# Signals and Systems

[TOC]

## Ch1 Signals and Systems

**Power and Energy of signals**
$$
E_{\infin} = \lim_{T\to \infin} \int_{-T}^{T} |x(t)|^2\mathrm d t\\
P_{\infin}=\lim_{T\to \infin} \frac{1}{2T}\int_{-T}^{T}|x(t)|^2\mathrm d t\\
E_{\infin} = \sum_{n=-\infin}^{\infin} |x[n]|^2\\
P_{\infin} = \lim_{N\to \infin}\frac{1}{2N+1}\sum_{n=-\infin}^{\infin}|x[n]|^2
$$
**Odd and Even signals**

**Odd and Even components of signals**

**Impulse signal**

- Sampling property: $x(t)\delta(t-t_0)=x(t_0)\delta(t-t_0)$.
- Scaling property: $\delta(at)=\delta(t)/|a|$.

**Unit step signal**

**Properties of systems**

- ***Linearity***: Scaling + Superposition

  - Scaling: If $x(t)\to y(t)(x[n]\to y[n])$, then
    $$
    kx(t)\to ky(t)(kx[n]\to ky[n])
    $$

  - Superposition: If $x_1(t)\to y_1(t),x_2(t)\to y_2(t)$, then
    $$
    x_1(t)+x_2(t)\to y_1(t)+y_2(t)
    $$

- ***Time Invariance***. If $x(t)\to y(t)$, then
  $$
  x(t-t_0)\to y(t-t_0)
  $$

- ***Causality***. If $x(t)=0,t<t_0$, then
  $$
  y(t)=0,t<t_0
  $$

- ***Stability***. A system is stable if the input to a stable system is bounded, then the output must also be bounded and therefore cannot diverge.
  $$
  |x(t)|<\infin \to |y(t)|<\infin
  $$

**Example**

- $y(t)=x(t-2)+x(2-t)$.

  Linear; Not Time Invariant; Non causal; Stable.
  $$
  x(t)\to x(t-2)+x(2-t)\\
  x(t-t_0)\to x(t-t_0-2)+{\color{red}x(2-t-t_0)}\\
  y(t-t_0)=x(t-t_0-2)+x(2-t+t_0)
  $$

- $y(t)=tx(t)$.

  Linear; Not Time Invariant; Causal; Unstable.

- $y(t)=\frac{\mathrm d ^2x(t)}{\mathrm d t^2}+3\frac{\mathrm d x(t)}{\mathrm d t}+2x(t)$.

  Linear; Time Invariant; Causal; Unstable（求导）.

- $\displaystyle y[n]=\begin{cases}x[n],&n\ge0\\-x[n],&n<0\end{cases}$.

  Linear; Not Time Invariant; Causal; Stable.

- $y[n]=nx[n-1]+x[n+1]$.

  Linear; Not Time Invariant; Non causal; Unstable.

## Ch2 - LTI Systems

离散时间信号可以看成单位脉冲序列 $\delta[n-k]$ 的加权求和（线性组合），即：
$$
x[n]=\sum_{k=-\infin}^{\infin} x[k]\delta[n-k]
$$
卷积的性质：交换律、结合律、函数卷积后的微分：
$$
\frac{\mathrm d }{\mathrm d t}[x(t)*h(t)]=x(t)*\frac{\mathrm d }{\mathrm d t}h(t)=\frac{\mathrm d }{\mathrm d t}x(t)*h(t)
$$

函数时移后的卷积：
$$
x(t-t_1)*h(t-t_2)=y(t-t_1-t_2)
$$

## Ch3 - Fourier Series

### CT

$$
\boxed{x(t)=\sum_{k=-\infin}^{\infin} a_k e^{jk\omega_0 t}}
$$

$$
\boxed{a_k={\color{red}\frac{1}{T}}\int_T x(t) e^{-jk\omega_0 t}\mathrm d t}
$$

**Convergence of FS**

Dirichlet Condition:

- Condition 1: $\displaystyle \int_T |x(t)|\mathrm d t<\infin$.
- Condition 2: In a finite time interval, $x(t)$ has a finite number of maxima and minima.

> Example: $x(t)=\displaystyle \sin \left(\frac{2\pi}{t}\right), 0<t\le 1$ violates condition 2.
>
> ![img](https://notes.sjtu.edu.cn/uploads/upload_d9db2022117e1625f6cbdfedd8b29fb4.png)

- Condition 3: In a finite time interval, $x(t)$ has only a finite number of discontinuities.

