Convolution Understanding Notes

Foreword: Signal Learning Understanding.

1. Convolution understanding:

1.1 Discrete Signal Representation

A discrete signal is represented as $x\left [ n \right ]$ , consider a simple discrete signal as shown in the figure below

1.2 Decomposition of discrete signals

Expressing it as a superposition of pulse signals is

$x[n]=x[-2]\delta [n+2]+x[-1]\delta [n+1]+x[1]\delta[n-1]+x[2]\delta[n-2]$

The signal can be understood as the superposition of multiple pulse signals (in fact, the discrete signal is formed by the superposition of multiple pulse signals with different amplitudes)

Next, it will $x[n]$ become the superposition of four signals $x[n]=x1[n]+x2[n]+x3[n]+x4[n]$ , and the four signals are expressed as follows

$x1[n]=x[-2]\delta [n+2]$

$x2[n]=x[-1]\delta [n+1]$

$x3[n]=x[1]\delta [n-1]$

$x4[n]=x[2]\delta [n-2]$

At this point, the signal is completely separated into four time-shifted pulse signals with different amplitudes

1.3 Unit impulse response

Next, analyze the response of the linear time-invariant system to the unit impulse signal

Due to the time invariance, let the response be $h[n]$ , if the input is the shift of the unit impulse signal $\delta [n-k]$ , then the output of the time invariant system is $h[n-k]$

Input the above four signals into the linear time-invariant system respectively, and $h$ the output can be obtained as follows:

$y1[n]=x[-2]h[n+2]$

$y2[n]=x[-1]h[n+1]$

$y3[n]=x[1]h[n-1]$

$y4[n]=x[2]h[n-2]$

According to the superposition of the linear system, the total output can be known $y[n]=y1[n]+y2[n]+y3[n]+y4[n]$

1.4 Convolution

Changing the above expression to another expression is convolution $y[n]=x[n]*h[n]=\sum_{k=-\infty }^{+\infty }x[k]h[n-k]$ , and the values of k corresponding to the above useful output are -2, -1, 1, 2.

1.5 Use diagrams to represent the entire process

Note: The pulse signal with an amplitude of 0 is not considered in the figure

2. Summary

A discrete signal can be represented by the superposition of an infinite number of time-shifted pulse signals with different amplitudes (the amplitude is considered to be 0 in undefined places), and a linear time-invariant system can be represented by a unit impulse response. The two Combined to get the convolution sum, the convolution integral is similar.

A simple understanding is to first decompose the signal into multiple pulse-like signals, and then input each pulse-like signal into the linear time-invariant system to obtain the corresponding output. According to the principle of superposition, the total output of all outputs superimposed together That is, the original signal is output through the system, which is convolution.

Convolution Understanding Notes

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