一、 e j ω 0 n {e^{j{\omega _0}n}} ejω0n的周期性
- 0 < ω 0 < 2 π 0 < {\omega _0} < 2\pi 0<ω0<2π，为序列的数字角频率
- 2 π ω 0 = N M \frac{ {2\pi }}{ { {\omega _0}}} = \frac{N}{M} ω02π=MN为有理数时，周期 N = M ( 2 π ω 0 ) N = M\left( {\frac{ {2\pi }}{ { {\omega _0}}}} \right) N=M(ω02π)，其中 ω 0 M = 2 π N \frac{ { {\omega _0}}}{M} = \frac{ {2\pi }}{N} Mω0=N2π称为基频。
二、单位冲激序列与单位阶跃序列
ε ( n ) = ∑ m = − ∞ n δ ( m ) = ∑ m = 0 ∞ δ ( n − m ) \varepsilon (n) = \sum\limits_{m = - \infty }^n {\delta (m)} = \sum\limits_{m = 0}^\infty {\delta (n - m)} ε(n)=m=−∞∑nδ(m)=m=0∑∞δ(n−m)
其中， ∑ m = − ∞ n δ ( m ) = ε ( n ) \sum\limits_{m = - \infty }^n {\delta (m)} = \varepsilon (n) m=−∞∑nδ(m)=ε(n)相当于积分。
三、 z z z变换
| ε ( n ) ↔ z z − 1 \varepsilon (n) \leftrightarrow \frac{z}{ {z - 1}} ε(n)↔z−1z | n ε ( n ) ↔ z ( z − 1 ) 2 n\varepsilon (n) \leftrightarrow \frac{z}{ { { {(z - 1)}^2}}} nε(n)↔(z−1)2z |
| a n ε ( n ) ↔ z z − a {a^n}\varepsilon (n) \leftrightarrow \frac{z}{ {z - a}} anε(n)↔z−az | n a n ε ( n ) ↔ z a ( z − a ) 2 n{a^n}\varepsilon (n) \leftrightarrow \frac{ {za}}{ { { {(z - a)}^2}}} nanε(n)↔(z−a)2za |
| cos ( β n ) ε ( n ) ↔ z 2 − z cos β z 2 − 2 z cos β + 1 \cos (\beta n)\varepsilon (n) \leftrightarrow \frac{ { {z^2} - z\cos \beta }}{ { {z^2} - 2z\cos \beta + 1}} cos(βn)ε(n)↔z2−2zcosβ+1z2−zcosβ | sin ( β n ) ε ( n ) ↔ z sin β z 2 − 2 z cos β + 1 \sin (\beta n)\varepsilon (n) \leftrightarrow \frac{ {z\sin \beta }}{ { {z^2} - 2z\cos \beta + 1}} sin(βn)ε(n)↔z2−2zcosβ+1zsinβ |
| ε ( − n − 1 ) ↔ − z z − 1 \varepsilon ( - n - 1) \leftrightarrow - \frac{z}{ {z - 1}} ε(−n−1)↔−z−1z | a n ε ( − n − 1 ) ↔ − z z − a {a^n}\varepsilon ( - n - 1) \leftrightarrow - \frac{z}{ {z - a}} anε(−n−1)↔−z−az |
| 单边 f ( n − m ) ε ( n − m ) ↔ z − m F ( z ) f(n - m)\varepsilon (n - m) \leftrightarrow {z^{ - m}}F(z) f(n−m)ε(n−m)↔z−mF(z) | f ( n − m ) ε ( n ) ↔ z − m [ F ( z ) + ∑ k = − m − 1 f ( k ) z − k ] f(n - m)\varepsilon (n) \leftrightarrow {z^{ - m}}\left[ {F(z) + \sum\limits_{k = - m}^{ - 1} {f(k){z^{ - k}}} } \right] f(n−m)ε(n)↔z−m[F(z)+k=−m∑−1f(k)z−k] |
| f ( n + m ) ε ( n ) ↔ z m [ F ( z ) − ∑ k = 0 m − 1 f ( k ) z − k ] f(n + m)\varepsilon (n) \leftrightarrow {z^m}\left[ {F(z) - \sum\limits_{k = 0}^{m - 1} {f(k){z^{ - k}}} } \right] f(n+m)ε(n)↔zm[F(z)−k=0∑m−1f(k)z−k] | a n f ( n ) ↔ F ( z a ) {a^n}f(n) \leftrightarrow F\left( {\frac{z}{a}} \right) anf(n)↔F(az) |