> Example:![img](https://notes.sjtu.edu.cn/uploads/upload_1e282c3ef0fd4e13f7bda8415c9b8d95.png)

$$
\mathrm{Dirichlet} \Rightarrow \mathrm{Converge}\\
\mathrm{Converge} \not\Rightarrow \mathrm{Dirichlet}
$$

**根据公式手推 Fourier Series**

| Property        | Signal                                             | Fourier Series                                               |
| --------------- | -------------------------------------------------- | ------------------------------------------------------------ |
| Time Shifting   | $x(t-t_0)$                                         | $a_ke^{-jk\omega_0 t_0}$                                     |
| Time Scaling    | $x(at)$                                            | $x(at)=\displaystyle  \sum_{k=-\infin}^{\infin} a_k e^{jk(a\omega_0) t}$ |
| Time Reversal   | $x(-t)$                                            | $a_{-k}$                                                     |
| Conjugation     | $x^*(t)$                                           | $a_{-k}^*$                                                   |
| Differentiation | $\displaystyle \frac{\mathrm d x(t)}{\mathrm d t}$ | $jk\omega_0 a_k$                                             |
| Integration     | $\displaystyle \int_{-\infin}^t x(t)$              | $\displaystyle \frac{1}{jk\omega_0} a_k$                     |
| Multiplication  | $x(t)y(t)$                                         | $c_k=a_k*b_k$                                                |

### DT

$$
\boxed{x[n]=\sum_{k=\left\langle N\right\rangle}a_k e^{jk\omega_0 n}}\\
\boxed{
a_k={\color{red}\frac{1}{N}}\sum_{n=\langle N\rangle} x[n] e^{-jk\omega_0 n}
}
$$

## Ch4 - Fourier Transform

$$
\underbrace{\mathrm{Periodic\ Signals}}_{\mathrm{Fourier\ Series}} \Rightarrow \underbrace{\mathrm{All \ Signals}}_{\mathrm{Fourier\ Transform}}
$$

对于特征函数，输入 $x(t)=e^{j\omega t}$，输出 $y(t)=H(j\omega)e^{j\omega t}$.

若输入 $x(t)=\sin \omega t$，则输出 $y(t)=|H(j\omega)|\sin(\omega t+\varphi(\omega))$.

### CT

**Analysis FT equation**:
$$
\boxed{X(j\omega)=\int_{-\infin}^{\infin} x(t) e^{-j\omega t}\mathrm d t}
$$
**Synthesis FT equation**:
$$
\boxed{x(t)={\color{red}\frac{1}{2\pi}} \int_{-\infin}^{\infin} X(j\omega) e^{j\omega t}\mathrm d \omega}
$$
**Basic CTFT pairs**

| Name                  | Function                                                     | Fourier Transform                                            | Equation        |
| --------------------- | ------------------------------------------------------------ | ------------------------------------------------------------ | --------------- |
| Impulse Function      | $\delta(t-t_0)$                                              | $e^{-j\omega t_0}$                                           | Analysis        |
| Impulse Function 2    | $e^{j\omega_0t}$                                             | $2\pi\delta(\omega-\omega_0)$                                | Synthesis       |
| Gate Function         | $$\begin{cases}1, & |t|\le T\\0, & else\end{cases}$$         | $\displaystyle \frac{2\sin (\omega T)}{\omega}=2T\operatorname{Sa}(\omega T)$ | Analysis        |
| Ideal low pass filter | $\displaystyle \frac{\sin Wt}{\pi t}=t\operatorname{Sa}(Wt)/\pi$ | $\begin{cases}1,&|\omega|\le W\\0,&else\end{cases} $         | Synthesis       |
| Single-side Exp       | $e^{-\alpha t}u(t)$                                          | $\displaystyle \frac{1}{a+j\omega}$                          | Analysis        |
| Double-side Exp       | $e^{-\alpha|t|}$                                             | $\displaystyle \frac{2\alpha}{\alpha^2+\omega^2}$            |                 |
| Sign Function         | $\operatorname{sgn}(t)$                                      | $\displaystyle \frac{2}{j\omega}$                            | Single-side Exp |
| Step Function         | $u(t)$                                                       | $\displaystyle \pi \delta(\omega)+\frac{1}{j\omega}$         | Sign            |
| Cosine Function       | $\cos (\omega_0 t)$                                          | $\pi[\delta(\omega+\omega_0)+\delta(\omega-\omega_0)]$       | Impulse         |
| Sine Function         | $\sin (\omega_0t)$                                           | $j\pi[\delta(\omega+\omega_0)-\delta(\omega-\omega_0)]$      | Impulse         |
| *Triangle* Function   | $$\begin{cases}1-|t|/T, & |t|\le T\\0, & else\end{cases}$$   | $\displaystyle \frac{4 \sin^2 (\omega T/2)}{\omega^2T}=T\operatorname{Sa}^2(\omega T/2)$ | Gate            |

