附录J 常用数学公式

J1 三角函数公式

\[ \cos^ {2} \theta = \frac {1}{2} (1 + \cos 2 \theta) \]

\[ \sin^ {2} \theta = \frac {1}{2} (1 - \cos 2 \theta) \]

\[ \cos^ {2} \theta + \sin^ {2} \theta = 1 \]

\[ \sin 2 \theta = 2 \sin \theta \cos \theta \]

\[ \cos 2 \theta = \cos^ {2} \theta - \sin^ {2} \theta \]

\[ \cos (\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \]

\[ \sin (\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \]

\[ \cos \alpha \cos \beta = \frac {1}{2} [ \cos (\alpha - \beta) + \cos (\alpha + \beta) ] \]

\[ \sin \alpha \sin \beta = \frac {1}{2} [ \cos (\alpha - \beta) - \cos (\alpha + \beta) ] \]

\[ \sin \alpha \cos \beta = \frac {1}{2} [ \sin (\alpha - \beta) + \sin (\alpha + \beta) ] \]

J2 欧拉公式

\[ \mathrm{e} ^ {\pm \mathrm{j} \theta} = \cos \theta \pm \mathrm{j} \sin \theta \quad \cos \theta = \frac {1}{2} (\mathrm{e} ^ {\mathrm{j} \theta} + \mathrm{e} ^ {- \mathrm{j} \theta}); \quad \sin \theta = \frac {1}{2 \mathrm{j}} (\mathrm{e} ^ {\mathrm{j} \theta} - \mathrm{e} ^ {- \mathrm{j} \theta}) \]

J3 对数的性质与运算法则

零与负数没有对数； \(\log_a a = 1\) \(\log_a 1 = 0\)

\[ \log_ {a} x y = \log_ {a} x + \log_ {a} y \quad \log_ {a} \frac {x}{y} = \log_ {a} x - \log_ {a} y \]

\[ \log_ {a} x ^ {\alpha} = \alpha \log_ {a} x \quad \log_ {a} b \cdot \log_ {b} a = 1 \]

换底公式 \(\log_{a}y = \log_{b}y / \log_{b}a\) 如 \(\log_{2}y = 3.32 \log_{10}y\)

J4 常用对数和自然对数

常用对数记作： \(\lg x = \log_{10}x\) ；自然对数记作： \(\ln x = \log_{e}x\)

两者关系： \(\mathrm{lgy} = \mathrm{lgelny}\approx 0.43\mathrm{lny};\mathrm{lny} = \mathrm{ln10}\cdot \mathrm{lgy}\approx 2.30\mathrm{lgy}\)

J5 冲激函数及其性质

\[ \delta (t) = \left\{ \begin{array}{l l} 0 & t \neq 0 \\ \infty & t = 0 \end{array} \right., \text {且} \int_ {- \infty} ^ {\infty} \delta (t) \mathrm{d} t = 1 \]

筛选特性 (抽样特性):

\[ f (t) \delta (t - t _ {0}) = f (t _ {0}) \delta (t - t _ {0}) \text {或} \int_ {- \infty} ^ {\infty} f (t) \delta (t _ {0} - t) \mathrm{d} t = f (t _ {0}) \]

搬移特性：

\[ f (t) * \delta (t - t _ {0}) = f (t - t _ {0}); F (\omega) * \delta (\omega - \omega_ {0}) = F (\omega - \omega_ {0}) \]

尺度变换性质：

\[ \delta (\omega) = \frac {1}{2 \pi} \delta (f), \quad \omega = 2 \pi f \]

傅里叶变换和反变换：

\[ \delta (t) \Leftrightarrow 1 \quad ; 1 \Leftrightarrow 2 \pi \delta (\omega) \text {或} \delta (f) \]

单位冲激序列及其变换：

\[ \delta_ {\mathrm{T}} (t) = \sum_ {n = - \infty} ^ {\infty} \delta (t - n T) \Longleftrightarrow \delta_ {\mathrm{T}} (\omega) = \frac {2 \pi}{T} \sum_ {n = - \infty} ^ {\infty} \delta (\omega - n \frac {2 \pi}{T}) \]

或 \(\delta_{\mathrm{T}}(t) = \sum_{n=-\infty}^{\infty}\delta(t-nT)\Leftrightarrow\delta_{\mathrm{T}}(f) = \frac{1}{T}\sum_{n=-\infty}^{\infty}\delta(f-n\frac{1}{T})\)

附录 J 常用数学公式