K4 RX dynamic range

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K4 RX dynamic range

Elecraft mailing list
Before putting my money up front for the first run of K4D I need to know what is the dynamic range
of the K4D RX at 2kHz spacing. K3 is 105 dB and K4D? If it is 20dB lower than K3 than it would be 85dB-correct?
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Re: K4 RX dynamic range

Lyle Johnson
Mark,

The "20 dB lower than a K3" figure is an estimate for 100 kHz Blocking
Dynamic Range rather than the 2 kHz Narrow Spaced Dynamic Range.

The K3 is listed at 140 to 150 dB (depending on model, synthesizer, etc)
on Sherwood's Receiver Test Data page.  The K4 series without the "HD"
option are estimated to be in the 120 to 130 dB range, typical of other
direct sampling SDR products (Flex, Apache, Icom, ...).

73,

Lyle KK7P

On 6/4/19 4:00 PM, mark roz via Elecraft wrote:
> Before putting my money up front for the first run of K4D I need to know what is the dynamic range
> of the K4D RX at 2kHz spacing. K3 is 105 dB and K4D? If it is 20dB lower than K3 than it would be 85dB-correct?
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K4: superhet vs. direct sampling

wayne burdick
Administrator
The superhet module buys a lot of BDR improvement. But also -- a subtlety I've failed to mention so far -- the superhet module is intended to somewhat improve 2 kHz IMDDR3 *and* make this figure more repeatable.

Q: Say what?

A: As Rob Sherwood noted many times before finally immortalizing this point in his must-read footnotes, A-to-D converters sharing the same part number are not all created equal. The long-time previous occupant of his Top Spot benefitted from a never-corroborated monotonicity in its ADC's LSBs. An act of god. The product of a very good day at the silicon foundry when, serendipitously, all the bunny suits were defect-free, and no one was exhaling molecules of grain alcohol or other substances from the night before.

That said, most ops can get by without the extra BDR and IMDDR3, because they're not situated in the RF equivalent of the Gulf Stream. Hence the different K4 models.

73,
Wayne
N6KR


> On Jun 4, 2019, at 5:10 PM, Lyle Johnson <[hidden email]> wrote:
>
> Mark,
>
> The "20 dB lower than a K3" figure is an estimate for 100 kHz Blocking Dynamic Range rather than the 2 kHz Narrow Spaced Dynamic Range.
>
> The K3 is listed at 140 to 150 dB (depending on model, synthesizer, etc) on Sherwood's Receiver Test Data page.  The K4 series without the "HD" option are estimated to be in the 120 to 130 dB range, typical of other direct sampling SDR products (Flex, Apache, Icom, ...).
>
> 73,
>
> Lyle KK7P
>
> On 6/4/19 4:00 PM, mark roz via Elecraft wrote:
>> Before putting my money up front for the first run of K4D I need to know what is the dynamic range
>> of the K4D RX at 2kHz spacing. K3 is 105 dB and K4D? If it is 20dB lower than K3 than it would be 85dB-correct?



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Re: K4: superhet vs. direct sampling

Bill K9YEQ
Wayne, all these options and modernizations, even though I love my K3S and really don't need to upgrade, I will be "forced" to break down and buy one.

72 & 73,
Bill
K9YEQ
FT'er for K2, KX1, KX3, KXPA100,  KAT500, W2, etc.

-----Original Message-----
From: [hidden email] <[hidden email]> On Behalf Of Wayne Burdick
Sent: Tuesday, June 4, 2019 9:19 PM
To: Elecraft Reflector <[hidden email]>
Subject: [Elecraft] K4: superhet vs. direct sampling

The superhet module buys a lot of BDR improvement. But also -- a subtlety I've failed to mention so far -- the superhet module is intended to somewhat improve 2 kHz IMDDR3 *and* make this figure more repeatable.

Q: Say what?

A: As Rob Sherwood noted many times before finally immortalizing this point in his must-read footnotes, A-to-D converters sharing the same part number are not all created equal. The long-time previous occupant of his Top Spot benefitted from a never-corroborated monotonicity in its ADC's LSBs. An act of god. The product of a very good day at the silicon foundry when, serendipitously, all the bunny suits were defect-free, and no one was exhaling molecules of grain alcohol or other substances from the night before.

