Demystifying Linear Convolution in Digital Signal Processing

When we talk about Digital Signal Processing (DSP) , one operation lurks at the heart of many fundamental systems— Linear Convolution . From filtering signals to analysing system responses, linear convolution stands as the mathematical backbone of discrete-time systems. But what is linear convolution? Why is it so crucial? And how do we perform it? Let’s dive deep into the red core of its logic. What is Linear Convolution? Linear convolution is a method used to determine the output of a Linear Time-Invariant (LTI) system when its input and impulse response are known. In simpler terms, if a system is known to behave predictably over time (i.e., linear and time-invariant), then convolution allows us to compute how it reacts to any input signal . Mathematically, if: x [ n ] x[n] x [ n ] is the input signal h [ n ] h[n] h [ n ] is the impulse response of the system Then the output y [ n ] y[n] y [ n ] is given by the convolution sum: y [ n ] = x [ n ] ∗ h [ n ] = ∑ k = −...