**Properties of CTFT**

> 先写下 Analysis 和 Synthesis Equation，然后填空

| Property                      | Signal                                                 | Fourier Transform                                            |
| ----------------------------- | ------------------------------------------------------ | ------------------------------------------------------------ |
| Time Shifting                 | $x(t-t_0)$                                             | $e^{-j\omega t_0}X(j\omega)$                                 |
| Frequency Shifting            | $x(t) e^{ j\omega_0 t}$                                | $X[j(\omega - \omega_0)]$                                    |
| Conjugation                   | $x^*(t)$                                               | $ X^*(-j\omega)$                                             |
| Time and frequency Scaling    | $x(at)$                                                | $\displaystyle \frac{1}{|a|}X\left(\frac{j\omega}{a}\right)$ |
| Time reverse                  | $x(-t)$                                                | $X(-j\omega)$                                                |
| Differentiation(in Time)      | $\displaystyle \frac{\mathrm d x(t)}{\mathrm d t}$     | $j\omega X(j\omega)$                                         |
| Integration                   | $\displaystyle \int_{-\infin}^t x(\tau)\mathrm d \tau$ | $\displaystyle \frac{1}{j\omega} X(j\omega)+ \pi X(0)\delta(\omega)$ |
| Differentiation(in frequency) | $tx(t)$                                                | $\displaystyle j\frac{\mathrm d }{\mathrm d \omega}X(j\omega)$ |
| Duality                       | $X(j\boxed{t})$                                        | $\boxed{2\pi} x(\boxed{-\omega})$                            |
| Convolution Property          | $h(t)*x(t)$                                            | $H(j\omega)X(j\omega)$                                       |
| Multiplication Property       | $h(t)x(t)$                                             | $\displaystyle \frac{1}{2\pi}H(j\omega)*X(j\omega)\\{\color{grey}Y(f)=H(f)*X(f)}$ |

> Notes:
>
> - $x(t)$ is real signal $\Rightarrow X(j\omega)=X^*(-j\omega)$.
> - $x(t)$ real and even $\Rightarrow X(-j\omega)=X(j\omega)\Rightarrow X(j\omega)=X^*(j\omega)\Rightarrow$ $X(j\omega)$ real.
> - $x(t)$ real and odd $\Rightarrow X(-j\omega)=-X(j\omega)\Rightarrow X(j\omega)=-X^*(j\omega)\Rightarrow$ $\operatorname{Re}\{X(j\omega)\}=0$.

**Example**
$$
\frac{\mathrm d }{\mathrm d t} f(at +b) \overset{\mathcal F}{\operatorname*{\longleftrightarrow}} \frac{j\omega}{|a|} F\left(\frac{j\omega}{a}\right) e^{j\omega b/a}
$$
**FT of CT Periodic Signal**
$$
x(t)=\sum_{k=-\infin}^{+\infin}a_k e^{jk\omega_0 t} \overset{\mathcal F}\Rightarrow \\
X(j\omega)=2\pi\sum_{k=-\infin}^{+\infin}  a_k\delta(\omega-k\omega_0)
$$
- 谱线位置：$\omega=k\omega_0$；周期信号的 $X_{T}(\omega)$ 只存在于 $\omega=k\omega_0$ 处。
- 谱线幅度：$2\pi a_k$（离散谱）

特殊函数：

- 梳状函数：$X_T(j\omega)=\omega_1 \delta_{\omega_1}(\omega)$.

