# EE 150: Signals and Systems {% file src="/files/-MY\_0GAAoIwEXpT9GgRg" %} Summary from UCB {% endfile %} ## Basic concepts ### Signals and Systems * **Signal**: a function of one or more independent variables (e.g., time and spatial variables); typically contains information about the behavior or nature of some physical phenomena * 信号是一系列独立变量 * **System**: responds to a particular signal input by producing another signal(output) * 系统是其一个信号进， 一个信号出 ![A kind of signal](/files/-MYTqI99Eo8gruKK5rPy) ### Transformation * Time reflection: x(t) ←→ x(−t), x\[n] ←→ x\[−n] * Time scaling: x(t) ←→ x(ct) * Time shift: x(t) ←→ x(t − t0), x\[n] ←→ x\[n − n0] * Usually **do shift then scaling** to avoid complex mathematics ### Even and Odd functions Every signal function is $$x(t) = Even(x) + Odd(t)$$ ,then $$x(-t) = Even(-t) + Odd(-t) = Even(t) - Odd(t)$$ $$Even(t) = \frac12 ( x(t) + x(-t) )$$ $$Odd(t) = \frac12 ( x(t) - x(-t) )$$ ### Periodic * Preodic: $$x(t) = x(t+mT)$$ or $$x\[t] = x\[t+mT]$$ * Fundamental period $$T$$: Smallest positive $$T$$ * 如果是合成的 signal, 其 Fundamental Period 是**最小公倍数** * Aperiodic: Non-preodic ### Eular's Formula $$e^{j \omega0 t} = \cos ( \omega0 t ) + j \cdot \sin (\omega\_0 t)$$ * Fundamental period $$T\_0 = 2 \pi / \mid \omega \_0 \mid$$ * $$A \cos (\omega0 t + \phi) = \frac A2 e^{j \phi} e^{j \omega\_0 t} + \frac A2 e^{-j \phi} e^{-j \omega\_0 t}$$ * 只要基础频率 $$\omega\_0$$ 一样， 指数和三角函数可以互相转化 * $$\begin{equation} \mathrm{e}^{-\mathrm{j} \omega t}=\cos (\omega t)-\mathrm{j} \sin (\omega t) \end{equation}$$ * $$\begin{equation} \cos (\omega t)=\frac{1}{2}\left(\mathrm{e}^{\mathrm{j} \omega t}+\mathrm{e}^{-\mathrm{j} \omega t}\right) \end{equation}$$ * $$\begin{equation} \sin (\omega t)=\frac{1}{2 \mathrm{j}}\left(\mathrm{e}^{\mathrm{j} \omega t}-\mathrm{e}^{-\mathrm{j} \omega t}\right) \end{equation}$$ ### Sin $$x(t) = A \cos (\omega \_0 t + \phi )$$ * unit: $$\omega\_0$$ * radians: $$\phi$$ * phase: $$\omega\_0t+\phi$$ ![Fundamantal Frequency， 角频率越大， 振荡越大](/files/-MYU-FwHLlGMB9A4Kyc6) ### Discrete Time Unit Step and Unit Impulse Sequence * $$\delta\[n]$$ * 只有 $$n=0$$有正值 ![](/files/-MYU2muKy-b7wzftzJXV) * $$u\[n]$$ * 只有 $$n \geq 0$$有正值 ![](/files/-MYU2q4iNayA7553bTrz) $$u\[n]$$相当于 $$\delta\[n]$$的积分 ### 常见信号与周期判断 ### 功率与能量 ![](/files/-MYoefiFvkQW4eFuWgDX) ### Other concepts * Energy and Power of Periodic Signals: 积分与积分后的处理 * 谐振: 同一个角频率的集合 * ## Properties of System ### Memory / Memoryless Output only depends on input **at the same time** ### Invertibility and Inverse System Distinct **input** leads to distinct **output** ### **Causality** {% hint style="info" %} **All memoryless are causal** {% endhint %} Output only depends on input **at the same time** or **before** ### **Stability** **Bounded input** gives **Bounded output** ### **T**ime-Invariance A **time-shift** in the input causes a **same time-shift** in the output $$x\[n] \to y\[n]$$ then $$x\[n-n\_0] \to y\[n-n\_0]$$ Example: ![](/files/-MYZz2RX-9hmhluTxAWo) ### Linearity Additivity and Scaling If $$x\[n] \to y\[n]$$ then $$ax\_1\[n] + bx\_2\[n] \to ay\_1\[n] + b y\_2 \[n]$$ {% hint style="info" %} If linear, zero input gives zero output $$x\[ n]=0 \to y\[n] = 2x\[n] = 0$$ {% endhint %} ## Convolution ### Begin We can construct any signal by discrete function $$y\[n] = x\[k]\delta\[n-k]$$, so that $$y\[n]$$can be valued only at $$k$$ and get $$x\[k]$$. By