Introduction - DSP Using MATLAB and Wavelets - page 21

Figure 1.7: Three glasses of water.
Suppose that we have a simple incrementer system that adds 1 to every input.
We will use x 1 [n] to represent the input signal, and y 1 [n] to represent the output. We
can write the output in terms of the input, i.e., y 1 [n] = x 1 [n]+1. If x 1 is the sequence
f1, 3, 7g, the output y 1 would be the sequencef2, 4, 8g. Now suppose we want to
get the original values of x 1 back. To do this, we need to \undo" the transform,
which we can do with the following \inverse" transform. Since we have given a
name to the reverse transform, we will call the rst transform (y 1 [n] = x 1 [n] + 1)
the \forward" transform.
The inverse transform for the incrementer system would be a decrementer, that
simply subtracts 1 from each value, i.e., y 2 [n] = x 2 [n]1. If we hooked these two
systems up in series, we would have the output of the rst system as the input to the
second, eectively assigning x 2 [n] = y 1 [n]. So y 2 [n] = x 2 [n]1, y 2 [n] = y 1 [n]1,
or y 2 [n] = (x 1 [n] + 1)1. It is easy to see that the two systems cancel each other
out, and the nal output (y 2 [n]) is what we started with originally (x 1 [n]).
Typically, we do not study systems that add or subtract constants.
A natural question is, \Why would we do this?" One answer is so that we can
analyze the transformed signal, or compress it. This does not follow from such a
simple example, but more complicated transforms (such as the discrete cosine trans-
form) have been used eectively to allow a compression program to automatically
alter the signal to store it in a compact manner.
Sometimes transforms are performed because things are easier to do in the trans-
formed domain. For example, suppose you wanted to nd the number of years
between the movies Star Wars (copyright MCMLXXVII) and Star Wars: Episode
II|Attack of the Clones (copyright MMII). Subtracting one number from another
is not a problem, but Roman numerals are dicult to work with [7]! The algorithm
we use of subtracting the right-most column digits rst, writing the dierence and
borrowing from the left, does not work. Instead, the easiest thing to do is convert
the Roman numerals to decimal numbers, perform the subtraction, and convert the
result back to Roman numerals. In this case, the transform would convert MCM-
LXXVII to 1977, and MMII to 2002. Subtracting 1977 from 2002 gives us 25, which
is the inverse-transformed to XXV. Performing the transform and later the inverse-