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
δΏ‘ε·ζ―δΈη³»εη¬η«ει
System: responds to a particular signal input by producing another signal(output)
η³»η»ζ―ε ΆδΈδΈͺδΏ‘ε·θΏοΌ δΈδΈͺδΏ‘ε·εΊ
Transformation
Time reflection: x(t) ββ x(βt), x[n] ββ x[βn]
\begin{equation} g(t)=\frac{d}{d t} \int x * h(t) d t=\ldots \end{equation}
ιεζ’
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
Orthonormal Basis
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}
\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.
\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!
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)
εζ’δΈιεζ’
Fourier series: \begin{equation} x(t)=\sum_{k=-\infty}^{\infty} a_{k} \mathrm{e}^{j k \omega_{0} t} \end{equation}
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
If x[n]βy[n] then ax1β[n]+bx2β[n]βay1β[n]+by2β[n]
x[n]=0βy[n]=2x[n]=0
We can construct any signal by discrete function y[n]=x[k]Ξ΄[nβk], so that y[n]can be valued only at k and get x[k]. By accumulation, the signal can be piled up. This is convolution, defined by x[n]=Ξ£k=ββββx[k]Ξ΄[nβk]=x[n]βh[n], which is called convolution sum.
Slide reversed x(t), for each g(t), is integral of multiplication
g(t)=β«dtdβxβh(t)dt=β¦
Consider the input to be x(t)=est, then the output is y(t)=β«h(Ο)es(tβΟ)dΟ=estβ«h(Ο)eβsΟdΟ
\begin{equation} H(s)=\int h(\tau) e^{-s \tau} d \tau \end{equation} is just a constant, i.e. eigenvalue for function est
When s 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 .