복소 지수 (Complex Exponential) Complex Exponential
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What is a Complex Exponential?
According to Euler’s formula,
\[e^{j\theta} = \cos\theta+j\sin\theta\]Therefore, $e^{j\theta}$ can be viewed as the point $(\cos\theta,\sin\theta)$ on the complex plane.
Since $\lvert e^{j\theta} \rvert=1$, the point always lies on the unit circle, and its phase is $\theta$. In other words, we can intuitively think of $e^{j\theta}$ as a vector rotated by an angle $\theta$ on the unit circle.
Discrete-time Complex Exponential
Now, let this angle vary with time.
In discrete time, time is represented by an integer-valued sample index $n$. If we set $\theta=\omega n$ so that the angle increases by $\omega$ radians per sample, we obtain
\[x[n] = e^{j\omega n}\]Using Euler’s formula, we can write
\[e^{j\omega n} = \cos(\omega n)+j\sin(\omega n)\]In other words, $e^{j\omega n}$ is a discrete-time signal that rotates by $\omega$ radians on the complex plane with each sample.
For example, if $\omega=\pi/4$, then it rotates by $45^\circ$ per sample.
Why Use Complex Exponential?
Then why do we use complex exponentials instead of just sine and cosine?
The first reason is that they provide a much more compact way to represent sinusoidal signals. For example,
\[\cos(\omega n) = \frac{1}{2} \left( e^{j\omega n}+e^{-j\omega n} \right)\]so sine and cosine can be expressed in terms of complex exponentials.
More importantly in DSP, copmlex exponentials are especially well suited for analyzing LTI systems.
Periodicity of Complex Exponentials
Fundamental Frequency and Harmonics
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What is a Complex Exponential?
According to Euler’s formula,
\[e^{j\theta} = \cos\theta + j\sin\theta\]Therefore, $e^{j\theta}$ can be viewed as the point $(\cos\theta,\sin\theta)$ on the complex plane. Since $\lvert e^{j\theta} \rvert=1$, the point always lies on the unit circle, and its phase is $\theta$. In other words, we can intuitively think of $e^{j\theta}$ as a vector rotated by an angle $\theta$ on the unit circle.
Discrete-time Complex Exponential
Now, let this angle vary with time.
In discrete time, time is represented by an integer-valued sample index $n$. If we set $\theta=\omega n$ so that the angle increases by $\omega$ radians per sample, we obtain
\[x[n] = e^{j\omega n}\]Using Euler’s formula, we can write
\[e^{j\omega n} = \cos(\omega n) + j\sin(\omega n)\]In other words, $e^{j\omega n}$ is a discrete-time signal that rotates by $\omega$ radians on the complex plane with each sample. For example, if $\omega=\pi/4$, then it rotates by $45^\circ$ per sample.
Why Use Complex Exponentials?
Then why do we use complex exponentials instead of just sine and cosine?
The first reason is that they provide a much more compact way to represent sinusoidal signals. For example,
\[\cos(\omega n) = \frac{1}{2} \left( e^{j\omega n} + e^{-j\omega n} \right)\]so sine and cosine can be expressed in terms of complex exponentials.
More importantly in DSP, complex exponentials are especially well suited for analyzing LTI systems. If $x[n]=e^{j\omega n}$ is applied to an LTI system, the output retains the same complex exponential form:
\[e^{j\omega n} \quad\longrightarrow\quad H(e^{j\omega})e^{j\omega n}\]In other words, the frequency remains unchanged, while the system only changes the magnitude and phase of that frequency component.
Therefore, if a complicated signal can be decomposed into a sum of complex exponentials, we can determine the overall output simply by analyzing how each frequency component is modified as it passes through the LTI system. This is one of the main reasons complex exponentials are used as the fundamental building blocks of Fourier analysis.
Periodicity of Complex Exponentials
For $x[n]$ to be periodic, there must exist a positive integer $N$ such that
\[x[n+N] = x[n]\]for all $n$. If we substitute $x[n]=e^{j\omega n}$, the left side becomes $e^{j\omega (n+N)}=e^{j\omega n}e^{j\omega N}$, so the condition is $e^{j\omega n}e^{j\omega N}=e^{j\omega n}$. Since $e^{j\omega n}$ is never zero, we can divide both sides by it. Therefore, $x[n]$ is periodic when
\[e^{j\omega N} = 1\]On the complex plane, this means that after $N$ samples, the complex exponential returns to its original position. In $N$ samples, it turns by a total angle of $\omega N$, so this happens only when the angle is a multiple of $2\pi$:
\[\omega N = 2\pi k \,,\qquad k\in\mathbb{Z}\]Dividing both sides by $2\pi N$ gives $\omega/2\pi = k/N$. Thus, a discrete-time complex exponential is periodic only when $\omega/2\pi$ is a rational number. This is different from a continuous-time complex exponential, which is periodic for any $\omega$.