That said, most ops can get by without the extra BDR and IMDDR3, because they're not situated in the RF equivalent of the Gulf Stream. Hence the different K4 models.

73,
Wayne
N6KR


> On Jun 4, 2019, at 5:10 PM, Lyle Johnson <[hidden email]> wrote:
>
> Mark,
>
> The "20 dB lower than a K3" figure is an estimate for 100 kHz Blocking Dynamic Range rather than the 2 kHz Narrow Spaced Dynamic Range.
>
> The K3 is listed at 140 to 150 dB (depending on model, synthesizer, etc) on Sherwood's Receiver Test Data page.  The K4 series without the "HD" option are estimated to be in the 120 to 130 dB range, typical of other direct sampling SDR products (Flex, Apache, Icom, ...).
>
> 73,
>
> Lyle KK7P
>
> On 6/4/19 4:00 PM, mark roz via Elecraft wrote:
>> Before putting my money up front for the first run of K4D I need to
>> know what is the dynamic range of the K4D RX at 2kHz spacing. K3 is 105 dB and K4D? If it is 20dB lower than K3 than it would be 85dB-correct?



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Re: K4: superhet vs. direct sampling

Clay Autery-2
Was thinking the very same thing....  but I am not so sure I will be
able to let my K3S go...  I may just be adding...  And then I can make
the K3S a traveling system.

______________________
Clay Autery, KY5G
(318) 518-1389

On 04-Jun-19 21:50, Bill Johnson wrote:

> Wayne, all these options and modernizations, even though I love my K3S and really don't need to upgrade, I will be "forced" to break down and buy one.
>
> 72 & 73,
> Bill
> K9YEQ
> FT'er for K2, KX1, KX3, KXPA100,  KAT500, W2, etc.
>
> -----Original Message-----
> From: [hidden email] <[hidden email]> On Behalf Of Wayne Burdick
> Sent: Tuesday, June 4, 2019 9:19 PM
> To: Elecraft Reflector <[hidden email]>
> Subject: [Elecraft] K4: superhet vs. direct sampling
>
> The superhet module buys a lot of BDR improvement. But also -- a subtlety I've failed to mention so far -- the superhet module is intended to somewhat improve 2 kHz IMDDR3 *and* make this figure more repeatable.
>
> Q: Say what?
>
> A: As Rob Sherwood noted many times before finally immortalizing this point in his must-read footnotes, A-to-D converters sharing the same part number are not all created equal. The long-time previous occupant of his Top Spot benefitted from a never-corroborated monotonicity in its ADC's LSBs. An act of god. The product of a very good day at the silicon foundry when, serendipitously, all the bunny suits were defect-free, and no one was exhaling molecules of grain alcohol or other substances from the night before.
>
> That said, most ops can get by without the extra BDR and IMDDR3, because they're not situated in the RF equivalent of the Gulf Stream. Hence the different K4 models.
>
> 73,
> Wayne
> N6KR
>
>
>> On Jun 4, 2019, at 5:10 PM, Lyle Johnson <[hidden email]> wrote:
>>
>> Mark,
>>
>> The "20 dB lower than a K3" figure is an estimate for 100 kHz Blocking Dynamic Range rather than the 2 kHz Narrow Spaced Dynamic Range.
>>
>> The K3 is listed at 140 to 150 dB (depending on model, synthesizer, etc) on Sherwood's Receiver Test Data page.  The K4 series without the "HD" option are estimated to be in the 120 to 130 dB range, typical of other direct sampling SDR products (Flex, Apache, Icom, ...).
>>
>> 73,
>>
>> Lyle KK7P
>>
>> On 6/4/19 4:00 PM, mark roz via Elecraft wrote:
>>> Before putting my money up front for the first run of K4D I need to
>>> know what is the dynamic range of the K4D RX at 2kHz spacing. K3 is 105 dB and K4D? If it is 20dB lower than K3 than it would be 85dB-correct?
>
>
> ______________________________________________________________
> Elecraft mailing list
> Home: http://mailman.qth.net/mailman/listinfo/elecraft
> Help: http://mailman.qth.net/mmfaq.htm
> Post: mailto:[hidden email]
>
> This list hosted by: http://www.qsl.net
> Please help support this email list: http://www.qsl.net/donate.html Message delivered to [hidden email]
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> Post: mailto:[hidden email]
>
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> Please help support this email list: http://www.qsl.net/donate.html
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Re: K4: superhet vs. direct sampling

Richard Corfield
In reply to this post by wayne burdick
I'd have thought that superhet would always have the advantage of improved
image rejection due to the IF band filtering. It would also allow ADCs and
associated processing to run slower or lower down in relation to its
Nyquist frequency so allowing for more detail in the sampled signal. Also
can narrow band analogue filtering beat the dynamic range of the ADC? If
its top 10 bits are taken up handling that huge strong signal in its input
passband then you've got fewer bits left for your signal of interest. So
both high bit depth and high sample rate in comparison to signal are harder
to achieve.

