傅立叶变换的性质及计算

常用信号的频谱

\[e^{-at} u(t) \overset{F}\longleftrightarrow \frac{1}{a+j\omega}\] \[\delta(t) \overset{F} \longleftrightarrow 1\] \[1 \overset{F} \longleftrightarrow 2 \pi \delta(\omega)\] \[u(t) \overset{F} \longleftrightarrow \pi \delta(\omega) + \frac{1}{j\omega}\] \[sgn(t) \overset{F} \longleftrightarrow \frac{2}{j\omega}\] \[\cos\omega_{0}t \overset{F} \longleftrightarrow \pi [\delta(\omega + \omega_{0}) + \delta(\omega - \omega_{0})]\] \[\sin\omega_{0}t \overset{F} \longleftrightarrow j\pi [\delta(\omega + \omega_{0}) - \delta(\omega - \omega_{0})]\] \[t \overset{F} \longleftrightarrow 2 \pi j \delta^{\prime}(\omega)\] \[x(t)\cos \omega_{0}t \overset{F} \longleftrightarrow \frac{1}{2}[X(\omega + \omega_{0}) + X(\omega - \omega_{0})]\]

性质

线性性

\[ax_{1}(t) + bx_{2}(t) \overset{F} \longleftrightarrow aX_{1}(j\omega) + bX_{2}(j\omega)\]

对称性

\[\text{若}x(t) \overset{F} \longleftrightarrow X(j \omega) \text{，则}X(jt) \overset{F} \longleftrightarrow 2 \pi x(-\omega)\]

时延性

\[\text{若}x(t) \overset{F} \longleftrightarrow X(j \omega) \text{，则}x(t-t_{0}) \overset{F} \longleftrightarrow e^{-j\omega t_{0}}X(j \omega) \text{，} e^{-j \omega_{0} t}x(t) \overset{F} \longleftrightarrow X(j(\omega + \omega_{0}))\]

尺度变换

\[\text{若}x(t) \overset{F} \longleftrightarrow X(j \omega) \text{，则} x(at) \overset{F} \longleftrightarrow \frac{1}{|a|}X(\frac{j \omega}{a})\]

微分性质

\[\text{若}x(t) \overset{F} \longleftrightarrow X(j \omega) \text{，则} x^{\prime}(t) \overset{F} \longleftrightarrow j \omega X(j \omega) \text{，} -jtx(t) \overset{F} \longleftrightarrow X^{\prime}(j \omega)\]

卷积性质

\[\text{若}x_{1}(t) \overset{F} \longleftrightarrow X(j \omega) \text{，}x_{1}(t) \overset{F} \longleftrightarrow X(j \omega) \text{，则} x_1(t) * x_2(t) \overset{F} \longleftrightarrow X_{1}(j \omega) \cdot X_{2}(j \omega) \text{，}x_1(t) \cdot x_2(t) \overset{F} \longleftrightarrow \frac{1}{2 \pi} [X_{1}(j \omega) * X_{2}(j \omega)]\]