Linearity, Causality, and BIBO Stability

2. Time-Invariance (Shift-Invariance)

Test: If $y[n] = \mathcal{T}{x[n]}$, then for any delay $k$: \(\mathcal{T}\{x[n-k]\} \stackrel{?}{=} y[n-k]\)

System	Test	Result
$y[n] = x[n] \cos(\omega_0 n)$	$y_1[n] = x[n-k]\cos(\omega_0 n) \ne y[n-k]$	Time-Variant
$y[n] = \cos[x(n)]$	$y_1[n] = \cos[x(n-k)] = y[n-k]$	Time-Invariant

Key insight: Explicit $n$-dependence (like $\cos(\omega_0 n)$) breaks time-invariance. Implicit $n$ through $x[n]$ is fine.

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3. Linearity (Superposition)

Test: For inputs $x_1[n], x_2[n]$ and constants $a, b$: \(\mathcal{T}\{a x_1[n] + b x_2[n]\} \stackrel{?}{=} a\mathcal{T}\{x_1[n]\} + b\mathcal{T}\{x_2[n]\}\)

Equivalently: Additivity + Homogeneity (Scaling)

System	Test	Result
$y[n] = 3x[n] + 2$	$3(ax_1+bx_2)+2 \ne a(3x_1+2)+b(3x_2+2)$	Nonlinear (affine)
$y[n] = \text{Re}{x[n]}$	$\text{Re}{ax_1+bx_2} = a\text{Re}{x_1}+b\text{Re}{x_2}$	Linear
$y[n] = x^2[n]$	$(ax_1+bx_2)^2 \ne a x_1^2 + b x_2^2$	Nonlinear

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4. Causality

Causal: Output at $n$ depends only on $x[n], x[n-1], x[n-2], \dots$ (present and past).

Non-Causal: Output depends on future inputs $x[n+1], x[n+2], \dots$

System	Causal?
$y[n] = x[n] + 2x[n-1]$	✓ Yes
$y[n] = x[n+1]$	✗ No (advance)
$y[n] = x[-n]$	✗ No (reversal)

Real-time systems MUST be causal. Non-causal systems are only realizable offline (stored signals) or with added delay.

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5. BIBO Stability

Bounded Input → Bounded Output (BIBO): If $|x[n]| \le B_x < \infty$ for all $n$, then $|y[n]| \le B_y < \infty$ for all $n$.

For LTI Systems: Impulse Response Condition

An LTI system is BIBO stable iff its impulse response $h[n]$ is absolutely summable:

\[\sum_{n=-\infty}^{\infty} |h[n]| < \infty\]

In the Z-Domain

LTI system with transfer function $H(z)$:

• Stable $\iff$ all poles of $H(z)$ lie inside the unit circle ($

  z

  < 1$)

• ROC must include the unit circle

$h[n]$	$\sum	h[n]	$	Stable?	Pole Location
$a^n u[n],	a	<1$	$\frac{1}{1-	a	}$	✓	$z=a$ (inside)
$a^n u[n],	a	>1$	$\infty$	✗	$z=a$ (outside)
$u[n]$	$\infty$	✗	$z=1$ (on circle)
$\delta[n]$	1	✓	None (FIR)

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