- 周期方波：周期 $T_1$，脉宽 $\tau$，高 $E$.
  $$
  X(j\omega)=\sum_{n=-\infin}^{\infin} E\tau \operatorname{Sa}\left(n\omega_1 \frac{\tau}{2}\right)\omega_1\delta(\omega-n\omega_1)
  $$



**Parseval's Relation**
$$
\int_{-\infin}^{\infin} |x(t)|^2 \mathrm d t = \frac{1}{2\pi} \int_{-\infin}^{\infin} |X(j\omega)|^2 \mathrm d \omega = \int_{-\infin}^{\infin} |X(f)|^2 \mathrm d f
$$
***Comparing with Laplace Transform***

**System Analysis by CTFT**

如果系统可以用微分方程的形式表示，则 $H(j\omega)$ 为有理函数。
$$
H(j\omega)=\frac{Y(j\omega)}{X(j\omega)}=\frac{\sum_{k=0}^M b_k (j\omega)^k}{\sum_{k=0}^{N} a_k (j\omega)^k}
$$

### DT

傅里叶变换时求和，逆变换是积分。
$$
\boxed{X(e^{j\omega})=\sum_{n=-\infin}^{\infin} x[n] e^{-j\omega n }}\\
\boxed{x[n]={\color{red}\frac{1}{2\pi}}\int_{2\pi} X(j\omega) e^{j\omega n}\mathrm d \omega}
$$

**Basic DTFT pairs**

| Name           | Signal                                            | Fourier Transform                                           |
| -------------- | ------------------------------------------------- | ----------------------------------------------------------- |
| Impulse        | $\delta[n-n_0]$                                   | $e^{-j\omega n_0}$                                          |
| Exp            | $u(n)a^n$                                         | $\displaystyle \frac{1}{1-ae^{-j\omega}}$                   |
| Gate           | $\begin{cases}1,|n|\le N_1\\0,|n|>N_1\end{cases}$ | $\displaystyle \frac{\sin \omega(N_1+1/2)}{\sin(\omega/2)}$ |
| Ideal low-pass | $\displaystyle \frac{\sin Wn}{\pi n}$             | $\begin{cases}1,&|\omega| \le W\\0,&else\end{cases}$        |

For DT, $e^{jk\omega_0n} \overset{\mathcal F}{\operatorname*{\longleftrightarrow}} \delta(\omega-k\omega_0)$,
$$
x[n]=\sum_{k=\langle N\rangle} a_k e^{jk\omega_0 n},\omega_0=\frac{2\pi}{N} \overset{\mathcal F}{\operatorname*{\longleftrightarrow}}\\
X(j\omega)=\sum_{k=-\infin}^{\infin} 2\pi a_k \sum_l \delta (\omega-k\omega_0-2\pi l)
$$
**Properties of DTFT**

> *Periodicity only for DTFT*
> $$
> X(e^{j\omega})=X(e^{j(\omega+2\pi)})
> $$

| Property                     | Signal                                  | Fourier Transform                                            |
| ---------------------------- | --------------------------------------- | ------------------------------------------------------------ |
| Time Shifting                | $x[n-n_0]$                              | $e^{-j\omega n_0} X(e^{j\omega})$                            |
| Frequency Shifting           | $x[n]e^{j\omega_0 n}$                   | $X(e^{j(\omega-\omega_0)})$                                  |
| Conjugation                  | $x^*[n]$                                | $X^*(e^{j(-\omega)})$                                        |
| Differencing                 | $x[n]-x[n-1]$                           | $(1-e^{-j\omega})X(e^{j\omega})$                             |
| Accumulation                 | $\displaystyle \sum_{k=-\infin}^n x[k]$ | $X(e^{j\omega})+\pi X(e^{j0})\sum_{l=-\infin}^n \delta[\omega-2\pi l]$ |
| Time Scaling                 | $x_{(k)}[n]$                            | $X(e^{jk\omega})$                                            |
| Time Reversing               | $x[-n]$                                 | $X(e^{-j\omega})$                                            |
| Differentiation in frequency | $nx[n]$                                 | $\displaystyle j\frac{X(e^{j\omega})}{\mathrm d \omega}$     |
| Convolution property         | $x[n]*h[n]$                             | $X(e^{j\omega})H(e^{j\omega})$                               |
| Multiplication property      | $x[n]h[n]$                              | $\displaystyle \frac{1}{2\pi}X(e^{j\omega})*H(e^{j\omega})$  |

