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John Wilkinson October 10th 05 08:48 PM

Receiver bandwidth
 
Hi,
I have bought a 45MHz crystal filter with a bandwidth of +/-10KHz,
according to the spec.
Now does this mean a real bandwidth of 20KHz?

If so when a receiver states a bandwidth of 7KHz, is that +/-7KHz, ie
14KHz?

If I am designing a good communications receiver from 6KHz-30MHz to receive
AM, SSB and CW and want 2 good filters, what are the bandwidths I should
consider?

Thanks,
John.

[email protected] October 10th 05 10:36 PM

On Mon, 10 Oct 2005 20:48:51 +0100, John Wilkinson
wrote:

Hi,
I have bought a 45MHz crystal filter with a bandwidth of +/-10KHz,
according to the spec.
Now does this mean a real bandwidth of 20KHz?


Likely the case. Some spec total bandwidth other specify 6db edge
as distance from the filter center frequency.

If so when a receiver states a bandwidth of 7KHz, is that +/-7KHz, ie
14KHz?


If they didn't put the +/- there then its 7khz total.

If I am designing a good communications receiver from 6KHz-30MHz to receive
AM, SSB and CW and want 2 good filters, what are the bandwidths I should
consider?


AM wide 12-16khz
Am narrow 6-10khz

SSB anywhere from 2-3khz with many around 2.4khz wide Any wider
than 3khz will be poor in crowded bands. I happen to prefer 2.1 to
2.3khz.

CW I've seen 1.4khz all the way down to 200hz most consider
400-600hz adaquate.

In all cases the skirt selectivity usually bandwitdth measured from
the 6 to 60db points are important indicators of filter quality and
any value of 2 or less is good enough and 1.4 would be excellent.
The idea is you'd ike to be able to put the offending signal outside
the bandpass and well attenuated.

What some builders do for CW is use the CW filter and use an peaked
audio filter to narrow the audio band pass. Not quite as effective
but often cheaper.


Allison
KB1GMX


Dale Parfitt October 11th 05 12:39 AM


wrote in message
...
On Mon, 10 Oct 2005 20:48:51 +0100, John Wilkinson
wrote:

Hi,
I have bought a 45MHz crystal filter with a bandwidth of +/-10KHz,
according to the spec.
Now does this mean a real bandwidth of 20KHz?


Likely the case. Some spec total bandwidth other specify 6db edge
as distance from the filter center frequency.

If so when a receiver states a bandwidth of 7KHz, is that +/-7KHz, ie
14KHz?


If they didn't put the +/- there then its 7khz total.

If I am designing a good communications receiver from 6KHz-30MHz to
receive
AM, SSB and CW and want 2 good filters, what are the bandwidths I should
consider?


AM wide 12-16khz
Am narrow 6-10khz

SSB anywhere from 2-3khz with many around 2.4khz wide Any wider
than 3khz will be poor in crowded bands. I happen to prefer 2.1 to
2.3khz.

CW I've seen 1.4khz all the way down to 200hz most consider
400-600hz adaquate.

In all cases the skirt selectivity usually bandwitdth measured from
the 6 to 60db points are important indicators of filter quality and
any value of 2 or less is good enough and 1.4 would be excellent.
The idea is you'd ike to be able to put the offending signal outside
the bandpass and well attenuated.

What some builders do for CW is use the CW filter and use an peaked
audio filter to narrow the audio band pass. Not quite as effective
but often cheaper.


Allison
KB1GMX

Allison's advice is right on the mark.
I recently built an amateur only band HF receiver and chose 6/2.5/0.5 for
the three modes.
I think what Allison meant to say about the peaked audio filter was to use
it in conjunction with the SSB filter for CW selectivity.

73,
Dale W4OP



Ken Scharf October 11th 05 01:20 AM

wrote:
On Mon, 10 Oct 2005 20:48:51 +0100, John Wilkinson
wrote:


Hi,
I have bought a 45MHz crystal filter with a bandwidth of +/-10KHz,
according to the spec.
Now does this mean a real bandwidth of 20KHz?



Likely the case. Some spec total bandwidth other specify 6db edge
as distance from the filter center frequency.


If so when a receiver states a bandwidth of 7KHz, is that +/-7KHz, ie
14KHz?



If they didn't put the +/- there then its 7khz total.


If I am designing a good communications receiver from 6KHz-30MHz to receive
AM, SSB and CW and want 2 good filters, what are the bandwidths I should
consider?



