State Space Representation and the State of Engineering Thinking

• Basics
• Academia / Research

Most, if not all, textbooks in signal processing (SP) thoroughly covers the frequency analysis of signals and systems alike, including the Fourier and the Z-transform that produce the well known Transfer Function. Another way of signal analysis, not as popular in signal processing though, is State Space representation. State space models describes the internal signals of the system or the process and how it affect the output, in contrast to the frequency representation that only describe the relation between the input and the output.

So why isn't state space representation popular in SP as it is in control systems? Even though the transfer function can be readily transformed into a state space form and vice-versa. I argue that the reason is how electrical engineers think. We simply do not need this level of details in many SP applications, so why bother to include them in the analysis. For example, a communication channel is almost always modelled by a frequency transfer function via Fourier transform or the Z-transform. Because we are interested in how the output of the channel changes with respect to the input, usually an impulse that is the most basic input signal, and not in how the input signal changes and evolves before reaching the output of the channel, e.g. how it might get reflected or scattered around in the wireless medium. On the other hand, in control systems we need know how the input evolves in order to control. As a result, state space representation is more appropriate in this case.

One case I found state space representation is used in SP, interestingly though, was in DSP filter implementation. Again, we need how the internal signals in the filter behaves so we can wisely introduce scaling to ultimately avoid overflow. An intuitive discussion of state space is found in the excellent book by Porat [1]. Another important example of the use of state space is in Kalman filtering widely use in tracking [2].

To conclude, it is noticed that electrical engineers use the models and representation that usually suits the application in terms of details offered y the model and required precision of the application. Is that efficient behaviour or just being lazy ;-)

[1] Porat, B., (1997). A Course in Digital Signal Processing. New York: John Wiley. 

[2] Kay, S., (1993). Fundamentals of Statistical Signal Processing. Upper Saddle River. Prentice-Hall PTR.

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