Re: swept sine accuracy-- Further question

[Piotr Majdak <piotr@xxxxxxxxxx>]

Dear list,

For a linear time invariant (LTI) system, the transfer function H(f) is
given by Y(f) / X(f) where is X is the input, Y is the output, and f is
the frequency. The inverse Fourier transform of H(f) is the well-known
impulse response h(t). The simplest way of measuring h(t) is to play an
impulse and record the response coming from the unknown system. Now, the
impulse can be considered as a very, very, really very short sweep. The
response to such a signal would still represent the system, provided
that the system response is recorded in its fully length. Thus, using a
sweep x(t) as the excitation signal, once the response y(t) has been
recorded, the transfer function is given by F[y(t)] / F[x(t)], where
F[.] is the Fourier transform.

This is valid as long as we speak about ideal LTI systems. In such a
case, any sweep will do. If you consider noise in the measurement then
you probably want to use a long excitation signal, which provides much
energy in a short amount of time. For example, one impulse is not as
good as one linear sweep because it has more energy. Best SNR is offered
by maximum length sequences [1]. If you consider your system as a
weakly-nonlinear system then an exponential sweep is better than MLS
[2-5], because it separates the harmonic distortions from the linear
part of the response [6].

The frequency resolution reported in [7] is an effect of using a
bandpass filter, which is essential in the implementation of the
time-delay spectrometry (TDS, see Sec. 5 in [7]). When another approach
is used (e.g. exponential sweeps or MLS), no filtering is required and
thus, the frequency resolution is 1/T where T is the measurement duration.

Hence, exponential sweeps or MLS provide the most efficient signal
identification, but they require storing and post processing of the
total system response. In the TDS, the processing is done in real-time
and the real and imaginary parts of the transfer function are directly
available.

I hope this helps a little...

br, Piotr Majdak

[1] D. D. Rife and J. Vanderkooy, (1989). “Transfer-Function Measurement
with Maximum-Length Sequences,” J. Audio Eng. Soc., vol. 37, pp. 419--444.
[2] C. Dunn and M. O. Hawksford, (1993). “Distortion Immunity of
MLS-Derived Impulse Response Measurements,” J. Audio Eng. Soc., vol. 41,
pp. 314--335
[3] D. D. Rife and C. Dunn, (1994). “Comments on Distortion Immunity of
MLS-Derived Impulse Response Measurements,” J. Audio Eng. Soc. (Letters
to the Editor), vol. 42, pp. 490--497
[4] J. Vanderkooy, (1994). “Aspects of MLS Measuring Systems,” J. Audio
Eng. Soc., vol. 42, pp. 219--231
[5] M. Wright and J. Vanderkooy, (1995). “Comments on ‘Aspects of MLS
Measuring Systems’,” J. Audio Eng. Soc. (Letters to the Editor), vol.
43, pp. 48--49.
[6] Majdak, P., Balazs, P., and Laback, B. (2007). "Multiple Exponential
Sweep Method for Fast Measurement of Head-Related Transfer Functions," J
Audio Eng Soc, 55, 623-637.
[7] J. Vanderkooy, (1986). "Another Approach to Time-Delay
Spectrometry," J. Audio Eng. Soc., vol. 34, pp. 523-538.


Nakajima wrote:
> Dear Dr. Beauchamp and Dear All,
>
> Thank you very much for the valuable information. Although I am a
> layperson in the field of signal processing, I need some guidelines
> for my auditory experiments. Vanderkooy's guideline happened to be the
> same as my personal guideline -- it can be derived easily from a
> simple dimensional analysis. Poletti's guideline seems basically the
> same.
>
> Now my problem comes. I may be wrong -- I am a layperson -- but we
> need both the frequency resolution and the temporal resolution, and we
> can never avoid the uncertainty between time and frequency. If the
> frequency resolution is at sqrt(sweep rate) Hz, then the temporal
> resolution should be at 1/sqrt(sweep rate) s. Thus, longer swept tones
> should not be always better even when we had an extremely fast
> computer. It seems like that we have to find an optimum sweep rate
> depending on our purpose. Considering the frequency resolution and the
> temporal resolution of the auditory system, I often utilize a sweep
> rate around 10,000 Hz/s. (We would be in a trouble if we used a rate
> of 1,000,000 Hz/s or 100 Hz/s.)
>
> I would be glad if someone -- either in the list or in the literature
> -- could formulate, or correct, what I am thinking.
>
> Best regards,
> Yoshitaka Nakajima
>
> ----- Original Message ----- From: "James W. Beauchamp"
> <jwbeauch@xxxxxxxxxxxxxxxxxxxxxx>
> To: <AUDITORY@xxxxxxxxxxxxxxx>
> Sent: Sunday, March 08, 2009 3:52 AM
> Subject: Re: swept sine accuracy
>
>
>> Thanks very much to everyone who responded to my question.
>>
>> This was actually for an undergraduate signal processing course
>> I've been teaching where we have just encountered frequency
>> response of linear time-invariant systems. The text discusses
>> transfer functions and implies that you need really long
>> duration sine waves of constant frequency to measure frequency
>> response and doesn't consider practical methods like the swept
>> sine method. I was explaining this method to the class and
>> mentioned that you can't sweep too fast, but I didn't have a
>> simple formula that captured how fast you could sweep based on
>> required resolution.
>>
>> I was not considering the possibilities of noise or nonlinearity.
>>
>> Two people offered formulas that I think would be useful:
>>
>> 1) frequency resolution (Hz) = sqrt(sweep rate (Hz/s))
>> This is based on
>> John Vanderkooy, "Another Approach to Time-Delay Spectrometry," JAES,
>> 1986 July/Aug. (thanks to Dan Mapes-Riordan)
>>
>> 2) sweep rate << 1/(pi*t^2), where t = duration of filter impulse
>> response.
>> This is based on
>> M. A. Poletti, "Linearly Swept Frequency Measurements, Time-Delay
>> Spectrometry, and the Wigner Distribution, JAES 36(6), 457-468, 1988.
>> (thanks to Christian Ciao)
>>
>> Also, this paper was frequently mentioned:
>> Swen Mテシllerand Paulo Massarani, "Transfer-Function Measurement with
>> Sweeps", JAES 49 (6), 443-471, June, 2001.
>>
>> Thanks again,
>>
>> Jim
>>