Signals and Systems

Signals and SystemsCh1 Signals and SystemsCh2 - LTI SystemsCh3 - Fourier SeriesCTDTCh4 - Fourier TransformCTDTCh5 - Application of Fourier TransformCh6 - Laplace TransformCh7 - Z Transform

Ch1 Signals and Systems

Power and Energy of signals

\begin{matrix} E_{\infty} = lim_{T \to \infty} \int_{- T}^{T} | x (t) |^{2} d t \\ P_{\infty} = lim_{T \to \infty} \frac{1}{2 T} \int_{- T}^{T} | x (t) |^{2} d t \\ E_{\infty} = \sum_{n = - \infty}^{\infty} | x [n] |^{2} \\ P_{\infty} = lim_{N \to \infty} \frac{1}{2 N + 1} \sum_{n = - \infty}^{\infty} | x [n] |^{2} \end{matrix}

Odd and Even signals

Odd and Even components of signals

Impulse signal

$x(t)\delta(t-t_0)=x(t_0)\delta(t-t_0)$ .
$\delta(at)=\delta(t)/|a|$ .

Unit step signal

Properties of systems

Linearity: Scaling + Superposition
- $x(t)\to y(t)(x[n]\to y[n])$ , then
  $k x (t) \to k y (t) (k x [n] \to k y [n])$
- $x_1(t)\to y_1(t),x_2(t)\to y_2(t)$ , then
  $x_{1} (t) + x_{2} (t) \to y_{1} (t) + y_{2} (t)$
Time Invariance $x(t)\to y(t)$ , then
$x (t - t_{0}) \to y (t - t_{0})$
Causality $x(t)=0,t<t_0$ , 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.
$| x (t) | < \infty \to | y (t) | < \infty$

Example

$y(t)=x(t-2)+x(2-t)$ .
Linear; Not Time Invariant; Non causal; Stable.
$\begin{matrix} x (t) \to x (t - 2) + x (2 - t) \\ x (t - t_{0}) \to x (t - t_{0} - 2) + x (2 - t - t_{0}) \\ y (t - t_{0}) = x (t - t_{0} - 2) + x (2 - t + t_{0}) \end{matrix}$
$y(t)=tx(t)$ .
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

$\delta[n-k]$ 的加权求和（线性组合），即：

x [n] = \sum_{k = - \infty}^{\infty} x [k] δ [n - k]

卷积的性质：交换律、结合律、函数卷积后的微分：

\frac{d}{d t} [x (t) * h (t)] = x (t) * \frac{d}{d t} h (t) = \frac{d}{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

x (t) = \sum_{k = - \infty}^{\infty} a_{k} e^{j k ω_{0} t}

a_{k} = \frac{1}{T} \int_{T} x (t) e^{- j k ω_{0} t} d t

Convergence of FS

Dirichlet Condition:

$\displaystyle \int_T |x(t)|\mathrm d t<\infin$ .
$x(t)$ has a finite number of maxima and minima.

$x(t)=\displaystyle \sin \left(\frac{2\pi}{t}\right), 0<t\le 1$ violates condition 2.

$x(t)$ has only a finite number of discontinuities.

Example:

\begin{matrix} Dirichlet \Rightarrow Converge \\ Converge ⇏ Dirichlet \end{matrix}

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

\begin{matrix} x [n] = \sum_{k = ⟨ N ⟩} a_{k} e^{j k ω_{0} n} \\ a_{k} = \frac{1}{N} \sum_{n = ⟨ N ⟩} x [n] e^{- j k ω_{0} n} \end{matrix}

Ch4 - Fourier Transform

\underset{Fourier Series}{\underset{⏟}{Periodic Signals}} \Rightarrow \underset{Fourier Transform}{\underset{⏟}{All Signals}}

$x(t)=e^{j\omega t}$ $y(t)=H(j\omega)e^{j\omega t}$ .

$x(t)=\sin \omega t$ $y(t)=|H(j\omega)|\sin(\omega t+\varphi(\omega))$ .

