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Signals and Systems: BIBO Stability Criterion
Bounded Signal
A signal whose magnitude is a finite value is called the bounded signal. A sine wave is an example of bounded signal.
BIBO Stable System
A system is called BIBO stable (or bounded-input, bounded-output stable) system, if and only if every bounded input to the system produces a bounded output.
BIBO Stability Criterion
For a system to be BIBO stable, the necessary condition is given by the expression,
$$\mathrm{\int_{-\infty }^{\infty}\left | h(t) \right |dt < \infty \; \;}\;\;...(1)$$
Where, h(t) is the impulse response of the system. The condition given in the expression (1) is called the BIBO stability criterion.
Proof
Consider an LTI (linear time-invariant) system with x(t) and y(t) as input and output respectively. Hence, the input and output of the system are related by the convolution integral, i.e.,
$$\mathrm{y(t)=\int_{-\infty }^{\infty}x(\tau )h\left ( t-\tau \right )d\tau \: \: } \;\;...(2)$$
Taking modulus (i.e., absolute value) on both side, we get,
$$\mathrm{\left | y(t) \right |=\left | \int_{-\infty }^{\infty}x(\tau )h\left ( t-\tau \right )d\tau \right | \: \: }\;\; ...(3)$$
By the triangle inequality, the absolute value of the integral of the product of two terms is always less than or equal to the integral of their absolute values. Hence, using this fact we get,
$$\mathrm{\left | \int_{-\infty }^{\infty}x(\tau )\; h\left ( t-\tau \right )d\tau \right |\leq\int_{-\infty }^{\infty }\left |x(\tau )\right |\;\left | h\left ( t-\tau \right ) \right |d\tau } $$
Now, if the input x(τ) of the system is bounded (or finite), i.e.,
$$\mathrm{\left | x(\tau ) \right |\leq K_{x}<\infty } $$
Where, 𝐾𝑥 is a real positive integer.
Then,
$$\mathrm{\left | \int_{-\infty }^{\infty}x(\tau )\; h\left ( t-\tau \right )d\tau \right |\leq\: K_{x}\int_{-\infty }^{\infty }\;\left | h\left ( t-\tau \right ) \right |d\tau } $$
$$\mathrm{\Rightarrow \left | y(t) \right |\leq\: K_{x}\int_{-\infty }^{\infty }\;\left | h\left ( t-\tau \right ) \right |d\tau } $$
Substituting the variables by 𝑢 = (𝑡 − 𝜏); 𝑑𝜏 = 𝑑𝑢. Then, the output of the system is bounded (i.e., 𝑦(𝑡) < ∞) if
$$\mathrm{\int_{-\infty }^{\infty }\;\left | h\left ( u \right ) \right |du<\infty} $$
By replacing u with t, we get,
$$\mathrm{\int_{-\infty }^{\infty }\;\left | h\left ( t \right ) \right |dt<\infty} $$
This is the necessary and sufficient condition for the BIBO stability of a system.
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