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
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]
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 x ( t ) = E v e n ( x ) + O d d ( t ) x(t) = Even(x) + Odd(t) x ( t ) = E v e n ( x ) + O dd ( t )
x ( β t ) = E v e n ( β t ) + O d d ( β t ) = E v e n ( t ) β O d d ( t ) x(-t) = Even(-t) + Odd(-t) = Even(t) - Odd(t) x ( β t ) = E v e n ( β t ) + O dd ( β t ) = E v e n ( t ) β O dd ( t )
E v e n ( t ) = 1 2 ( x ( t ) + x ( β t ) ) Even(t) = \frac12 ( x(t) + x(-t) ) E v e n ( t ) = 2 1 β ( x ( t ) + x ( β t ))
O d d ( t ) = 1 2 ( x ( t ) β x ( β t ) ) Odd(t) = \frac12 ( x(t) - x(-t) ) O dd ( t ) = 2 1 β ( x ( t ) β x ( β t ))
Periodic
Preodic: x ( t ) = x ( t + m T ) x(t) = x(t+mT) x ( t ) = x ( t + m T ) x [ t ] = x [ t + m T ] x[t] = x[t+mT] x [ t ] = x [ t + m T ]
Fundamental period T T T T T T
ε¦ζζ―εζη signal, ε
Ά Fundamental Period ζ―ζε°ε
¬εζ°
Eular's Formula
e j Ο 0 t = cos β‘ ( Ο 0 t ) + j β
sin β‘ ( Ο 0 t ) e^{j \omega0 t} = \cos ( \omega0 t ) + j \cdot \sin (\omega_0 t) e jΟ 0 t = cos ( Ο 0 t ) + j β
sin ( Ο 0 β t )
Fundamental period T 0 = 2 Ο / β£ Ο 0 β£ T_0 = 2 \pi / \mid \omega _0 \mid T 0 β = 2 Ο / β£ Ο 0 β β£
A cos β‘ ( Ο 0 t + Ο ) = A 2 e j Ο e j Ο 0 t + A 2 e β j Ο e β j Ο 0 t A \cos (\omega0 t + \phi) = \frac A2 e^{j \phi} e^{j \omega_0 t} + \frac A2 e^{-j \phi} e^{-j \omega_0 t} A cos ( Ο 0 t + Ο ) = 2 A β e j Ο e j Ο 0 β t + 2 A β e β j Ο e β j Ο 0 β t
εͺθ¦εΊη‘ι’η Ο 0 \omega_0 Ο 0 β
\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
x ( t ) = A cos β‘ ( Ο 0 t + Ο ) x(t) = A \cos (\omega _0 t + \phi ) x ( t ) = A cos ( Ο 0 β t + Ο )
phase: Ο 0 t + Ο \omega_0t+\phi Ο 0 β t + Ο
Discrete Time Unit Step and Unit Impulse Sequence
Ξ΄ [ n ] \delta[n] Ξ΄ [ n ]
εͺζ n = 0 n=0 n = 0
u [ n ] u[n] u [ n ]
εͺζ n β₯ 0 n \geq 0 n β₯ 0
u [ n ] u[n] u [ n ] Ξ΄ [ n ] \delta[n] Ξ΄ [ n ]
εΈΈθ§δΏ‘ε·δΈε¨ζε€ζ
εηδΈθ½ι
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
T ime-Invariance
A time-shift in the input causes a same time-shift in the output
x [ n ] β y [ n ] x[n] \to y[n] x [ n ] β y [ n ] x [ n β n 0 ] β y [ n β n 0 ] x[n-n_0] \to y[n-n_0] x [ n β n 0 β ] β y [ n β n 0 β ]
Example:
Linearity
Additivity and Scaling
If x [ n ] β y [ n ] x[n] \to y[n] x [ n ] β y [ n ] a x 1 [ n ] + b x 2 [ n ] β a y 1 [ n ] + b y 2 [ n ] ax_1[n] + bx_2[n] \to ay_1[n] + b y_2 [n] a x 1 β [ n ] + b x 2 β [ n ] β a y 1 β [ n ] + b y 2 β [ n ]
If linear, zero input gives zero output
x [ n ] = 0 β y [ n ] = 2 x [ n ] = 0 x[ n]=0 \to y[n] = 2x[n] = 0 x [ n ] = 0 β y [ n ] = 2 x [ n ] = 0
Convolution
Begin
We can construct any signal by discrete function y [ n ] = x [ k ] Ξ΄ [ n β k ] y[n] = x[k]\delta[n-k] y [ n ] = x [ k ] Ξ΄ [ n β k ] y [ n ] y[n] y [ n ] k k k x [ k ] x[k] x [ k ] piled up . This is convolution, defined by x [ n ] = Ξ£ k = β β β x [ k ] Ξ΄ [ n β k ] = x [ n ] β h [ n ] x[n] =\Sigma _{k = - \infty} ^ {\infty} x[k]\delta[n-k] = x[n] * h[n] x [ n ] = Ξ£ k = β β β β x [ k ] Ξ΄ [ n β k ] = x [ n ] β h [ n ] convolution sum .
