## Keywords

1. Harmonic structure of sound
2. Parson code of music
3. Linear time-invariant theory
4. Autocorrelation
5. Noise
6. Chirps
7. DCT compression
8. Discrete Fourier transform
9. filtering
10. convolution

## 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.

1. Linear: $f(a X_1(t) + b X_2(t)) = a f(X_1(t)) + b f(X_2(t))$
2. 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.

## Impulse Response

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.

$$\begin{equation} Y(t) = \int d\tau h(\tau) X(t-\tau). \end{equation}$$

## Transfer Function

For the impulse response, the transfer function is obtained through the Laplace transform of the response,

$$\begin{equation} \tilde h(s) = \mathscr L_s [ h(t) ]. \end{equation}$$

With the response function, we know that the response with some other input that is set in the same environment is

$$\begin{equation} \tilde Y(s) = \tilde h(s) \tilde X(s). \end{equation}$$

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