AM wide 12-16khz
Am narrow 6-10khz

SSB anywhere from 2-3khz with many around 2.4khz wide Any wider
than 3khz will be poor in crowded bands. I happen to prefer 2.1 to
2.3khz.

CW I've seen 1.4khz all the way down to 200hz most consider
400-600hz adaquate.

In all cases the skirt selectivity usually bandwitdth measured from
the 6 to 60db points are important indicators of filter quality and
any value of 2 or less is good enough and 1.4 would be excellent.
The idea is you'd ike to be able to put the offending signal outside
the bandpass and well attenuated.

What some builders do for CW is use the CW filter and use an peaked
audio filter to narrow the audio band pass. Not quite as effective
but often cheaper.


Allison
KB1GMX

That 45mhz would make a good 'roofing' filter to use ahead of a 455khz
if filter. Normally using a first if of 45mhz and a second if of 455khz
would result in 'second order' images leaking in, but with a
good roofing filter ahead of the second mixer the problem is solved.

Some years ago, I bought a bunch of 9mhz if filters at the Dayton
Hamvention. They are 3.2khz BW (at the 6db points) filters made by
CF Networks for the Gonset Sidewinder transciever. I think this was
a vhf rig, which would account for the wider bandwidth. However, these
are 8 pole filters, with a shape factor of 1.08 @ 15db down (rises
to 1.23 @ 45db down). With this shape factor, these filters probably
have similar rejection of off frequency qrm as would a 2.4khz filter
with a 2.0 shape factor. Since I have a bunch of them, I could use
2 or even 3 of them in the IF stage (one after the mixer, one before
the detector, and one between stages). This would decrease the
apparent shape factor even more. The nominal carier frequencies for
these filters were 8998.0 khz and 9001.7 khz. With the 3.2khz
bandwidth this put the cariers 250hz outside the stated bandwidth
(-15db point for these filters).

I wonder just how these filters would actually work out in a rig
today. That 3.2khz bandwidth DOES seem a bit wide, but the crazy
shape factor seems to make it worthwhile to try.


[email protected] October 11th 05 12:56 PM

For VHF 3.2 would likely work well enough except if there is a
sporatic E opening and people tend to pile up. Then it's really wide!

For filters I use microprocessor crystals in the ladder configuration.
With the correct shunt C and 4-8 crystals you can make a very fine
crystal filter with bandwisths from 2-400hz to as wide as you may
want. The design process is documented elsewhere (see EMRFD
and the handbook as well as internet). I build for 6 and 2m and have
built very nice ladder filters this way. Also since microprocessor
crystals in the range of 4-20mhz are dirt cheap it's also a help.
Also the higher the frequency the less likely dual conversion is
required to avoid images and allows the selectivity to be closer
to the antenna (better overload performance).

My latest 6m rig uses 12mhz crystals, 8 of them for a 2.3khz
bandwidth at 6db and 3.9khz at 60db with symetrical skirts.


Allison
KB!GMX




On Mon, 10 Oct 2005 20:20:05 -0400, Ken Scharf
wrote:

wrote:
On Mon, 10 Oct 2005 20:48:51 +0100, John Wilkinson
wrote:


Hi,
I have bought a 45MHz crystal filter with a bandwidth of +/-10KHz,
according to the spec.
Now does this mean a real bandwidth of 20KHz?



Likely the case. Some spec total bandwidth other specify 6db edge
as distance from the filter center frequency.


If so when a receiver states a bandwidth of 7KHz, is that +/-7KHz, ie
14KHz?



If they didn't put the +/- there then its 7khz total.


If I am designing a good communications receiver from 6KHz-30MHz to receive
AM, SSB and CW and want 2 good filters, what are the bandwidths I should
consider?



AM wide 12-16khz
Am narrow 6-10khz

SSB anywhere from 2-3khz with many around 2.4khz wide Any wider
than 3khz will be poor in crowded bands. I happen to prefer 2.1 to
2.3khz.

CW I've seen 1.4khz all the way down to 200hz most consider
400-600hz adaquate.

In all cases the skirt selectivity usually bandwitdth measured from
the 6 to 60db points are important indicators of filter quality and
any value of 2 or less is good enough and 1.4 would be excellent.
The idea is you'd ike to be able to put the offending signal outside
the bandpass and well attenuated.

What some builders do for CW is use the CW filter and use an peaked
audio filter to narrow the audio band pass. Not quite as effective
but often cheaper.