CT

Analysis FT equation:

X (j ω) = \int_{- \infty}^{\infty} x (t) e^{- j ω t} d t

Synthesis FT equation:

x (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} X (j ω) e^{j ω t} d ω

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	$h(t)x(t)$	$\displaystyle \frac{1}{2\pi}H(j\omega)X(j\omega)\\{\color{grey}Y(f)=H(f)X(f)}$

Example

\frac{d}{d t} f (a t + b) \overset{F}{⟷} \frac{j ω}{| a |} F (\frac{j ω}{a}) e^{j ω b / a}

FT of CT Periodic Signal

\begin{matrix} x (t) = \sum_{k = - \infty}^{+ \infty} a_{k} e^{j k ω_{0} t} \overset{F}{\Rightarrow} \\ X (j ω) = 2 π \sum_{k = - \infty}^{+ \infty} a_{k} δ (ω - k ω_{0}) \end{matrix}

$\omega=k\omega_0$ $X_{T}(\omega)$ $\omega=k\omega_0$ 处。
$2\pi a_k$ （离散谱）

特殊函数：

$X_T(j\omega)=\omega_1 \delta_{\omega_1}(\omega)$ .
$T_1$ $\tau$ $E$ .
$X (j ω) = \sum_{n = - \infty}^{\infty} E τ Sa (n ω_{1} \frac{τ}{2}) ω_{1} δ (ω - n ω_{1})$

Parseval's Relation

\int_{- \infty}^{\infty} | x (t) |^{2} d t = \frac{1}{2 π} \int_{- \infty}^{\infty} | X (j ω) |^{2} d ω = \int_{- \infty}^{\infty} | X (f) |^{2} d f

Comparing with Laplace Transform

System Analysis by CTFT

$H(j\omega)$ 为有理函数。

H (j ω) = \frac{Y (j ω)}{X (j ω)} = \frac{\sum_{k = 0}^{M} b_{k} (j ω)^{k}}{\sum_{k = 0}^{N} a_{k} (j ω)^{k}}

DT

傅里叶变换时求和，逆变换是积分。

\begin{matrix} X (e^{j ω}) = \sum_{n = - \infty}^{\infty} x [n] e^{- j ω n} \\ x [n] = \frac{1}{2 π} \int_{2 π} X (j ω) e^{j ω n} d ω \end{matrix}

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	$\displaystyle \frac{\sin Wn}{\pi n}$	$\begin{cases}1,&\|\omega\| \le W\\0,&else\end{cases}$

$e^{jk\omega_0n} \overset{\mathcal F}{\operatorname*{\longleftrightarrow}} \delta(\omega-k\omega_0)$ ,

\begin{matrix} x [n] = \sum_{k = ⟨ N ⟩} a_{k} e^{j k ω_{0} n}, ω_{0} = \frac{2 π}{N} \overset{F}{⟷} \\ X (j ω) = \sum_{k = - \infty}^{\infty} 2 π a_{k} \sum_{l} δ (ω - k ω_{0} - 2 π l) \end{matrix}

Properties of DTFT

Periodicity only for DTFT
$X (e^{j ω}) = X (e^{j (ω + 2 π)})$

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 = - \infty}^{\infty} | x [n] |^{2} = \frac{1}{2 π} \int_{2 π} | X (e^{j ω}) |^{2} d ω

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。
$1 \overset{\mathcal F}{\operatorname*{\longleftrightarrow}} 2\pi \delta(\omega)$ .
$\delta(t-t_0) \overset{\mathcal F}{\operatorname*{\longleftrightarrow}} e^{-j\omega t_0}$ .
FT of CT Periodic Signal
Gate function or delta function.

Ch5 - Application of Fourier Transform

无失真传输条件

H (j ω) = K e^{- j ω t_{0}} \Rightarrow h (t) = K^{'} δ (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：
$τ_{g} = - \frac{d φ (ω)}{d ω}$
$\tau_g(\omega)$ 为常数。

Sampling Theorem

$x(t)$ $\displaystyle p(t)=\sum_{n=-\infin}^{+\infin} \delta(t-nT)$ $x_p(t)=x(t)\cdot p(t)$ ；在频域内，有

P (j ω) = ω_{s} \sum_{n = - \infty}^{+ \infty} δ (ω - k ω_{s})

X_{p} (j ω) = \frac{1}{2 π} X (j ω) * P (j ω)

X_{p} (j ω) = \frac{1}{T} \sum_{k = - \infty}^{+ \infty} X (j (ω - k ω_{s}))

$x_p(t)$ $X_p(j\omega)$ $1/T$ $X(j\omega)$ $T$ $\omega_m$ $\omega_s-\omega_m$ $x(t)$ .

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).

Practical Filters

Ch6 - Laplace Transform

The Bilateral Laplace transform is defined as

x (t) \overset{L}{⟷} X (s) = \int_{- \infty}^{\infty} x (t) e^{- s t} d t = \int_{- \infty}^{\infty} [x (t) e^{- σ t}] e^{- j ω t} d t

$s=\sigma+j\omega$ $X(s)=\mathcal F\{x(t)e^{-\sigma t}\}$ .

