1.4 Frequency definitions and periodicity

Frequency definitions

In signal processing we use several types of frequencies. This may seem confusing at first, but it is really not that difficult.

Analog frequency

The frequency of an analog signal is the easiest to understand. A trigonometric function with argument $t=2\pi Ft$ generates a periodic function with

• a single frequency F.
• period T
• the relation $T=\frac{1}{F}$

Digital frequency

The digital frequency is defined as $f=\frac{F}{F_s}$ , where $F_s$ is the sampling frequency. The sampling interval is the inverse of the sampling frequency, $T_s=\frac{1}{F_s}$ . A discrete time signal with digital frequency f therefore has a frequency given by $F=f{F}_s$ if the samples are spaced at $T_s=\frac{1}{F_s}$ .

Consequences

In design of digital sinusoids we do not have to settle for a physical frequency. We can associate any physical frequency F with the digital frequency f, by choosing the appropriate samplingfrequency $F_s$ . (Using the relation $f=\frac{F}{F_s}$ )

According to the relation $T_s=\frac{1}{F_s}$ choosing an appropriate sampling frequency is equivivalent to choosing a sampling interval, which implies that digital sinusoids canbe designed by specifying the sampling interval.

Angular frequencies

The angular frequencies are obtained by multiplying the frequencies by the factor $2\pi$ :

• Angular frequency

  $=2\pi F$

• Digital angular frequency

  $=2\pi f$

Signal periodicity

Any analog sine or cosine function is periodic. So it may seem surprising that discrete trigonometric signals not necessarily are periodic. Let us defineperiodicity mathematically.

If for all $k\in$ we have

• Analog signals

  $x(t)=x(n+k{T}_0)$ , then $x(t)$ is periodic with period $T_0$ .

• Discrete time signals

  $x(n)=x(n+kN)$ , then $x(n)$ is periodic with period N.

Conclusion

For a time discrete trigonometric signal to be periodic its digital frequency has to be a rational number , i.e. given by the ratio of two integers. Contrast this to analog trigonometric signals.

Matlab file

periodicity.m

Take a look at

• Introduction
• Discrete time signals
• Analog signals
• Discrete vs Analog signals
• Energy&Power
• Exercises