Exponential Signal

Introduction

The exponential signal is arguably the most fundamental building block in signals and systems theory. Why? Because exponential functions are eigenfunctions of linear time-invariant (LTI) systems — when you feed an exponential into an LTI system, you get an exponential back out (possibly scaled). This property makes exponentials the natural basis for analyzing any linear system, and it is the reason we use Laplace transforms, Fourier transforms, and z-transforms — all of which decompose signals into complex exponentials.

Think of it this way: every signal you will encounter in this course can be expressed as a combination of exponentials. Understanding them deeply is like understanding the alphabet before you start reading.

The Real Exponential Signal

Continuous-Time Real Exponential

The continuous-time real exponential signal is defined as:

$$x(t) = Ae^{at}$$

where $A$ is the initial amplitude (value at $t = 0$) and $a$ is the rate parameter that determines the behavior:

• If $a > 0$: The signal grows exponentially (used in modeling population growth, nuclear chain reactions, unstable systems)
• If $a < 0$: The signal decays exponentially (RC circuit discharge, radioactive decay, stable system transients)
• If $a = 0$: The signal is a constant ($x(t) = A$ for all $t$)

The Time Constant

For a decaying exponential $x(t) = Ae^{-t/\tau}$, the time constant $\tau = -1/a$ is the time it takes for the signal to decay to $1/e \approx 36.8\%$ of its initial value. This is an incredibly useful concept:

• After $1\tau$: signal has decayed to 36.8% of $A$
• After $2\tau$: signal has decayed to 13.5% of $A$
• After $3\tau$: signal has decayed to 5.0% of $A$
• After $5\tau$: signal has decayed to 0.67% of $A$ (practically zero)

In circuit analysis, the time constant of an RC circuit is $\tau = RC$, and for an RL circuit, $\tau = L/R$.

Energy and Power

For a decaying exponential $x(t) = e^{-at}u(t)$ with $a > 0$:

$$E = \int_0^{\infty} |x(t)|^2 dt = \int_0^{\infty} e^{-2at} dt = \frac{1}{2a}$$

Since the energy is finite, this is an energy signal (not a power signal).

The Complex Exponential Signal

Continuous-Time Complex Exponential

The complex exponential is defined as:

$$x(t) = Ae^{st}$$

where $s = \sigma + j\omega$ is a complex number. Expanding:

$$x(t) = Ae^{(\sigma + j\omega)t} = Ae^{\sigma t} \cdot e^{j\omega t}$$

This is a sinusoidal signal ($e^{j\omega t}$) multiplied by a real exponential envelope ($e^{\sigma t}$):

• $\sigma < 0$: Damped sinusoid (decaying oscillation) — natural response of underdamped circuits
• $\sigma > 0$: Growing sinusoid — indicates instability
• $\sigma = 0$: Pure sinusoid with constant amplitude — the steady-state response

Euler's Formula: The Bridge Between Exponentials and Sinusoids

Euler's formula is the most important identity in signal processing:

$$e^{j\omega t} = \cos(\omega t) + j\sin(\omega t)$$

This means every sinusoid can be expressed as a combination of complex exponentials:

$$\cos(\omega t) = \frac{e^{j\omega t} + e^{-j\omega t}}{2}$$

$$\sin(\omega t) = \frac{e^{j\omega t} - e^{-j\omega t}}{2j}$$

This identity is the foundation of Fourier analysis — we decompose signals into complex exponentials precisely because of this relationship.

The Discrete-Time Exponential Signal

Real Discrete Exponential

$$x[n] = A \cdot r^n$$

where $r$ is the base:

• $|r| < 1$: Decaying sequence (converges to zero)
• $|r| > 1$: Growing sequence (diverges)
• $|r| = 1$: Constant magnitude
• $r > 0$: All samples have the same sign
• $r < 0$: Samples alternate in sign (oscillating decay/growth)

Complex Discrete Exponential

$$x[n] = A \cdot e^{(\sigma_0 + j\omega_0)n} = A \cdot e^{\sigma_0 n} \cdot e^{j\omega_0 n}$$

The key difference from continuous time: discrete-time complex exponentials are periodic in frequency with period $2\pi$. That is, $e^{j\omega_0 n} = e^{j(\omega_0 + 2\pi)n}$ for all integer $n$. This is why the discrete-time Fourier transform only needs the range $-\pi$ to $\pi$ (or $0$ to $2\pi$).

Eigensignal Property of Complex Exponentials

The most profound property: if the input to an LTI system with impulse response $h(t)$ is $x(t) = e^{st}$, the output is:

$$y(t) = H(s) \cdot e^{st}$$

where $H(s) = \int_{-\infty}^{\infty} h(\tau)e^{-s\tau} d\tau$ is the system's transfer function (the Laplace transform of $h(t)$).

The complex exponential goes in, and the same complex exponential comes out, scaled by $H(s)$. This eigenfunction property is THE reason we use Laplace and Fourier transforms — they decompose signals into these eigenfunctions, making system analysis a simple multiplication.

Worked Examples

Example 1: Find the time constant and sketch $x(t) = 5e^{-2t}u(t)$.

Solution: Comparing with $Ae^{-t/\tau}$: $1/\tau = 2$, so $\tau = 0.5$ seconds. The signal starts at 5 and decays to $5/e \approx 1.84$ after 0.5 seconds. It is essentially zero after $5\tau = 2.5$ seconds.

Example 2: Express $x(t) = 3e^{-t}\cos(4t)u(t)$ in terms of complex exponentials.

Solution: Using Euler's formula: $$x(t) = \frac{3}{2}e^{-t}(e^{j4t} + e^{-j4t})u(t) = \frac{3}{2}(e^{(-1+j4)t} + e^{(-1-j4)t})u(t)$$

This is a sum of two complex exponentials with $s = -1 \pm j4$.

Example 3: Determine if $x[n] = (0.8)^n u[n]$ is an energy signal.

Solution: $E = \sum_{n=0}^{\infty} |0.8^n|^2 = \sum_{n=0}^{\infty} 0.64^n = \frac{1}{1-0.64} = \frac{1}{0.36} \approx 2.78$

Since energy is finite, it is an energy signal.

Applications

• RC and RL circuits: Natural response is always exponential — $v(t) = V_0 e^{-t/RC}$
• Radioactive decay: $N(t) = N_0 e^{-\lambda t}$ (half-life = $\ln 2/\lambda$)
• Control systems: Stability is determined by whether exponential modes decay or grow
• Signal analysis: Laplace and z-transforms decompose signals into exponential components
• Digital filters: IIR filter responses are sums of discrete exponentials

Key Takeaways

• Real exponentials model growth ($a>0$) or decay ($a<0$), with time constant $\tau = 1/|a|$
• Complex exponentials combine sinusoidal oscillation with exponential envelopes
• Euler's formula $e^{j\omega t} = \cos\omega t + j\sin\omega t$ bridges exponentials and sinusoids
• Complex exponentials are eigenfunctions of LTI systems — the foundation of transform methods
• Discrete-time exponentials have periodic frequency behavior (period $2\pi$)
• After $5\tau$, a decaying exponential is effectively zero (0.67% remaining)