Complex functions of a real variable

Cartesian Form

Polar or Exponential Form

Euler's Identities

Conjugate

Operations on two complex numbers

nth power and nth roots of a complex number

Functions of a complex variable

Complex Functions of a real variable

Magnitude and Phase Plot

Examples and References

Introduction

You may be used to dealing with functions of a variable such as $y = f(x)$, where $x$ is called the independent variable and $y$ is called the dependent variable and typically, $y$ takes real values when $x$ takes real values. In this course, we will be interested in complex functions of a real variable. Often the real variable will represent time or frequency. Such a function, normally denoted as $x(t)$ or $X(\omega)$ is a function which takes a complex value for every real value of the independent variable $t$ or $\omega$. Pay attention to the notation carefully - $t$ or $\omega$ now becomes the independent variable and $x(t)$ or $X(\omega)$ now becomes the dependent variable. We can also think of the complex function as the combination of two real functions of the independent variable, one for the real part of $x(t)$ and one for the imaginary part of $x(t)$.

When dealing with real functions of a real variable, you may be used to plotting the function $x(t)$ as a function of $t$. However, when $x(t)$ is a complex function, there is a problem in plotting this function since for every value of $t$, we need to plot a complex number. In this case, we do one of two things - either we plot the real part of $x(t)$ versus $t$ and plot the imaginary part of $x(t)$ versus $t$, or we plot $|x(t)|$ versus $t$ and $\mbox{arg}(x(t))$ versus $t$. Either of these is fine, but we do need two plots to effectively understand how $x(t)$ changes with $t$.

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Examples

Example 1

Consider the function $x(t) = e^{j2\pi t} = \cos {2\pi t} + j \sin {2\pi t}$ for all real values of $t$.
This is clearly a complex function of a real variable $t$. $\Re\{x(t)\}, \Im\{x(t)\}, |x(t)|, \mbox{arg}(x(t))$ are all real functions of the real variable $t$. Hence, we can plot $\Re\{x(t)\}$ versus $t$ and $\Im\{x(t)\}$ versus $t$ or we can plot $|x(t)$ versus $t$ and $\mbox{arg}(x(t))$ versus $t$ as shown in figure.

Example 2

Consider the function $H(\omega) = \frac{1}{1+j\omega}$, where $\omega$ is a real variable. Roughly sketch the magnitude and phase of $H(\omega)$ as a function of $\omega$.

(1)

\begin{eqnarray} \nonumber H(\omega) &=& \frac{1}{1+j\omega}\\ \nonumber |H(\omega)| &=& \frac{1}{\sqrt{1+\omega^2}}\\ \nonumber \angle(H(\omega)) &=& 0-\tan^{-1}\omega \end{eqnarray}

A plot of $|H(\omega)|$ versus $\omega$ and $\angle(H(\omega))$ versus $\omega$ is shown in Figure.

Example 3

Consider the function $X(\omega) = \frac{j \omega}{1+j\omega}$, where $\omega$ is a real variable. Roughly sketch the magnitude and phase of $X(\omega)$ as a function of $\omega$.

(2)

\begin{eqnarray} \nonumber X(\omega) &=& \frac{j\omega}{1+j\omega}\\ \nonumber |X(\omega)| &=& \frac{|\omega|}{\sqrt{1+\omega^2}}\\ \nonumber \angle(X(\omega)) &=& \left\{\begin{array}{ll} -\frac{\pi}{2}-\tan^{-1}\omega &,\omega <0 \\ \frac{\pi}{2}-\tan^{-1}\omega &,\omega >0\end{array} \right. \end{eqnarray}

A plot of $|X(\omega)|$ versus $\omega$ and $\angle(X(\omega))$ versus $\omega$ is shown in figure.

MATLAB code for Example 3

%Defining Variables
omega = linspace(-10*pi,10*pi,1000);
x = (j*omega)./(1+j*omega);

% Subplot for Magnitude
subplot(2,1,1);
plot(omega,abs(x));
xlabel('w');
ylabel('|x(j w)|');

% Subplot for Phase
subplot(2,1,2);
plot(omega,phase(x));
xlabel('w');
ylabel('phase(x(j w))');

Output plots for Example 3

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