Which one of the following is an eigen function of the class of all continuous-time, linear, time invariant systems (u(t) denotes the unit-step function)?

Eigen Functions of LTI Systems Explained

In the realm of signals and systems, understanding eigen functions is crucial, especially for continuous-time, linear, time-invariant (LTI) systems. An eigen function is a special input signal that, when passed through a system, produces an output which is simply a scaled version of the same input signal. The scaling factor is called the eigenvalue.

Understanding LTI Systems

A continuous-time system is considered linear if it satisfies both the additivity and homogeneity properties. This means if inputs $x_1(t)$ and $x_2(t)$ produce outputs $y_1(t)$ and $y_2(t)$ respectively, then an input $a{x}_1(t)+b{x}_2(t)$ will produce an output $a{y}_1(t)+b{y}_2(t)$, where $a$ and $b$ are constants.

A system is time-invariant if a time shift in the input signal results in an identical time shift in the output signal. That is, if $x(t)$ produces $y(t)$, then $x(t-t_0)$ will produce $y(t-t_0)$ for any time shift $t_0$.

The behavior of a continuous-time LTI system is completely characterized by its impulse response, denoted as $h(t)$.

What is an Eigen Function?

For an LTI system, an input signal $x(t)$ is an eigen function if the output $y(t)$ is given by:

$y(t)=\lambda x(t)$

where $\lambda$ is a complex constant called the eigenvalue. This means the shape of the signal remains unchanged, only its amplitude and phase might be altered.

Why Complex Exponentials are Eigen Functions

The most important class of eigen functions for all continuous-time LTI systems are the complex exponentials of the form $e^{j{\omega }_{0}t}$. Let's demonstrate this:

Consider an LTI system with impulse response $h(t)$. If the input to this system is $x(t)=e^{j{\omega }_{0}t}$, the output $y(t)$ is given by the convolution integral:

$y(t)={\int }_{-\infty }^{\infty }h(\tau )x(t-\tau )d\tau$

Substituting $x(t-\tau )=e^{j{\omega }_0(t-\tau )}$ into the integral:

$y(t)={\int }_{-\infty }^{\infty }h(\tau )e^{j{\omega }_0(t-\tau )}d\tau$

We can separate the term $e^{j{\omega }_{0}t}$ from the integral as it does not depend on $\tau$:

$y(t)=e^{j{\omega }_{0}t}{\int }_{-\infty }^{\infty }h(\tau )e^{-j{\omega }_0\tau }d\tau$

The integral term ${\int }_{-\infty }^{\infty }h(\tau )e^{-j{\omega }_0\tau }d\tau$ is precisely the Fourier Transform of the impulse response $h(t)$ evaluated at frequency ${\omega }_0$. This is represented as $H(j{\omega }_0)$, which is a complex constant for a given ${\omega }_0$.

Therefore, the output becomes:

$y(t)=H(j{\omega }_0)e^{j{\omega }_{0}t}$

Comparing this with the definition $y(t)=\lambda x(t)$, we see that $e^{j{\omega }_{0}t}$ is indeed an eigen function, and the corresponding eigenvalue is $H(j{\omega }_0)$, which is the system's frequency response at the input frequency ${\omega }_0$.

Analyzing the Options for Eigen Functions

Let's evaluate each given option to determine which one fits the definition of an eigen function for all continuous-time LTI systems.

Option 1: Causal Complex Exponential

- The signal is $e^{j{\omega }_{0}t}u(t)$, where $u(t)$ is the unit-step function.
- The presence of $u(t)$ makes the signal zero for $t<0$. This means the signal is not defined for all time, and its characteristics begin abruptly at $t=0$.
- While an LTI system can process this signal, it is not a fundamental eigen function in the same sense as $e^{j{\omega }_{0}t}$ for all LTI systems. The step function breaks the time-invariance property for the eigen function characteristic, as a time-shifted version of this input would not necessarily produce a scaled time-shifted output of the same form.

Option 2: Cosine Function

- The signal is $\cos {\omega }_{0}t$.
- This can be expressed using Euler's formula as $\cos {\omega }_{0}t=\frac{1}{2}(e^{j{\omega }_{0}t}+e^{-j{\omega }_{0}t})$.
- Since an LTI system is linear, if $e^{j{\omega }_{0}t}$ and $e^{-j{\omega }_{0}t}$ are eigen functions, then their sum (or linear combination) will produce an output that is a linear combination of their scaled versions: $y(t)=\frac{1}{2}H(j{\omega }_0)e^{j{\omega }_{0}t}+\frac{1}{2}H(-j{\omega }_0)e^{-j{\omega }_{0}t}$.
- For the output to be a scaled version of $\cos {\omega }_{0}t$, we would need $H(j{\omega }_0)$ and $H(-j{\omega }_0)$ to relate in a specific way (e.g., for real systems, $H(-j{\omega }_0)=H^{\ast }(j{\omega }_0)$). While $\cos {\omega }_{0}t$ often behaves like an eigen function for real LTI systems (producing a scaled and phase-shifted cosine), it is fundamentally a sum of two complex exponentials, not a single one. The core eigen functions are the complex exponentials themselves.

Option 3: Pure Complex Exponential

- The signal is $e^{j{\omega }_{0}t}$.
- As proven above, when $e^{j{\omega }_{0}t}$ is input to any continuous-time LTI system, the output is $H(j{\omega }_0)e^{j{\omega }_{0}t}$.
- This perfectly matches the definition of an eigen function, where the original signal $e^{j{\omega }_{0}t}$ is scaled by the eigenvalue $H(j{\omega }_0)$. This property holds true for all continuous-time LTI systems.

Option 4: Sine Function

- The signal is $\sin {\omega }_{0}t$.
- This can be expressed as $\sin {\omega }_{0}t=\frac{1}{2j}(e^{j{\omega }_{0}t}-e^{-j{\omega }_{0}t})$.
- Similar to the cosine function, the sine function is a linear combination of two complex exponentials. While it often behaves predictably with LTI systems, it is not a fundamental eigen function in its own right, but rather a composition of the true eigen functions.

Conclusion on Eigen Functions

Based on the fundamental properties of continuous-time LTI systems, the complex exponential $e^{j{\omega }_{0}t}$ is the universal eigen function. Its unique characteristic of passing through any LTI system and emerging merely scaled by the system's frequency response at that specific frequency makes it exceptionally important in system analysis, particularly in Fourier analysis.