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# LTI Systems ## Definition A system is said to be linear time-invariant (LTI) if it satisfies the properties of both linearity and time-invariance. ### Linearity A system is linear if it satisfies the superposition principle. That is, if $y_1(t)$ is the response of the system to an input $x_1(t...

# LTI Systems ## Definition A system is said to be linear time-invariant (LTI) if it satisfies the properties of both linearity and time-invariance. ### Linearity A system is linear if it satisfies the superposition principle. That is, if $y_1(t)$ is the response of the system to an input $x_1(t)$, and $y_2(t)$ is the response of the system to an input $x_2(t)$, then for any constants $a$ and $b$, the response to the input $ax_1(t) + bx_2(t)$ is $ay_1(t) + by_2(t)$. ### Time-Invariance 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 $y(t)$ is the response of the system to an input $x(t)$, then the response to the input $x(t - t_0)$ is $y(t - t_0)$ for any time shift $t_0$. ## Properties of LTI Systems ### Convolution The output $y(t)$ of an LTI system with impulse response $h(t)$ to an input $x(t)$ is given by the convolution of $x(t)$ and $h(t)$: $y(t) = x(t) * h(t) = \int_{-\infty}^{\infty}x(\tau)h(t - \tau)d\tau$ ### Frequency Response The frequency response $H(f)$ of an LTI system is the Fourier transform of its impulse response $h(t)$: $H(f) = \int_{-\infty}^{\infty}h(t)e^{-j2\pi ft}dt$ The output $Y(f)$ of the system in the frequency domain is the product of the input $X(f)$ and the frequency response $H(f)$: $Y(f) = X(f)H(f)$ ### Eigenfunctions The eigenfunctions of LTI systems are complex exponentials $e^{j2\pi ft}$. When a complex exponential is input to an LTI system, the output is the same complex exponential multiplied by the frequency response of the system at that frequency: $e^{j2\pi ft} \rightarrow H(f)e^{j2\pi ft}$ ## Stability An LTI system is stable if every bounded input produces a bounded output (BIBO stability). A continuous-time LTI system is stable if and only if its impulse response $h(t)$ is absolutely integrable: $\int_{-\infty}^{\infty}|h(t)|dt < \infty$ A discrete-time LTI system is stable if and only if its impulse response $h[n]$ is absolutely summable: $\sum_{n = -\infty}^{\infty}|h[n]| < \infty$ ## Causality An LTI system is causal if the output at any time depends only on present and past values of the input. A continuous-time LTI system is causal if and only if its impulse response $h(t)$ is zero for $t < 0$: $h(t) = 0$ for $t < 0$ A discrete-time LTI system is causal if and only if its impulse response $h[n]$ is zero for $n < 0$: $h[n] = 0$ for $n < 0$