Allison
KB1GMX

That 45mhz would make a good 'roofing' filter to use ahead of a 455khz
if filter. Normally using a first if of 45mhz and a second if of 455khz
would result in 'second order' images leaking in, but with a
good roofing filter ahead of the second mixer the problem is solved.

Some years ago, I bought a bunch of 9mhz if filters at the Dayton
Hamvention. They are 3.2khz BW (at the 6db points) filters made by
CF Networks for the Gonset Sidewinder transciever. I think this was
a vhf rig, which would account for the wider bandwidth. However, these
are 8 pole filters, with a shape factor of 1.08 @ 15db down (rises
to 1.23 @ 45db down). With this shape factor, these filters probably
have similar rejection of off frequency qrm as would a 2.4khz filter
with a 2.0 shape factor. Since I have a bunch of them, I could use
2 or even 3 of them in the IF stage (one after the mixer, one before
the detector, and one between stages). This would decrease the
apparent shape factor even more. The nominal carier frequencies for
these filters were 8998.0 khz and 9001.7 khz. With the 3.2khz
bandwidth this put the cariers 250hz outside the stated bandwidth
(-15db point for these filters).

I wonder just how these filters would actually work out in a rig
today. That 3.2khz bandwidth DOES seem a bit wide, but the crazy
shape factor seems to make it worthwhile to try.



Tim Shoppa October 11th 05 02:00 PM

AM wide 12-16khz
Am narrow 6-10khz


Maybe it's my aging male ears but I've never heard an AM transmission
where a 10kHz filter was needed or even desirable. Whenever I switch
on the wider filter I just hear more hiss and static, and no more
fidelity.

I can hear a small difference between 6kHz, 8kHz, and 10kHz with local
AM BCB stations. My preference has always been to abhor hiss so I tend
to go towards the narrower side. I don't think my ears are as
sensitive to hiss as when I was younger but it still bothers me
especially with extended listening.

I am told that the 16kc wide filter on my R-390A was used largely for
when multiple channels were being multiplexed (e.g. multi-RTTY or
multi-voice channels).

Tim.


William E. Sabin October 11th 05 02:28 PM


"Tim Shoppa" wrote in message
oups.com...
AM wide 12-16khz
Am narrow 6-10khz


Maybe it's my aging male ears but I've never heard an AM transmission
where a 10kHz filter was needed or even desirable. Whenever I switch
on the wider filter I just hear more hiss and static, and no more
fidelity.


In my homebrew receiver (see QRZ.COM), for SSB reception using headphones, I
toggle-switch in a modified audio response increase in the upper range of
speech frequencies, 3 dB at 3000 Hz. This greatly improves speech
intelligibility at low audio volume, and this saves my aging ears from
overload, especially when listening to weak signals. It is important to
protect hearing, at any age.

Bill W0IYH



[email protected] October 11th 05 02:39 PM

On 11 Oct 2005 06:00:50 -0700, "Tim Shoppa"
wrote:

AM wide 12-16khz
Am narrow 6-10khz


Maybe it's my aging male ears but I've never heard an AM transmission
where a 10kHz filter was needed or even desirable. Whenever I switch
on the wider filter I just hear more hiss and static, and no more
fidelity.

I can hear a small difference between 6kHz, 8kHz, and 10kHz with local
AM BCB stations. My preference has always been to abhor hiss so I tend
to go towards the narrower side. I don't think my ears are as
sensitive to hiss as when I was younger but it still bothers me
especially with extended listening.


That is because most AM broadcast station use brick wall filters at
around 4-5khz. Use to be a time back in the analog days the roll off
was both higher and gentler. Of course the Beach Boys surf music
was current then.

I also listed two AM filters for that reason. Completeness only.
Though some of the 75M AM window folks (3885khz) insist that
a wide filter is useful.

I am told that the 16kc wide filter on my R-390A was used largely for
when multiple channels were being multiplexed (e.g. multi-RTTY or
multi-voice channels).


That may be but AM used ot be much better fidelity at one time.

Allison
KB!GMX

Martin Potter October 12th 05 02:45 AM


John Wilkinson ) writes:

If I am designing a good communications receiver from 6KHz-30MHz to receive
AM, SSB and CW and want 2 good filters, what are the bandwidths I should
consider?



John, if you really want to listen to CW and FSK signals in the VLF
band (8-30 kHz) then you will find a 100 Hz filter very useful.

