EE 150: Signals and Systems
ShanghaiTech University - Spring 2021
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
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 ,then
Periodic
Preodic: or
Fundamental period : Smallest positive
如果是合成的 signal, 其 Fundamental Period 是最小公倍数
Aperiodic: Non-preodic
Eular's Formula
Fundamental period
只要基础频率 一样, 指数和三角函数可以互相转化
Sin
unit:
radians:
phase:
Discrete Time Unit Step and Unit Impulse Sequence
只有 有正值
只有 有正值
相当于 的积分
常见信号与周期判断
功率与能量
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
All memoryless are causal
Output only depends on input at the same time or before
Stability
Bounded input gives Bounded output
Time-Invariance
A time-shift in the input causes a same time-shift in the output
then
Example:
Linearity
Additivity and Scaling
If then
If linear, zero input gives zero output
Convolution
Begin
We can construct any signal by discrete function , so that can be valued only at and get . By accumulation, the signal can be piled up. This is convolution, defined by , which is called convolution sum.
Example:
Properties of Convolution
Commutative:
交换律
Bi-linear:
Shift:
Identity: δ(t) is the identity signal,
Identity is unique:
Associative:
Smooth derivative:
Properties of L.T.I System
Memoryless: when
Invertibility: A system is invertible only if an inverse system exists.
Causality: when
Stability:
Convolving with itself
Calculation
Sliding window:
Reverse the simpler one
Record reversed 's jumping points
Slide reversed , for each , is integral of multiplication
逆变换
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
Consider the input to be , then the output is
is just a constant, i.e. eigenvalue for function
Orthonormal Basis
When purely imaginary , is orthonormal and standard among different .
Definition of inner-product of perioidic functions:
Fourier Analysis
CT & DT
CT:
纯虚数
DT:
Periodic signals & Fourier Series Expansion
may be expressed as a Fourier series:
,
sum of sinusoids whose frequencies are multiple of ω0, the “fundamental frequency”.
Where can be obtained by
Case 0 is often special!
controls a constant
And,
Odd / Even
Approximation
Linearity
Time-shift
Time-reverse
Time-scaling
Multiplication
which is Convolution
conjugation & conjugate symmetry
7
Parseval's identity
Proof:
Continuous-Time Fourier Transform (CTFT)
变换与逆变换
Fourier series:
Inverse fourier transform:
Fourier Transform Pair
Fourier Transform:
For periodic signals, , i.e. the Fourier Series
is called the "Spectrum" of
Inverse F.T.
Dual Porperty
Means that when we put the in time domain, it will produce times waveform in freq domain
Properties
Normal CT Fourier Pairs
REF
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