In signal processing, most transform basis functions look like \( e^{(\text{something})} \) — such as \( e^{j\omega t} \), \( e^{-st} \), or \( e^{-j2\pi kt/N} \). This is not a coincidence. Exponentials have deep mathematical properties that make them ideal for representing and analyzing signals.
1. Exponentials Are Eigenfunctions of LTI Systems
If you feed an LTI system a complex exponential:
x(t) = e^{st}the output has the same shape:
y(t) = H(s) e^{st}This makes exponentials extremely convenient because they diagonalize LTI systems and turn convolution into simple multiplication.
2. Sines and Cosines Are Special Cases of Complex Exponentials
Euler’s identity:
e^{j\omega t} = \cos(\omega t) + j\sin(\omega t)Using exponentials automatically includes all sinusoidal basis functions used in Fourier methods.
3. Exponentials Are Solutions of Linear Differential Equations
Many physical systems obey differential equations whose solutions are sums of exponentials:
x(t) = C_1 e^{s_1 t} + C_2 e^{s_2 t} + \dotsSo exponentials naturally model real-world oscillations and decays.
4. Exponentials Are Orthonormal Over Many Domains
For the Fourier transform:
\int e^{-j\omega t} e^{j\omega' t} dt = 0 \quad \text{(when } \omega \neq \omega' \text{)}This orthogonality makes them ideal for building complete, non-redundant bases.
5. They Simplify Mathematics
Exponentials turn complex operations into simple ones:
- Differentiation \( \rightarrow \) multiplication
- Convolution \( \rightarrow \) multiplication
- Modulation \( \rightarrow \) shifts
- Shifts \( \rightarrow \) multiplication by \( e^{-s\tau} \)
6. They Form a Complete Basis for Signals
Exponentials can represent:
- Periodic signals \( \rightarrow \) Fourier series
- Aperiodic signals \( \rightarrow \) Fourier transform
- Discrete signals \( \rightarrow \) DFT/FFT
- Exponential decay systems \( \rightarrow \) Laplace transform
This universality is why they appear everywhere.
Summary
Most signal-processing transforms use exponential basis functions because complex exponentials are eigenfunctions of LTI systems, making analysis simple, physically meaningful, and widely applicable.