Why must $N$ be an integer in discrete time?
Samples exist only at integer values of $n$. So what happens if the period is not an integer?
What if $N=2.5$? ($\omega/2\pi$ is rational)
Take the signal that makes one turn every $2.5$ samples, $e^{j(2\pi/2.5)n}=e^{j0.8\pi n}$.
But there is no sample at $n=2.5$, so the signal cannot come back there. Each sample turns it by $144^\circ$, so the angle first comes back to $0^\circ$ at $n=5$ ($5\times 144^\circ=720^\circ$, two full turns). So its period is $N=5$, not $2.5$.What if $N=2\pi$? ($\omega/2\pi$ is irrational)
Take a signal that makes one turn every $2\pi$ samples, $e^{jn}$.
Since $2\pi$ is irrational, the angle never comes back to exactly $0^\circ$, no matter how many samples we wait. So the signal never repeats. It has no period at all.
To sum up, the following three conditions say the same thing:
\[x[n+N] = x[n] \text{ for all } n \quad\iff\quad e^{j\omega N} = 1 \quad\iff\quad \omega N = 2\pi k\]Solving for $\omega$ gives $\omega = 2\pi k/N$. This is why an $N$-periodic complex exponential is commonly written as
\[e^{jk\frac{2\pi}{N}n}\]Fundamental Period
A periodic signal does not have just one period. For example, if $x[n+4]=x[n]$ for all $n$, then $4$ is a period of $x[n]$. It follows that
\[x[n+8] = x[n] \,,\qquad x[n+12] = x[n]\]so $8$, $12$, and other positive multiples of $4$ are also periods.
Among all the periods of a signal, the smallest positive one is called the fundamental period, written $N_0$. So when we say that a signal is $N$-periodic, we only mean that $N$ is one of its periods. $N$ is not necessarily the fundamental period.
Harmonics
Consider the complex exponential $e^{j(2\pi/N)n}$, whose fundamental period is $N$. Its angular frequency is called the fundamental angular frequency:
\[\omega_0=\frac{2\pi}{N}\]Its integer multiples are
\[\omega_k=k\omega_0=\frac{2\pi k}{N}\]These are called harmonic frequencies, and the corresponding complex exponentials are $e^{j(2\pi k/N)n}$.
All of these complex exponentials repeat every $N$ samples, but some of them repeat in fewer than $N$ samples. For example, when $N=8$ and $k=4$,
\[e^{j\frac{2\pi \cdot 4}{8}n} = e^{j\pi n} = (-1)^n\]which repeats every $2$ samples. So $8$ is a period of this signal, but its fundamental period is $N_0=2$.
With $N=8$, every harmonic repeats every $8$ samples. For some $k$, the signal already repeats after fewer samples, so its fundamental period $N_0$ is shorter than $8$.
Low and High Frequencies
In continuous time, $e^{j\omega t}$ turns by $\omega$ radians per unit time, so a larger $\omega$ always means a faster oscillation. In discrete time, this is no longer true. Since $n$ is an integer, $e^{j2\pi n}=1$. Therefore,
\[e^{j(\omega+2\pi)n} = e^{j\omega n} \cdot e^{j2\pi n} = e^{j\omega n}\]Thus, the frequencies $\omega$ and $\omega+2\pi$ produce exactly the same signal. A discrete-time complex exponential has distinct frequencies only within an interval of length $2\pi$, which is usually taken to be $0\leq\omega<2\pi$ or $-\pi\leq\omega<\pi$.
This changes what low and high frequency mean. As $\omega$ increases from $0$, the signal oscillates faster and faster, but only until $\omega=\pi$:
\[e^{j\pi n} = (-1)^n\]This signal changes sign at every sample, and a sequence of samples cannot vary any faster than that. Beyond $\omega=\pi$, the oscillation slows down again. For example, rotating by $\omega=7\pi/4$ ($315^\circ$) per sample is equivalent to the same point as rotating by $-\pi/4$ ($-45^\circ$), so the signal is a slow rotation in the opposite direction. At $\omega=2\pi$, the signal is constant again, just as it is at $\omega=0$.
Therefore, a larger $\omega$ does not necessarily mean a higher frequency. The frequencies near $\omega=0$ and $\omega=2\pi$ are low, and the highest frequency is $\omega=\pi$.
Move $\omega$ from $0$ to $2\pi$. For $\omega \leq \pi$, the dashed and red curves are the same. For $\omega > \pi$, the dashed curve keeps getting faster, but a slower cosine (red) passes through the same samples.