.... Martin VE3OAT



[email protected] October 12th 05 08:55 PM

From: "Tim Shoppa" on Tues 11 Oct 2005 06:00

AM wide 12-16khz
Am narrow 6-10khz


Maybe it's my aging male ears but I've never heard an AM transmission
where a 10kHz filter was needed or even desirable. Whenever I switch
on the wider filter I just hear more hiss and static, and no more
fidelity.

I can hear a small difference between 6kHz, 8kHz, and 10kHz with local
AM BCB stations. My preference has always been to abhor hiss so I tend
to go towards the narrower side. I don't think my ears are as
sensitive to hiss as when I was younger but it still bothers me
especially with extended listening.

I am told that the 16kc wide filter on my R-390A was used largely for
when multiple channels were being multiplexed (e.g. multi-RTTY or
multi-voice channels).


As a point of historical reference, the "390" series was intended
for fixed-point to fixed-point communications reception as well as
in the land-forces' AN/GRC-26 truck-mobile HF hut. The wide
bandwidth was intended to be used with "commercial-government"
SSB that had a 12 KHz wide bandwidth to accomodate four 3 KHz
"voice grade" channels frequency multiplexed. Each voice channel
could handle 4 to 8 TTY circuits, each using tone-pairs (one for
Mark, another for Space) via auxilliary "carrier" equipment. A
common arrangement was to have one voice channel as the "order
wire" or command channel, a second voice channel as an "overseas
radiotelephone" circuit, the remaining two voice channels used for
TTY circuits. Depending on the carrier and auxilliary equipment,
TTY tone pairs could be doubled to take care of selective fading
effects common on HF long paths.

For some illustration of actual use of R-390s at a large receiver
site, go to http://kauko.hallikainen.org/history/equipment and look
down the page at "stations." Click on the 1962 communications
station link for a repro of a booklet produced by the Japan Signal
Service Battalion about Army station ADA in Tokyo. ADA shared the
receiver site with the USAF and the antenna field was over dozens
of acres of small farms northwest of Tokyo. PDF is about 10 MB.
I was assigned to the transmitter and control sites at ADA.

The "commercial" 12 KHz wide SSB was standard practice on HF from
the 1930s. Not many left in service now although there are a few
unmistakable continuous-roaring-multi-tone commercial SSB signals
to be heard outside of ham bands. The "roaring" sound comes from
the TTY tone pairs (at least 8 circuits per station).




Ken Scharf October 14th 05 10:16 PM

wrote:
For VHF 3.2 would likely work well enough except if there is a
sporatic E opening and people tend to pile up. Then it's really wide!

For filters I use microprocessor crystals in the ladder configuration.
With the correct shunt C and 4-8 crystals you can make a very fine
crystal filter with bandwisths from 2-400hz to as wide as you may
want. The design process is documented elsewhere (see EMRFD
and the handbook as well as internet). I build for 6 and 2m and have
built very nice ladder filters this way. Also since microprocessor
crystals in the range of 4-20mhz are dirt cheap it's also a help.
Also the higher the frequency the less likely dual conversion is
required to avoid images and allows the selectivity to be closer
to the antenna (better overload performance).

My latest 6m rig uses 12mhz crystals, 8 of them for a 2.3khz
bandwidth at 6db and 3.9khz at 60db with symetrical skirts.


Allison
KB!GMX

I managed to buy over 400 pcs of 8.3886mhz crystals on ebay, for
just pennies each. I am planning on trying to build ladder filters
with them. These crystals are in the larger HC6/u size holders which
are supposed to work better than the miniature size used in the micro-
processor crystals.

First step would be to build the DDS vfo for
the radio since I can program the DDS to function as a sweep generator
for aligning the filter. With the DDS sweeping the output frequency
while providing a sawtooth ramp to drive the scope sweep in step
with the frequency sweep I could see the actual bandwith plotted on
the scope.

[email protected] October 14th 05 11:14 PM

On Fri, 14 Oct 2005 17:16:21 -0400, Ken Scharf
wrote:

wrote:
For filters I use microprocessor crystals in the ladder configuration.
With the correct shunt C and 4-8 crystals you can make a very fine
Allison
KB!GMX

I managed to buy over 400 pcs of 8.3886mhz crystals on ebay, for
just pennies each. I am planning on trying to build ladder filters
with them. These crystals are in the larger HC6/u size holders which
are supposed to work better than the miniature size used in the micro-
processor crystals.


