Does anyone know how sidelobe falloff rate in Table 1 is calculated?

Harris: "On the Use of Windows for Harmonic Analysis with the Discrete Fourier
Transform"

I cannot seem to get these values.
How many sidelobes are used? Is it a linear regression to the first few side lobes?  

Thanks in advance, Jim

On Oct 1, 2:46 pm, Jim <james.goldf...@gmail.com> wrote:
> Does anyone know how sidelobe falloff rate in Table 1 is calculated?
>
> Harris: "On the Use of Windows for Harmonic Analysis with the Discrete Fourier
Transform"
>
> I cannot seem to get these values.
> How many sidelobes are used? Is it a linear regression to the first few side lobes?
>
> Thanks in advance, Jim


The following might help:

Albert H. Nuttal, “Some Windows with Very Good Sidelobe Behavior,”
IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol.
ASSP-29, No. 1, February 1981, pps. 84-91.

He points out a number of corrections/additions to the Harris paper.
As for the sidelobe levels, see eq. (18) and the text in the above
paper.

Kevin McGee

(P.S. I might add more later)

On Monday, October 1, 2012 11:46:21 AM UTC-7, Jim wrote:
> Does anyone know how sidelobe falloff rate in Table 1 is calculated?

Sidelobe rolloff values are the asymptotic rates for sidelobes distant from the mainlobe. Some
papers give both a rolloff and an offset. See the "bin 1 offset" in the tables in:

http://www.compdsp.com/presentations/Dalrymple/dbd.pdf

Page 7 has a figure for the von Hann window with the line drawn in black from the first bin from
DC to the peaks of the distant sidelobes.

Dale B. Dalrymple

On 10/1/2012 11:46 AM, Jim wrote:
> Does anyone know how sidelobe falloff rate in Table 1 is calculated?
>
> Harris: "On the Use of Windows for Harmonic Analysis with the Discrete Fourier
Transform"
>
> I cannot seem to get these values.
> How many sidelobes are used? Is it a linear regression to the first few side lobes?
>
> Thanks in advance, Jim
>

Not a direct answer to your question but certainly connected and maybe 
helpful:

A general mathematical relationship according to Taylor has the sidelobe 
falloff rate as 1/f, 1/f^N, etc. according to how many derivatives of 
the window are zero at the edges.

Another observation is that if you use combinations of sincs as the 
basis set for constructing a filter [or in this case, the FT of a window 
- such as with a Remez algorithm]  that are weighted by the binomial 
coefficients then the decay of the sum becomes more rapid as the number 
of terms is increased.
[One can use sincs as basis functions instead of cosines - which is 
central to the windowing method of filter design].
As you increase the number of terms for each of the composite basis 
functions, the number of degrees of freedom for the design is decreased 
in exchange for the rate of decay.

Fred

On 10/2/12 12:52 AM, dbd wrote:
> On Monday, October 1, 2012 11:46:21 AM UTC-7, Jim wrote:
>> Does anyone know how sidelobe falloff rate in Table 1 is calculated?
>
> Sidelobe rolloff values are the asymptotic rates for sidelobes distant from the mainlobe.
Some papers give both a rolloff and an offset. See the "bin 1 offset" in the tables
in:
>
> http://www.compdsp.com/presentations/Dalrymple/dbd.pdf
>
> Page 7 has a figure for the von Hann window with the line drawn in black from the first bin
from DC to the peaks of the distant sidelobes.
>

so Dale, taking a look at your paper makes me regret i missed the last 
comp.dsp conference (i actually missed both of them).

here is an observation and a sorta rhetorical question for y'all.  it 
has to do with the slope, in dB per octave or dB per decade of the 
envelope or "skirt" of the sidelobes of a window's frequency response 
(in dB vs. log frequency).

note that both the Rectangular and Hamming window has a slope of -6 
dB/oct (or -20 dB/dec) although the Hamming window has a much better 
constant dB drop of the envelope than the rectangular.  what do the 
Rectangular and Hamming windows have in common in their time-domain 
properties that might be behind this?

