2.3: Sinusoidal Time Functions

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James Kirtley
Massachusetts Institute of Technology via MIT OpenCourseWare

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A sinusoidal function of time might be written in at least two ways:

\[\ f(t)=A \cos (\omega t+\phi)\label{17} \]

\[\ f(t)=B \cos (\omega t)+C \sin (\omega t)\label{18} \]

A third way of writing this time function is as the sum of two complex exponentials:

\[\ f(t)=\underline{X} e^{j \omega t}+\underline{X}^{*} e^{-j \omega t}\label{19} \]

Note that the form of equation 19, in which complex conjugates are added together, guarantees that the resulting function is real.

Now, to relate equation 19 with the other forms of the sinusoidal function, equations 17 and 18, see that \(\ \underline{X}\) may be expressed as:

\[\ \underline{X}=|\underline{X}| e^{j \psi}\label{20} \]

Then equation 19 becomes:

\[\ f(t)=|\underline{X}| e^{j \psi} e^{j \omega t}+|\underline{X}|^{*} e^{-j \psi} e^{-j \omega t}\label{21} \]

\[\ =|\underline{X}| e^{j(\psi+\omega t)}+|\underline{X}|^{*} e^{-j(\psi+\omega t)}\label{22} \]

\[\ =2|\underline{X}| \cos (\omega t+\psi)\label{23} \]

Then, the coefficients in equation 17 are related to those of equation 19 by:

\[\ |\underline{X}|=\frac{A}{2}\label{24} \]

\[\ \psi=\phi\label{25} \]

Alternatively, we could write

\[\ \underline{X}=x+j y\label{26} \]

in which the real and imaginary parts of \(\ \underline{X}\) are:

\[\ x=|\underline{X}| \cos (\psi)\label{27} \]

\[\ y=|\underline{X}| \sin (\psi)\label{28} \]

Then the time function is written:

\[\ f(t)=x\left(e^{j \omega t}+e^{-j \omega t}\right)+j y\left(e^{j \omega t}-e^{-j \omega t}\right)\label{29} \]

\[\ =2 x \cos (\omega t)-2 y \sin (\omega t)\label{30} \]

Thus:

\[\ A=2 x\label{31} \]

\[\ B=-2 y\label{32} \]

\[\ X=\frac{A}{2}-j \frac{B}{2}\label{33} \]

It is also possible to write equation 19 in the form:

\[\ f(t)=R e\left(2 \underline{X} e^{j \omega t}\right)\label{34} \]

While both expressions (19 and 34) are equivalent, it is advantageous to use one or the other of them, according to circumstances. The first notation (equation 19) is the full representation of that sinusoidal signal and may be used under any circumstances. It is, however, cumbersome, so that the somewhat more compact version(equation 34) is usually used. Chiefly when nonlinear products such as power are involved, it is necessary to be somewhat cautions in its use, however, as we will see later on.