frequency analysis
> windowing (or even filter design) is not the same thing as "Raised
> Cosine" as applied to pulse-shaping for Nyquist filters (i.e., matched
> filters) for communication systems.
>
> Wikipedia is useful to illustrate the difference. For window
> functions, the most relevant windows are the Hann and Hamming:
>
> http://en.wikipedia.org/wiki/Window_function
>
> Note that the idea is just to take a cycle of a cosine function and
> adjust the amplitude and DC offset. That becomes the "Raised Cosine"
> window function.
>
> For a comm system pulse-shaping application the function, and
> interpretation, is not the same:
>
> http://en.wikipedia.org/wiki/Raised-cosine_filter
>
> Note that the argument of the cosine function is very different, and
> contains beta, the so-called "rolloff factor" or "excess bandwidth"
> metric. This beta function takes the center passband of the filter
> and stretches it out so that there is a flat portion in the middle.
> This most resembles the Tukey window in the first Wiki link on window
> functions, but it's still not quite the same thing.
>
> In the pulse-shaping example the "Raised Cosine" function is directly
> describing the spectrum of the signal and has nothing to do with
> resolution or sidelobe control. So the basic use of the function
> also differs from the spectral windowing application.
>
> The idea of the spectral and time-domain dualities still holds for
> both cases, and there is some conceptual overlap, but that's
> fundamental for pretty much anything.
>
> It's clear that in both applications "Raised Cosine" is not an
> inappropriate name for the functions. It is just unfortunate that
> the same name being applied for two different things (gosh, that never
> happens otherwise in DSP ;) ) causes regular confusion.
>
>
> Eric Jacobsen
> Minister of Algorithms
> Abineau Communications
> http://www.abineau.com

Eric,

Yes. I understood that once you'd mentioned it. It's a good point that
I'd never particularly focused on.

Fred
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