Scale characteristics of convolution

 

▌Convolution operation▌

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1. Convolution definition

Convolution operation is an important operation in mathematics, and it is widely used in signal processing and system analysis. It is very close to the form of correlation operations . For two continuous time signals $x\left(t\right),and\left(t\right)$ , the convolution operation between them is defined as:$$x\left(t\right)\ast and\left(t\right)={\int }_{-\infty }^{\infty }x\left(\tau \right)and\left(t-\tau \right)d\tau$$

For discrete sequences, the corresponding convolution and operations can also be defined :

$$x\left[n\right]\ast and\left[n\right]=\sum _{m=-\infty }^{\infty }x\left[m\right]and\left[n-m\right]$$

The convolution (convolution sum) operation satisfies the commutative law, associative law, and distribution rate .

2. Schematic diagram of convolution

The following figure shows the process of convolving two signals to obtain the result. You can select one of them for de -folding, translation, and then multiplying, integrating with another signal , or the final result:

 

▌Scaling changes▌

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The scale change of the signal refers to the independent variable of the signal multiplied by a constant, such as the signal $f\left(t\right)$ , choose a constant$a$ , corresponding to the signal$f\left(at\right)$ Compared with the original signal, the scale has changed.

当$a>When\; 1$ , the corresponding signal waveform shrinks; when$a<The$ corresponding waveform stretches at$1$ o'clock. The figure below shows$\mathrm{without}c(t)\; The$ signal is divided by2 n − 1 2n − 1as the independent variable$2n-$The factor of$1$ changes and the corresponding waveform changes.

 

▌Convolution scale change▌

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Normally, two signals are convolved, and if the scale is changed for any one of them, there is not much relationship between the obtained result and the original result.

1. Scale properties

But when the two signals undergo the same scale change, the result obtained will have the same scale change between the convolution results of the original two signals. such as:

$$and\left(t\right)=x\left(t\right)\ast h\left(t\right)$$
那么：$$and\left(\frac{t}{2}\right)=\frac{1}{2}x\left(\frac{t}{2}\right)\ast h\left(\frac{t}{2}\right)$$

2. Proof

This property can be proved by variable substitution:

3. Examples

For any signal $f\left(t\right)$，都有：
$$f\left(t\right)=f\left(t\right)\ast d\left(t\right)$$

那么：$$f\left(\frac{t}{2}\right)\ast d\left(\frac{t}{2}\right)=f\left(\frac{t}{2}\right)\ast \left[2d\left(t\right)\right]=2f\left(\frac{t}{2}\right)$$