DiscreteTime
Frequency
The concept of frequency (in radians/second)
of a continuoustime sinusoid, for example,
_{}
is
familiar from courses in physics and calculus. Such a signal is periodic for
any frequency _{}, and increasing the (absolute value of) frequency results in
a sinusoid that “oscillates faster.”
The
situation is similar for a continuoustime phasor, a complexvalued
signal of the form
_{}
We
can view this as a unitlength vector in the complex plane that rotates as t
increases, counterclockwise for _{}, and clockwise for _{}. The
phasor is periodic regardless of the value of _{} and the fundamental
period, _{}, decreases as the (absolute value of) frequency increases.
The projections of this vector on the real axis and imaginary axis yield the
trigonometric signals _{}, the real part of the phasor, and _{}, the imaginary part. Increasing the frequency causes the
vector to rotate faster, and the corresponding trigonometric signals to
oscillate faster.
Interpretations
of frequency are somewhat different in discrete time. A discretetime phasor,
defined for integer index n,
_{}
is
periodic if and only if the frequency _{} (in radians per
indexnumber) is a rational multiple of _{}. That is, if and only if for some integer m and some
positive integer N we have _{}. The fundamental period then is the least integer N
such that this expression for _{} holds.
The
applet below produces discretetime, periodic phasors, with frequencies
specified by values of the integers m and N, and displays the
real and imaginary parts for the range of index values _{}. Two phasors can be produced at once to facilitate
comparison. The applet can be used to explore the following features of
frequency in discrete time.
A
discretetime phasor may not rotate “faster” and the period may not decrease as
the frequency increases. In particular, increasing or decreasing the frequency
by _{} does not change the
signal. This is easily verified by viewing the applet with frequencies _{} and _{}, for example. In
general, the phenomenon follows from the fact that, for any integer n,
_{}
Therefore, we often restrict the range of discretetime frequencies to _{} or _{}.
For frequencies that are within such a range, there can be an interpretation in terms of faster or slower oscillation of the corresponding trigonometric signals. For example play the applet with _{} and _{}. Then use the frequencies _{} and _{} to see that this interpretation is not always valid.
Another feature of discretetime phasors is that, unlike the continuoustime case, there may be no apparent, visual direction of rotation that depends on the sign of _{}. For example, play the applet with the frequencies _{} and _{}, then again with _{} and _{}.
To
improve your understanding of these issues, you are invited to take a quiz. Solutions to the quiz questions are
available here.
First version by Marina Smelyansky, final version by Andrea Dunham
