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
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System: responds to a particular signal input by producing another signal(output)
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Transformation
Time reflection: x(t) ββ x(βt), x[n] ββ x[βn]
Time scaling: x(t) ββ x(ct)
Time shift: x(t) ββ x(t β t0), x[n] ββ x[n β 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
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Aperiodic: Non-preodic
Eular's Formula
Fundamental period
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\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
unit:
radians:
phase:
Discrete Time Unit Step and Unit Impulse Sequence
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εͺζ ζζ£εΌ
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Other concepts
Energy and Power of Periodic Signals: η§―εδΈη§―εεηε€η
θ°ζ―: εδΈδΈͺθ§ι’ηηιε
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:
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Bi-linear:
Shift:
Identity: Ξ΄(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
\begin{equation} y[n]=\sum_{k=-\infty}^{n} x[k] h[n-k] \end{equation}
Stability:
Convolving with itself
Calculation
Sliding window:
Reverse the simpler one
Record reversed 's jumping points
Slide reversed , for each , is integral of multiplication
\begin{equation} g(t)=\frac{d}{d t} \int x * h(t) d t=\ldots \end{equation}
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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
\begin{equation} H(s)=\int h(\tau) e^{-s \tau} d \tau \end{equation} is just a constant, i.e. eigenvalue for function
Orthonormal Basis
When 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} <x_{1}(t), x_{2}(t)>=\frac{1}{T_{0}} \int_{T_{0}} x_{1}(t) x_{2}^{*}(t) d t \end{equation}
Fourier Analysis
CT & DT
CT:
\begin{equation} \mathrm{e}^{\mathrm{j} \omega \mathrm{t}} \longrightarrow \mathrm{H}(\mathrm{j} \omega) \mathrm{e}^{\mathrm{j} \omega \mathrm{t}} \end{equation}
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DT:
\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
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 Ο0, the βfundamental frequencyβ.
Where 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!
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)
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Fourier series: \begin{equation} x(t)=\sum_{k=-\infty}^{\infty} a_{k} \mathrm{e}^{j k \omega_{0} t} \end{equation}
Inverse fourier transform:
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
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 times waveform in freq domain
Properties
Normal CT Fourier Pairs
REF
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