LTI Systems: Impulse Response, Convolution, and Correlation

Why LTI Systems Are Special

When a system is both Linear and Time-Invariant, it unlocks a complete mathematical toolkit:

Tool	What It Gives You
Impulse Response $h[n]$	Complete characterization: $y[n] = x[n] * h[n]$
Convolution	Output for any input via $y[n] = \sum x[k]h[n-k]$
Frequency Response $H(e^{j\omega})$	Eigenfunction: $e^{j\omega n} \to H(e^{j\omega})e^{j\omega n}$
Z-Transform $H(z)$	Algebraic analysis, stability, filter design
Difference Equation	$y[n] + \sum a_k y[n-k] = \sum b_k x[n-k]$
Pole-Zero Plot	Visual stability, frequency response shaping

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Standard LTI Operators and Their Impulse Responses

System	Difference Equation	$h[n]$	Notes
Identity	$y[n] = x[n]$	$\delta[n]$	Pass-through
Unit Delay	$y[n] = x[n-1]$	$\delta[n-1]$	$z^{-1}$
Unit Advance	$y[n] = x[n+1]$	$\delta[n+1]$	Non-causal
Moving Average (M)	$y[n] = \frac{1}{M}\sum_{k=0}^{M-1} x[n-k]$	$\frac{1}{M}[u[n]-u[n-M]]$	FIR, lowpass

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Convolution: The LTI Output Formula

\[y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k]\]

Numerical Example

$x[n] = {1, 2, 3}$ (at $n=0,1,2$) $h[n] = {1, 1}$ (at $n=0,1$) — 2-point moving average

       k:  0   1   2
x[k]:      1   2   3

n=0:  x[0]h[0] = 1·1 = 1
n=1:  x[0]h[1] + x[1]h[0] = 1·1 + 2·1 = 3
n=2:  x[1]h[1] + x[2]h[0] = 2·1 + 3·1 = 5
n=3:  x[2]h[1] = 3·1 = 3

y[n] = {1, 3, 5, 3}

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Correlation: Measuring Similarity

Auto-Correlation

\(R_{xx}[k] = \sum_n x[n] x[n-k] = x[k] * x[-k]\)

• $R_{xx}[0]$ = energy
• Peak at $k=0$, symmetric: $R_{xx}[-k] = R_{xx}[k]$
• Periodicity detection, pitch estimation

Cross-Correlation

\(R_{xy}[k] = \sum_n x[n] y[n-k]\)

• Time delay estimation (radar, sonar, GPS)
• Pattern matching, template detection
• $R_{xy}[k] \ne R_{yx}[k]$ but $R_{xy}[k] = R_{yx}[-k]$

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Comprehensive Classification Summary

System	Static?	TI?	Linear?	Causal?	Stable?	LTI?
$y[n] = x[n]$	✓	✓	✓	✓	✓	✓
$y[n] = x[n-1]$	✗	✓	✓	✓	✓	✓
$y[n] = x[n+1]$	✗	✓	✓	✗	✓	✓*
$y[n] = 3x[n]+2$	✓	✓	✗	✓	✓	✗
$y[n] = x^2[n]$	✓	✓	✗	✓	Depends	✗
$y[n] = nx[n]$	✓	✗	✓	✓	✗	✗
$y[n] = x[-n]$	✗	✗	✓	✗	Depends	✗

*Non-causal LTI (realizable offline)

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