BIBO Stability of Discrete-Time Systems

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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,

$$\mathrm{\sum_{n=0}^{\infty}\:|h(n)|\:\lt\:\infty}$$

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.,

$$\mathrm{h(n)\:=\:0; \:\:\text{for }\:n\:\lt\:0}$$

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 x(n) is a bounded input sequence satisfying $\mathrm{|x(n)|\:\leq\:M_{x}\:\leq\:\infty}$, and h(n) is the impulse response of the system, then the output y(n) of the system can be determined using the convolution sum, i.e.,

$$\mathrm{y(n) \:=\: \sum_{k=-\infty}^{\infty}\: x(k) h(n\:-\:k) \:=\: \sum_{k=-\infty}^{\infty}\: h(k) x(n\:-\:k)}$$

The magnitude of the output sequence is given by,

$$\mathrm{|y(n)| \:=\: \left| \sum_{k=-\infty}^{\infty}\: h(k) x(n\:-\:k) \right| \:=\: \sum_{k=-\infty}^{\infty}\: |h(k)\: x(n\:-\:k)|}$$

Since the magnitude of the sum of terms is less than or equal to the sum of the magnitudes, i.e.,

$$\mathrm{|y(n)| \:=\: \left| \sum_{k=-\infty}^{\infty}\: h(k) \:x(n\:-\:k) \right| \:\leq \:\sum_{k=-\infty}^{\infty}\: |h(k)| |x(n\:-\:k)|}$$

Now, consider the bounded value of the input is equal to M, then the above expression becomes,

$$\mathrm{|y(n)| \:\leq\: M \:\sum_{k=-\infty}^{\infty}\: |h(k)|}$$

For the system to be stable,

$$\mathrm{|y(n)| \:\lt\: \infty}$$

This condition will be satisfied when,

$$\mathrm{\sum_{k=-\infty}^{\infty} \:|h(k)| \:\lt\: \infty}$$

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

$$\mathrm{\sum_{n=0}^{\infty} \:|h(n)| \:\lt\: \infty}$$

And the system transfer function of causal LTI discrete-time system is given by,

$$\mathrm{H(z) \:=\: \sum_{n=0}^{\infty} \:h(n)\: z^{-n}}$$

The magnitude of the transfer function is given by,

$$\mathrm{|H(z)| \:=\: \sum_{n=0}^{\infty} \:\left| h(n)\: z^{-n} \right|}$$

$$\mathrm{\Rightarrow\: |H(z)| \:\leq\: \sum_{n=0}^{\infty}\: |h(n)|\: |z^{-n}|}$$

Thus, the evaluation of the magnitude of the transfer function |H(z)| on unit circle (for the unit circle |z| = 1) results,

$$\mathrm{|H(z)| \:\leq \:\sum_{n=0}^{\infty}\: |h(n)| \:\lt\: \infty}$$

Hence, it shows that for a stable system, the ROC of the system transfer function includes the unit circle.