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A signal whose magnitude is a finite value is called the bounded signal. A sine wave is an example of bounded signal.

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.

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**.

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, 𝐾_{𝑥 }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.

- Related Questions & Answers
- Signals and Systems – BIBO Stability of Discrete-Time Systems
- Signals and Systems – Causality and Paley-Wiener Criterion for Physical Realization
- Signals and Systems: Multiplication of Signals
- Signals and Systems: Even and Odd Signals
- Signals and Systems: Periodic and Aperiodic Signals
- Signals and Systems: Energy and Power Signals
- Signals and Systems: Classification of Systems
- Signals and Systems: Addition and Subtraction of Signals
- Signals and Systems: Real and Complex Exponential Signals
- Signals and Systems: Amplitude Scaling of Signals
- Signals and Systems – Classification of Signals
- Signals and Systems: Linear and Non-Linear Systems
- Signals and Systems: Invertible and Non-Invertible Systems
- Signals and Systems: Linear Time-Invariant Systems
- Signals and Systems: Time Variant and Time-Invariant Systems

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