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Analog Signals

2 June, 2016 - 15:17

Analog signals are usually signals defined over continuous independent variable(s). Modeling the Speech Signal is produced by your vocal cords exciting acoustic resonances in your vocal tract. The result is pressure waves propagating in the air, and the speech signal thus corresponds to a function having independent variables of space and time and a value corresponding to air pressure: s (x, t) (Here we use vector notation x to denote spatial coordinates). When you record someone talking, you are evaluating the speech signal at a particular spatial location, x0 say. An example of the resulting waveform s (x0,t) is shown in this Figure 1.1.

Speech Example

Figure 1.1 Speech Example
A speech signal's amplitude relates to tiny air pressure variations. Shown is a recording of the vowel "e" (as in "speech").  

Photographs are static, and are continuous-valued signals defined over space. Black-and-white images have only one value at each point in space, which amounts to its optical refection properties. In Figure 1.2, an image is shown, demonstrating that it (and all other images as well) are functions of two independent spatial variables.

Figure 1.2 Lena
On the left is the classic Lena image, which is used ubiquitously as a test image. It contains straight and curved lines, complicated texture, and a face. On the right is a perspective display of the Lena image as a signal: a function of two spatial variables. The colors merely help show what signal values are about the same size. In this image, signal values range between 0 and 255; why is that?  

Color images have values that express how reflectivity depends on the optical spectrum. Painters long ago found that mixing together combinations of the so-called primary colors red, yellow and blue can produce very realistic color images. Thus, images today are usually thought of as having three values at every point in space, but a different set of colors is used: How much of red, green and blue is present. Mathematically, color pictures are multivalued vector-valued signals: s (x)=(r (x) ,g (x) ,b (x))T .

Interesting cases abound where the analog signal depends not on a continuous variable, such as time, but on a discrete variable. For example, temperature readings taken every hour have continuous analog values, but the signal's independent variable is (essentially) the integers.