Example:
Properties of Convolution
Commutative: x ( t ) β h ( t ) = h ( t ) β x ( t ) x(t)βh(t) =h(t)βx(t) x ( t ) β h ( t ) = h ( t ) β x ( t )
Bi-linear: ( a x 1 ( t ) + b x 2 ( t ) ) β h ( t ) = a ( x 1 β h ) + b ( x 2 β h ) , x β ( a h 1 + b h 2 ) = a ( x β h 1 ) + b ( x β h 2 ) (ax_1(t) +bx_2(t))βh(t) =a(x_1βh) +b(x_2βh),xβ(ah_1+bh_2) =a(xβh_1) +b(xβh_2) ( a x 1 β ( t ) + b x 2 β ( t )) β h ( t ) = a ( x 1 β β h ) + b ( x 2 β β h ) , x β ( a h 1 β + b h 2 β ) = a ( x β h 1 β ) + b ( x β h 2 β )
Shift: x ( t β Ο ) β h ( t ) = x ( t ) β h ( t β Ο ) x(tβΟ)βh(t) =x(t)βh(tβΟ) x ( t β Ο ) β h ( t ) = x ( t ) β h ( t β Ο )
Identity: Ξ΄(t) is the identity signal, x β Ξ΄ = x = Ξ΄ β x xβΞ΄=x=Ξ΄βx x β Ξ΄ = x = Ξ΄ β x
Identity is unique: i ( t ) = i ( t ) β Ξ΄ ( t ) = Ξ΄ ( t ) i(t) =i(t)βΞ΄(t) =Ξ΄(t) i ( t ) = i ( t ) β Ξ΄ ( t ) = Ξ΄ ( t )
Associative: x 1 β ( x 2 β x 3 ) = ( x 1 β x 2 ) β x 3 x_1β(x_2βx_3) = (x_1βx_2)βx_3 x 1 β β ( x 2 β β x 3 β ) = ( x 1 β β x 2 β ) β x 3 β
Smooth derivative: y ( n ) β² = x ( n ) β² β h ( n ) = x ( n ) β h ( n ) β² y(n)' = x(n)'*h(n) = x(n)*h(n)' y ( n ) β² = x ( n ) β² β h ( n ) = x ( n ) β h ( n ) β²
Properties of L.T.I System
Memoryless: x [ n ] β 0 x[n] \neq 0 x [ n ] ξ = 0 n = 0 n = 0 n = 0
Invertibility: A system is invertible only if an inverse system exists.
