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I have a practical question regarding the various applications of excitation signals in identifying Linear Time-Invariant (LTI) systems. Specifically, I am curious about the usage of the following signals:

  • Stepped sinusoids
  • Compact pulses
  • Spread spectrum
  • Chirp (I could imagine automotive radar applications)
  • Gaussian noise

I know the basic difference among. For e.g., mainly depending on the requirement for specific SNR we use different excitation signal. But I believe there should be more technical aspects behind. Could someone provide some insights into the practical applications of other signals for LTI system identification?

Disclaimer: I dont have a signal processing background.

Thank you.

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3 Answers 3

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The basic principle of system identification is Fourier division (in one shape or another). If we excite with a signal spectrum $X[k]$ and receive the spectrum $Y[k]$ we can calculate the transfer function as

$$H[k] = \frac{Y[k]}{X[k]} \tag{1}$$

No system is noise free so we typically have additional noise in the measured signal and so in practice we are looking at something like

$$H'[k] = \frac{Y[k]+N[k]}{X[k]} = \frac{Y[k]}{X[k]} + \frac{N[k]}{X[k]} = H[k] + \frac{N[k]}{X[k]} \tag{2} $$

where $H'[k]$ is the estimated transfer function and $N[k]$ is the noise spectrum. We can see that the measurement error (due to additive noise) is

$$E[k] = H'[k] - H[k] = \frac{N[k]}{X[k]} \tag{3}$$

We can therefore choose the excitation spectrum to adjust the measurement error to our liking. For example: if we want an error that's constant with frequency, we would choose an excitation spectrum that's similar to the noise spectrum.

The other way to minimize the error is to crank up the gain: make $X[k]$ as large as possible. Most systems are constrained by a maximum amplitude (at some point something is going to clip or become non-linear). A good measurement signal has a low "Crest Factor" (ratio of peak to RMS). i.e. maximizes the power for a given maximum peak amplitude.

So criteria for a good measurement signal are

  1. Has a spectral shape that results in the desired error shape. Optimizes the frequency depended SNR to our specific requirements
  2. Has a low Crest factor
  3. Is easily invertible. At some point we have to calculate $1/X[k]$ and that's hard to do if $X[k] = 0$ even at frequencies we don't care about.
  4. Is not overly sensitive to non-linear distortions (to the extent they occur)
  5. Accommodates all system constraints (power, thermal, peak, short term spectrum etc).

At this point its clear that the "best" choice of excitation signal depends a lot on the specific properties of the system and the requirements of the measurement. There is no "one size fits all".

Let's go through the list

  • Stepped sinusoids: I'm not a big fan of those since the don't give you a complete transfer function and typically under-samples in the frequency domain.
  • Compact pulses:: Tend to have very high Crest factor. Only a good option for very low noise situations.
  • Spread spectrum: Seems needlessly complicated, but I have never used it for measurements.
  • Chirp: That's simply the white version of the general class of sweep signals. These tend to be a good choice in many cases. Standard sweeps (linear, log) or easy to generate and have very low crest factor. You can even generate sweeps with arbitrary spectra but this is a bit more complicated. On the downside, sweeps are "short term" narrow band, i.e. at any given point in time all the measurement energy is concentrated in a fairly narrow band and some systems are sensitive to this which limits the measurement power. Sweeps are also fairly sensitive to harmonic distortions: these can manifest themselves as artificial "reflections" in the time domain.
  • Gaussian noise: Very good choice overall. Again that's a subset of noise signals. Specifically "pseudo-random" noise signals are quite useful (see below). While technically Gaussian noise isn't amplitude limited, amplitudes of more than 5 times the standard deviation are so rare, that clipping is them is usually fine: the resulting error is typical tiny as compared to all other error mechanisms.

