On 8/11/2010 9:54 PM, Steve Pope wrote:> Fred Marshall<fmarshall_xremove_the_xs@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. �����������������������������������������������������������������������
FFT Windowing and Transmit Pulse Shaping
Started by ●August 11, 2010
Reply by ●August 11, 20102010-08-11
Reply by ●August 12, 20102010-08-12
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. > > JerryJerry, 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
Reply by ●August 12, 20102010-08-12
On Thu, 12 Aug 2010 12:43:08 -0700, Fred Marshall <fmarshall_xremove_the_xs@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? > >FredI'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
Reply by ●August 12, 20102010-08-12
Eric Jacobsen wrote:> On Thu, 12 Aug 2010 12:43:08 -0700, Fred Marshall > <fmarshall_xremove_the_xs@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.comEric, Yes. I understood that once you'd mentioned it. It's a good point that I'd never particularly focused on. Fred