Causality: h [ n ] = 0 h[n] = 0 h [ n ] = 0 n < 0 n < 0 n < 0
y [ n ] = β k = 0 β h [ k ] Γ [ n β k ] y[n]=\sum_{k=0}^{\infty} h[k] \times[n-k] y [ n ] = β k = 0 β β h [ k ] Γ [ n β k ]
\begin{equation} y[n]=\sum_{k=-\infty}^{n} x[k] h[n-k] \end{equation}
Stability:
( β k = β β β β£ h [ k ] β£ < β ) ( \sum_{k=-\infty}^{\infty}|h[k]|<\infty ) ( β k = β β β β β£ h [ k ] β£ < β )
Convolving Ξ΄ ( t ) Ξ΄(t) Ξ΄ ( t )
( Ξ΄ ( t ) β Ξ΄ ( t ) ) = Ξ΄ ( t ) ( \delta(t) * \delta(t)) = \delta(t) ( Ξ΄ ( t ) β Ξ΄ ( t )) = Ξ΄ ( t )
Calculation
Sliding window:
Reverse the simpler one x ( t ) x(t) x ( t )
Record reversed x ( t ) x(t) x ( t )
Slide reversed x ( t ) x(t) x ( t ) g ( t ) g(t) g ( t )
g ( t ) = β« d d t x β h ( t ) d t = β¦ g(t)=\int \frac{d}{d t} x * h(t) d t=\ldots g ( t ) = β« d t d β x β h ( t ) d t = β¦
\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
Consider the input to be x ( t ) = e s t x(t) = e^{st} x ( t ) = e s t y ( t ) = β« h ( Ο ) e s ( t β Ο ) d Ο = e s t β« h ( Ο ) e β s Ο d Ο y(t)=\int h(\tau) e^{s(t-\tau)} d \tau=e^{s t} \int h(\tau) e^{-s \tau} d \tau y ( t ) = β« h ( Ο ) e s ( t β Ο ) d Ο = e s t β« h ( Ο ) e β s Ο d Ο
\begin{equation} H(s)=\int h(\tau) e^{-s \tau} d \tau \end{equation} is just a constant, i.e. eigenvalue for function e s t e^{st} e s t
Orthonormal Basis
When s s s \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: e j Ο t e^{j\omega t } e jΟ t
\begin{equation} \mathrm{e}^{\mathrm{j} \omega \mathrm{t}} \longrightarrow \mathrm{H}(\mathrm{j} \omega) \mathrm{e}^{\mathrm{j} \omega \mathrm{t}} \end{equation}
DT: e j Ο n e^{j \omega n } e jΟn
\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
x ( t ) x(t) x ( t )
\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 a k a_k a k β
\begin{equation} a_{k}=\frac{1}{T_{0}} \int_{T_{0}} x(\tau) e^{-j k \omega_{0} \tau} d \tau \end{equation}
a 0 a_0 a 0 β
And,
x ( t ) = β k = β β β a k e j k Ο 0 t = β k = β β β ( a k cos β‘ ( k Ο 0 t ) + j a k sin β‘ ( k Ο 0 t ) ) = a 0 + β k > 0 ( ( a k + a β k ) cos β‘ ( k Ο 0 t ) + j ( a k β a β k ) sin β‘ ( k Ο 0 t ) ) \begin{aligned} x(t) &=\sum_{k=-\infty}^{\infty} a_{k} e^{j k \omega_{0} t} \\ &=\sum_{k=-\infty}^{\infty}\left(a_{k} \cos \left(k \omega_{0} t\right)+j a_{k} \sin \left(k \omega_{0} t\right)\right) \\ &=a_{0}+\sum_{k>0}\left(\left(a_{k}+a_{-k}\right) \cos \left(k \omega_{0} t\right)+j\left(a_{k}-a_{-k}\right) \sin \left(k \omega_{0} t\right)\right) \end{aligned} x ( t ) β = k = β β β β β a k β e jk Ο 0 β t = k = β β β β β ( a k β cos ( k Ο 0 β t ) + j a k β sin ( k Ο 0 β t ) ) = a 0 β + k > 0 β β ( ( a k β + a β k β ) cos ( k Ο 0 β t ) + j ( a k β β a β k β ) sin ( k Ο 0 β t ) ) β Odd / Even
Approximation
Linearity
