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求图中两种周期信号的傅里叶级数。

大学 2022-01-06 00:31:23 级数,周期,信号已帮助人

求图中两种周期信号的傅里叶级数。

热搜：傅里叶级数展开公式a0怎么求

答案解析

难度★★☆ 搜索

图(a)，f(t)的三角形式傅里叶级数为

$latex.action?latex=ZlxsZWZ0KCB0IFxyaWdodCk9YV97MH0rXHN1bV97bj0xfV574oiefVxsZWZ0WyBhX3tufWNvc1xs%0D%0AZWZ0KCBuXG9tZWdhIF97MX10IFxyaWdodCkrYl97bn1zaW5cbGVmdCggblxvbWVnYSBfezF9dCBc%0D%0AcmlnaHQpIFxyaWdodF0%3D$
$latex.action?latex=XGxlZnQoIFxvbWVnYSBfezF9PVxmcmFjezJccGkgfXtUfSBccmlnaHQp$
$latex.action?latex=YV97MH09XGZyYWN7MX17VH1caW50X3stXGZyYWN7XHBpIH17NH19XntcZnJhY3szVH17NH19Zlxs%0D%0AZWZ0KCB0IFxyaWdodClkdD1cZnJhY3sxfXtUfVxsZWZ0WyBcaW50X3stXGZyYWN7VH17NH19Xntc%0D%0AZnJhY3tUfXs0fX1cbGVmdCggXGZyYWN7RX17Mn0rXGZyYWN7MkV9e1R9dCBccmlnaHQpZHQrXGlu%0D%0AdF97XGZyYWN7VH17NH19XntcZnJhY3szVH17NH19XGxlZnQoIFxmcmFjezNFfXsyfS1cZnJhY3sy%0D%0ARX17VH10IFxyaWdodClkdCBccmlnaHRd$
$latex.action?latex=PVxmcmFjezF9e1R9XGxlZnRbIFxmcmFje0V9ezJ9XHRpbWVzIFxmcmFje1R9ezJ9KzArXGZyYWN7%0D%0AM0V9ezJ9XHRpbWVzIFxmcmFje1R9ezJ9LVxpbnRfe1xmcmFje1R9ezR9fV57XGZyYWN7M1R9ezR9%0D%0AfVxmcmFjezJFfXtUfXRkdCBccmlnaHRd$
令t=u+ $latex.action?latex=XGZyYWN7VH17Mn0%3D$ ，则 $latex.action?latex=YV97MH09XGZyYWN7MX17VH1cbGVmdFsgRVQtXGludF97LVxmcmFje1R9ezR9fV57XGZyYWN7VH17%0D%0ANH19XGZyYWN7MkV9e1R9XGxlZnQoIHUrXGZyYWN7VH17Mn0gXHJpZ2h0KWR0IFxyaWdodF09XGZy%0D%0AYWN7MX17VH1cbGVmdFsgRVQtMC1FXHRpbWVzIFxmcmFje1R9ezJ9IFxyaWdodF09XGZyYWN7RX17%0D%0AMn0%3D$
$latex.action?latex=YV97bn09XGZyYWN7Mn17VH1caW50X3stXGZyYWN7VH17NH19XntcZnJhY3szVH17NH19ZlxsZWZ0%0D%0AKCB0IFxyaWdodCljb3NcbGVmdCggblxvbWVnYSBfezF9dCBccmlnaHQpZHQ9XGZyYWN7Mn17VH1c%0D%0AbGVmdFsgXGludF97LVxmcmFje1R9ezR9fV57XGZyYWN7VH17NH19XGxlZnQoIFxmcmFje0V9ezJ9%0D%0AK1xmcmFjezJFfXtUfXQgXHJpZ2h0KWNvc1xsZWZ0KCBuXG9tZWdhIF97MX10IFxyaWdodClkdCBc%0D%0AcmlnaHRd$
