Re: swept sine accuracy

[Bob Masta <audio@xxxxxxxxxxx>]

On 5 Mar 2009 at 19:13, James W. Beauchamp wrote:

> Guys,
> 
> This is a not strictly an auditory question, but it could be
> useful for people doing acoustic measurements. If you use a
> swept sine wave to measure the frequency response of a linear
> system, what is the limitation on the speed of the sweep in
> terms of how accurate the result would be? I imagine it has
> something to do with how smooth the actual frequency response 
> is. If it has some pronounced bumps, they could be smoothed
> out if the sweep is too fast.
> 
> In practice, you could sweep at some arbitrary rate, and then
> slow it by a factor of two, and if the result is the same
> (within an acceptable tolerance) you could say that you've
> converged on the solution.
> 
> But I'd like to have a theoretical result.
> 
> Jim Beauchamp
> Univ. of Illinois at Urbana-Champaign

The sweep needs to be slow enough that the system response 
does not change significantly (whatever is significant to 
you) during the analysis time window. With modern FFT 
methods the time window can be known exactly, but there's 
more to it:  You need to use a peak-hold sweep to only see 
the response peaks, AND you need to use a Flat-Top spectral 
window function to prevent spectral leakage (at signal 
frequencies that give a non-integer number of cycles per 
analysis sample set) from causing response bounce.  (All of 
the more usual window functions have lots of bounce, which 
is simply a peak error at "wrong" frequencies.)

But there is another approach that works much better, IMHO: 
Don't use a sweep, use a stepped frequency.  Better yet, if 
the same system is producing the driving signal and 
analyzing the response, you can set the frequency steps to 
fall exactly on spectral lines, such that you don't need 
any window function.  This gives a "perfect" spectrum in 
the minimum amount of time.  

The only caveat is that since you are stepping frequency, 
the system under test must have time to respond to each 
step before you make the measurement.  In practice, since 
the steps are small (only one spectral line apart), this 
turns out to not be a problem.

I have Application Notes on Frequency Response Measurement 
at <http://www.daqarta.com/dw_0a00.htm>.  These are 
specifically oriented toward my Daqarta software, but 
should be generally applicable.

Best regards,

Bob Masta
 
            D A Q A R T A
Data AcQuisition And Real-Time Analysis
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