# [time-nuts] power spectrum of hard limiter output

Magnus Danielson magnus at rubidium.dyndns.org
Mon Jan 24 21:19:15 UTC 2011

```On 24/01/11 21:32, jimlux wrote:
> On 1/24/11 11:44 AM, Magnus Danielson wrote:
>> On 24/01/11 07:39, Bruce Griffiths wrote:
>>> jimlux wrote:
>>>> On 1/23/11 10:01 PM, Magnus Danielson wrote:
>>>>> Jim,
>>>>>
>>>>> On 24/01/11 02:35, jimlux wrote:
>>>>>> I'm looking for a reference that gives the power spectrum of the
>>>>>> output
>>>>>> of a hard limiter (1 bit thresholder) with band limited noise and a
>>>>>> single sinusoid.
>>>>>>
>>>>>> At high SNR, the output of the limiter is basically a square wave
>>>>>> at at
>>>>>> the input frequency, but as the SNR decreases, it starts to act
>>>>>> like a
>>>>>> soft limiter with a gaussian characteristic, so what is the power
>>>>>> spectrum of the output?
>>>>>
>>>>> It goes towards sine as I recall it. The gaussian noise rubs of
>>>>> overtones. Gardner describes this in his PLL book. Setting up a nice
>>>>> sawtooth detector is no good when seeing bad noise, as it will degrade
>>>>> into a sine-detector anyways, so using a multiplier is better for
>>>>> those
>>>>> conditions as you get a more stable property.
>>>>>
>>>>> Another approach of understanding is to consider that when the
>>>>> gaussian
>>>>> noise is sufficiently high it will start interpolate on the slope of
>>>>> the
>>>>> sine and as you add more noise more and more of the sine would
>>>>> linearize
>>>>> until you come to the point where it is linear. Soft-clipping will
>>>>> indeed be similar.
>>>>>
>>>>> I haven't seen a spectrum plot, but simulation in spice should be
>>>>> trivial. Setting up a sine + noise, comparator and then a low-pass
>>>>> filter should be a trivial SPICE setup. It does not take much
>>>>> imagination to see that the spectrum will migrate from that of a
>>>>> square
>>>>> over to that of a sine. It will loose power in those overtones.
>>>>>
>>>>
>>>> oh, yes.. I did the simulation, and modeled the aliasing of the
>>>> overtones and all..
>>>>
>>>> I was looking for a reference to cite (you know how it is.. measure it
>>>> yourself and it's something *you* did.. but cite someone who ground
>>>> through it before, and it's worth a lot more...)
>>>>
>>> See:
>>>
>>> http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=4101285
>>
>> http://archive.numdam.org/ARCHIVE/RSMUP/RSMUP_1969__42_/RSMUP_1969__42__1_0/RSMUP_1969__42__1_0.pdf
>>
>>
>>
>>
>>
>>
>> ... found using "hard limiter autocorrelation sine" in Google.
>>
>
> Excellent.. I had found the Aronson paper (wherein he effectively says
> "there's no simple analytical expression".. that's nice and justifies
> doing it by numerical modeling). ANd the Jain paper (DTIC), which deals
> with multiple sines. I hadn't run across the one by Greenhall (who still
> works at JPL)...
>
> Thanks a bunch to the nuts..

Actually, if you think of it... for you to be able to even consider a
"waveform" you will need to do filtering, but then... the shape of that
filter will also have great influence on the overtone spectra... ruling
out analytic expressions. Noise degradation of hard limiter is analyzed
before, See chapter 6.5 Limiters in Gardner.

What are you *really* trying to achieve? 1-bit ADC at the end of a noisy
channel?

Cheers,
Magnus

```