**Parseval's relationship**
$$
\sum_{n=-\infin}^{\infin} |x[n]|^2 = \frac{1}{2\pi} \int_{2\pi} |X(e^{j\omega})|^2 \mathrm d \omega
$$
***Comparing with Z transform***

| Time Domain | Frequency Domain |
| ----------- | ---------------- |
| Continuous  | Aperiodic        |
| Discrete    | Periodic         |
| Periodic    | Discrete         |
| Aperiodic   | Continuous       |

> 注，可以通过形象的例子加深理解。
>
> 1. $1 \overset{\mathcal F}{\operatorname*{\longleftrightarrow}} 2\pi \delta(\omega)$.
> 2. $\delta(t-t_0) \overset{\mathcal F}{\operatorname*{\longleftrightarrow}} e^{-j\omega t_0}$.
> 3. FT of CT Periodic Signal
> 4. Gate function or delta function.

## Ch5 - Application of Fourier Transform

**无失真传输条件**
$$
H(j\omega)=Ke^{-j\omega t_0 }\Rightarrow h(t) =K'\delta(t-t_0)
$$

- 幅频特性为一常数。

- 相频特性为一条过原点的负斜率直线。

- Group delay：
  $$
  \tau_g=-\frac{\mathrm d \varphi(\omega)}{\mathrm d \omega}
  $$
  无相位失真时，$\tau_g(\omega)$ 为常数。

**Sampling Theorem**

对于连续时间信号 $x(t)$，若用冲激串 $\displaystyle p(t)=\sum_{n=-\infin}^{+\infin} \delta(t-nT)$ 对其进行采样，则在时域内有 $x_p(t)=x(t)\cdot p(t)$；在频域内，有
$$
P(j\omega)=\omega_s \sum_{n=-\infin}^{+\infin} \delta(\omega-k\omega_s)
$$

$$
X_p(j\omega)=\frac{1}{2\pi} X(j\omega) * P(j\omega)
$$

$$
X_p(j\omega)={\color{red}\frac{1}{T}}\sum_{k=-\infin}^{+\infin} X(j(\omega-k\omega_s))
$$

> 结论：采样所得信号 $x_p(t)$ 的频谱函数 $X_p(j\omega)$ 是由无穷多个平移了的、同时幅度上有 $1/T$ 的尺度变换的 $X(j\omega)$ 叠加构成的。通过增益为 $T$ 的低通滤波器，截止频率大于 $\omega_m$ 但是小于 $\omega_s-\omega_m$ 的低通滤波器，即可还原 $x(t)$.

**The Modulation Property**（经过低通滤波，幅值为二）

画图表示













**Ideal Filters**

- Lowpass, Highpass(1-low).
- Bandpass(low-low), Bandstop(1-bandpass).

![](https://notes.sjtu.edu.cn/uploads/upload_50f2c81574f5dc4a4f2a0f0a2391f751.png)

**Practical Filters**

## Ch6 - Laplace Transform

The ***Bilateral Laplace transform*** is defined as
$$
x(t) \overset{\mathcal L}{\operatorname*{\longleftrightarrow}} X(s)=\int_{-\infin}^{\infin} x(t) e^{-st}\mathrm d t=\boxed{\int_{-\infin}^{\infin} [x(t) e^{-\sigma t}]e^{-j\omega t}\mathrm d t}
$$
where $s=\sigma+j\omega$. Notice that $X(s)=\mathcal F\{x(t)e^{-\sigma t}\}$.

The ***Inverse Laplace Transform*** can be derived from the inverse FT.
$$
\boxed{x(t)={\color{red}\frac{1}{2\pi j}}\int_{\sigma-j\infin}^{\sigma+j\infin} X(s)e^{st}\mathrm d s}
$$
**Laplace Pairs**
$$
\frac{1}{s+a} \overset{\mathcal L}{\operatorname*{\longleftrightarrow}}
 \begin{cases}
 e^{-at}u(t),&\operatorname{Re}\{s\} >-\operatorname{Re}\{a\}\\
 -e^{-at}u(-t),&\operatorname{Re}\{s\} <-\operatorname{Re}\{a\}
\end{cases}
$$
**Properties of the Laplace Transform**