Rect:

    w(t)  =  rect(t/2)

Hamming:

    w(t) = ( 0.54 + 0.46 cos(pi*t) ) rect(t/2)


then note that the slope of the sidelobe envelope for the Hann window is 
-18 dB/oct (or -60 dB/dec).


Hann:

    w(t) = ( 1/2 + 1/2 cos(pi*t) ) rect(t/2)


now, i have proposed a window to be used for splicing that i informally 
dubbed the "Flattened Hann" window that is defined as

Flattened Hann:

    w(t) = ( 1/2 + 9/8 cos(pi*t) - 1/8 cos(3*pi*t) ) rect(t/2)


Now, can you guys guess what the slope of the sidelobe rolloff envelope 
would be for this window?  i will postulate -30 dB/oct or -100 dB/dec. 
can you guess why (from a time-domain property) and how to extend the 
concept to get steeper rolloffs?  and what the tradeoff is?



-- 

r b-j                  r...@audioimagination.com

"Imagination is more important than knowledge."

On 10/2/2012 11:39 AM, robert bristow-johnson wrote:
> On 10/2/12 12:52 AM, dbd wrote:
>> On Monday, October 1, 2012 11:46:21 AM UTC-7, Jim wrote:
>>> Does anyone know how sidelobe falloff rate in Table 1 is calculated?
>>
>> Sidelobe rolloff values are the asymptotic rates for sidelobes distant
>> from the mainlobe. Some papers give both a rolloff and an offset. See
>> the "bin 1 offset" in the tables in:
>>
>> http://www.compdsp.com/presentations/Dalrymple/dbd.pdf
>>
>> Page 7 has a figure for the von Hann window with the line drawn in
>> black from the first bin from DC to the peaks of the distant sidelobes.
>>
>
> so Dale, taking a look at your paper makes me regret i missed the last
> comp.dsp conference (i actually missed both of them).
>
> here is an observation and a sorta rhetorical question for y'all.  it
> has to do with the slope, in dB per octave or dB per decade of the
> envelope or "skirt" of the sidelobes of a window's frequency response
> (in dB vs. log frequency).
>
> note that both the Rectangular and Hamming window has a slope of -6
> dB/oct (or -20 dB/dec) although the Hamming window has a much better
> constant dB drop of the envelope than the rectangular.  what do the
> Rectangular and Hamming windows have in common in their time-domain
> properties that might be behind this?
>
> Rect:
>
>     w(t)  =  rect(t/2)
>
> Hamming:
>
>     w(t) = ( 0.54 + 0.46 cos(pi*t) ) rect(t/2)
>
>
> then note that the slope of the sidelobe envelope for the Hann window is
> -18 dB/oct (or -60 dB/dec).
>
>
> Hann:
>
>     w(t) = ( 1/2 + 1/2 cos(pi*t) ) rect(t/2)
>
>
> now, i have proposed a window to be used for splicing that i informally
> dubbed the "Flattened Hann" window that is defined as
>
> Flattened Hann:
>
>     w(t) = ( 1/2 + 9/8 cos(pi*t) - 1/8 cos(3*pi*t) ) rect(t/2)
>
>
> Now, can you guys guess what the slope of the sidelobe rolloff envelope
> would be for this window?  i will postulate -30 dB/oct or -100 dB/dec.
> can you guess why (from a time-domain property) and how to extend the
> concept to get steeper rolloffs?  and what the tradeoff is?
>
>
>

In general, it's what I said.
The rectangular and the Hamming have the same slope properties at the 
edges; so, they have the same sidelobe decay rate.
The Hann has a double zero at the edges.
etc.

One way to think of it is this:
There is a sinusoid in the FT of the window funcion that is represented 
by the samples at the edges of the window.
That sinusoid is the basis for the sidelobes.
If the sample is zero at the edges then that sinusoid has zero value but 
its time derivative has value which shows up as a decayed version.
The more zero derivatives there are, the faster the decay.

Taylor figured out the actual relationships going from rect > 1/f
then on to the others....