The HC6 parts work fine as do the HC18, 49 and so on. The real trick
is doing the work to measure and check the crystals for use and then
calculate the capacitors and termination impedence based on that.
For a little work you get fine filters dirt cheap.

First step would be to build the DDS vfo for
the radio since I can program the DDS to function as a sweep generator
for aligning the filter. With the DDS sweeping the output frequency
while providing a sawtooth ramp to drive the scope sweep in step
with the frequency sweep I could see the actual bandwith plotted on
the scope.


While I have a DDS to do that with I found that using the first "high"
crystal in a VXO that gets calibrated worked as well with a lot less
fuss. Then I can use the same osc to sweep the filter later to test
it by adding a varicap doide.

Allison

Ken Scharf October 15th 05 12:38 AM

wrote:
On Fri, 14 Oct 2005 17:16:21 -0400, Ken Scharf
wrote:


wrote:

For filters I use microprocessor crystals in the ladder configuration.
With the correct shunt C and 4-8 crystals you can make a very fine
Allison
KB!GMX


I managed to buy over 400 pcs of 8.3886mhz crystals on ebay, for
just pennies each. I am planning on trying to build ladder filters
with them. These crystals are in the larger HC6/u size holders which
are supposed to work better than the miniature size used in the micro-
processor crystals.



The HC6 parts work fine as do the HC18, 49 and so on. The real trick
is doing the work to measure and check the crystals for use and then
calculate the capacitors and termination impedence based on that.
For a little work you get fine filters dirt cheap.


First step would be to build the DDS vfo for
the radio since I can program the DDS to function as a sweep generator
for aligning the filter. With the DDS sweeping the output frequency
while providing a sawtooth ramp to drive the scope sweep in step
with the frequency sweep I could see the actual bandwith plotted on
the scope.



While I have a DDS to do that with I found that using the first "high"
crystal in a VXO that gets calibrated worked as well with a lot less
fuss. Then I can use the same osc to sweep the filter later to test
it by adding a varicap doide.

That would work fine, but with the DDS, I can program the actual
frequency range to be swept and probably be able to calibrate the
scope face to read the actual frequency 'break' points on the filter.
Using the vxo method will get you a working filter quickly no
doubt, but I will still need the DDS vfo for the finished rig, so
I just figured I'd do that first.

How many 'rocks' did you use in an SSB filter? I've seen some designs
on the web with 6 crystals, would the shape factor be any better with
8 or more? (with over 400 crystals in the junk box I can go crazy,
but I'd still have to find the capacitors :-).

Dale Parfitt October 15th 05 01:24 AM


"Ken Scharf" wrote in message
.. .
wrote:
On Fri, 14 Oct 2005 17:16:21 -0400, Ken Scharf
wrote:


wrote:

For filters I use microprocessor crystals in the ladder configuration.
With the correct shunt C and 4-8 crystals you can make a very fine
Allison
KB!GMX

I managed to buy over 400 pcs of 8.3886mhz crystals on ebay, for
just pennies each. I am planning on trying to build ladder filters
with them. These crystals are in the larger HC6/u size holders which
are supposed to work better than the miniature size used in the micro-
processor crystals.



The HC6 parts work fine as do the HC18, 49 and so on. The real trick
is doing the work to measure and check the crystals for use and then
calculate the capacitors and termination impedence based on that.
For a little work you get fine filters dirt cheap.


First step would be to build the DDS vfo for
the radio since I can program the DDS to function as a sweep generator
for aligning the filter. With the DDS sweeping the output frequency
while providing a sawtooth ramp to drive the scope sweep in step
with the frequency sweep I could see the actual bandwith plotted on
the scope.



Just sweeping a filter designed for SSB would be fine I suppose. I recently
had an opportunity to hear the difference between a stock FT-1000 CW filter
and one homebrewed with attention paid to group delay- the difference was
very clear to hear- in favor of the homebrew filter.
\
Dale



[email protected] October 15th 05 02:50 AM

On Fri, 14 Oct 2005 19:38:37 -0400, Ken Scharf
wrote:


That would work fine, but with the DDS, I can program the actual
frequency range to be swept and probably be able to calibrate the
scope face to read the actual frequency 'break' points on the filter.
Using the vxo method will get you a working filter quickly no
doubt, but I will still need the DDS vfo for the finished rig, so
I just figured I'd do that first.


Makes sense. I found the other way easy when DDS chips were 55$ each!
The varicap sweep is calabrated to the scope so that was not an issue
for sweeping the filter. I found that using wideband noise and a
sound card was better.