The Inverse Laplace Transform can be derived from the inverse FT.

x (t) = \frac{1}{2 π j} \int_{σ - j \infty}^{σ + j \infty} X (s) e^{s t} d s

Laplace Pairs

\begin{matrix} \frac{1}{s + a} \overset{L}{⟷} {\begin{cases} e^{- a t} u (t), & Re {s} > - Re {a} \\ - e^{- a t} u (- t), & Re {s} < - Re {a} \end{cases} \end{matrix}

Properties of the Laplace Transform

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

Initial- and Final-value Theorems

$x(t)=0$ $t<0$ $x(t)$ no singularities $t=0$ , then

x (0^{+}) = lim_{s \to \infty} s X (s)

分母的阶次高于分子的阶次，不能有常数值。

Final-value theorem

$x(t)=0$ $t<0$ $x(t)$ a finite limit $t\to \infin$ , then

lim_{t \to \infty} x (t) = lim_{s \to 0} s X (s)

Geometric evaluation of FT from pole-zero

CT system function properties

stable $\Leftrightarrow$ $h(t)$ $\Leftrightarrow$ $\displaystyle \int_{-\infin}^{\infin} |h(t)|\mathrm d t<\infin$ $\Leftrightarrow$ $H(s)$ $j\omega$ axis.
Causality $\Rightarrow$ $h(t)$ $\Rightarrow$ $H(s)$ is a right-half plane.
特别注意 causal 和 right-sided 的区别。
$\displaystyle H(s)=\frac{e^{sT}}{s+1}, \operatorname{Re}\{s\} >-1$
$H(s)$ 有理时，系统因果和 ROC 在最右侧极点的右侧等价。
$H(s)$ 有理且因果时，系统稳定等价于虚轴在 ROC 中，等价于所有极点都在左半平面。
Steady State Response.

Block Diagram

Unilateral Laplace Transform

X (s) = \int_{0^{-}}^{\infty} x (t) e^{- s t} d t = UL {x (t)}

$H(s)=\mathcal H(s)$ .

\frac{d x (t)}{d t} \overset{L}{⟷} s X (s) - x (0^{-})

\frac{d^{2} x (t)}{d t^{2}} \overset{L}{⟷} s^{2} X (s) - s x (0^{-}) - x^{'} (0^{-})

\frac{d^{3} x (t)}{d t^{3}} \overset{L}{⟷} s^{3} X (s) - s^{2} x (0^{-}) - s x^{'} (0^{-}) - x^{″} (0^{-})

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

Ch7 - Z Transform

The Bilateral Z transform is defined as

x [n] \overset{Z}{⟷} X (z) = \sum_{n = - \infty}^{\infty} x [n] z^{- n} = Z {x [n]}

The relationship between Z transform and DTFT:

X (z) = \sum_{n = - \infty}^{\infty} x [n] r^{- n} e^{- j ω n} = F {x [n] r^{- n}}

Inverse Z transform

x [n] = \frac{1}{2 π j} \oint_{| z | = r} X (z) z^{n - 1} d z

Other method:

X (z) = \sum_{n = 0}^{\infty} x (n) z^{- n}

Rational form:

Steps:

$z$ $\displaystyle \frac{X(z)}{z}=\frac{N(z)}{D(z)}$ .（Add one pole to the function）
$\displaystyle \frac{X(z)}{z}\cdot z$ .
Inverse Z transform.
$\begin{matrix} \frac{z}{z - a} \overset{Z}{⟷} {\begin{cases} a^{n} u (n), & | z | > | a | \\ - a^{n} u (- n - 1), & | z | < | a | \end{cases} \end{matrix}$

Properties of Z Transform

Property	Signal	Z Transform	ROC
Linearity	$ax_1[n]+bx_2[n]$	$aX_1(z)+bX_2(z)$	$R_1\cap R_2$
Time shifting	$x[n-n_0]$	$z^{-n_0}X(z)$	$R$
$z$ -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	$x_1[n]*x_2[n]$	$X_1(z)X_2(z)$	$R_1\cap R_2$
First Difference	$x[n]-x[n-1]$	$(1-z^{-1})X(z)$	$R\cap \|z\|>0$
Accumulation	$\sum_{k=-\infin}^n x[k]$	$\dfrac{X(z)}{1-z^{-1}}$	$R\cap \|z\|>1$
$z$ -domain	$nx[n]$	$\displaystyle -z\frac{\mathrm d X(z)}{\mathrm d z}$	$R$

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

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	$\displaystyle \sum_{k=0}^n x[k]$	$\displaystyle \frac{1}{1-z^{-1}}\mathcal X(z)$
$z$ -domain	$nx[n]$	$-\displaystyle z\frac{\mathrm d \mathcal X(z)}{\mathrm d z}$

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