z ( t ) = Ξ± x ( t ) + Ξ² y ( t ) β΅ S β· βΆ Ξ± a k + Ξ² b k z(t)=\alpha x(t)+\beta y(t) \underset{\longleftrightarrow}{\longleftarrow S}{\longrightarrow} \alpha a_{k}+\beta b_{k} z ( t ) = Ξ±x ( t ) + Ξ² y ( t ) β· β΅ S β βΆ Ξ± a k β + Ξ² b k β Time-shift
x ( t β t 0 ) β· F S e β j k Ο 0 t 0 a k x\left(t-t_{0}\right) \stackrel{F S}{\longleftrightarrow} e^{-j k \omega_{0} t_{0}} a_{k} x ( t β t 0 β ) β· FS β e β jk Ο 0 β t 0 β a k β Time-reverse
x ( β t ) β΅ F S a β k x(-t) \stackrel{F S}{\longleftarrow} a_{-k} x ( β t ) β΅ FS β a β k β Time-scaling
x ( Ξ± t ) = β k = β β β a k e j k ( Ξ± Ο 0 ) t x(\alpha t)=\sum_{k=-\infty}^{\infty} a_{k} e^{j k\left(\alpha \omega_{0}\right) t} x ( Ξ± t ) = k = β β β β β a k β e jk ( Ξ± Ο 0 β ) t Multiplication
x ( t ) y ( t ) β· S β· , h k = β l = β β β a l b k β 1 x(t) y(t) \underset{\longleftrightarrow}{\longleftrightarrow S}, h_{k}=\sum_{l=-\infty}^{\infty} a_{l} b_{k-1} x ( t ) y ( t ) β· β· S β , h k β = l = β β β β β a l β b k β 1 β which is Convolution
conjugation & conjugate symmetry
7
d x ( t ) d t β· F S , j k Ο 0 a k \frac{d x(t)}{d t} \longleftrightarrow F S, j k \omega_{0} a_{k} d t d x ( t ) β β· FS , jk Ο 0 β a k β
β« β β t x ( Ο ) d Ο Β FSΒ β· , a k j k Ο 0 \int_{-\infty}^{t} x(\tau) d \tau \underset{\longleftrightarrow}{\text { FS }}, \frac{a_{k}}{j k \omega_{0}} β« β β t β x ( Ο ) d Ο β· Β FSΒ β , jk Ο 0 β a k β β Parseval's identity
1 T β« T β£ x ( t ) β£ 2 d t = β k = β β β β£ a k β£ 2 \frac{1}{T} \int_{T}|x(t)|^{2} d t=\sum_{k=-\infty}^{\infty}\left|a_{k}\right|^{2} T 1 β β« T β β£ x ( t ) β£ 2 d t = k = β β β β β β£ a k β β£ 2 Proof:
1 T β« T β£ x ( t ) β£ 2 d t = 1 T β« T β k 1 , k 2 a k 1 a k 2 β e j ( k 1 β k 2 ) Ο 0 t d t = β k 1 , k 2 a k 1 a k 2 β Ξ΄ [ k 1 β k 2 ] = β k β£ a k β£ 2 \begin{aligned} \frac{1}{T} \int_{T}|x(t)|^{2} d t &=\frac{1}{T} \int_{T} \sum_{k_{1}, k_{2}} a_{k_{1}} a_{k_{2}}^{*} e^{j\left(k_{1}-k_{2}\right) \omega_{0} t} d t \\ &=\sum_{k_{1}, k_{2}} a_{k_{1}} a_{k_{2}}^{*} \delta\left[k_{1}-k_{2}\right] \\ &=\sum_{k}\left|a_{k}\right|^{2} \end{aligned} T 1 β β« T β β£ x ( t ) β£ 2 d t β = T 1 β β« T β k 1 β , k 2 β β β a k 1 β β a k 2 β β β e j ( k 1 β β k 2 β ) Ο 0 β t d t = k 1 β , k 2 β β β a k 1 β β a k 2 β β β Ξ΄ [ k 1 β β k 2 β ] = k β β β£ a k β β£ 2 β
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}
Inverse fourier transform: x ( t ) = 1 2 Ο β« β β X ( j Ο ) e j Ο t Β d Ο x(t)=\frac{1}{2 \pi} \int_{\infty}^{\infty} X(j \omega) \mathrm{e}^{j \omega t} \mathrm{~d} \omega x ( t ) = 2 Ο 1 β β« β β β X ( jΟ ) e jΟ t Β d Ο
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 x ( t ) x(t) x ( t )
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 2 Ο 2\pi 2 Ο x ( β t ) x(-t) x ( β t ) freq domain
Properties
Normal CT Fourier Pairs
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