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BIBO Stability of Discrete-Time Systems
Stability and Causality
The necessary and sufficient condition for a causal linear time invariant (LTI) discrete-time system to be BIBO stable is given by,
$$\mathrm{\mathit{\sum_{n=\mathrm{0}}^{\infty }\left|h\left ( n \right ) \right|< \infty }}$$
Therefore, if the impulse response of an LTI discrete-time system is absolutely summable, then the system is BIBO stable.
Also, for the system to be causal, the impulse response of the system must be equal to zero for 𝑛 < 0, i.e.,
$$\mathrm{\mathit{h\left ( n \right )=\mathrm{0};\; \; \mathrm{for}\: n< \mathrm{0}}}$$
In other words, if the given LTI discrete-time system is causal, then the region of convergence (ROC) for H(z) will be outside the outermost pole.
Therefore, for a causal LTI discrete-time system, all the poles of H(z) must lie inside the unit circle in the z-plane, i.e., the ROC of the system transfer function must include the unit circle.
Time Domain Condition for the Stability of LTI Discrete-Time Systems
For a system, when the bounded input sequence always produces a bounded output sequence, then the system is said to be stable system. On the other hand, if the output sequence is unbounded for a bounded sequence, then the system is said to be unstable system.
Now, consider $\mathrm{\mathit{x\left ( n \right )}}$ is a bounded input sequence satisfying $\mathrm{\mathit{\left|x\left ( n \right ) \right|\leq M_{x}\leq \infty }}$, and $\mathrm{\mathit{h\left ( n \right )}}$ is the impulse response of the system, then the output y(n) of the system can be determined using the convolution sum, i.e.,
$$\mathrm{\mathit{y\left ( n \right )\mathrm{\,=\,}\sum_{k\mathrm{\,=\,}-\infty }^{\infty }x\left ( k \right )h\left ( n-k \right )\mathrm{\,=\,}\sum_{k\mathrm{\,=\,}-\infty }^{\infty }h\left ( k \right )x\left ( n-k \right )}}$$
The magnitude of the output sequence is given by,
$$\mathrm{\mathit{\left|y\left ( n \right ) \right|\mathrm{\,=\,}\left|\sum_{k\mathrm{\,=\,}-\infty }^{\infty }h\left ( k \right )x\left ( n-k \right ) \right|\mathrm{\,=\,}\sum_{k\mathrm{\,=\,}-\infty }^{\infty }\left|h\left ( k \right )x\left ( n-k \right ) \right|}}$$
Since the magnitude of the sum of terms is less than or equal to the sum of the magnitudes, i.e.,
$$\mathrm{\mathit{\left|y\left ( n \right ) \right|\mathrm{\,=\,}\left|\sum_{k\mathrm{\,=\,}-\infty }^{\infty }h\left ( k \right )x\left ( n-k \right ) \right|\leq \sum_{k\mathrm{\,=\,}-\infty }^{\infty }\left|h\left ( k \right )\right|\left|x\left ( n-k \right ) \right|}}$$
Now, consider the bounded value of the input is equal to M, then the above expression becomes,
$$\mathrm{\mathit{\left|y\left ( n \right ) \right|\leq M\sum_{k\mathrm{\,=\,}-\infty }^{\infty }\left|h\left ( k \right ) \right|}}$$
For the system to be stable,
$$\mathrm{\mathit{\left|y\left ( n \right ) \right|< \infty }}$$
This condition will be satisfied when,
$$\mathrm{\mathit{\sum_{k\mathrm{\,=\,}-\infty }^{\infty }\left|h\left ( k \right ) \right|< \infty }}$$
i.e., an LTI system is BIBO stable if its impulse response is absolutely summable. This is the necessary and sufficient time domain condition of the stability of LTI discrete-time systems.
Explanation – For a stable system, the ROC of a system transfer function includes the unit circle −
Since the necessary and sufficient condition for a causal LTI discrete-time system to be BIBO stable is
$$\mathrm{\mathit{\sum_{n\mathrm{\,=\,}\mathrm{0} }^{\infty }\left|h\left ( n \right ) \right|< \infty }}$$
And the system transfer function of causal LTI discrete-time system is given by,
$$\mathrm{\mathit{H\left ( z \right )\mathrm{\,=\,}\sum_{n\mathrm{\,=\,}\mathrm{0}}^{\infty }h\left ( n \right )z^{-n}}}$$
The magnitude of the transfer function is given by,
$$\mathrm{\mathit{\left|H\left ( z \right ) \right|\mathrm{\,=\,}\sum_{n\mathrm{\,=\,}\mathrm{0}}^{\infty }\left|h\left ( n \right )z^{-n}\right|}}$$
$$\mathrm{\mathit{\Rightarrow \left|H\left ( z \right ) \right|\leq \sum_{n\mathrm{\,=\,}\mathrm{0}}^{\infty }\left|h\left ( n \right ) \right|\left|z^{-n}\right|}}$$
Thus, the evaluation of the magnitude of the transfer function $\mathrm{\mathit{\left|H\left ( z \right ) \right|}}$ on unit circle (for the unit circle |𝑧| = 1) results,
$$\mathrm{\mathit{\left|H\left ( z \right ) \right|\leq \sum_{n\mathrm{\,=\,}\mathrm{0}}^{\infty }\left|h\left ( n \right )\right|< \infty }}$$
Hence, it shows that for a stable system, the ROC of the system transfer function includes the unit circle.
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