LTI Systems

LTIs are a way to model the behavior of many processes and systems in science and technology.

LTI stands for

• Linear
• Time-invariant

In reality, no physical system is truely linear and time-invariant - they are usually simplified to meed these criteria. In DSP, most building blocks - like filters - can be both linear and time-invariant.

An LTI is described by its impulse response $h[n]$. It processes an input $x[n]$ and produces the output $y[n]$ through convolution:

$$ y[n] = x[n] * h[n]$$

This is usually visualized with the following block diagram in DSP:

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LTI Properties¶

To qualify as linear and time-invariant, a system (or function) $S(x)$ has to meet two criteria:

• linearity
• time invariance

Linearity¶

Superposition - adding two signals before being processed is equivalent to adding the results of two processed signals:

$$ S(x_1 + x_2) = S(x_1) + S(x_2) $$

Homogeneity - scaling an input of an LTI has the same result as scaling the output:

$$ S(a x) = a S(x)$$

Time Invariance¶

$$ S(x - \tau(t)) = S(h(t − \tau )) $$

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Causality¶

LTI systems are causal, if their impulse response is causal:

$$ h[n] = 0 ~ \mathrm{for} ~ n < 0$$

For causal systems, the output depends only on present and past inputs - not on future inputs. This is the case for all physical systems - digital systems, however can be non-causal.