- Harmonic structure of sound
- Parson code of music
- Linear time-invariant theory
- DCT compression
- Discrete Fourier transform
Linear Time-Invariant System
We describe the system with $Y(t) = f(X(t))$, where $X(t)$ is the input, and $Y(t)$ is the output.
- Linear: $f(a X_1(t) + b X_2(t)) = a f(X_1(t)) + b f(X_2(t))$
- Time-invariant: input $X(t+\Delta t)$ will produce the shifted signal $Y(t+\Delta t)$.
LTI systems are memory systems, casual, real, and stable. Stable means the output won’t reach infinite if the input is finite. It’s bounded.
Suppose we have a impulse $X(t) = I(t)$, and output $h(t)$.
Now we have another input $X(t)$, we can ask that what would the output be if we put the input in the same environment as the previous impulse.
For the impulse response, the transfer function is obtained through the Laplace transform of the response,
With the response function, we know that the response with some other input that is set in the same environment is