Eigenfunction of LTI system

A complex exponential $e^{j2\pi f_{0}t}$ is an eigenfunction of every LTI system. Feed one in, the system outputs the same complex exponential multiplied by a complex constant — same function out, just scaled.

The derivation

Feed $x(t)=e^{j2\pi f_{0}t}$ into an LTI system with impulse response $h(t)$, and compute the output by convolution:

$y(t)={\int }_{-\infty }^{\infty }h(\tau )e^{j2\pi f_0(t-\tau )}d\tau =e^{j2\pi f_{0}t}{\int }_{-\infty }^{\infty }h(\tau )e^{-j2\pi f_0\tau }d\tau .$

The factor $e^{j2\pi f_{0}t}$ came out of the integral because it doesn’t depend on the dummy variable $\tau$. The remaining integral doesn’t depend on $t$ — it is a single complex number that depends only on $h$ and the frequency $f_0$. Call it $H(f_0)$. Then

$y(t)=H(f_0)e^{j2\pi f_{0}t}.$

The output is the same function as the input, multiplied by a complex constant. This is exactly what the word eigenfunction means in linear algebra: a vector (here, a function) that the operator (here, the LTI system) maps to a scalar multiple of itself.

What H(f₀) is

The complex constant $H(f_0)$ — the scaling factor at frequency $f_0$ — is the value of the system’s frequency response at $f_0$. Equivalently, it’s the Fourier transform of $h$ evaluated at $f=f_0$, and the transfer function $H(s)$ evaluated at $s=j2\pi f_0$.

Three names, same object:

$H(f_0)=\mathcal{F}\{h\}(f_0)=H(s){|}_{s=j2\pi f_0}.$

The last equality presumes the system is stable, so that the imaginary axis lies inside the Laplace ROC of $H(s)$. For unstable systems the Fourier transform of $h$ doesn’t exist and only the Laplace description is well-defined; substituting $s=j2\pi f_0$ then doesn’t give a meaningful frequency response.

Why this is enormous

When we represent a signal as a sum of complex exponentials (a Fourier series for periodic signals, a Fourier transform for aperiodic ones), we know what the system does to each exponential by inspection: it multiplies by $H(f)$ at that frequency. Summing the outputs (legal by linearity) gives the output corresponding to the summed input.

In short: if we can decompose a signal into complex exponentials, we can compute the system’s response without doing convolution. Convolution in time becomes multiplication in frequency. This is the founding observation of all of frequency-domain analysis.

Real sinusoids and what the system does

For a real input $\cos (2\pi f_{0}t)$, the eigenfunction decomposition gives the output

$y(t)=|H(f_0)|\cos (2\pi f_{0}t+\angle H(f_0)).$

The amplitude gets scaled by $|H(f_0)|$, the phase gets shifted by $\angle H(f_0)$. This single sentence is the entire content of frequency-response analysis: once you know $H(f)$ — magnitude and phase as functions of frequency — you know what the system does to every sinusoid, and (by superposition) to every signal.

That’s why Bode plots (plots of $|H|$ in dB and $\angle H$ in degrees versus frequency) are the universal language for describing LTI systems.