How many 'rocks' did you use in an SSB filter? I've seen some designs
on the web with 6 crystals, would the shape factor be any better with
8 or more? (with over 400 crystals in the junk box I can go crazy,
but I'd still have to find the capacitors :-).


I'd say 4 is a useable minimum. With that I'll add the skirts at 40db
down are not very good though. I've used 6-8 to get a good 6-60db
shape (under 2:1). There is a problem if you go for too many. The
filter can have enough group delay that while it's shape is good, the
sound has a hollowness.

The caps, once you figured the qalues you will likely end up using
parallel values. IE: 232pf may be a 220+12pf or a 220 and a 4-20pf
trimmer.

Allison

[email protected] October 15th 05 02:55 AM

On Sat, 15 Oct 2005 00:24:39 GMT, "Dale Parfitt"
wrote:

Just sweeping a filter designed for SSB would be fine I suppose. I recently
had an opportunity to hear the difference between a stock FT-1000 CW filter
and one homebrewed with attention paid to group delay- the difference was
very clear to hear- in favor of the homebrew filter.
\
Dale


My prefered filter has a gausian to 6db shape for less ringing and
group delay. I work for that goal.

However, try the KK7B Phasing rigs for sound. They are direct
conversion SSB (image rejecting) so all the selectivity is in the
audio bandpass. I use a miniR2 and T2 pair on 6m and filter
artifacts like group delay aren't there. Transparentcy is a good
word to describe it.


Allison
KB1GMX


William E. Sabin October 15th 05 03:52 PM

A common mistake in years past was to try to put all of the selectivity into
a single super-deluxe crystal, mechanical or digital filter. These filters
quite often have "raspy" noise interference at the edges of the passband
(especially in CW mode) due to the enhancements of spectral noise peaks at
the very sharp band edges, caused by the conversion of the phase statistics
of noise to amplitude statistics (each edge of the filter acts like a phase
disciminator). This raspy noise interferes with weak signals.

The filter can be equalized for group delay, as mentioned, with improvement
in the problem. The band edges can be softened, with good results. A much
better way is to use two or more intermediate-performance filters in
cascade. This method softens the edges so that the effect is greatly
reduced. It also improves overall shape factor. A cascade in this manner
of identical bandpass or audio lowpass filters tends in the limit toward the
Bessel or even the Gaussian response. Digital filters can also use a method
called Transition Band Sampling (see Oppenheim and Schafer 1975 or Oppenheim
and Willsky 1983). All of these results are related in principle to the
Central Limit Theorem of statistics.

The cascaded filter approach is also very beneficial in other respects, in
particular the reduction of wideband noise in high-gain IF amplifiers. This
noise degrades AGC performance and adds audio frequency noise to the product
detector output.

Bill W0IYH


"Dale Parfitt" wrote in message
news:b%X3f.301$W32.225@trnddc06...

Just sweeping a filter designed for SSB would be fine I suppose. I
recently had an opportunity to hear the difference between a stock FT-1000
CW filter and one homebrewed with attention paid to group delay- the
difference was very clear to hear- in favor of the homebrew filter.
\
Dale




William E. Sabin October 16th 05 01:19 PM

Receiver bandwidth .. dayton filter find!
 

"William E. Sabin" wrote in message
news:6J84f.437250$x96.418250@attbi_s72...

A couple of improvements in the following paragraph:

The filter can be equalized for group delay, as mentioned, with
improvement in the problem. The band edges can be softened, with good
results. A much better way is to use two or more intermediate-performance
filters in cascade. This method softens the edges so that the effect is
greatly reduced. It also improves overall shape factor. A cascade in
this manner of identical bandpass or audio lowpass filters tends in the
limit toward the Bessel or even the Gaussian response. Digital filters
can also use a method called Transition Band Sampling (see Oppenheim and
Schafer 1975)


Delete the incorrect second Reference to Oppenheim and Willsky 1983.

All of these results are related in principle to the Central Limit
Theorem of statistics.


See http://mathworld.wolfram.com/CentralLimitTheorem.html for this
interesting topic. The general idea that the theorem alludes to in this
example is that as sharp-cornered filters are cascaded the passband response
becomes noticeably more rounded at the corners, similar to Bessel and
Gaussian filters.

Smoothing and Windowing methods can be used to reduce sharp corners in
discrete sequences such as digital filters (see Oppenheim and Schafer 1975
and many other sources).

Bill W0IYH




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