accumulation, the signal can be **piled up**. This is convolution, defined by $$x\[n] =\Sigma \_{k = - \infty} ^ {\infty} x\[k]\delta\[n-k] = x\[n] \* h\[n]$$, which is called **convolution sum**. Example: ![Origin Signal](/files/-MY_49tLDlPK5tMQk-QQ) ![pile up](/files/-MY_4HFDscC2pmJHTI8s) ### Properties of Convolution * Commutative: $$x(t)∗h(t) =h(t)∗x(t)$$ * 交换律 * Bi-linear: $$(ax\_1(t) +bx\_2(t))∗h(t) =a(x\_1∗h) +b(x\_2∗h),x∗(ah\_1+bh\_2) =a(x∗h\_1) +b(x∗h\_2)$$ * Shift: $$x(t−τ)∗h(t) =x(t)∗h(t−τ)$$ * Identity: δ(t) is the identity signal, $$x∗δ=x=δ∗x$$ * Identity is unique: $$i(t) =i(t)∗δ(t) =δ(t)$$ * Associative: $$x\_1∗(x\_2∗x\_3) = (x\_1∗x\_2)∗x\_3$$ * Smooth derivative: $$y(n)' = x(n)'\*h(n) = x(n)\*h(n)'$$ ### Properties of L.T.I System * Memoryless: $$x\[n] \neq 0$$ when $$n = 0$$ * Invertibility: A system is invertible only if an inverse system exists. * Causality: $$h\[n] = 0$$ when $$n < 0$$ * $$y\[n]=\sum\_{k=0}^{\infty} h\[k] \times\[n-k]$$ * $$\begin{equation} y\[n]=\sum\_{k=-\infty}^{n} x\[k] h\[n-k] \end{equation}$$ * Stability: * $$( \sum\_{k=-\infty}^{\infty}|h\[k]|<\infty )$$ * Convolving $$δ(t)$$ with itself * $$( \delta(t) \* \delta(t)) = \delta(t)$$ ### Calculation Sliding window: 1. Reverse the simpler one $$x(t)$$ 2. Record reversed $$x(t)$$ 's jumping points 3. Slide reversed $$x(t)$$, for each $$g(t)$$, is integral of multiplication $$g(t)=\int \frac{d}{d t} x \* h(t) d t=\ldots$$ $$\begin{equation} g(t)=\frac{d}{d t} \int x \* h(t) d t=\ldots \end{equation}$$ ### 逆变换 ![](/files/-MYogUne2CCUeZiDQ_7V) ## Eigen-function of L.T.I ### Eigen-functions * A signal for which the system's output is just a constant (possibly complex) times the input. * **Eigen Basis**: If an input signal can be decomposed to a weighted sum of eigenfunctions (eigen basis), then the output can be easily found. So goal: Getting an **Eigen basis** of an L.T.I system. ### e^{st} as eigenfunction of L.T.I 1. Consider the input to be $$x(t) = e^{st}$$, then the output is $$y(t)=\int h(\tau) e^{s(t-\tau)} d \tau=e^{s t} \int h(\tau) e^{-s \tau} d \tau$$ 2. $$\begin{equation} H(s)=\int h(\tau) e^{-s \tau} d \tau \end{equation}$$ is just a constant, i.e. eigenvalue for function $$e^{st}$$ ### Orthonormal Basis When $$s$$ purely imaginary $$\begin{equation} j k \omega\_{0} \end{equation}$$, $$\begin{equation} e^{j k \omega\_{0} t} \end{equation}$$ is orthonormal and standard among different . * Definition of inner-product of perioidic functions: $$\begin{equation} \=\frac{1}{T\_{0}} \int\_{T\_{0}} x\_{1}(t) x\_{2}^{\*}(t) d t \end{equation}$$ ## Fourier Analysis ### CT & DT * CT: $$e^{j\omega t }$$ * $$\begin{equation} \mathrm{e}^{\mathrm{j} \omega \mathrm{t}} \longrightarrow \mathrm{H}(\mathrm{j} \omega) \mathrm{e}^{\mathrm{j} \omega \mathrm{t}} \end{equation}$$ * 纯虚数 * DT: $$e^{j \omega n }$$ * $$\begin{equation} \mathrm{e}^{j \omega n} \rightarrow H\left(\mathrm{e}^{j \omega}\right) \mathrm{e}^{j \omega n} \end{equation}$$ ### Periodic signals & Fourier Series Expansion $$x(t)$$ may be expressed as a Fourier series: $$\begin{equation} x(t)=\sum\_{k=-\infty}^{\infty} a\_{k} \cdot e^{j k \omega\_{0} t} \end{equation}$$, $$\begin{equation} x(t) \leftarrow{ }^{F . S .