Chapter 2 - Signal Properties¶

Periodicity¶

Find the fundamental period of a sine wave with angular frequency \(\omega_{0}\) and amplitude \(A\).

\[ f(t) = A\sin{\left(\omega_{0} t\right)} \]

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\[ T = \frac{2\pi}{\omega_{0}} \]

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Using the property

\[ f(t) = f(t + T) \]

We get

\[ A\sin{\left(\omega_{0}t\right)} = A\sin{\left(\omega_{0}\left(t + T\right)\right)} \]

Now expanding out

\[ A\sin{\left(\omega_{0}\left(t + T\right)\right)} = A\sin{\left(\omega_{0}t + \omega_{0} T\right)} \]

Using the trigonometric identity:

\[ \sin{(x)} = \sin{\left(x + 2\pi k\right)} \]

We can compare corresponding terms and get:

\[ \omega_{0}T = 2\pi k \]

Then solving for \(T\), we have:

\[ T = \frac{2\pi k}{\omega_{0}} \]

The smallest positive \(T\) would be when \(k=1\), so the fundamental period for the signal is:

\[ \Rightarrow T = \frac{2\pi}{\omega_{0}} \]