[time-nuts] power spectrum of hard limiter output

[Magnus Danielson]

On 26/01/11 04:32, jimlux wrote:
> On 1/25/11 10:47 AM, Magnus Danielson wrote:
>> Jim,
>>
>> On 25/01/11 14:53, jimlux wrote:
>>> On 1/24/11 1:19 PM, Magnus Danielson wrote:
>>>
>>>> What are you *really* trying to achieve? 1-bit ADC at the end of a
>>>> noisy
>>>> channel?
>>>>
>>>
>>>
>>> I have a GPS receiver front end (sampler) that normally one tests by
>>> running GPS signals through it, acquiring and tracking the signals and
>>> deriving SNR estimates, etc. , but we're in a situation where we don't
>>> have either the back end processing or the GPS signals. We *do* have a
>>> signal generator, so I was looking for some analytical expression(s)
>>> that say, if you put in a tone with X SNR, this is what you should see
>>> coming out of the sampler.
>>>
>>> It's easy to do a sort of qualitative test (put in a big signal, see if
>>> you get a square wave out), but it would be nice to be able to have a
>>> way to make a quantitative measurement, particularly of the noise figure
>>> & gain of the receiver. People have done a sort of ad hoc measurement
>>> (hooking up a spectrum analyzer to the single bit digital output of the
>>> sampler), but I was looking for something a bit more rigorous, but not
>>> to the point where *I* wanted to grind out the pages of equations.. I
>>> was hoping that someone else (e.g. Aronson) had gone through the
>>> exercise.
>>>
>>> The interesting thing is that there *is* a fair amount of analysis of
>>> the bandlimited signal(s) and noise into a hard/soft limiter into a
>>> filter. However, there's not much on systems where there is a sampling
>>> process as well (which aliases all those harmonics down, of course). The
>>> more recent literature I was able to find tends to be of a more
>>> empirical nature (e.g. the modeling/simulation/experimental results).
>>>
>>> And that's fine (after all, Aronson says that simple closed form
>>> solutions probably don't exist). I can crank out models with the best of
>>> them, but, philosophically, if there is a nice *simple* analytical
>>> approximation, that's nicer.
>>
>> What you can do... is try different amplitudes and different SNRs. By
>> monitoring the compression that the added noise provides for various
>> sine amplitudes you can derive the internal noise and hence noise factor.
>
> yes.. in fact, I did some simulations this morning and figured it all
> out. For what it's worth, it's sort of like trying to measure No by
> working from measured BER to Eb/No, where you know Eb. You need to be in
> a particular range of SNR to have it work well.. too high, and the noise
> is so small that you need to run zillions of samples to get a decent
> measurement precision. Too low and you can't see the sine wave in the
> noise unless you integrate over many samples. So, for a given number of
> "bits" out of the limiter, there's an optimum range of SNRs.
>
> Interesting stuff.

Indeed. It's just like No measurement and where I was inspired.

You need to be in the same range as your No with the added No since 
that's when No+Ni changing Ni will give best sensitivity.

>>
>> I'm sure you can borrow a GPS simulator if you really need to. If you
>> only can record the bit-stream for post-processing, any of several
>> software GPS softwares would be able to decode the stream. Even my hack
>> would be able to do it. Maybe only doing FFT-based locking would suffice
>> for you.
>
> Oh.. doing it with recorded bits and a software GPS processor is
> straightforward (and actually how they usually test these things), but
> we were looking for a way to use a RF signal generator and no GPS
> signals. Those GPS simulators are a pretty pricey piece of gear,
> especially if you want L1,L2, and L5.

True, but even the big guys get L1 simulators for their bulk testing. 
You can get many test-cases with a simple L1 C/A instrument. Then you 
can use the big iron test generator for those test which really requires it.

Cheers,
Magnus