| Property                   | Signal                                                  | Laplace Transform                                      | ROC                                            |
| -------------------------- | ------------------------------------------------------- | ------------------------------------------------------ | ---------------------------------------------- |
| Linearity                  | $ax_1(t)+bx_2(t)$                                       | $aX_1(s)+bX_2(s)$                                      | At least $R_1 \cap R_2$                        |
| Time Shifting              | $x(t-t_0)$                                              | $e^{-st_0}X(s)$                                        | $R$                                            |
| Shifting $s$               | $e^{s_0 t}x(t)$                                         | $X(s-s_0)$                                             | $R+s_0$                                        |
| Time Scaling               | $x(at)$                                                 | $\displaystyle \frac{1}{|a|}X\left(\frac{s}{a}\right)$ | $aR$                                           |
| Conjugation                | $x^*(t)$                                                | $X^*(s^*)$                                             | $R$                                            |
| Convolution                | $x_1(t)*x_2(t)$                                         | $X_1(s)X_2(s)$                                         | At least $R_1 \cap R_2$                        |
| D in Time Domain           | $\mathrm d x(t)/\mathrm d t$                            | $sX(s)$                                                | At least $R$                                   |
| D in $s$-Domain            | $-tx(t)$                                                | $\mathrm d X(s)/\mathrm d s$                           | $R$                                            |
| Integration in Time Domain | $\displaystyle \int_{-\infin}^{t}x(\tau)\mathrm d \tau$ | $\displaystyle \frac{1}{s}X(s)$                        | At least $R\cap \{\operatorname{Re}\{s\} >0\}$ |

**Initial- and Final-value Theorems**

If $x(t)=0$ for $t<0$ and $x(t)$ contains *no singularities* at $t=0$, then
$$
x(0^+) = \lim_{s\to \infin} sX(s)
$$
> 分母的阶次高于分子的阶次，不能有常数值。

**Final-value theorem**

If $x(t)=0$ for $t<0$ and $x(t)$ has *a finite limit* as $t\to \infin$, then
$$
\lim_{t\to \infin} x(t) =\lim_{s\to 0} sX(s)
$$

**Geometric evaluation of FT from pole-zero**

**CT system function properties**

1. System is **stable** $\Leftrightarrow $ $h(t)$ is stable $\Leftrightarrow$ $\displaystyle \int_{-\infin}^{\infin} |h(t)|\mathrm d t<\infin$ $\Leftrightarrow$ ROC of $H(s)$ includes $j\omega$ axis.

2. **Causality** $\Rightarrow$ $h(t)$ right-sided signal $\Rightarrow$ ROC of $H(s)$ is a right-half plane.

   > 特别注意 causal 和 right-sided 的区别。

   ROC right-half plane 但系统不 Causal 的反例：$\displaystyle H(s)=\frac{e^{sT}}{s+1}, \operatorname{Re}\{s\} >-1$

   但是当且仅当 $H(s)$ <u>有理</u>时，<u>系统因果和 ROC 在最右侧极点的右侧等价</u>。

   当 $H(s)$ 有理且因果时，<u>系统稳定</u> 等价于虚轴在 ROC 中，<u>等价于所有极点都在左半平面</u>。
   
3. Steady State Response.

**Block Diagram**

**Unilateral Laplace Transform**
$$
\mathcal X(s)=\int_{0^-} ^{\infin } x(t) e^{-st}\mathrm d t=\mathcal {UL} \{x(t)\}
$$
If system is causal, then $H(s)=\mathcal H(s)$.


$$
\boxed{\frac{\mathrm d x(t)}{\mathrm d t} \overset{\mathcal L}{\operatorname*{\longleftrightarrow}}
 s\mathcal X(s)-x(0^{-})}
$$

$$
\boxed{\frac{\mathrm d^2 x(t)}{\mathrm d t^2} \overset{\mathcal L}{\operatorname*{\longleftrightarrow}}
 s^2\mathcal X(s)-sx(0^{-})-x'(0^{-})}
$$

$$
\boxed{\frac{\mathrm d ^3 x(t)}{\mathrm d t^3} \overset{\mathcal L}{\operatorname*{\longleftrightarrow}}s^3 \mathcal X(s)-s^2 x(0^-)-sx'(0^-)-x''(0^-)}
$$

- ZIR: Response for zero input. $x(t)=0,\mathcal X(s)=0$.
- ZSR: Response for zero state input. (Bilateral)

- Total Response: ZIR+ZSR

## Ch7 - Z Transform

The ***Bilateral Z transform*** is defined as
$$
x[n] \overset{\mathcal Z}{\operatorname*{\longleftrightarrow}}

