When an electrical engineering student graduates, by necessity he or she has had to focus on either digital electronics or RF/microwave theory, to be able to master the breadth and volume of material. As one's industry experience broadens however, the lines become blurred, and the two worlds often merge. Microwave signals can coexist with high transmission digital serial busses.

Digital engineers involved in high-speed communications are at home analyzing signals as a function of time, in the "time domain". However, signals can also be analyzed as a function of frequency, in the "frequency domain". As communications scale up to a million bits per second, digital engineers must be able to utilize frequency domain measurements to interpret time domain behavior. Likewise, radio frequency/microwave engineers sometimes must analyze signals in digital applications, so they must master time domain measurement.

To be a more effective engineer, enhancing one's comfort level in both realms allows for more creative options in design, and far more efficient troubleshooting. This fluency in time domain and frequency domain can be expanded even more if one adds the modal domain to one's analysis toolkit. The modal domain examines the frequencies at which structures, fluids or signals can behave erratically, and their changing shapes in instantaneous dynamic response to high vibration. One application of the modal domain would be testing systems and structures under the vibrational excitation of an earthquake.

The time, frequency and modal domains are three distinctly different vantage points from which to analyze signals. It could be compared to using three sets of eye glasses with convex, concave, and fish-eye lenses. All the same information is present, but by seeing the data distributed differently, new correlations are noted. By being able to separate out the variables and components, one notices relationships between them. These three domains of signal analysis allow the engineer to "map" the signal's characteristics.

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## First Vantage Point: Time Domain

When the unit of measurement is recorded in seconds or its multiples (minutes, hours), then the analysis is in the time domain. Signal sampling taken over time renders a representation of time as measured by a periodic change in the signal or data. For example, data showing the progression of amplitude over a specific time period, would be "amplitude given time". An electrical signal can be displayed as a voltage versus time waveform on an oscilloscope, which draws a graph of the instantaneous signal voltage over time.

In the time domain, the value will always be in real numbers. We analyze signals, mathematical functions, or perhaps scientific data, as measured in sequential time samples. Signal sources and interference are also defined in the time domain.

*Waveform graphs can be helpful when not knowing how to model domains appropriately. *

## Second Vantage Point: Frequency Domain

When analysis concerns frequency/energy units such as Hertz, then the analysis is in the frequency domain. The unit hertz (Hz) was once called cps, or cycles per second. Frequency is the number of times each event has occurred during the recording period.

Here in the frequency domain, we can observe amplitude versus frequency. The amplitude of a wave or vibration is expressed in positive numbers, with the peak amplitude as a measure of deviation from its central value. The same signal can also be displayed in a power versus frequency format. This would be displayed on a spectrum analyzer, which is capable of both time and frequency domain analysis.

With frequency domain analysis one can figure out the key points in the total data set, rather than examining every variation which occurs in the time domain. A frequency domain graph shows either the phase shift or magnitude of a signal at each frequency that it exists at. It shows how much of the signal lies within each given frequency band over a range of frequencies.

A signal could be described as the sum of many sine waves ("Fourier series") that have differing pulses, phases and amplitudes. Switching between the time domain and the frequency domain and back again, is accomplished by performing mathematical integration using the "Fourier Transform" equations. Fourier transforms (FTs) take a signal and express it in terms of the frequencies of the waves that make up that signal.

## Third Vantage Point: Modal Domain

Changing one's vantage point to the modal domain allows for measurements of the status of the network or structure as a specific signal traverses through it. Modal analysis is the study of the dynamic (changing) properties of systems in the frequency domain.

Both in the time and frequency domains, a signal corresponds to a specific vibration "mode", or shape taken up by the network, system or structure during vibration. From the vantage point of the modal domain, the total set of vibrating "modes" of a signal will reveal the signal's characteristics, such as the frequencies at which the structure will amplify the effect of a load. A modal analysis can indicate the limits of a system, such as at which frequency the structure will absorb all the energy applied to it, and what the shape, or "mode" of that frequency looks like.

It is important to know the frequencies at which structures can behave erratically, such as in predicting potential earthquake damage. In product design, the modal domain is used to determine if a structure is meshed properly, and if there are any possibilities of resonance.

The equivalence between the modal, time and frequency domains is not quite as strong as that between the time and frequency domains. This is because in order to minimize the effects of noise and small experimental errors, mathematical curve fitting is used in transforming from frequency measurements to the modal domain. No information is lost in this curve fitting -- all three domains still contain the same information -- but with varying levels of noise.

*Wave representations of a sound signal*

To sum up, by analyzing signals and systems from the time domain, the frequency domain, and the modal domain, we are able to gain glimpses into signal aspects that were not visible from the time domain only. It is as if we are looking at the same three-dimensional graph but looking from different angles, extracting unique information from each. By becoming fluent in all three domains, the savvy engineer is equipped for greater certainty in design, in-depth signal analysis capabilities, and efficient troubleshooting.

Since high-speed signals cross PCB boundaries, effective signal analysis must include the signal source, destination, and the return path.

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