EE 150: Signals and Systems
ShanghaiTech University - Spring 2021
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
- 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
- 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
Fundamantal Frequency， 角频率越大， 振荡越大
- 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
Additivity and Scaling
If linear, zero input gives zero output
We can construct any signal by discrete function
, so that
can be valued only at
. By accumulation, the signal can be piled up. This is convolution, defined by
, which is called convolution sum.
- Identity: δ(t) is the identity signal,
- Identity is unique:
- Smooth derivative:
- Invertibility: A system is invertible only if an inverse system exists.
- Convolvingwith itself
- 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
is orthonormal and standard among different .
- Definition of inner-product of perioidic functions:
may be expressed as a Fourier series:
sum of sinusoids whose frequencies are multiple of ω0, the “fundamental frequency”.
can be obtained by
- Case 0 is often special!
- controls a constant
which is Convolution
Inverse fourier transform:
- For periodic signals,, i.e. the Fourier Series
- is called the "Spectrum" of
Means that when we put the
in time domain, it will produce
waveform in freq domain