The opposing view being the cost of achieving a high quality Superhet
conversion? If direct sampling and high speed signal processing (FPGA?) can
achieve the results so much more cheaply and simply and reliably? Like the
Class D amplifier in reverse.

 - Richard (M0RJC)


On Wed, 5 Jun 2019 at 03:19, Wayne Burdick <[hidden email]> wrote:

> The superhet module buys a lot of BDR improvement. But also -- a subtlety
> I've failed to mention so far -- the superhet module is intended to
> somewhat improve 2 kHz IMDDR3 *and* make this figure more repeatable.
>
> Q: Say what?
>
> A: As Rob Sherwood noted many times before finally immortalizing this
> point in his must-read footnotes, A-to-D converters sharing the same part
> number are not all created equal. The long-time previous occupant of his
> Top Spot benefitted from a never-corroborated monotonicity in its ADC's
> LSBs. An act of god. The product of a very good day at the silicon foundry
> when, serendipitously, all the bunny suits were defect-free, and no one was
> exhaling molecules of grain alcohol or other substances from the night
> before.
>
> That said, most ops can get by without the extra BDR and IMDDR3, because
> they're not situated in the RF equivalent of the Gulf Stream. Hence the
> different K4 models.
>
> 73,
> Wayne
> N6KR
>
>
> > On Jun 4, 2019, at 5:10 PM, Lyle Johnson <[hidden email]> wrote:
> >
> > Mark,
> >
> > The "20 dB lower than a K3" figure is an estimate for 100 kHz Blocking
> Dynamic Range rather than the 2 kHz Narrow Spaced Dynamic Range.
> >
> > The K3 is listed at 140 to 150 dB (depending on model, synthesizer, etc)
> on Sherwood's Receiver Test Data page.  The K4 series without the "HD"
> option are estimated to be in the 120 to 130 dB range, typical of other
> direct sampling SDR products (Flex, Apache, Icom, ...).
> >
> > 73,
> >
> > Lyle KK7P
> >
> > On 6/4/19 4:00 PM, mark roz via Elecraft wrote:
> >> Before putting my money up front for the first run of K4D I need to
> know what is the dynamic range
> >> of the K4D RX at 2kHz spacing. K3 is 105 dB and K4D? If it is 20dB
> lower than K3 than it would be 85dB-correct?
>
>
>
> ______________________________________________________________
> Elecraft mailing list
> Home: http://mailman.qth.net/mailman/listinfo/elecraft
> Help: http://mailman.qth.net/mmfaq.htm
> Post: mailto:[hidden email]
>
> This list hosted by: http://www.qsl.net
> Please help support this email list: http://www.qsl.net/donate.html
> Message delivered to [hidden email]
>
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Re: K4: superhet vs. direct sampling

Elecraft mailing list
In reply to this post by Clay Autery-2
Hi Clay, Presently I have a pair of K3's. One is my main rig and one is my backup and traveling rig. I will sell my one K3 and keep the other as the backup and traveling rig when the K4 arrives.
73,
N2TK, Tony

-----Original Message-----
From: [hidden email] <[hidden email]> On Behalf Of Clay Autery
Sent: Wednesday, June 5, 2019 12:05 AM
To: [hidden email]
Subject: Re: [Elecraft] K4: superhet vs. direct sampling

Was thinking the very same thing....  but I am not so sure I will be able to let my K3S go...  I may just be adding...  And then I can make the K3S a traveling system.

