Convolution Properties I

Convolution computations can be simplified by utilizing their inherent properties.

The commutative property reveals that the input and the impulse response of an LTI (Linear Time-Invariant) system can be interchanged without affecting the output:

The associative property suggests that the merged convolution of three functions remains unchanged regardless of the sequence of convolution. For instance, for three functions x(t), h(t), and g(t) is written as,

When two LTI systems with impulse responses are connected in series, their respective equations can be combined using the associative property to derive an equivalent joint impulse response, which is the convolution of their individual impulse responses.

The distributive property enables the convolution operation on the sum of multiple input signals, allowing complex impulse responses to be broken down into simpler components. Mathematically, this is represented as:

The time-shift property implies that delaying the input of a time-invariant system results in the output being delayed by the same amount. Similarly, if the system has a built-in delay, the output is delayed by the sum of the input delay and system delay. For a time shift t₀:

Computationally, this property allows signals to be delayed or advanced, leveraging their symmetry or causality to simplify the convolution operation.

These properties — commutative, associative, distributive, and time-shift — are fundamental tools for simplifying convolution operations in LTI systems, making complex signal processing tasks more manageable and efficient.