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Categories of Signals

Determining the best bandwidth limiting conditions for a signal is not necessarily a trivial task. The example above provided a simple and efficient way of limiting the bandwidth of the signal and is very often exactly how it is performed. Nevertheless, some additional factors must sometimes be taken into consideration to optimize the bandwidth of a signal in order to obtain the best possible conversion results. These factors are usually related to general characteristics of the signal to be digitized.

Periodic Signals

Periodic signals are in the same category as the square wave in the example above. But the Fourier Transform representation of a square wave, defined as a series of pure odd-multiple sine wave extending to infinity is, strictly speaking, only valid for a signal having started at the moment of the Big Bang and continuing until the end of eternity. Needless to say, such signals do not actually exist in real life. Signals always change over time and their bandwidth can rarely be expressed as a sum of static frequency components. It is usually more convenient to consider the bandwidth of a signal in terms of “typical frequency distribution” or “typical spectral envelope”.

Quasi-Periodic Signals

Quasi-periodic signals usually consist of a basic periodic signal that has some of it frequency characteristics modified by another signal. For example, this is the case of most telecom signals that are based on a central radio carrier frequency signal for which the amplitude, frequency and/or phase can be modified by a modulating signal, resulting on an altered frequency spectrum centered on the carrier component. A detailed description of these types of modulated signals is completely outside the scope of this discussion, but we can nevertheless use them to illustrate the general spectral characteristics of quasi-periodic signals. A typical spectral distribution of quasi-periodic signals is shown in the following figure.

A typical quasi-periodic signal consists of a central frequency component (FC) and a series of lower and upper “sideband” components created on each of FC by the modulating signal. The profile of those sidebands is never static and constantly changes according to the fluctuation of the modulating signal. Because of this, it is usually more convenient to consider the sidebands in terms of a “spectral envelope” that encompasses all possible components according to their typical modulating and statistical distribution.

There are at least three different ways could be used to digitize such quasi-periodic signals. The first method is shown in the following figure.

This is the “classic” (or “brute force”) method and is the one that was previously presented for the square wave. We simply determine a Nyquist frequency (FN) just above the highest frequency component of the signal, and we then ensure that the ADC is clocked at least at twice FN. This method is not necessarily the most efficient, as most of the signal’s frequency components only occupy only a small fraction of the ADC’s potential sampling bandwidth.
The second method, shown in the following figure, is often the one most commonly used for quasi-periodic signals.

With this method, the modulated signal is shifted in frequency using what is known as a “mixer”, a device that multiplies the signal with a constant frequency sine wave generated by a “local oscillator”. The multiplication operation creates two copies of the signal, one shifted down (and mirrored) and the other shifted up in frequency. A lowpass (or bandpass) filter is used to remove the upper frequency components. By properly selecting the frequency of the local oscillator, it is possible to relocate the signal’s spectrum at the lowest and most convenient location in the spectrum in order to minimize the value of FN and facilitate the selection of the most appropriate and performing ADC.
The third method, shown in the following figure, is much less common but can nevertheless be valuable to consider.

We’ve discussed in the previous post (“From Analog to Digital – Part 3: Signal Sampling”) how signals present in the upper Nyquist zones were folded back (directly for odd Nyquist zones, or mirrored for even Nyquist zones) into the first Nyquist zone. Even though this would normally be undesirable, it can sometimes be beneficial to do so and this method provides a very good example.
This method is known as undersampling because the signal is not sampled rapidly enough. It provides the same benefits as the previously mentioned method, but without requiring the use of a mixer. But there are nevertheless some drawbacks to using it. The first one is that the signal must be completely enclosed in one of the Nyquist zones. If not, some frequency components would be aliased in unpredictable ways and the resulting signal would be impossible to interpret properly. Secondly, the sampling rate must be chosen to be an exact multiple of the signal’s bandwidth, so the signal will fit perfectly within a single Nyquist zone. Lastly, this process puts a lot of pressure on the ADC and its ability to sample signals that are much faster than what it was designed and optimized for. Some degradation in performance will necessarily occur and we will be discussing this in posts to come.

Aperiodic or Transient Signals

Aperiodic (or transient) signals are usually unpredictable and random in nature. They often correspond to events that possess a relatively well-defined short-term behaviour in the time-domain (such as pulses), but for which the long-term behaviour cannot be determined. The frequency spectrum (Fourier Transform) of such signals doesn’t correspond to a series of discrete frequency components and can only be estimated as a function of the spectral envelope of the expected short-term signal transients.
An ideal (and extreme) example for this would be for a signal defined by the function

y=sin⁡(x)/x

This signal corresponds to a single pulse of a dampened sine wave function. Graphs of the signal in the time-domain and in the frequency domain are shown below.

As we can observe, the Fourier Transform of such a signal is an ideal bandwidth limited spectral envelope, also known as a “brick wall”. Real life transient signals are much less ideal and their spectral envelope can only approximate the frequency profile of the sin(x)/x function.
A rule-of-thumb method to approximate the frequency spectrum of short pulse signals based on their characteristics in the time-domain was already discussed in a previous blog (“Optimizing the Number of Analog-To-Digital Converters in Small Animal Imaging (SAI) Systems”). This method uses the “rise time” (Tr) of the pulse (the time taken by the pulse to rise from 10% to 90% of its maximum amplitude) to evaluate the bandwidth of the signal.


Bandwidth=0.35/Tr
Once this evaluation of the bandwidth is obtained, the “classic” sampling method is normally used, with the pulse signals sampled at greater than twice their bandwidth using an acceptable sampling rate given by:

Sampling Rate≥0.7/Tr

After adding some tolerance margin:

Sampling Rate≥1/Tr

Non-Periodic or Random Signals

Non-periodic signals are completely random in nature and their bandwidth cannot be predicted using any analytical or approximation method. An example of this would be if the digitization of audio (music) signals, where the contents can vary from total silence to a complete coverage of all frequencies in the audio spectrum simultaneously. Proper bandwidth evaluation of such signals can only be determined empirically and based on an estimation of where the useful frequencies of interest are located in the relevant spectrum (for audio signals, the frequencies detectable by the human ear vary from 20 Hz to 20KHz). Once the appropriate bandwidth has been estimated, specifically designed filters are used to limit the bandwidth of the signal prior to the digitization process. An example of this, for audio signals, is illustrated by the following figure.

Conclusion

We’ve discussed the capability of considering a signal from both the time-domain and frequency-domain perspective, which is a fundamental principle in the understanding of the certain characteristics and limitations of ADCs, and of which we will discuss in blogs to come.
We’ve also examined various types of signals and discussed how to properly determine their bandwidth and the types of sampling techniques that are most appropriate for each of them.