$latex.action?latex=PVxmcmFjezJ9e1R9XGxlZnRbIFxmcmFje0V9ezJ9XGludF97LVxmcmFje1R9ezR9fV57XGZyYWN7%0D%0AVH17NH19Y29zXGxlZnQoIG5cb21lZ2EgX3sxfXQgXHJpZ2h0KWR0K1xpbnRfe1xmcmFje1R9ezR9%0D%0AfV57XGZyYWN7M317NH1UfVxsZWZ0KCBcZnJhY3szRX17Mn0tXGZyYWN7MkV9e1R9dCBccmlnaHQp%0D%0AY29zXGxlZnQoIG5cb21lZ2EgX3sxfXQgXHJpZ2h0KWR0IFxyaWdodF0%3D$
在第二个积分表达式中，令t=u+ $latex.action?latex=XGZyYWN7VH17Mn0%3D$ ，则
$latex.action?latex=YV97bn09XGZyYWN7Mn17VH1cbGVmdFsgRVxpbnRfezB9XntcZnJhY3tUfXs0fX1jb3NcbGVmdCgg%0D%0AblxvbWVnYSBfezF9dCBccmlnaHQpZHQrXGludF97LVxmcmFje1R9ezR9fV57XGZyYWN7VH17NH19%0D%0AXGxlZnQoIFxmcmFjezNFfXsyfS1cZnJhY3syRX17VH1cbGVmdCggdStcZnJhY3tUfXsyfSBccmln%0D%0AaHQpIFxyaWdodCljb3NcbGVmdCggblxvbWVnYSBfezF9dStuXHBpICBccmlnaHQpZHUgXHJpZ2h0%0D%0AXQ%3D%3D$
$latex.action?latex=PVxmcmFjezJ9e1R9XGxlZnRbIEVcaW50X3swfV57XGZyYWN7VH17NH19Y29zXGxlZnQoIG5cb21l%0D%0AZ2EgX3sxfXQgXHJpZ2h0KWR0K1xmcmFje0V9ezJ9XGludF97LVxmcmFje1R9ezR9fV57XGZyYWN7%0D%0AVH17NH19Y29zXGxlZnQoIG5cb21lZ2EgX3sxfXUrblxwaSAgXHJpZ2h0KWR1IFxyaWdodF09MA%3D%3D$
$latex.action?latex=Yl97bn09XGZyYWN7Mn17VH1caW50X3tcZnJhY3szVH17NH19XnstXGZyYWN7VH17NH19ZlxsZWZ0%0D%0AKCB0IFxyaWdodClzaW5cbGVmdCggblxvbWVnYSBfezF9dCBccmlnaHQpZHQ9XGZyYWN7Mn17VH1c%0D%0AbGVmdFsgXGludF97LVxmcmFje1R9ezR9fV57XGZyYWN7VH17NH19XGxlZnQoIFxmcmFje0V9ezJ9%0D%0AK1xmcmFjezJFfXtUfXQgXHJpZ2h0KXNpblxsZWZ0KCBuXG9tZWdhIF97MX10IFxyaWdodClkdCtc%0D%0AaW50X3tcZnJhY3tUfXs0fX1ee1xmcmFjezN9ezR9VH1cbGVmdCggXGZyYWN7M0V9ezJ9LVxmcmFj%0D%0AezJFfXtUfXQgXHJpZ2h0KXNpblxsZWZ0KCBuXG9tZWdhIF97MX10IFxyaWdodClkdCBccmlnaHRd%0D%0A$