______________________
Clay Autery, KY5G
(318) 518-1389

On 04-Jun-19 21:50, Bill Johnson wrote:

> Wayne, all these options and modernizations, even though I love my K3S and really don't need to upgrade, I will be "forced" to break down and buy one.
>
> 72 & 73,
> Bill
> K9YEQ
> FT'er for K2, KX1, KX3, KXPA100,  KAT500, W2, etc.
>
> -----Original Message-----
> From: [hidden email]
> <[hidden email]> On Behalf Of Wayne Burdick
> Sent: Tuesday, June 4, 2019 9:19 PM
> To: Elecraft Reflector <[hidden email]>
> Subject: [Elecraft] K4: superhet vs. direct sampling
>
> The superhet module buys a lot of BDR improvement. But also -- a subtlety I've failed to mention so far -- the superhet module is intended to somewhat improve 2 kHz IMDDR3 *and* make this figure more repeatable.
>
> Q: Say what?
>
> A: As Rob Sherwood noted many times before finally immortalizing this point in his must-read footnotes, A-to-D converters sharing the same part number are not all created equal. The long-time previous occupant of his Top Spot benefitted from a never-corroborated monotonicity in its ADC's LSBs. An act of god. The product of a very good day at the silicon foundry when, serendipitously, all the bunny suits were defect-free, and no one was exhaling molecules of grain alcohol or other substances from the night before.
>
> That said, most ops can get by without the extra BDR and IMDDR3, because they're not situated in the RF equivalent of the Gulf Stream. Hence the different K4 models.
>
> 73,
> Wayne
> N6KR
>
>
>> On Jun 4, 2019, at 5:10 PM, Lyle Johnson <[hidden email]> wrote:
>>
>> Mark,
>>
>> The "20 dB lower than a K3" figure is an estimate for 100 kHz Blocking Dynamic Range rather than the 2 kHz Narrow Spaced Dynamic Range.
>>
>> The K3 is listed at 140 to 150 dB (depending on model, synthesizer, etc) on Sherwood's Receiver Test Data page.  The K4 series without the "HD" option are estimated to be in the 120 to 130 dB range, typical of other direct sampling SDR products (Flex, Apache, Icom, ...).
>>
>> 73,
>>
>> Lyle KK7P
>>
>> On 6/4/19 4:00 PM, mark roz via Elecraft wrote:
>>> Before putting my money up front for the first run of K4D I need to
>>> know what is the dynamic range of the K4D RX at 2kHz spacing. K3 is 105 dB and K4D? If it is 20dB lower than K3 than it would be 85dB-correct?
>
>
> ______________________________________________________________
> Elecraft mailing list
> Home: http://mailman.qth.net/mailman/listinfo/elecraft
> Help: http://mailman.qth.net/mmfaq.htm
> Post: mailto:[hidden email]
>
> This list hosted by: http://www.qsl.net Please help support this email
> list: http://www.qsl.net/donate.html Message delivered to
> [hidden email]
> ______________________________________________________________
> Elecraft mailing list
> Home: http://mailman.qth.net/mailman/listinfo/elecraft
> Help: http://mailman.qth.net/mmfaq.htm
> Post: mailto:[hidden email]
>
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> list: http://www.qsl.net/donate.html Message delivered to
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Re: K4: superhet vs. direct sampling

W2xj
In reply to this post by Richard Corfield
Direct sampling has no image issues.

Sent from my iPad

> On Jun 5, 2019, at 3:23 AM, Richard Corfield <[hidden email]> wrote:
>
> I'd have thought that superhet would always have the advantage of improved
> image rejection due to the IF band filtering. It would also allow ADCs and
> associated processing to run slower or lower down in relation to its
> Nyquist frequency so allowing for more detail in the sampled signal. Also
> can narrow band analogue filtering beat the dynamic range of the ADC? If
> its top 10 bits are taken up handling that huge strong signal in its input
> passband then you've got fewer bits left for your signal of interest. So
> both high bit depth and high sample rate in comparison to signal are harder
> to achieve.
>
> The opposing view being the cost of achieving a high quality Superhet
> conversion? If direct sampling and high speed signal processing (FPGA?) can
> achieve the results so much more cheaply and simply and reliably? Like the
> Class D amplifier in reverse.
>
> - Richard (M0RJC)
>
>
>> On Wed, 5 Jun 2019 at 03:19, Wayne Burdick <[hidden email]> wrote:
>>
>> The superhet module buys a lot of BDR improvement. But also -- a subtlety
>> I've failed to mention so far -- the superhet module is intended to
>> somewhat improve 2 kHz IMDDR3 *and* make this figure more repeatable.
>>
>> Q: Say what?
>>
>> A: As Rob Sherwood noted many times before finally immortalizing this
>> point in his must-read footnotes, A-to-D converters sharing the same part
>> number are not all created equal. The long-time previous occupant of his
>> Top Spot benefitted from a never-corroborated monotonicity in its ADC's
>> LSBs. An act of god. The product of a very good day at the silicon foundry
>> when, serendipitously, all the bunny suits were defect-free, and no one was
>> exhaling molecules of grain alcohol or other substances from the night
>> before.
>>
>> That said, most ops can get by without the extra BDR and IMDDR3, because
>> they're not situated in the RF equivalent of the Gulf Stream. Hence the
>> different K4 models.
>>
>> 73,
>