| n f ( n ) ↔ − z d F ( z ) d z nf(n) \leftrightarrow - z\frac{ {dF(z)}}{ {dz}} nf(n)↔−zdzdF(z) | f ( n a ) ↔ F ( z a ) f\left( {\frac{n}{a}} \right) \leftrightarrow F\left( { {z^a}} \right) f(an)↔F(za) |
| f ( n ) n + m ↔ z m ∫ z ∞ x − ( m + 1 ) F ( x ) d x \frac{ {f(n)}}{ {n + m}} \leftrightarrow {z^m}\int_z^\infty { {x^{ - (m + 1)}}F(x)dx} n+mf(n)↔zm∫z∞x−(m+1)F(x)dx | 时域部分和实质 f ( n ) ⊗ ε ( n ) = ∑ i = − ∞ ∞ f ( i ) ε ( n − i ) = ∑ i = − ∞ n f ( i ) f(n) \otimes \varepsilon (n) = \sum\limits_{i = - \infty }^\infty {f(i)\varepsilon (n - i)} = \sum\limits_{i = - \infty }^n {f(i)} f(n)⊗ε(n)=i=−∞∑∞f(i)ε(n−i)=i=−∞∑nf(i) |
| z z z域反转 f ( − k ) ↔ F ( z − 1 ) f( - k) \leftrightarrow F({z^{ - 1}}) f(−k)↔F(z−1) | 尺度变换 e j ω 0 n x ( n ) ↔ X ( e − j ω 0 ⋅ z ) {e^{j{\omega _0}n}}x(n) \leftrightarrow X\left( { {e^{ - j{\omega _0}}} \cdot z} \right) ejω0nx(n)↔X(e−jω0⋅z) |
| f ( 0 ) = lim z → ∞ F ( z ) f(0) = \mathop {\lim }\limits_{z \to \infty } F(z) f(0)=z→∞limF(z) | lim n → ∞ f ( n ) = lim z → 1 ( z − 1 ) F ( z ) \mathop {\lim }\limits_{n \to \infty } f(n) = \mathop {\lim }\limits_{z \to 1} (z - 1)F(z) n→∞limf(n)=z→1lim(z−1)F(z) |
其中， z z z域反转的收敛域为 α < ∣ z ∣ < β ↔ 1 β < ∣ z ∣ < 1 α \alpha < \left| z \right| < \beta \leftrightarrow \frac{1}{\beta } < \left| z \right| < \frac{1}{\alpha } α<∣z∣<β↔β1<∣z∣<α1，求终值 lim n → ∞ f ( n ) \mathop {\lim }\limits_{n \to \infty } f(n) n→∞limf(n)时，先判断其是否存在，且 ∣ z ∣ ⩾ 1 \left| z \right| \geqslant 1 ∣z∣⩾1， F ( z ) F(z) F(z)极点在单位圆内， z = 1 z=1 z=1处有一阶极点。
四、拉普拉斯变换
五、采样定理
- 时域 T s T_s Ts采样，频域 2 π T s \frac{2\pi}{T_s} Ts2π延拓，幅度为 × 1 T s \times \frac{1}{T_s} ×Ts1
- 时域 T T T延拓，频域 2 π T \frac{2\pi}{T} T2π采样，幅度为 × 2 π T \times \frac{2\pi}{T} ×T2π
δ T s ( t ) = ∑ k = − ∞ ∞ δ ( t − k T s ) ↔ 2 π T s ∑ k = − ∞ ∞ δ ( ω − k ω s ) {\delta _{ {T_s}}}(t) = \sum\limits_{k = - \infty }^\infty {\delta (t - k{T_s})} \leftrightarrow \frac{ {2\pi }}{ { {T_s}}}\sum\limits_{k = - \infty }^\infty {\delta (\omega - k{\omega _s})} δTs(t)=k=−∞∑∞δ(t−kTs)↔Ts2πk=−∞∑∞δ(ω−kωs)
1 ↔ 2 π δ ( ω ) ↓ ↓ 采样 延拓 \begin{aligned} &\begin{array}{rr} 1 & & \leftrightarrow 2 \pi \delta(\omega) \\ \downarrow & & \downarrow \end{array}\\ &\text { 采样 } \quad \quad \text { 延拓 } \end{aligned} 1↓↔2πδ(ω)↓ 采样 延拓
六、傅里叶变换对
七、罗斯准则
八、系统响应
