EE 150: Signals and Systems
ShanghaiTech University - Spring 2021
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Summary from UCB
- 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
- 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
Every signal function is
,then
- 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
- Fundamental period
-
- 只要基础频率一样, 指数和三角函数可以互相转化
-
-
-
- unit:
- radians:
- phase:

Fundamantal Frequency, 角频率越大, 振荡越大
-
- 只有有正值

-
- 只有有正值

相当于
的积分

- 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
-
Output only depends on input at the same time
Distinct input leads to distinct output
All memoryless are causal
Output only depends on input at the same time or before
Bounded input gives Bounded output
A time-shift in the input causes a same time-shift in the output
then
Example:

Additivity and Scaling
If
then
If linear, zero input gives zero output
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:

Origin Signal

pile up
- Commutative:
- 交换律
- Bi-linear:
- Shift:
- Identity: δ(t) is the identity signal,
- Identity is unique:
- Associative:
- Smooth derivative:
- Memoryless:when
- Invertibility: A system is invertible only if an inverse system exists.
- Causality:when
-
-
- Stability:
- Convolvingwith itself
-
Sliding window:
- 1.Reverse the simpler one
- 2.Record reversed's jumping points
- 3.Slide reversed, for each, is integral of multiplication

- 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.
- 1.Consider the input to be, then the output is
- 2.is just a constant, i.e. eigenvalue for function
When
purely imaginary
,
is orthonormal and standard among different .
- Definition of inner-product of perioidic functions:
- CT:
-
- 纯虚数
- DT:
-