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## Constructing a Signal out of Components

One view of the input signal illustrated above in figure 4.2, is that it is made up of a number of contributing signals - mathematically, we can consider any reasonable set of orthogonal signals as components, but the easiest ones to use are sine functions.

One extreme case often used to illustrate this is the square wave signal. In figure 4.4, we show a square wave. If this was made out of a number of sine waves with different frequencies, the contribution of each frequency would be as illustrated in figure 4.5.

The way then that we build up the square wave constructively out of a set of sine waves of different frequenies can be seen in the progression of figures,

It may seem odd that a simple on-off'' signal takes a lot of contributions, but then the point is that this method of representing the continuous signal is general, and can represent any input signal.

Input data can be transformed in a number of ways to make it easier to apply certain compression techniques. The most common transform in current techniques is the Discrete Cosine Transform. This is a variant of the Discrete Fourier Transform, which is in turn, the digital (discrete) version of the Continuous Fourier Transform.

As described earlier, any signal (whether a video or audio signal) can be considered a periodic wave. If we think of a sequence of sounds, they are a modulation of an audio wave; similarly, the scan over an image or scene carried out by a camera conveys a wave which has periodic features in time (in the time frame of the scene, as well as over multiple video frames in the moving picture). It is possible to convert from the original signal as a function of time, to the fourier series, which is the sum of a set of terms, each being a particular frequency (or wavelength). You can think of these terms or coefficients as being the contribution of a set of base pure sine-wave'' frequencies (also known as the spectral density), that together make up the actual signal.

where K1 = 1

and

where K1 = pi / 2

You can imagine these sweeping through a typical audio signal as shown in figure 4.10, and pulling out'' a spectrum (see figure 4.11, or set of coefficients that represent the contribution of each frequency to that part of the signal strength.

Next: Lossless Data Compression Up: Nature of the Signal Previous: Analog to Digital Conversion:
Jon CROWCROFT
1998-12-03