Is a sinusoidal signal the only one not distorted by an LTI system?

Question

In a signals and systems class in engineering school, I was shown a proof that a sinusoidal signal goes through a linear time-invariant (LTI) system undistorted; its shape and frequency are not changed - only its amplitude and phase (potentially) are.

In other words, if the input is of the form $v_i = V\sin\left(\omega t + \phi\right)$, then in general the output is of the form $v_o = V^{\prime}\sin\left(\omega t + \phi^\prime\right)$.

This was an "aha! moment" for me in understanding the usefulness and power of the Fourier series/transform.

Now I wonder, however, is a sinusoidal signal the only type of (periodic) signal that goes through an LTI system undistorted?

• If so, I would love to be directed to a proof, and to hear any insight about why this is so.
• If not, is it possible to define a new transform based on the alternative kernel?

Answer 1

Complex exponential function $x(t)=e^{i\omega_0 t}$ is the eigenfunction of an LTI system. That is, the output of an LTI system to this input is an scaled version of it. The scaling factor is the system function when evaluated at $\omega_0$. That is: $$y(t)=H(\omega_0)e^{i\omega_0t} \tag{1}$$ The proof is not so difficult. Just substitute the input $x(t)=e^{i\omega_0 t}$ in the differential equation that describes the system, take the Fourier transform, and then simplify. Note that the given equation in the question is also derived from $(1)$.

Answer 2

Quick answer: You are correct but to a certain point.

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The LTI system, in simple terms, is such a system that conforms to the superposition theorem.

1. One possibility is that the system output is independent from the input frequency.

   $$ y(t) = k x(t) $$

In this kind of system, if $ x(t) = A \sin (\omega t + \phi) $, then clearly, for any value of $ \omega $ the amplitude is equally affected. Thus in this kind of system (assuming it can be realizable), any signal passes undistorted. This is because, the time function is not dependent on the source frequency.

2. Asume that the system's transfer function is frequency dependent. For example

$$ H(s) = \frac {1}{1+ \frac {s}{\omega _c}} $$

Then, as you infer, any input signal whose frequency is beyond $ \frac{\omega_c}{2\pi}$ will be attenuated (and clearly some signals like an square wave get distorted.) But if the input signal is composed only of frequencies that fall well below $ \frac{\omega_c}{2\pi} $, we can say that the signal passes undistorted. For example, if $ \frac{\omega_c}{2\pi} = 25000Hz $, if the input signal only has components in the audible spectrum, the output signal is undistorted.

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This does not intend to disprove your insight (in fact I believe you are in the right track.) But as you are in the engineering school, you should be open to how the mathematics you learn will be applied to practical problems.

Answer 3

The eigenvalues and eigenfunctions of LTI systems are given by the Laplace transform

• In the continuous case : not only the sines but also any exponential. With an input $f(t) = e^{s t}$ where $s \in \mathbb{C}$ the ouput of your LTI system is $$f \ast h(t) = \int_{-\infty}^\infty e^{s (t-\tau)}h(\tau)d\tau = H(s) e^{st}$$ where $H(s)=\int_{-\infty}^\infty h(t) e^{-st}dt$ is the (bilateral) Laplace transform of the impulse response $h(t)$.

• In the discrete case : not only the sines but also any discrete exponential. With an input $f(n) = z^n$ where $z \in \mathbb{C}$ the ouput is $$f \ast h(n) = \sum_{n=-\infty}^\infty z^{n-m}h(m) = H(z) z^n$$ where $H(z)=\sum_{-\infty}^\infty h(n)z^{-n}$ is the Z-transform of the impulse response $h(n)$.

And the only other eigenfunctions are the linear combination of such, if associated to the same eigenvalue $H(s),H(z)$.

Now in the context of signal processing, you'll accept slight modifications of the input signal (some sort of approximate eigenfunctions). Studying those is the purpose of filters theory, one of the most important problem being the approximation of the ideal low-pass filter.