Pseudo-Random Noise

That's my personal favorite. It' easy enough to generate by applying a random phase to your target spectrum and performing an inverse FFT. This will result in a Gaussian distribution with a Crest factor of 4-5 (for a reasonable signal length). If that's a problem you can further decrease the crest factor through iterative optimizers. You can knock down a reasonable wide band signal to a Crest factor of about 1.5 which is on par with sweeps.

Pseudo Random noise measurements are also relatively insensitive to non-linear distortions. They just show up as uncorrelated noise in both the time and frequency domain.

You can also generate as many uncorrelated versions as you want simply by using different seeds for the random phase generator. This can be quite helpful in doing simultaneous measurements in multi-channel systems.

Measurement gain

Setting the gain properly is often overlooked. It's a simple optimization process but can yield a lot of extra SNR. If the gain is too low, you have excess additive noise. If the gain is too high, you are overdriving and see distortion or compression. The optimization is simple enough: Vary the gain, calculate the SNR and find the maximum in the curve.

Coherent averaging

Another way to improve SNR is to use a periodic excitation and average the received periods. This will reduce uncorrelated noise by about $10\log N$, where N is the number of periods. This works well if

  1. The period of the excitation signal is at least as long as the impulse response of the system
  2. The system stationary over the entire measurement time.
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  • $\begingroup$ Using Maximum Length Sequences (a.k a. Galois Linear Feedback Shift Register sequences) you get a crest factor of 1. 3 dB better than sinusoidal or chirp excitation signals. But when there is a nonlinearity in the system, there's som truly goofy behavior you get with MLS. The cool thing is that, except for a tiny DC component, MLS is guarateed white. $\endgroup$ Apr 18 at 18:25
  • $\begingroup$ @robertbristow-johnson: in my world "white" is a bad thing. Most acoustical background noise is pink or brown, so a white excitation results in very poor SNR at low frequencies. $\endgroup$
    – Hilmar
    Apr 19 at 17:33
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    $\begingroup$ Yeah, it probably can be anything as long as it's not zero at too many frequencies. Then it's just a two-channel FFT (at least that's what it was called in the previous century). Use music, whatever $\endgroup$ Apr 20 at 0:39
  • $\begingroup$ Is "spread spectrum" not synonymous with pseudo-random noise? $\endgroup$
    – DavidG25
    May 6 at 14:59
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The two main issues are the amount of energy in the excitation signal and where that energy is in the spectrum.

Stepped sinusoids : This is good as far as energy is concerned (no need to have a large amplitude), but only looks at the system to be identified at a discrete set of frequencies. Depending on how those frequencies are selected and the system to be identified, this might not capture all the interesting parts of the response.

Compact pulses : The problem with pulses is the large amplitude necessary to identify the "stop band" of the system to be identified. The nice thing about pulses is that their energy is distributed over a wide range of frequencies.

Spread spectrum : Depending on how it's generated, this keeps the amplitude down and also covers a reasonable part of the spectrum.

Chirp : Depending on how it's generated, this keeps the amplitude down and also covers a reasonable part of the spectrum.

Gaussian noise : Gaussian noise is both the best sort of noise (if it's white, it covers the entire spectrum) and the worst sort of noise (its amplitude is unbounded).


Where the amplitude is bounded, system identification makes up for it by integrating the effect of the signal over a longer time so the same amount of energy can be sent, just not instantaneously.

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Hilmar's answer is very good, but in practice the instantaneous bandwidth and other factors can be important. For example Vector Network Analyzers essentially are doing stepped sinusoids. This simplifies a number of aspects of the instrument design, as well as improving some specifications. Using other wideband signals would theoretically improve measurement speed, but practically I would imagine calibrations becoming difficult among other practical problems.

Another practical problem that Hilmar's answer did not cover is when there is interference. This is typical in cellular networks where they use orthogonal sequences so that you can estimate the channel from different cell towers.

Here are some examples.

https://en.wikipedia.org/wiki/Zadoff%E2%80%93Chu_sequence

https://en.wikipedia.org/wiki/Maximum_length_sequence

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