PLL noise bandwidth

Hi,
Can someone tell me in a few words what does noise bandwidth in a PLL mean
and also suggest me book or papers where I can know more about it. I want
to write an algorithm for the code tracking loop of a DSSS system and the
book 'Software defined GPS and Galileo receiver' that I am referring now
discusses this topic.

Thanks

On 01/15/2011 01:19 PM, johnlovestohate wrote:
> Hi,
> Can someone tell me in a few words what does noise bandwidth in a PLL mean
> and also suggest me book or papers where I can know more about it. I want
> to write an algorithm for the code tracking loop of a DSSS system and the
> book 'Software defined GPS and Galileo receiver' that I am referring now
> discusses this topic.

"Noise bandwidth" is an interesting term, because it's difficult to 
describe clearly, yet it's a simple concept once you know what it means.

Note that the term applies to any filter, not just phase-locked loops.

So: the noise bandwidth of a filter is the equivalent bandwidth of that 
filter for white noise.  I.e., it's the bandwidth that the filter would 
need to have if it were perfectly rectangular in frequency and passed 
exactly the same amount of noise power as the filter in question when 
the filter is presented with white noise.

Is in clear yet?  No?  Good -- that means you're human.

Consider this example: take a first-order lowpass filter:

H(s) = w_0 / (s + w_0).

This filter has a 3dB cutoff frequency of f_0 = w_0 / (2 * pi).  Usually 
when we talk about the "bandwidth" of a first-order lowpass filter, we'd 
use the term as a synonym of "3dB cutoff frequency".

But consider what happens when we run white noise through this filter, 
with spectral density N_0 watts/hz.  The average power at the output of 
the filter is w_0 / 4.  This is larger than the filter cutoff frequency 
by a factor of pi/2.  So the first-order lowpass filter _acts_ like a 
perfectly rectangular filter with bandwidth f_n = w_0 / 4 (or f_n = f_0 
* pi / 2).  This 'effective bandwidth with respect to noise power' is 
called the filter's noise bandwidth.

I hope this helps.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html

Ok. That was a succint explanation and with that I looked around a few more
articles and understood more. Thanks Tim.

On continuation of the same topic how does one decide the noise bandwidth
of a PLL? In the book I mentioned earlier the author provides an example of
a PLL with 3 different noise bandwidths. The acquisition program determines
the carrier with an offset of 21Hz and the PLL tracks fastest with very
little nois when noise bandwidth is 30Hz, tracks faster but with more noise
when noise bandwidth is 60Hz and doesn't accurately track with a lot of
noise when noise bandwidth is 10Hz.

So how does one determine what value of NB must be used? I am not familiar
with this topic so any suggestion of a good book is welcome.

Thanks

On 01/15/2011 06:25 PM, johnlovestohate wrote:
> Ok. That was a succint explanation and with that I looked around a few more
> articles and understood more. Thanks Tim.
>
> On continuation of the same topic how does one decide the noise bandwidth
> of a PLL? In the book I mentioned earlier the author provides an example of
> a PLL with 3 different noise bandwidths. The acquisition program determines
> the carrier with an offset of 21Hz and the PLL tracks fastest with very
> little nois when noise bandwidth is 30Hz, tracks faster but with more noise
> when noise bandwidth is 60Hz and doesn't accurately track with a lot of
> noise when noise bandwidth is 10Hz.
>
> So how does one determine what value of NB must be used? I am not familiar
> with this topic so any suggestion of a good book is welcome.

The books on PLL design that I'm personally acquainted with are from the 
End of the Analog Days, and don't mention how to do it digitally.

I'm not quite sure why the author is using noise bandwidth here -- I 
think the 'noise' in the 10Hz case isn't random noise, however, but the 
deterministic (if erroneous) behavior of a loop that's not in lock.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html

> On continuation of the same topic how does one decide the noise bandwidth
> of a PLL? In the book I mentioned earlier the author provides an example of
> a PLL with 3 different noise bandwidths. The acquisition program determines
> the carrier with an offset of 21Hz and the PLL tracks fastest with very
> little nois when noise bandwidth is 30Hz, tracks faster but with more noise
> when noise bandwidth is 60Hz and doesn't accurately track with a lot of
> noise when noise bandwidth is 10Hz.
>
> So how does one determine what value of NB must be used? I am not familiar
> with this topic so any suggestion of a good book is welcome.
>
> Thanks

So this appears to be a carrier recovery question, the PLL is being
used to recover the carrier in a digital comm.

In this case the modulation is a form of noise and you want the PLL BW
to be narrow to reject that noise and return a pure carrier.  From
this point of view, the narrower the better.  But...
this slows acquisition time. You can use a two speed approach, wide BW
to acquire , then narrow.

The other more interesting problem is that the carrier in the Tx may
not be perfect.  It may have jitter i.e. phase noise and maybe doppler
it may move around and the downconverter in the Rx may not be perfect
either.  If the Rx PLL loop BW is too narrow, it won't be able to
track this well.

so the overall answer as it is to any interesting question is.... it
depends.

You want the PLL BW as narrow as possible but not so narrow that it
doesn't track the Tx well.

A rule of thumb is the carrier recovery PLL BW should be about 1/100
to 1/1000 the symbol rate.

Mark