在第二个积分表达式中，令u= $latex.action?latex=XGZyYWN7VH17Mn0%3D$ =t，则
$latex.action?latex=Yl97bn09XGZyYWN7Mn17VH1caW50X3stXGZyYWN7VH17NH19XntcZnJhY3szVH17NH19ZlxsZWZ0%0D%0AKCB0IFxyaWdodClzaW5cbGVmdCggblxvbWVnYSBfezF9dCBccmlnaHQpZHQ9XGZyYWN7Mn17VH1c%0D%0AbGVmdFsgXGludF97LVxmcmFje1R9ezR9fV57XGZyYWN7VH17NH19XGxlZnQoIFxmcmFje0V9ezJ9%0D%0AK1xmcmFjezJFfXtUfXQgXHJpZ2h0KXNpblxsZWZ0KCBuXG9tZWdhIF97MX10IFxyaWdodClkdCtc%0D%0AaW50X3tcZnJhY3tUfXs0fX1ee1xmcmFjezN9ezR9VH1cbGVmdCggXGZyYWN7M0V9ezJ9LVxmcmFj%0D%0AezJFfXtUfXQgXHJpZ2h0KXNpblxsZWZ0KCBuXG9tZWdhIF97MX10IFxyaWdodClkdCBccmlnaHRd%0D%0A$
n为偶数时，bn=0；n为奇数时
$latex.action?latex=Yl97bn09XGZyYWN7Mn17VH1caW50X3swfV57XGZyYWN7VH17NH19XGZyYWN7OEV9e1R9dHNpblxs%0D%0AZWZ0KCBuXG9tZWdhIF97MX10IFxyaWdodClkdD1cZnJhY3sxNkV9e1ReezJ9fVxsZWZ0WyAtdFxm%0D%0AcmFje2Nvc1xsZWZ0KCBuXG9tZWdhIF97MX10IFxyaWdodCl9e25cb21lZ2EgX3sxfX1cbWlkIF97%0D%0AMH1ee1xmcmFje1R9ezR9fStcaW50X3swfV57XGZyYWN7VH17NH19XGZyYWN7Y29zXGxlZnQoIG5c%0D%0Ab21lZ2EgX3sxfXQgXHJpZ2h0KX17blxvbWVnYSBfezF9fWR0IFxyaWdodF0%3D$
$latex.action?latex=PVxmcmFjezE2RX17VF57Mn19XGxlZnRbIDArXGZyYWN7c2luXGZyYWN7blxwaSB9ezJ9fXtcbGVm%0D%0AdCggblxwaSAgXHJpZ2h0KV57Mn19IFxyaWdodF09XGZyYWN7NEV9e1xsZWZ0KCBuXHBpICBccmln%0D%0AaHQpXnsyfX1zaW5cZnJhY3tuXHBpIH17Mn0%3D$
f(t)的三角形式傅里叶级数为 $latex.action?latex=ZlxsZWZ0KCB0IFxyaWdodCk9XGZyYWN7RX17Mn0rXHN1bV97bj0xfV574oiefVxmcmFjezRFfXtc%0D%0AbGVmdCggblxwaSAgXHJpZ2h0KV57Mn19c2luXGZyYWN7blxwaSB9ezJ9c2luXGxlZnQoIG5cb21l%0D%0AZ2EgX3sxfXQgXHJpZ2h0KQ%3D%3D$ ，其中n=1，3，5，…
图(b)，由图知f(t)为偶数，所以有bn=0，而

