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Frequency definitions and periodicity

Module by: Anders Gjendemsjø

Summary: Overview of frequency definitons, both analog and digital. 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 Ωt=2πFt Ω t 2 F t generates a periodic function with

  • a single frequency F.
  • period T
  • the relation T=1F T 1 F
Frequency is then interpreted as how many periods there are per time unit. If we choose seconds as our time unit, frequency will be measured in Hertz, which is most common.

Digital frequency

The digital frequency is defined as f=FFs f F Fs , where FsFs is the sampling frequency. The sampling interval is the inverse of the sampling frequency, Ts=1Fs Ts 1 Fs . A discrete time signal with digital frequency f therefore has a frequency given by F=fFs F f Fs if the samples are spaced at Ts=1Fs Ts 1 Fs .

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 sampling frequency FsFs. (Using the relation f=FFs f F Fs )

According to the relation Ts=1Fs Ts 1 Fs choosing an appropriate sampling frequency is equivivalent to choosing a sampling interval, which implies that digital sinusoids can be designed by specifying the sampling interval.

Angular frequencies

The angular frequencies are obtained by multiplying the frequencies by the factor 2π2:

  • Angular frequency - Ω=2πF Ω 2 F
  • Digital angular frequency - ω=2πf ω 2 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 define periodicity mathematically.

If for all k∈ℤ k ℤ we have

  • Analog signals - xt=xn+kT0 x t x n k T0 , then xtxt is periodic with period T0T0.
  • Discrete time signals - xn=xn+kN x n x n k N , then xnxn is periodic with period N.

Example 1

Consider the signal xt=sin2πFt x t 2 F t which obviously is periodic. You can check by using the periodicity definition and some trigonometric identitites.

Example 2

Consider the signal xn=sin2πfn x n 2 f n . Q:Is this signal periodic?

A: To check we will use the periodicity definition and some trigonometric identities.

Periodicity is obtained if we can find an N which leads to xn=xn+kN x n x n k N for all k∈ℤ k ℤ . Let us expand sin2πfn+kN 2 f n k N .

sin2πfn+kN=sin2πfncos2πfkN+cos2πfnsin2πfkN 2 f n k N 2 f n 2 f k N 2 f n 2 f k N (1)
To make the right hand side of Equation 1 equal to sin2πfn 2 f n , we need to impose a restriction on the digital frequency f. According to Equation 1 only fN=m f N m will yield periodicity, m∈ℤ m ℤ .

Example 3

Consider the following signals xt=cos2π18t x t 2 1 8 t and xn=cos2π18n x n 2 1 8 n , as shown in Figure 1.

Figure 1
Subfigure 1.1: a) cos2π18t 2 1 8 t Subfigure 1.2: b) cos2π18n 2 1 8 n
Subfigure 1.1 (periodicity1.png)Subfigure 1.2 (periodicity2.png)
Are the signals periodic, and if so, what are the periods?

Both the physical and digital frequency is 1/8 so both signals are periodic with period 8.

Example 4

Consider the following signals xt=cos2π23t x t 2 2 3 t and xn=cos2π23n x n 2 2 3 n , as shown in Figure 2.

Figure 2
Subfigure 2.1: a) cos2π23t 2 2 3 t Subfigure 2.2: b) cos2π23n 2 2 3 n
Subfigure 2.1 (periodicity3.png)Subfigure 2.2 (periodicity4.png)
Are the signals periodic, and if so, what are the periods?

The frequencies are 2/3 in both cases. The analog signal then has period 3/2. The discrete signal has to have a period that is an integer, so the smallest possible period is then 3.

Example 5

Consider the following signals xt=cos2t x t 2 t and xn=cos2n x n 2 n , as shown in Figure 3.

Figure 3
Subfigure 3.1: a) cos2t 2 t Subfigure 3.2: b) cos2n 2 n
Subfigure 3.1 (periodicity5.png)Subfigure 3.2 (periodicity6.png)
Are the signals periodic, and if so, what are the periods?

The frequencies are 1/π in both cases. The analog signal then has period π. The discrete signal is not periodic because the digital frequency is not a rational number.

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

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