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Re: K4: superhet vs. direct sampling

Richard Corfield
I was getting confused momentarily with Direct Conversion :-)

On Wed, 5 Jun 2019 at 13:23, W2xj <[hidden email]> wrote:

> Direct sampling has no image issues.
>
> Sent from my iPad
>
> > On Jun 5, 2019, at 3:23 AM, Richard Corfield <[hidden email]>
> wrote:
> >
> > I'd have thought that superhet would always have the advantage of
> improved
> > image rejection due to the IF band filtering. It would also allow ADCs
> and
> > associated processing to run slower or lower down in relation to its
> > Nyquist frequency so allowing for more detail in the sampled signal. Also
> > can narrow band analogue filtering beat the dynamic range of the ADC? If
> > its top 10 bits are taken up handling that huge strong signal in its
> input
> > passband then you've got fewer bits left for your signal of interest. So
> > both high bit depth and high sample rate in comparison to signal are
> harder
> > to achieve.
> >
> > The opposing view being the cost of achieving a high quality Superhet
> > conversion? If direct sampling and high speed signal processing (FPGA?)
> can
> > achieve the results so much more cheaply and simply and reliably? Like
> the
> > Class D amplifier in reverse.
> >
> > - Richard (M0RJC)
> >
> >
> >> On Wed, 5 Jun 2019 at 03:19, Wayne Burdick <[hidden email]> wrote:
> >>
> >> The superhet module buys a lot of BDR improvement. But also -- a
> subtlety
> >> I've failed to mention so far -- the superhet module is intended to
> >> somewhat improve 2 kHz IMDDR3 *and* make this figure more repeatable.
> >>
> >> Q: Say what?
> >>
> >> A: As Rob Sherwood noted many times before finally immortalizing this
> >> point in his must-read footnotes, A-to-D converters sharing the same
> part
> >> number are not all created equal. The long-time previous occupant of his
> >> Top Spot benefitted from a never-corroborated monotonicity in its ADC's
> >> LSBs. An act of god. The product of a very good day at the silicon
> foundry
> >> when, serendipitously, all the bunny suits were defect-free, and no one
> was
> >> exhaling molecules of grain alcohol or other substances from the night
> >> before.
> >>
> >> That said, most ops can get by without the extra BDR and IMDDR3, because
> >> they're not situated in the RF equivalent of the Gulf Stream. Hence the
> >> different K4 models.
> >>
> >> 73,
> >
>
>
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Re: K4: superhet vs. direct sampling

David Woolley (E.L)
In reply to this post by W2xj
On 05/06/2019 13:23, W2xj wrote:
> Direct sampling has no image issues.

That's because they are called aliasing issues!

--
David Woolley

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Re: K4: superhet vs. direct sampling

Richard Corfield
Which is interesting. We were taught that you can recover signals up to the
Nyquist frequency (half sampling frequency) before aliasing becomes an
issue. People low pass filter well before then to avoid aliasing issues,
but it means you need a sampling frequency sufficiently more than twice the
highest frequency you want to record. So presumably for the 6m band that's
sufficiently greater than 110,000,000 samples per second.

What gets me though, is when you look at the waveforms and the sampling
waveforms, a signal at the Nyquist frequency could have magnitude from 0 to
full size depending on its phase with respect to the sampling. Assuming
regular sampling (do any dither the sampling?). Move just below Nyquist and
you'll see a beat frequency come in as it moves in and out of phase. I
guess the maths is based on infinite time - Fourier being the integral from
-infinity to +infinity - but we've not got infinite time to listen to an
infinitely long SSB signal.

