Berkeley Notes
  • Introduction
  • EE120
    • Introduction to Signals and Systems
    • The Fourier Series
    • The Fourier Transform
    • Generalized transforms
    • Linear Time-Invariant Systems
    • Feedback Control
    • Sampling
    • Appendix
  • EE123
    • The DFT
    • Spectral Analysis
    • Sampling
    • Filtering
  • EECS126
    • Introduction to Probability
    • Random Variables and their Distributions
    • Concentration
    • Information Theory
    • Random Processes
    • Random Graphs
    • Statistical Inference
    • Estimation
  • EECS127
    • Linear Algebra
    • Fundamentals of Optimization
    • Linear Algebraic Optimization
    • Convex Optimization
    • Duality
  • EE128
    • Introduction to Control
    • Modeling Systems
    • System Performance
    • Design Tools
    • Cascade Compensation
    • State-Space Control
    • Digital Control Systems
    • Cayley-Hamilton
  • EECS225A
    • Hilbert Space Theory
    • Linear Estimation
    • Discrete Time Random Processes
    • Filtering
  • EE222
    • Real Analysis
    • Differential Geometry
    • Nonlinear System Dynamics
    • Stability of Nonlinear Systems
    • Nonlinear Feedback Control
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  • Types of Signals
  • Properties of the Unit Impulse
  • Signal transformations
  • Convolution
  • Systems and their properties
  • Exponential Signals

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  1. EE120

Introduction to Signals and Systems

Types of Signals

Definition 1

A signal is a function of one or more variables

Definition 2

A continuous signal x(t)x(t)x(t) maps R→R\mathbb{R} \rightarrow \mathbb{R}R→R

Definition 3

A discrete signal x[n]x[n]x[n] maps Z→R\mathbb{Z} \rightarrow \mathbb{R}Z→R

Properties of the Unit Impulse

Definition 4

The unit impulse in discrete time is defined as

δ[n]={1if n=00else \delta[n] = \begin{cases} 1 & \text{if } n = 0\\ 0 & \text{else } \end{cases}δ[n]={10​if n=0else ​

  • f[n]δ[n]=f[0]δ[n]f[n]\delta[n] = f[0]\delta[n]f[n]δ[n]=f[0]δ[n]

  • f[t]δ[n−N]=f[N]δ[n−N]f[t]\delta[n-N] = f[N]\delta[n-N]f[t]δ[n−N]=f[N]δ[n−N]

Definition 5

The unit impulse in continuous time is the dirac delta function

δ(t)=limΔ→0δΔ(t)δΔ(t)={1Δ,if 0≤t<Δ0else.\delta(t)=lim_{\Delta\rightarrow 0}\delta_{\Delta}(t) \qquad \delta_{\Delta}(t)=\begin{cases} \frac{1}{\Delta}, & \text{if } 0 \leq t < \Delta \\ 0 & \text{else.} \end{cases}δ(t)=limΔ→0​δΔ​(t)δΔ​(t)={Δ1​,0​if 0≤t<Δelse.​

  • f(t)δ(t)=f(0)δ(t)f(t)\delta(t) = f(0)\delta(t)f(t)δ(t)=f(0)δ(t)

  • f(t)δ(t−τ)=f(τ)δ(t−τ)f(t)\delta(t-\tau) = f(\tau)\delta(t-\tau)f(t)δ(t−τ)=f(τ)δ(t−τ)

  • δ(at)=1∣a∣δ(t)\delta(at) = \frac{1}{|a|}\delta(t)δ(at)=∣a∣1​δ(t)

Definition 6

The unit step is defined as

u[n]={1if n≥00else.u[n] = \begin{cases} 1 & \text{if } n \geq 0\\ 0 & \text{else.} \end{cases}u[n]={10​if n≥0else.​

Signal transformations

Signals can be transformed by modifying the variable.

  • x(t−τ)x(t - \tau)x(t−τ): Shift a signal right by τ\tauτ steps.

