Discrete-Time Frequency

 

The concept of frequency (in radians/second) of a continuous-time 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 continuous-time phasor, a complex-valued signal of the form

���������������������������������

We can view this as a unit-length 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 discrete-time phasor, defined for integer index n,

�������������������������������

is periodic if and only if the frequency �(in radians per index-number) 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 discrete-time, 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 discrete-time 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 discrete-time 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 discrete-time phasors is that, unlike the continuous-time 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.