You can buy a reasonably cheap digital oscilloscope (Tektronix TBS1000 -
£600) which boasts 1GS/s so a Nyquist frequency of 500MHz. They claim 50MHz
analogue bandwidth.  (If you're thinking of buying this double check the
figures first rather than take my work for it. The marketing material was a
little opaque).

A 2nd order filter (cheap, simple) with a 3dB point at 50MHz and
40dB/decade would be about 40dB down by the Nyquist frequency. In the lower
bands, say 10MHz, the first aliases are from 990MHz which is 55dB down if I
remember my maths right. I'd assume someone would use better than a 2nd
order filter or a faster sampling ADC.



On Thu, 6 Jun 2019 at 10:38, David Woolley <[hidden email]>
wrote:

> On 05/06/2019 13:23, W2xj wrote:
> > Direct sampling has no image issues.
>
> That's because they are called aliasing issues!
>
> --
> David Woolley
>
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Re: K4: superhet vs. direct sampling

WILLIE BABER
In reply to this post by wayne burdick
In my experience the value of high-level blocking and low phase noise is in the ability to hear very weak signals next to  strong ones.  When I first got into So2r years ago, I discovered an entire layer of very weak signals when I switched to usingTenTec Omni V and Omni VI; this was related directly to lower phase noise in so2r where the other radio was not a TenTec.  Also, high level blocking allows you to hear those weak ones in S & P that you would otherwise roll right over especially when the band is full of strong signals. If you are not into cw contesting, and in so2r in particular, then what I just said doesn't matter and lots of radios become good radios. I have used nothing in so2r better than 2 x k3, with Omni VI, Orion and Tentec Eagle almost as good.  After those, there is Kenwood ts590s.

The thing is, you cannot notice the signals that you are not hearing if you have two radios that perform equally poor, and especially with respect to phase noise.

73, will, wj9b

CWops #1085
CWA Advisor levels II and III
http://cwops.org/

--------------------------------------------
On Tue, 6/4/19, Wayne Burdick <[hidden email]> wrote:

 Subject: [Elecraft] K4: superhet vs. direct sampling
 To: "Elecraft Reflector" <[hidden email]>
 Date: Tuesday, June 4, 2019, 8:18 PM
 
 The superhet module buys a lot of
 BDR improvement. But also -- a subtlety I've failed to
 mention so far -- the superhet module is intended to
 somewhat improve 2 kHz IMDDR3 *and* make this figure more
 repeatable.
 
 Q: Say
 what?
 
 A: As Rob Sherwood
 noted many times before finally immortalizing this point in
 his must-read footnotes, A-to-D converters sharing the same
 part number are not all created equal. The long-time
 previous occupant of his Top Spot benefitted from a
 never-corroborated monotonicity in its ADC's LSBs. An
 act of god. The product of a very good day at the silicon
 foundry when, serendipitously, all the bunny suits were
 defect-free, and no one was exhaling molecules of grain
 alcohol or other substances from the night before.
 
 That said, most ops can get by
 without the extra BDR and IMDDR3, because they're not
 situated in the RF equivalent of the Gulf Stream. Hence the
 different K4 models.
 
 73,
 Wayne
 N6KR
 
 
 >
 On Jun 4, 2019, at 5:10 PM, Lyle Johnson <[hidden email]>
 wrote:
 >
 > Mark,
 >
 > The "20 dB
 lower than a K3" figure is an estimate for 100 kHz
 Blocking Dynamic Range rather than the 2 kHz Narrow Spaced
 Dynamic Range.
 >
 >
 The K3 is listed at 140 to 150 dB (depending on model,
 synthesizer, etc) on Sherwood's Receiver Test Data
 page.  The K4 series without the "HD" option are
 estimated to be in the 120 to 130 dB range, typical of other
 direct sampling SDR products (Flex, Apache, Icom, ...).
 >
 > 73,
 >
 > Lyle KK7P
 >
 > On 6/4/19 4:00 PM,
 mark roz via Elecraft wrote:
 >> Before
 putting my money up front for the first run of K4D I need to
 know what is the dynamic range
 >> of
 the K4D RX at 2kHz spacing. K3 is 105 dB and K4D? If it is
 20dB lower than K3 than it would be 85dB-correct?
 
 
 
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