  • x(−t)x(-t)x(−t): Rotate a signal about the t=0t=0t=0

  • x(kt)x(kt)x(kt): Stretch a signal by a factor of kkk

These operations can be combined to give more complex transformations. For example, y(t)=x(τ−t)=x(−(t−τ))y(t) = x(\tau - t) = x(-(t-\tau))y(t)=x(τ−t)=x(−(t−τ)) flips xxx and shifts it right by τ\tauτ timesteps. This is equivalent to shifting xxx left by τ\tauτ timesteps and then flipping it.

Convolution

Definition 7

The convolution of two signals in discrete time

(x∗h)[n]=∑k=−∞∞x[k]h[n−k](x*h)[n] = \sum_{k=-\infty}^{\infty}{x[k]h[n-k]}(x∗h)[n]=∑k=−∞∞​x[k]h[n−k]

Definition 8

The convolution of two signals in continuous time

(x∗h)(t)=∫−∞∞x(τ)h(t−τ)dτ(x*h)(t) = \int_{-\infty}^{\infty}{x(\tau)h(t-\tau)d\tau}(x∗h)(t)=∫−∞∞​x(τ)h(t−τ)dτ

While written in discrete time, these properties apply in continuous time as well.

  • (x∗δ)[n]=x[n](x*\delta)[n] = x[n](x∗δ)[n]=x[n]

  • x[n]∗δ[n−N]=x[n−N]x[n]*\delta[n-N]=x[n-N]x[n]∗δ[n−N]=x[n−N]

  • (x∗h)[n]=(h∗x)[n](x*h)[n] = (h*x)[n](x∗h)[n]=(h∗x)[n]

  • x∗(h1+h2)=x∗h1+x∗h2x * (h_1 + h_2) = x*h_1 + x*h_2x∗(h1​+h2​)=x∗h1​+x∗h2​

  • x∗(h1∗h2)=(x∗h1)∗h2x * (h_1 * h_2) = (x * h_1) * h_2x∗(h1​∗h2​)=(x∗h1​)∗h2​

Systems and their properties

Definition 9

A system is a process by which input signals are transformed to output signals

Definition 10

A memoryless system has output which is only determined by the input's present value

Definition 11

A causal system has output which only depends on input at present or past times

Definition 12

A stable system produces bounded output when given a bounded input. By extension, this means an unstable system is when ∃\exists∃a bounded input that makes the output unbounded.

Definition 13

A system is time-invariant if the original input x(t)x(t)x(t) is transformed to y(t)y(t)y(t), then x(t−τ)x(t-\tau)x(t−τ) is transformed to y(t−τ)y(t-\tau)y(t−τ)

Definition 14

A system f(x)f(x)f(x) is linear if and only if for the signals y1(t)=f(x1(t)),y2(t)=f(x2(t))y_1(t) = f(x_1(t)), y_2(t) = f(x_2(t))y1​(t)=f(x1​(t)),y2​(t)=f(x2​(t)), then scaling (f(ax1(t))=ay(t)f(a x_1(t)) = a y(t)f(ax1​(t))=ay(t) and superposition (f(x1(t)+x2(t))=y1(t)+y2(t)f(x_1(t) + x_2(t)) = y_1(t) + y_2(t)f(x1​(t)+x2​(t))=y1​(t)+y2​(t)) hold.

Notice: The above conditions on linearity require that x(0)=0x(0) = 0x(0)=0 because if a=0a = 0a=0, then we need y(0)=0y(0) = 0y(0)=0 for scaling to be satisfied

Definition 15

The impulse response of a system f[x]f[x]f[x] is h[n]=f[δ[n]]h[n] = f[\delta[n]]h[n]=f[δ[n]], which is how it response to an impulse input.

Definition 16

A system has a Finite Impulse Response (FIR) if h[n]h[n]h[n]decays to zero in a finite amount of time

Definition 17

A system has an Infinite Impulse Response (IIR) if h[n]h[n]h[n]does not decay to zero in a finite amount of time

Exponential Signals

Exponential signals are important because they can succinctly represent complicated signals using complex numbers. This makes analyzing them much easier.

x(t)=est,x[n]=zn(s,z∈C)x(t) = e^{st}, x[n] = z^n (s, z \in \mathbb{C})x(t)=est,x[n]=zn(s,z∈C)

Definition 18

The frequency response of a system is how a system responds to a purely oscillatory signal

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