Summary: Overview of frequency definitons, both analog and digital. Periodicity.
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In signal processing we use several types of frequencies. This may seem confusing at first, but it is really not that difficult.
The frequency of an analog signal is the easiest to understand.
A trigonometric function with argument
The digital frequency is defined as
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
According to the relation
The angular frequencies are obtained by multiplying the
frequencies by the factor
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
Consider the signal
Consider the signal
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
Consider the following signals
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Both the physical and digital frequency is 1/8 so both signals are periodic with period 8.
Consider the following signals
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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.
Consider the following signals
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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.
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.
Take a look at • Introduction; • Discrete time signals; • Analog signals; • Discrete vs Analog signals; • Energy & Power; • Exercises ?