定義
普通函式
衝激函式的定義為:
![衝激函式](/img/2/467/wZwpmL2ADNxczMxgjNwMzM1UTM1QDN5MjM5ADMwAjMwUzL4YzLxEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
單位衝擊函式
![衝激函式](/img/2/30b/wZwpmL1EDMzIjMzcDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL3gzLyAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
通常,單位衝激函式
滿足:![衝激函式](/img/c/87c/wZwpmL1gDM3UDMxAjNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwYzLwEzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![衝激函式](/img/0/5df/wZwpmL2IzM0gDM5kTNwMzM1UTM1QDN5MjM5ADMwAjMwUzL5UzL0UzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(1)當 時,
![衝激函式](/img/a/f87/wZwpmLxgDMwETN3kTNwMzM1UTM1QDN5MjM5ADMwAjMwUzL5UzL0UzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(2)
![衝激函式](/img/2/30b/wZwpmL1EDMzIjMzcDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL3gzLyAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![衝激函式](/img/d/23e/wZwpmLxATO4EzMxMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLyIzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
單位衝激函式 又稱Dirac函式或者 函式。
![衝激函式](/img/2/30b/wZwpmL1EDMzIjMzcDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL3gzLyAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
註:單位衝激函式 並不是經典意義下的函式,而是一個廣義函式(或者奇異函式),它不能用通常意義下的“值的對應關係”來理解和使用,而是通過它的性質來使用。
廣義函式
![衝激函式](/img/6/47d/wZwpmL1gTM3AzMxAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczL3gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/6/47d/wZwpmL1gTM3AzMxAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczL3gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/d/3fe/wZwpmL3gzM4MTN3kDMxMzM1UTM1QDN5MjM5ADMwAjMwUzL5AzL4AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/6/47d/wZwpmL1gTM3AzMxAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczL3gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/f/fed/wZwpmLzMDO1ADO4QzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL2IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/d/3fe/wZwpmL3gzM4MTN3kDMxMzM1UTM1QDN5MjM5ADMwAjMwUzL5AzL4AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/6/47d/wZwpmL1gTM3AzMxAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczL3gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/8/9ed/wZwpmLzcTM4IjN2cTOwMzM1UTM1QDN5MjM5ADMwAjMwUzL3kzL2QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
選擇一類性能良好的函式 ,稱為檢驗函式 (它相當於定義域),一個廣義函式 對檢驗函式空間中的每個函式 賦予一個數值 的映射,該數與廣義函式 和檢驗函式 有關,記作 。廣義函式可寫為
![衝激函式](/img/4/f09/wZwpmL1cDN3AzNyEzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLxczL0czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![衝激函式](/img/2/30b/wZwpmL1EDMzIjMzcDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL3gzLyAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![衝激函式](/img/6/47d/wZwpmL1gTM3AzMxAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczL3gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/2/64e/wZwpmL0UTNzEzNwAjNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwYzL3UzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![衝激函式](/img/e/58f/wZwpmL3cjNxcTM2YzNxMzM1UTM1QDN5MjM5ADMwAjMwUzL2czL1czLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![衝激函式](/img/6/47d/wZwpmL1gTM3AzMxAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczL3gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/e/58f/wZwpmL3cjNxcTM2YzNxMzM1UTM1QDN5MjM5ADMwAjMwUzL2czL1czLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![衝激函式](/img/2/30b/wZwpmL1EDMzIjMzcDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL3gzLyAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
衝擊函式 與檢驗函式的作用效果是從 中篩選出它在 時刻的函式值 ,這常稱為衝擊函式的取樣性質(或篩選性質)。簡言之,能從檢驗函式 中篩選出函式值 的廣義函式就稱為衝擊函式 。
性質
篩選性質
![衝激函式](/img/7/42a/wZwpmL1ITNwYDM0EzNxMzM1UTM1QDN5MjM5ADMwAjMwUzLxczL2IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/7/82f/wZwpmLxgTN0kDOxMDOwMzM1UTM1QDN5MjM5ADMwAjMwUzLzgzLwMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
如果信號 是一個在 處連續的普通函式,則有
