In my battle with transfer function estimation, ed I have been dealing with many noise problems lately and have come across this application note from TI regarding the calculation of noise figures for basic op-amp circuits. This noise figure deals with the ratio of circuit signal-to-noise ratio (SNR) at the input versus the output. The article goes through the derivation of noise analysis equations due to thermal noise in resistive elements and due to rms noise figures of the active device and goes on to quantify the noise figure as a function of temperature, resistances and op-amp parameters. This can be useful in determining performance properties of circuits given a set of passive components and can be used to define a “best case scenario”.
This all well and good, but then one might ask what noise has to do with system identification (transfer function estimation)? The simple answer is that the frequency-domain transfer function can be determined by passing “noise” through a system and comparing the spectral properties of the output versus the input. The idea is that white noise has a flat spectra (over infinite time) so the transfer function can be accurately determined for all frequencies (again, given infinite time). If infinity is too long a time to wait, one trick is using something called periodic random noise to give a well defined spectral distribution in finite time. A Gaussian random number generator can create white noise, however, an inverse Fourier transform is used to to generate the periodic noise.
Essentially, enough sinusoids are added together to cover the frequency range of interest with equal amplitudes and randomized phases that are distributed over +/- pi. The amplitude will relate to the desired resulting rms value for the noise and the number of summed sinusoids. The frequencies of choice should line up with the sampled frequencies in the following FFT that will be computed to compare the spectra of the input and output signals. The signal will now look like noise and will be “random”, however, all of the frequency domain components will maintain their amplitude and phase through the whole procedure leading to less variance in the FFTs.
( slyt094.pdf )