Digital Signal Processing Reference
In-Depth Information
2.1.3
Digital Frequency
With respect to the oscillatory behavior captured by the complex exponen-
tial, a note on the concept of “frequency” is in order. In the continuous-time
world (the world of textbook physics, to be clear), where time is measured in
seconds, the usual unit of measure for frequency is the Hertz which is equiv-
alent to 1
l
g
r
,
y
i
d
.
,
©
,
L
s
second. In the discrete-time world, where the index
n
represents
a dimensionless time, “digital” frequency is expressed in radians which is
itself a dimensionless quantity.
(1)
The best way to appreciate this is to con-
sider an algorithm to generate successive samples of a discrete-time sinu-
soid at a digital frequency
/
ω
0
:
ω
←
0;
initialization
φ
←
initial phase value;
repeat
x
(
ω
+
φ
)
←
sin
;
compute next value
ω
←
ω
+
ω
0
;
update phase
until
done
At each iteration,
(2)
the argument of the trigonometric function is incre-
mented by
0
and a new output sample is produced. With this in mind, it is
easy to see that
the highest frequency manageable by a discrete-time system
is
ω
-periodicity of
the trigonometric functions “maps back” the output values to a frequency
between 0 and 2
ω
max
=
2
π
; for any frequency larger than this, the inner 2
π
π
. This can be expressed as an equation:
sin
n
π
)+
φ
=
(
ω
+
(
ω
+
φ
)
2
k
sin
n
(2.11)
for all values of
k
-equivalence of digital frequencies is a perva-
sive concept in digital signal processing and it has many important conse-
quences which we will study in detail in the next Chapters.
∈
.This2
π
(1)
An angle measure in radians is dimensionless since it is defined in terms of the ratio of
two lengths, the radius and the arc subtended by the measured angle on an arbitrary
circle.
(2)
Here is the algorithmwritten in C:
extern double omega0;
extern double phi;
static double omega = 0;
double GetNextValue()
{
omega += omega0;
return sin(omega + phi);
}
