4.3: The Function e^jθ and the Unit Circle

Let's try to extend our definitions of the function $$e^x$$ to the argument $$x=jΘ$$. Then $$e^{jΘ}$$ is the function

$e^{jθ}=lim_{n→∞}(1+j\frac θ n)^n$

The complex number $$1+j\frac θ n$$ is illustrated in the Figure. The radius to the point $$1+j\frac θ n$$ is $$r=(1+\frac {θ^2} {n^2})^{1/2}$$ and the angle is $$φ=\tan^{−1}\frac θ n$$

This means that the nth power of $$1+j\frac θ n$$ has radius $$r^n=(1+\frac {θ^2} {n^2})^{n/2}$$ and angle $$nφ=n\;\tan^{−1}\frac θ n$$ (Recall our study of powers of z.)

Therefore the complex number $$(1+j\frac θ n)^n$$ may be written as

$(1+j\frac θ n)^n=(1+\frac {θ^2} {n^2})^{n/2}[\cos(n\;\tan^{−1}\frac θ n)+j\sin(n\tan^{−1}\frac θ n)]$

For $$n$$ large, $$(1+\frac {θ^2} {n^2})^{n/2}2≅1$$, and $$n\tan^{−1}\frac θ n≅n \frac θ n=θ$$. Therefore $$(1+j\frac θ n)^n$$ is approximately

$(1+j\frac θ n)^n=1(\cosθ+j\sinθ)$

This finding is consistent with our previous definition of $$e^{jθ}$$ !

The series expansion for $$e^{jθ}$$ is obtained by evaluating Taylor's formula at $$x=jθ$$:

$e^{jθ}=∑_{n=0}^∞\frac 1 {n!}(jθ)n$

When this series expansion for $$e^{jθ}$$ is written out, we have the formula

$e^{jθ}=∑^∞_{n=0}\frac 1 {(2n)!} (jθ)^{2n}+∑^∞_{n=0}\frac 1 {(2n+1)!}(jθ)^{2n+1} = ∑^∞_{n=0}\frac {(−1)^n} {(2n)!} θ^{2n}+j∑^∞_{n=0}\frac {(−1)^n}{(2n+1)!} θ^{2n+1}$

It is now clear that $$\cosθ$$ and $$\sinθ$$ have the series expansions

$\cosθ=∑^∞_{n=0}\frac {(−1)^n} {(2n)!} θ^{2n}$

$\sinθ=∑_{n=0}^∞\frac {(−1)^n} {(2n+1)!} θ^{2n+1}$

When these infinite sums are truncated at N−1, then we say that we have N-term approximations for $$\cosθ$$ and $$\sinθ$$:

$\cosθ≅∑^{N−1}_{n=0} \frac {(−1)^n} {(2n)!} θ^{2n}$

$\sinθ≅∑_{n=0}^{N−1} \frac {(−1)^n} {(2n+1)!} θ^{2n+1}$

The ten-term approximations to $$\cosθ$$ and $$\sinθ$$ are plotted over exact expressions for $$\cosθ$$ and $$\sinθ$$ in the Figure. The approximations are very good over one period $$(0≤θ≤2π)$$, but they diverge outside this interval. For more accurate approximations over a larger range of θ′s, we would need to use more terms. Or, better yet, we could use the fact that $$cosθ$$ and $$sinθ$$ are periodic in θ. Then we could subtract as many multiples of $$2π$$ as we needed from θ to bring the result into the range $$[0,2π]$$ and use the ten-term approximations on this new variable. The new variable is called θ-modulo $$2π$$.

Exercise $$\PageIndex{1}$$

Write out the first several terms in the series expansions for $$\cosθ$$ and $$\sinθ$$.

Demo 2.1 (MATLAB)

Create a MATLAB file containing the following demo MATLAB program that computes and plots two cycles of $$\cosθ$$ and $$\sinθ$$ versus θ. You should observe Figure. Note that two cycles take in $$2(2π)$$ radians, which is approximately 12 radians.

Code $$\PageIndex{1}$$ (MATLAB):

clg; j = sqrt(-1); theta = 0:2*pi/50:4*pi; s = sin(theta); c = cos(theta); plot(theta,s); elabel('theta in radians'); ylabel('sine and cosine'); hold on plot(theta,c); hold off

Exercise $$\PageIndex{2}$$

(MATLAB) Write a MATLAB program to compute and plot the ten-term approximations to $$\cosθ$$ and $$\sinθ$$ for θ running from 0 to $$2(2π)$$ in steps of $$2π/50$$. Compute and overplot exact expressions for $$\cosθ$$ and $$\sinθ$$. You should observe a result like the Figure.

The Unit Circle

The unit circle is defined to be the set of all complex numbers z whose magnitudes are 1. This means that all the numbers on the unit circle may be written as $$z=e^{jθ}$$. We say that the unit circle consists of all numbers generated by the function $$z=e^{jθ}$$ as θ varies from 0 to $$2π$$. See below Figure.

A Fundamental Symmetry

Let's consider the two complex numbers $$z_1$$ and $$\frac 1 {z^∗_1}$$, illustrated in Figure. We call $$\frac 1 {z^∗_1}$$the “reflection of z through the unit circle” (and vice versa). Note that $$z_1=r_1e^{jθ_1}$$ and $$\frac 1 {z^∗_1} = \frac 1 {r_1e^{jθ_1}}$$. The complex numbers $$z_1−e^{jθ}$$ and $$\frac 1 {z^*_1} −e^{jθ}$$ are illustrated in the Figure below. The magnitude squared of each is

$|z_1−e^{jθ}|^2=(z_1−e^{jθ})(z^∗_1−e^{−jθ})$

$|\frac 1 {z^*_1} −e^{jθ}|^2=(\frac 1 {z^*_1−e^{jθ}})(\frac 1 {z_1} −e^{−jθ})$

The ratio of these magnitudes squared is

$β^2=\frac {(z_1−e^{jθ})(z^∗_1−e^{−jθ})} {(\frac 1 {z^*_1}−e^{jθ})(\frac 1 {z_1} −e^{−jθ})}$

This ratio may be manipulated to show that it is independent of θ, meaning that the points $$z_1$$ and $$\frac 1 {z^∗_1}$$ maintain a constant relative distance from every point on the unit circle:

$β^2=\frac {e^{jθ}(e^{−jθ}z_1−1)(z^∗_1e^{jθ}−1)e^{−jθ}} {\frac {1} {zi} (1−e^{jθ}z^∗_1)(1−z_1e^{−jθ}) \frac 1 z_1} = |z_1|^2 \;,\;\mathrm{independent} \;\mathrm{of}\;θ!$

This result will be of paramount importance to you when you study digital filtering, antenna design, and communication theory.

Exercise $$\PageIndex{3}$$

Write the complex number $$z−e^{jθ}$$ as $$re^{jφ}$$. What are $$r$$ and $$φ$$?