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# m01 - An Overview of Continuous-Time Signals

Module by: C. Sidney Burrus. E-mail the author

Summary: Many phenomena are modeled as continuous-time signals. The description and analysis uses functions of a real variable (often time or distance) and Fourier theory is used to give a frequency domain description. Wavelets are a modern tool for signal analysis.

## [Continuous-Time Signals]Continuous-Time Signals

Signals occur in a wide range of physical phenomenon. They might be human speech, blood pressure variations with time, seismic waves, radar and sonar signals, pictures or images, stress and strain signals in a building structure, stock market prices, a city's population, or temperature across a plate. These signals are often modeled or represented by a real or complex valued mathematical function of one or more variables. For example, speech is modeled by a function representing air pressure varying with time. The function is acting as an mathematical analogy to the speech signal and, therefore, is called an analog signal. For these signals, the independent variable is time and it changes continuously so that the term continuous-time signal is also used. In our discussion, we talk of the mathematical function as the signal even though it is really a model or representation of the physical signal.

The description of signals in terms of their sinusoidal frequency content has proven to be one of the most powerful tools of continuous and discrete-time signal description, analysis, and processing. For that reason, we will start the discussion of signals with a development of Fourier transform methods. We will first review the continuous-time methods of the Fourier series (FS), the Fourier transform or integral (FT), and the Laplace transform (LT). Next the discrete-time methods will be developed in more detail with the discrete Fourier transform (DFT) applied to finite length signals followed by the discrete-time Fourier transform (DTFT) for infinitely long signals and ending with the Z-transform which allows the powerful tools of complex variable theory to be applied.

More recently, a new tool has been developed for the analysis of signals. Wavelets and wavelet transforms [1][2][3][4][5] are another more flexible expansion system that also can describe continuous and discrete-time finite or infinite duration signals. We will very briefly introduce the ideas behind wavelet-based signal analysis.

## References

1. Barbara Burke Hubbard. (1996). The World According to Wavelets. [Second Edition 1998]. Wellesley, MA: A K Peters.
2. C. Sidney Burrus, Ramesh A. Gopinath and Haitao Guo. (1998). Introduction to Wavelets and the Wavelet Transform. Upper Saddle River, NJ: Prentice Hall.
3. Ingrid Daubechies. (1992). Ten Lectures on Wavelets. [Notes from the 1990 CBMS-NSF Conference on Wavelets and Applications at Lowell, MA]. Philadelphia, PA: SIAM.
4. Martin Vetterli and Jelena Kova\vcević. (1995). Wavelets and Subband Coding. Upper Saddle River, NJ: Prentice-Hall.
5. Gilbert Strang and T. Nguyen. (1996). Wavelets and Filter Banks. Wellesley, MA: Wellesley-Cambridge Press.

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