Fred

Fred

On Tue, 02 Oct 2012 14:39:10 -0400, robert bristow-johnson
<r...@audioimagination.com> wrote:

>note that both the Rectangular and Hamming window has a slope of -6 
>dB/oct (or -20 dB/dec) although the Hamming window has a much better 
>constant dB drop of the envelope than the rectangular.  what do the 
>Rectangular and Hamming windows have in common in their time-domain 
>properties that might be behind this?

Discontinuity in the "0th" derivative.

>Now, can you guys guess what the slope of the sidelobe rolloff envelope 
>would be for this window?  i will postulate -30 dB/oct or -100 dB/dec. 
>can you guess why (from a time-domain property) and how to extend the 
>concept to get steeper rolloffs?  and what the tradeoff is?

Discontinuity in the 1st derivative (triangular window) -> 12dB/oct.

Discontinuity in the 2nd derivative (Hann) -> 18dB/oct.

Etc.

On Oct 2, 4:43 pm, Greg Berchin <gjberc...@chatter.net.invalid> wrote:
> On Tue, 02 Oct 2012 14:39:10 -0400, robert bristow-johnson
>
> <r...@audioimagination.com> wrote:
> >note that both the Rectangular and Hamming window has a slope of -6
> >dB/oct (or -20 dB/dec) although the Hamming window has a much better
> >constant dB drop of the envelope than the rectangular.  what do the
> >Rectangular and Hamming windows have in common in their time-domain
> >properties that might be behind this?
>
> Discontinuity in the "0th" derivative.
>
> >Now, can you guys guess what the slope of the sidelobe rolloff envelope
> >would be for this window?  i will postulate -30 dB/oct or -100 dB/dec.
> >can you guess why (from a time-domain property) and how to extend the
> >concept to get steeper rolloffs?  and what the tradeoff is?
>
> Discontinuity in the 1st derivative (triangular window) -> 12dB/oct.
>
> Discontinuity in the 2nd derivative (Hann) -> 18dB/oct.
>
> Etc.

Papoulis provides a derivation of this property in "Signal Analysis"
under Section 3-B Asymptotic Properties of Fourier Transforms. It
states that if the derivatives of order up to n exist and are of
bounded variation in the interval (-inf,inf) then the Fourier
transform tends to zero at least as fast as 1/f^n+1 as |f|-> inf

Some notes:
1) It's derived for continuous systems (not sampled)
2) It states at least as fast - so it could be faster
3) It is asymptotic - it doesn't say how close to infinity you have to
be to see this.

I believe there was also a paper in the IEEE that discussed this type
of thing, but I don't have the reference handy.

Cheers,
David

On 10/2/12 4:43 PM, Greg Berchin wrote:
> On Tue, 02 Oct 2012 14:39:10 -0400, robert bristow-johnson
> <r...@audioimagination.com>  wrote:
>
>> note that both the Rectangular and Hamming window has a slope of -6
>> dB/oct (or -20 dB/dec) although the Hamming window has a much better
>> constant dB drop of the envelope than the rectangular.  what do the
>> Rectangular and Hamming windows have in common in their time-domain
>> properties that might be behind this?
>
> Discontinuity in the "0th" derivative.
>
>> Now, can you guys guess what the slope of the sidelobe rolloff envelope
>> would be for this window?  i will postulate -30 dB/oct or -100 dB/dec.
>> can you guess why (from a time-domain property) and how to extend the
>> concept to get steeper rolloffs?  and what the tradeoff is?
>
> Discontinuity in the 1st derivative (triangular window) ->  12dB/oct.
>
> Discontinuity in the 2nd derivative (Hann) ->  18dB/oct.
>
> Etc.

so here's the first of the Etc.:


    w(t) = ( 1/2 + 9/8 cos(pi*t) - 1/8 cos(3*pi*t) ) rect(t/2)


BTW, i think the wavelet/filter-bank people look at this same function 
in the frequency-domain as a complementary filter prototype which is an 
FIR with 5 taps; -1/16, 9/16, 1/2, 9/16, -1/16 .

but the reason i liked it as a window is because it's continuous through 
the 2nd derivative.  that seems to me to make for a good splice and the 
effect was in the frequency domain is that a frequency component has 
little splatter from the splice at distant frequencies.

-- 

r b-j                  r...@audioimagination.com

"Imagination is more important than knowledge."