$latex.action?latex=YV97bn09XGZyYWN7NH17VH1caW50X3swfV57XGZyYWN7VH17Mn19ZlxsZWZ0KCB0IFxyaWdodClj%0D%0Ab3NcbGVmdCggblxvbWVnYSBfezF9dCBccmlnaHQpZHQ%3D$ $latex.action?latex=XGxlZnQoIFxvbWVnYSBfezF9PVxmcmFjezJccGkgfXtUfSBccmlnaHQp$
f(t)的表达式为 $latex.action?latex=ZlxsZWZ0KCB0IFxyaWdodCk9XGxlZnRce1xiZWdpbnthcnJheX17Y2N9XGZyYWN7NEV9e1R9dCAm%0D%0AIHRcaW4gXGxlZnRbIDAsXGZyYWN7VH17NH0gXHJpZ2h0XVxcRSAgJiB0XGluIFxsZWZ0WyBcZnJh%0D%0AY3tUfXs0fSxcZnJhY3tUfXsyfSBccmlnaHRdXGVuZHthcnJheX0%3D$ ，所以有
$latex.action?latex=YV97bn09XGZyYWN7NH17VH1cbGVmdFsgXGludF97MH1ee1xmcmFje1R9ezR9fXRjb3NcbGVmdCgg%0D%0AblxvbWVnYSBfezF9dCBccmlnaHQpZHQrXGludF97XGZyYWN7VH17NH19XntcZnJhY3tUfXsyfX1F%0D%0AY29zXGxlZnQoIG5cb21lZ2EgX3sxfXQgXHJpZ2h0KWR0IFxyaWdodF0%3D$
$latex.action?latex=PVxmcmFjezR9e1R9XGxlZnRbIFxmcmFjezRFfXtUfVxsZWZ0WyBcZnJhY3t0c2luXGxlZnQoIG5c%0D%0Ab21lZ2EgX3sxfXQgXHJpZ2h0KX17blxvbWVnYSBfezF9fVxtaWQgX3swfV57XGZyYWN7VH17NH19%0D%0ALVxpbnRfezB9XntcZnJhY3tUfXs0fX1cZnJhY3tzaW5cbGVmdCggblxvbWVnYSBfezF9dCBccmln%0D%0AaHQpfXtuXG9tZWdhIF97MX19ZHQgXHJpZ2h0XStFXGZyYWN7c2luXGxlZnQoIG5cb21lZ2EgX3sx%0D%0AfXQgXHJpZ2h0KX17blxvbWVnYSBfezF9fVxtaWQgX3tcZnJhY3tUfXs0fX1ee1xmcmFje1R9ezJ9%0D%0AfSBccmlnaHRd$
$latex.action?latex=PVxmcmFjezR9e1R9XGxlZnRbIFxmcmFjezRFfXtUfVxsZWZ0WyBcZnJhY3tUfXs0fVxjZG90IFxm%0D%0AcmFje3NpblxmcmFje25ccGkgfXsyfX17blxvbWVnYSBfezF9fStcZnJhY3tjb3NcbGVmdCggblxv%0D%0AbWVnYSBfezF9dCBccmlnaHQpfXtcbGVmdCggblxvbWVnYSBfezF9IFxyaWdodCleezJ9fVxtaWQg%0D%0AX3swfV57XGZyYWN7VH17NH19IFxyaWdodF0rXGZyYWN7RX17blxvbWVnYSBfezF9fVxsZWZ0KCAt%0D%0Ac2luXGZyYWN7blxwaSB9ezJ9IFxyaWdodCkgXHJpZ2h0XQ%3D%3D$
$latex.action?latex=PVxmcmFjezR9e1R9XGxlZnRbIFxmcmFje2Nvc1xsZWZ0KCBcZnJhY3tuXHBpIH17Mn0gXHJpZ2h0%0D%0AKS0xfXtcbGVmdCggblxvbWVnYSBfezF9IFxyaWdodCleezJ9fVx0aW1lcyBcZnJhY3s0RX17VH0g%0D%0AXHJpZ2h0XT1cZnJhY3s0RX17XGxlZnQoIG5ccGkgIFxyaWdodCleezJ9fVxsZWZ0WyBjb3NcbGVm%0D%0AdCggXGZyYWN7blxwaSB9ezJ9LTEgXHJpZ2h0KSBccmlnaHRd$
$latex.action?latex=YV97MH09XGZyYWN7Mn17VH1caW50X3swfV57XGZyYWN7VH17Mn19ZlxsZWZ0KCB0IFxyaWdodClk%0D%0AdD1cZnJhY3syfXtUfVxsZWZ0WyBcaW50X3swfV57XGZyYWN7VH17NH19XGZyYWN7NEV9e1R9dGR0%0D%0AK1xpbnRfe1xmcmFje1R9ezR9fV57XGZyYWN7VH17Mn19RWR0IFxyaWdodF09XGZyYWN7Mn17VH1c%0D%0AbGVmdFsgXGZyYWN7NEV9e1R9XHRpbWVzIFxmcmFjezF9ezJ9XGxlZnQoIFxmcmFje1R9ezR9IFxy%0D%0AaWdodCleezJ9K0VcdGltZXMgXGZyYWN7VH17NH0gXHJpZ2h0XT1cZnJhY3szfXs0fUU%3D$
f(t)的三角形式傅里叶级数为 $latex.action?latex=ZlxsZWZ0KCB0IFxyaWdodCk9XGZyYWN7M317NH1FK1xzdW1fe249MX1ee%2BKInn1cZnJhY3s0RX17%0D%0AXGxlZnQoIG5ccGkgIFxyaWdodCleezJ9fVxsZWZ0KCBjb3NcbGVmdCggXGZyYWN7blxwaSB9ezJ9%0D%0AIFxyaWdodCktMSBccmlnaHQpY29zXGxlZnQoIG5cb21lZ2EgX3sxfXQgXHJpZ2h0KQ%3D%3D$ 。