Digital Signal Processing Reference
In-Depth Information
For a fractional decimal number, multiply by 2 and keep the whole part. Then
repeat with the remaining fractional part until it becomes zero (though it is possible
for it to repeat forever, just as 1/3 does in decimal). When nished, read the whole
parts back from the top down. For example, say we want to convert .375 to binary:
:3752 = 0 plus :75
:752 = 1 plus :5
:52 = 1 plus 0:
Our answer for this is .011 binary. We can put the answers together and conclude
that 4.375 equals 0100.011 in binary.
1.1.5
Complex Numbers
Complex numbers are important to digital signal processing. For example, functions
like the Fourier transform (fft and ifft) return their results as complex numbers.
This topic is covered in detail later in this topic, but for the moment you can think
of the Fourier transform as a function that converts data to an interesting form.
Complex numbers provide a convenient way to store two pieces of information,
either x and y coordinates, or a magnitude (length of a 2D vector) and an angle
(how much to rotate the vector). This information has a physical interpretation in
some contexts, such as corresponding to the magnitude and phase angle of a sinusoid
for a given frequency, when returned from the Fourier transform.
Most likely, you rst ran into complex numbers in a math class, factoring roots
with the quadratic formula. That is, an equation such as
2x
2
+ 4x30 = 0
could be factored as (xroot
1
)(xroot
2
), where
b
p
b
2
4ac
2a
root
1;2
=
:
If the numbers work out nicely, then the quadratic formula results in a couple
of real roots. In this case, it would be 3;5, to make (x3)(x + 5) = 0. A problem
arises when b
2
4ac is negative because the square root of a negative quantity does
not exist. The square root operator is supposed to return the positive root. How
can we take a positive number, multiply it by itself, and end up with a negative
number? For every \real" value, even the negative ones, the square is positive. To
deal with this, we have the imaginary quantity j. (Some people prefer i instead.)
