Digital Signal Processing Reference
In-Depth Information
measurements, we understand intuitively that the temperature has values between
readings, and we do not know what values these would be. If a cold wind blows, the
temperature goes down, and if the sun shines through the clouds, then it goes up.
For example, suppose we measure the temperature every hour. By doing this, we
are choosing to ignore the temperature for all time except for the hourly readings.
This is an important idea: the signal may vary over time, but when we take periodic
readings of the signal, we are left with only a representation of the signal.
A signal can be thought of as a (continuous or discrete) sequence of (continuous
or discrete) values. That is, a continuous signal may have values at any arbitrary
index value (you can measure the temperature at noon, or, if you like, you can
measure it at 0.0000000003 seconds after noon). A discrete signal, however, has
restrictions on the index, typically that it must be an integer. For example, the
mass of each planet in our solar system could be recorded, numbering the planets
according to their relative positions from the sun. For simplicity, a discrete signal is
assumed to have an integer index, and the relationship between the index and time
(or whatever parameter) must be given. Likewise, the values for the signal can be
with an arbitrary precision (continuous), or with a restricted precision (discrete).
That is, you could record the temperature out to millionths of a degree, or you
could restrict the values to something reasonable like one digit past the decimal.
Discrete does not mean integer, but rather that the values could be stored as a
rational number (an integer divided by another integer). For example, 72.3 degrees
Fahrenheit could be thought of as 723/10. What this implies is that irrational
numbers cannot be stored in a computer, but only approximated. is a good
example. You might write 3.14 for , but this is merely an approximation. If you
wrote 3.141592654 to represent , this is still only an approximation. In fact, you
could write out to 50 million digits, but it would still be only an approximation!
It is possible to consider a signal whose index is continuous and whose values
are discrete, such as the number of people who are in a building at any given time.
The index (time) may be measured in fractions of a second, while the number of
people is always a whole number. It is also possible to have a signal where the index
is discrete, and the values are continuous; for example, the time of birth of every
person in a city. Person #4 might have been born only 1 microsecond before person
#5, but they technically were not born at the same time. That does not mean that
two people cannot have the exact same birth time, but that we can be as precise as
we want with this time. Table 1.1 gives a few example signals, with continuous as
well as discrete indices and quantities measured.
For the most part, we will concentrate on continuous signals (which have a
continuous index and a continuous value), and discrete signals (with an integer index
and a discrete value). Most signals in nature are continuous, but signals represented