![衝激函式](/img/9/7fb/wZwpmL1YTN5UzNyMjNwMzM1UTM1QDN5MjM5ADMwAjMwUzLzYzL4czLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![衝激函式](/img/7/42a/wZwpmL1ITNwYDM0EzNxMzM1UTM1QDN5MjM5ADMwAjMwUzLxczL2IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/7/42a/wZwpmL1ITNwYDM0EzNxMzM1UTM1QDN5MjM5ADMwAjMwUzLxczL2IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/7/82f/wZwpmLxgTN0kDOxMDOwMzM1UTM1QDN5MjM5ADMwAjMwUzLzgzLwMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![衝激函式](/img/0/59c/wZwpmL4cjM3UTM5gDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL4gzLyEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![衝激函式](/img/7/82f/wZwpmLxgTN0kDOxMDOwMzM1UTM1QDN5MjM5ADMwAjMwUzLzgzLwMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![衝激函式](/img/7/42a/wZwpmL1ITNwYDM0EzNxMzM1UTM1QDN5MjM5ADMwAjMwUzLxczL2IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/0/59c/wZwpmL4cjM3UTM5gDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL4gzLyEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
上式表明,信號 與衝激函式相乘,篩選出連續時間信號 在 時的函式值 ,可以理解為衝激函式在 時刻對函式 的一瞬間的作用,其值是衝激函式和 相乘的結果,瞬間趨於無窮大 。
取樣性質
![衝激函式](/img/7/42a/wZwpmL1ITNwYDM0EzNxMzM1UTM1QDN5MjM5ADMwAjMwUzLxczL2IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/7/82f/wZwpmLxgTN0kDOxMDOwMzM1UTM1QDN5MjM5ADMwAjMwUzLzgzLwMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
如果信號 是一個在 處連續的普通函式,則有
![衝激函式](/img/9/cc7/wZwpmL4YzN3EzNzgzNwMzM1UTM1QDN5MjM5ADMwAjMwUzL4czL4MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![衝激函式](/img/7/42a/wZwpmL1ITNwYDM0EzNxMzM1UTM1QDN5MjM5ADMwAjMwUzLxczL2IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/9/fed/wZwpmL0MDM1ADN0QTMxMzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL0IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![衝激函式](/img/7/42a/wZwpmL1ITNwYDM0EzNxMzM1UTM1QDN5MjM5ADMwAjMwUzLxczL2IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/7/82f/wZwpmLxgTN0kDOxMDOwMzM1UTM1QDN5MjM5ADMwAjMwUzLzgzLwMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![衝激函式](/img/0/59c/wZwpmL4cjM3UTM5gDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL4gzLyEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![衝激函式](/img/7/42a/wZwpmL1ITNwYDM0EzNxMzM1UTM1QDN5MjM5ADMwAjMwUzLxczL2IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/7/42a/wZwpmL1ITNwYDM0EzNxMzM1UTM1QDN5MjM5ADMwAjMwUzLxczL2IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/d/ca7/wZwpmL0IzMzYTNzUjMzEzM1UTM1QDN5MjM5ADMwAjMwUzL1IzL0gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![衝激函式](/img/0/59c/wZwpmL4cjM3UTM5gDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL4gzLyEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
衝激信號的取樣特性表明,一個連續時間信號 與衝激函式相乘,並在時間域 上積分,其結果為信號 在 時的函式值 。該式可以理解為衝激函式作用於函式 ,趨於穩態時最終作用的結果,即得到信號 在 時刻的值 。
導數性質
衝激函式的導數性質如下:
![衝激函式](/img/a/cea/wZwpmL0ETMygTOzITOwMzM1UTM1QDN5MjM5ADMwAjMwUzLykzL4MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
其證明如下:
![衝激函式](/img/1/3b4/wZwpmLyAzN5ETNwUTMxMzM1UTM1QDN5MjM5ADMwAjMwUzL1EzL1gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![衝激函式](/img/8/93b/wZwpmL3EDNyYzNyQTOwMzM1UTM1QDN5MjM5ADMwAjMwUzL0kzLzgzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/c/dbc/wZwpmLzgTN3kTN5ITOwMzM1UTM1QDN5MjM5ADMwAjMwUzLykzL1czLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
尺度變換
衝激函式的尺度變換性質如下:
![衝激函式](/img/8/50b/wZwpmL3cTM1gjMykDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL5gzLxgzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
其推論明如下:
![衝激函式](/img/f/cf6/wZwpmL3AzN0ITO3cjNwMzM1UTM1QDN5MjM5ADMwAjMwUzL3YzLwMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
(1)
![衝激函式](/img/5/363/wZwpmLyIjMyYjMykDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL5gzL1QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(2)
![衝激函式](/img/a/794/wZwpmL0EjN4YzM0QTOwMzM1UTM1QDN5MjM5ADMwAjMwUzL0kzLwIzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![衝激函式](/img/1/41f/wZwpmL1UzM0cjMxUjMxMzM1UTM1QDN5MjM5ADMwAjMwUzL1IzLwYzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(3) 當 時,
![衝激函式](/img/9/089/wZwpmL0QTM4ITM2cjNxMzM1UTM1QDN5MjM5ADMwAjMwUzL3YzL4IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![衝激函式](/img/2/30b/wZwpmL1EDMzIjMzcDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL3gzLyAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
(4) ,為偶函式
![衝激函式](/img/0/244/wZwpmL1gjN3gDO1kTOwMzM1UTM1QDN5MjM5ADMwAjMwUzL5kzL1EzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![衝激函式](/img/5/740/wZwpmL1YTN1UDM2czMxMzM1UTM1QDN5MjM5ADMwAjMwUzL3MzL3YzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
(5) ,為奇函式
套用
衝激函式可用於信號處理,通過衝激函式來表示複雜的信號,可以簡化對複雜信號的一些特性的研究。衝激函式及其延時衝激函式的線性組合來表示或逼近,再利用系統的迭加原理,可以通過簡單的信號如單位衝激函式的頻譜,以及頻域特性來討論比較複雜信號的頻譜。從而減少計算複雜信號頻譜的難度 。