LTIjärjestelmät

LTIjärjestelmät, or Linear Time-Invariant systems, are fundamental concepts in signal processing, control theory, and many other fields of engineering and science. An LTI system is characterized by two key properties: linearity and time-invariance.

Linearity means that the system obeys the superposition principle. If an input signal is a sum of two other signals, the output of the system will be the sum of the outputs produced by each individual signal. Furthermore, if an input signal is scaled by a constant, the output will also be scaled by the same constant. Mathematically, if $y_1(t)$ is the output for input $x_1(t)$ and $y_2(t)$ is the output for input $x_2(t)$, then for any constants $a$ and $b$, the output for input $ax_1(t) + bx_2(t)$ is $ay_1(t) + by_2(t)$.

Time-invariance means that the system's behavior does not change over time. If an input signal is applied at a certain time, the output produced will be the same as if the identical input signal was applied at any other time, just shifted in time. In other words, if $y(t)$ is the output for input $x(t)$, then the output for a time-shifted input $x(t-\tau)$ is simply the original output shifted by the same amount, $y(t-\tau)$.

The mathematical representation of LTI systems often involves differential or difference equations. The impulse response, denoted as $h(t)$ for continuous-time systems or $h[n]$ for discrete-time systems, is a crucial characteristic. It is the output of the system when the input is an impulse function. Convolution is the mathematical operation that describes how the impulse response combines with an arbitrary input signal to produce the output signal. For continuous-time LTI systems, the output $y(t)$ is given by the convolution of the input $x(t)$ and the impulse response $h(t)$: $y(t) = x(t) * h(t)$. Similarly, for discrete-time systems, $y[n] = x[n] * h[n]$. The properties of LTI systems, such as their frequency response and stability, can be readily analyzed using tools like the Fourier transform and Laplace transform.