Signal Processing Insight

Signal processing is an interesting engineering and mathematical discipline. And it's rapidly invading many fields of engineering, science and even our daily lives. This may be attributed to the high level of mathematical abstraction since signals can be obtained from any source such as acoustic, seismic, medical , etc, which facilitates using signal processing as a vital tool to solve many problems in engineering and science.

To get a clear insight into signal processing we need to understand the title. A signal is simply a carrier of information whereas processing is the mathematical operations that the signal undergoes to achieve a certain task. This brief explanation conceals underneath it the reason why such operations are used in signal processing instead of others, e.g. Fourier analysis instead of Taylor expansion which is widely used in mechanics, which will be clarified later on.

As human beings we are interested in informations, varying informations to be more precise. For example a constant signal is not very informative since we can extract the level solely, as it is the case for a single sinusoid (having three constant parameters; amplitude, frequency and phase), a varying signal would carry more information to us such as AM or FM signal.

From mathematical point of view we have an infinite vector space of signal and we need a basis to describe such vector space. Such vector space must conform to our need to describe signal variations, an appropriate basis function would none other than the sinusoid, different frequencies represent different change rates, more formally this is the famous Fourier analysis. From engineering point view, signals with high variations in time consists of high frequency components and thus require high bandwidth.

But like everything in engineering there must be a compromise. The time-frequency representation is inversely promotional, i.e. a wide frequency support (high frequency resolution) results in a narrow time support, meaning when represent a signal accurately in the frequency domain yields a poor time representation. Do not despair, there is an alternative that ensures a satisfactory time-frequency trade off, this alternative is wavelets. Wavelets is a class of function that forms a basis to represent signals, wavelet functions can be tailored to the required time-frequency trade off in both resolutions. Wavelets are still mathematically intensive from practical point of view, it will be a while till the use of wavelet spread into applications or even replace the Fourier analysis in certain fields. Actually it depends on the theoretical improvement in simplifying the computation of (like the fast Fourier transform) and the advance in computational horse power.