Mathematically based method to increase bandwidth of ADC

[Abstract Dissonance]

When Tek designed the 7104 1 GHz analog scope, they considered
splitting up the input into frequency bands, amplifying each, and
recombinimg at the CRT. They opted for brute-force silicon plus other
tricks, in the end. The story is in one of Jim Williams' books.

Yeah, this is what I'm basically doing is splitting up the bands but then
all bands are frequency shifted into the same frequency range so the ADC can
handle it. I suppose that adds an additional complication that is
impractical ;/

 

[John Larkin]

Yeah, this is what I'm basically doing is splitting up the bands but then
all bands are frequency shifted into the same frequency range so the ADC can
handle it. I suppose that adds an additional complication that is
impractical ;/

That's just what LeCroy is doing, but with a separate ADC (actually, a
bank of interleaved ADCs) per band.

John

 

[Abstract Dissonance]

I'm going to explain it in a practical way so some of you understand.

Lets suppose you have a signal s(t),

you apply a LP filter on s(t) to get s_0(t) = LP(s(t))

You construct another signal s_1(t) = BP(s(t) - s_0(t)).

The idea is that we now have subtracted those frequencies that are in s_0(t)
out of s_1(t) since they are already in s_0(t). This is true because the
fourier transform is linear. (BP = Band pass and means to filter the signal
appropriately)


In general,

s_k(t) = BP(s(t) - s_(k-1)(t))

The problem is that each s_k(t) has higher and higher frequency bands...
but each of them is disjoint..

|
|---|---|---|
| 1 | 2 | 3 |...
| | | |
|---------------

The numbers represent the bands and the frequency range of the function s_k.
Each band is disjoint from any others(just show here all together).

so the bandwidth of s_(k+1) is disjoint from s_k but right next to it but in
higher frequency.


This is in effect exactly what I've done. Nothing special.

Now, the "trick" is to use the fact that we can shift the bands all into the
lowest frequency range(or any range we want)..

so we have the frequencies for s_k,

|
|---|
| k |
| |
|---------------

(but notices this doesn't mean s_k = s_(k+1) but just that they both have
the same bandwidth)



Now we can use ADC(s)'s that have the bandwidth to sample the s_k's. Once we
do this we will have there "digital" versions and we can then "unshift" the
functions back to there original locations, i.e.,

|
|---|---|---|
| 1 | 2 | 3 |...
| | | |
|---------------

and then just sum all the functions back up to get the original s(t).

------
I just made this up at the moment so I'm not sure if its mathematically
solid. The above equations seem to work but are not like the original
convolution I had before. (although I think it might be the same but is
simplified).
------

 

[Abstract Dissonance]

When Tek designed the 7104 1 GHz analog scope, they considered
splitting up the input into frequency bands, amplifying each, and
recombinimg at the CRT. They opted for brute-force silicon plus other
tricks, in the end. The story is in one of Jim Williams' books.

I wrote a new post that explains my method in terms like this(its very
similar)... I'm not sure if they are doing what I said or not but it sounds
similar.

 

[Abstract Dissonance]

That's just what LeCroy is doing, but with a separate ADC (actually, a
bank of interleaved ADCs) per band.

yeah, I think the interleaved method is the same. By delaying the signal
slightly you are, in effect, picking up higher frequencies(relative to the
first)... the longer the delay the lower the frequencies you get. Because a
"low frequency" won't be able to change fast enough between two consecutive
samples as it would in 3. I'm not sure the math behind it but I think its
very similar to what I'm doing if one were able to represent it
mathematically(basicaly instead of shifting the frequency spectrums you are
shifting the clock... but they are equivilent mathematically)

 

[PetePope]

yeah, I think the interleaved method is the same. By delaying the signal
slightly you are, in effect, picking up higher frequencies(relative to the
first)... the longer the delay the lower the frequencies you get. Because a
"low frequency" won't be able to change fast enough between two consecutive
samples as it would in 3. I'm not sure the math behind it but I think its
very similar to what I'm doing if one were able to represent it
mathematically(basicaly instead of shifting the frequency spectrums you are
shifting the clock... but they are equivilent mathematically)

There are two methods currently in use - one is old and the other is
new.

The old method is temporal (or time) interleaving in which multiple
digitizers are "interleaved".
This method is utilized to increase the effective sample rate of the
digitizer and often can
increase the memory length in a scope (since usually an individual
digitizer drives its own
memory). The amount of interleaving ranges from 160 digitizers running
at 125 Ms/s to
12 in Agilent scopes to 16 digitizers running at 1.25 Gs/s in Tek
Scopes. You will not find
individual digitizing elements running much above 1.5 Gs/s, so anything
faster is time
interleaved (this is for real-time digitizers - don't get confused with
equivalent time or sampling
scopes which behave completely differently). While interleaving seems
easy, it is very
difficult in that that the frequency response of each digitizer must be
matched precisely.

The new method is one which shifts frequencies. This method has the
added advantage
of increasing bandwidth. The following white paper explains this for
the layman.

http://www.lecroy.com/tm/Library/WhitePapers/PDF/DBI_Explained.pdf

The frequency shifting method has been used successfully in the
development
of an 11 GHz, 40 Gs/s scope (the LeCroy SDA 11000) and an 18 GHz, 60
Gs/s
scope (the LeCroy SDA 18000) which is the fastest real-time waveform
digitizer
on the planet.

There are many issues with the development of such an instrument. No
one would
attempt such a design without the DSP capability to recover and "fix"
the waveform
after running the gauntlet of the microwave circuitry required to
separate, downconvert
and acquire the waveforms. LeCroy has this expertise and is the world
expert in
waveform processing and analysis.

The phase is affected by the local oscillator used for the mixing
action and the phase of the
local oscillator must be known or recovered in order to accomplish the
desired result -
something you might not have considered. Additionally, the sharp
filters utilized create
quite a bit of phase distortion at the band edges that must be
corrected.

search the patent and publication database at www.uspto.gov for more
details.

P.S. Don't believe all of the Tek stories - they'll tell you they
invented everything - especially
if you're in the market for a scope

Pete Pupalaikis
www.LeCroy.com