Why a Sine Wave is an Eigenfunction of an LTI System
1. What is an Eigenfunction?
A function \( x(t) \) is called an eigenfunction of a system if:
\[ \text{System}\{x(t)\} = \lambda x(t) \]
Where:
- The output has the same shape as the input
- It is only scaled (and possibly phase shifted)
- \( \lambda \) is the eigenvalue
2. LTI System Description
An LTI (Linear Time-Invariant) system is completely described by its impulse response \( h(t) \).
The output is given by convolution:
\[ y(t) = x(t) * h(t) \]
\[ y(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau \]
3. Use a Complex Exponential Input
Consider the input:
\[ x(t) = e^{j\omega t} \]
Substitute into convolution:
\[ y(t) = \int_{-\infty}^{\infty} e^{j\omega \tau} h(t-\tau) d\tau \]
Rewrite:
\[ e^{j\omega \tau} = e^{j\omega t} e^{-j\omega (t-\tau)} \]
\[ y(t) = e^{j\omega t} \int_{-\infty}^{\infty} h(t-\tau) e^{-j\omega (t-\tau)} d\tau \]
Let \( u = t-\tau \), then:
\[ y(t) = e^{j\omega t} \int_{-\infty}^{\infty} h(u) e^{-j\omega u} du \]
The integral is the Fourier Transform of \( h(t) \):
\[ H(j\omega) \]
Therefore:
\[ y(t) = H(j\omega) e^{j\omega t} \]
4. Final Result
\[ \boxed{ \text{LTI}\{e^{j\omega t}\} = H(j\omega) e^{j\omega t} } \]
This means:
- Input = complex exponential
- Output = same complex exponential
- Only multiplied by \( H(j\omega) \)
Eigenfunction: \( e^{j\omega t} \)
Eigenvalue: \( H(j\omega) \)
5. Why Sine is Also an Eigenfunction
\[ \sin(\omega t) = \frac{e^{j\omega t} - e^{-j\omega t}}{2j} \]
Because LTI systems are linear:
\[ \sin(\omega t) \rightarrow |H(j\omega)| \sin(\omega t + \angle H(j\omega)) \]
- Frequency does not change
- Only amplitude and phase change
6. Intuitive Explanation
- LTI systems cannot create new frequencies
- LTI systems cannot shift frequency
- They only scale and phase-shift each frequency component
Since a sine wave contains only one frequency, it passes through unchanged except for gain and phase shift.
Conclusion
\[ \boxed{ \text{Complex exponentials (and hence sine waves) are eigenfunctions of all LTI systems.} } \]
Because convolution with \( h(t) \) simply multiplies them by \( H(j\omega) \).