} \rightarrow a\_{k} \end{equation}$$ sum of sinusoids whose frequencies are multiple of ω0, the “fundamental frequency”. Where $$a\_k$$ can be obtained by $$\begin{equation} a\_{k}=\frac{1}{T\_{0}} \int\_{T\_{0}} x(\tau) e^{-j k \omega\_{0} \tau} d \tau \end{equation}$$ * Case 0 is often special! * $$a\_0$$ controls a constant And, $$ \begin{aligned} x(t) &=\sum\_{k=-\infty}^{\infty} a\_{k} e^{j k \omega\_{0} t} \ &=\sum\_{k=-\infty}^{\infty}\left(a\_{k} \cos \left(k \omega\_{0} t\right)+j a\_{k} \sin \left(k \omega\_{0} t\right)\right) \ &=a\_{0}+\sum\_{k>0}\left(\left(a\_{k}+a\_{-k}\right) \cos \left(k \omega\_{0} t\right)+j\left(a\_{k}-a\_{-k}\right) \sin \left(k \omega\_{0} t\right)\right) \end{aligned} $$ #### Odd / Even ![奇偶特性](/files/-MYeKhUViQBZ-BXS3APT) #### Approximation ![](/files/-MYeLE5KRbRPJsOXz6TV) #### Linearity $$ z(t)=\alpha x(t)+\beta y(t) \underset{\longleftrightarrow}{\longleftarrow S}{\longrightarrow} \alpha a\_{k}+\beta b\_{k} $$ #### Time-shift $$ x\left(t-t\_{0}\right) \stackrel{F S}{\longleftrightarrow} e^{-j k \omega\_{0} t\_{0}} a\_{k} $$ #### Time-reverse $$ x(-t) \stackrel{F S}{\longleftarrow} a\_{-k} $$ #### Time-scaling $$ x(\alpha t)=\sum\_{k=-\infty}^{\infty} a\_{k} e^{j k\left(\alpha \omega\_{0}\right) t} $$ #### Multiplication $$ x(t) y(t) \underset{\longleftrightarrow}{\longleftrightarrow S}, h\_{k}=\sum\_{l=-\infty}^{\infty} a\_{l} b\_{k-1} $$ which is **Convolution** #### conjugation & conjugate symmetry ![](/files/-MYjnXuW3wVdg8eCt8xg) #### 7 $$ \frac{d x(t)}{d t} \longleftrightarrow F S, j k \omega\_{0} a\_{k} $$ $$ \int\_{-\infty}^{t} x(\tau) d \tau \underset{\longleftrightarrow}{\text { FS }}, \frac{a\_{k}}{j k \omega\_{0}} $$ #### Parseval's identity $$ \frac{1}{T} \int\_{T}|x(t)|^{2} d t=\sum\_{k=-\infty}^{\infty}\left|a\_{k}\right|^{2} $$ Proof: $$ \begin{aligned} \frac{1}{T} \int\_{T}|x(t)|^{2} d t &=\frac{1}{T} \int\_{T} \sum\_{k\_{1}, k\_{2}} a\_{k\_{1}} a\_{k\_{2}}^{*} e^{j\left(k\_{1}-k\_{2}\right) \omega\_{0} t} d t \ &=\sum\_{k\_{1}, k\_{2}} a\_{k\_{1}} a\_{k\_{2}}^{*} \delta\left\[k\_{1}-k\_{2}\right] \ &=\sum\_{k}\left|a\_{k}\right|^{2} \end{aligned} $$ ## Continuous-Time Fourier Transform (CTFT) ### 变换与逆变换 Fourier series: $$\begin{equation} x(t)=\sum\_{k=-\infty}^{\infty} a\_{k} \mathrm{e}^{j k \omega\_{0} t} \end{equation}$$ Inverse fourier transform: $$x(t)=\frac{1}{2 \pi} \int\_{\infty}^{\infty} X(j \omega) \mathrm{e}^{j \omega t} \mathrm{\~d} \omega$$ ### Fourier Transform Pair Fourier Transform: $$\begin{equation} \mathcal{F}: X(j \omega)=\int\_{-\infty}^{\infty} x(t) e^{-j \omega t} d t \end{equation}$$ * For periodic signals, $$\begin{equation} X(j \omega)=2 \pi \sum\_{-\infty}^{\infty} a\_{k} \delta\left(\omega-k \omega\_{0}\right) \end{equation}$$, i.e. the Fourier Series * $$\begin{equation} X(j \omega) \end{equation}$$is called the "Spectrum" of $$x(t)$$ Inverse F.T. $$\begin{equation} \mathcal{F}^{-1}: x(t)=\frac{1}{2 \pi} \int\_{-\infty}^{\infty} X(j \omega) e^{j \omega t} d \omega \end{equation}$$ ### Dual Porperty $$\begin{equation} \mathcal{F}(\mathcal{F}(x(t)))=2 \pi \cdot x(-t) \end{equation}$$ Means that when we put the $$\begin{equation} X(j \omega) \end{equation}$$ in **time** domain, it will produce $$2\pi$$times $$x(-t)$$ waveform in **freq** domain ### Properties ![](/files/-MYjlT8Z0nqLVQiOtJWx) ![](/files/-MYjpd8BxI0HvzK9HkEZ) ### Normal CT Fourier Pairs ![](/files/-MYjlbCCghmS1h27c3IS) ![](/files/-MZSVf6BQJTjFBfO2J2p) ## REF * --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. 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