 X(z)=\sum_{n=-\infin}^{\infin} x[n] z^{-n} =Z \{x[n]\}
$$
The relationship between Z transform and DTFT:
$$
X(z)=\sum_{n=-\infin}^{\infin} x[n]r^{-n} e^{-j\omega n}=\mathcal F \{x[n]r^{-n}\}
$$
***Inverse Z transform***
$$
x[n]=\frac{1}{2\pi j}\oint_{|z|=r} X(z)z^{n-1}\mathrm d z
$$
Other method:
$$
X(z)=\sum_{n=0}^{\infin} x(n)z^{-n}
$$
Rational form:

Steps:

1. Divide by $z$: $\displaystyle \frac{X(z)}{z}=\frac{N(z)}{D(z)}$.（Add one pole to the function）

2. Using Partial fractional expansion: $\displaystyle \frac{X(z)}{z}\cdot z$.

3. Inverse Z transform.
   $$
   \frac{z}{z-a} \overset{\mathcal Z}{\operatorname*{\longleftrightarrow}}
   
   \begin{cases}
   a^n u(n),&|z|>|a|\\
   -a^{n} u(-n-1),&|z|<|a|
   \end{cases}
   $$

**Properties of Z Transform**

| Property                          | Signal                    | Z Transform                                          | ROC                    |
| --------------------------------- | ------------------------- | ---------------------------------------------------- | ---------------------- |
| Linearity                         | $ax_1[n]+bx_2[n]$         | $aX_1(z)+bX_2(z)$                                    | At least $R_1\cap R_2$ |
| Time shifting                     | $x[n-n_0]$                | $z^{-n_0}X(z)$                                       | $R$                    |
| Scaling in the $z$-domain         | $z_0^nx[n]$               | $X(z/z_0)$                                           | $z_0R$                 |
| Time reversal                     | $x[-n]$                   | $X(z^{-1})$                                          | $R^{-1}$               |
| Time expansion                    | $x_{(k)}[n]$              | $X(z^k)$                                             | $R^{1/k}$              |
| Conjugation                       | $x^*[n]$                  | $X^*(z^*)$                                           | $R$                    |
| Convolution                       | $x_1[n]*x_2[n]$           | $X_1(z)X_2(z)$                                       | At least $R_1\cap R_2$ |
| First Difference                  | $x[n]-x[n-1]$             | $(1-z^{-1})X(z)$                                     | At least $R\cap |z|>0$ |
| Accumulation                      | $\sum_{k=-\infin}^n x[k]$ | $\dfrac{X(z)}{1-z^{-1}}$                             | At least $R\cap |z|>1$ |
| **Differentiation in $z$-domain** | $nx[n]$                   | $\displaystyle -z\frac{\mathrm d X(z)}{\mathrm d z}$ | $R$                    |

*Initial Value Theorem*: If $x[n]=0$ for $n<0$, then $x[0]=\lim_{z\to \infin} X(z)$.

- Causal?
- Stability?

**Properties of Unilateral Z Transform**

| Property                      | Signal                            | Uni ZT                                                       |
| ----------------------------- | --------------------------------- | ------------------------------------------------------------ |
| Linearity                     | $ax_1[n]+bx_2[n]$                 | $a\mathcal X_1(z)+b\mathcal X_2(z)$                          |
| Time Delay                    | $x[n-1]$                          | $z^{-1}\mathcal X(z)+x[-1]$                                  |
| Time Delay 2                  | $x[n-2]$                          | $z^{-2}\mathcal X(z)+z^{-1}x[-1]+x[-2]$                      |
| Time Advance                  | $x[n+1]$                          | $zX(z)-zx[0]$                                                |
| Time Advance 2                | $x[n+2]$                          | $z^2X(z)-z^2x[0]-zx[1]$                                      |
| First Difference              | $x[n]-x[n-1]$                     | $(1-z^{-1})\mathcal X(z)-x[-1]$                              |
| Accumulation                  | $\displaystyle \sum_{k=0}^n x[k]$ | $\displaystyle \frac{1}{1-z^{-1}}\mathcal X(z)$              |
| Differentiation in $z$-domain | $nx[n]$                           | $-\displaystyle z\frac{\mathrm d \mathcal X(z)}{\mathrm d z}$ |

*Initial Value Theorem*: $x[0]=\lim_{z\to \infin} \mathcal X(z)$.