Signals and Systems¶
terminology
linear
system which follows principle of
superpositionlaw of additivity
\(x_1(t)+x_2(t) \longrightarrow \boxed{system} \longrightarrow y_1(t)+y_2(t)\)
law of homogenity
\(kx(t) \longrightarrow \boxed{system} \longrightarrow ky(t)\)
time invariant
\(x(t-t_o) \longrightarrow \boxed{system} \longrightarrow y(t-t_o)\)
causal: output of system is independent of future values of inputanti-causal: output of system depends on only future values of inputstatic: output of system depends on only present values of inputdynamic: output of system depends on past or future values of input at any instant\(\delta\)
\(\displaystyle\int _{-\infty}^{\infty}x(t) \frac{d^n}{d t^n}\delta(t-t_1) dt=(-1)^n\frac{d^n}{d t^n}x(t) \Big| _{t=t_1}\) only if \(x(t) \Big|_{t=\infty} = finite\)
\(x(t)\delta (t-t_1) =x(t_1) \delta (t-t_1)\)
average
\(\text{average} =\displaystyle\frac{1}{T_o} \int_{\frac{-T_o}{2}}^{\frac{T_o}{2}} x(t) dt\)
energy
\(\rm{E=\displaystyle \int_{-\infty}^{\infty}|x(t)|^2dt}\)
\(\rm{E=\displaystyle \sum_{-\infty}^{\infty}|x[n]|^2}\)
power
\(power =\frac{1}{T_o} \displaystyle\int_{\frac{-T_o}{2}}^{\frac{T_o}{2}}\mid x(t)\mid^2 dt = MSV(x(t))=\overline {x^2(t)}\)
\(power = \begin{cases} \rm{P=\displaystyle \dfrac{1}{N} \sum_{n=N}|x[n]|^2} &\text{if periodic} \\ \rm{P=\lim \limits_{x\rightarrow \infty}\displaystyle \dfrac{1}{2N+1} \sum_{n=-N}^{N}|x[n]|^2} &\text{if non-periodic} \end{cases}\)
half wave symmetry
\(x(t)=-x(t\pm \frac{T}{2})\)
conjugate symmetric
\(x(t)=x^*(-t)\)
conjugate anti-symmetric
\(x(t)=-x^*(-t)\)
\(x(t)=x_{CS}(t)+x_{CAS}(t)\)
\(|A+B|^2 = |A|^2+|B|^2+AB^*+A^*B\)
parseval’s theorem
\(\displaystyle \boxed{E_{x(t)}=\frac{1}{2\pi}\int_{-\infty}^{+\infty}|X(j\omega)|^2d\omega = \int_{-\infty}^{+\infty}|X(f)|^2df = \int_{-\infty}^{+\infty}|x(t)|^2dt}\)
\(\displaystyle \boxed{E_{x[n]}=\sum \limits_{n=-\infty}^{+\infty}|x[n]|^2 =\frac{1}{2\pi}\int_{-\pi}^{+\pi}|X(e^{j\omega})|^2d\omega }\)
energy of standard signals
\(-k\cdot x(-at+b) \rightarrow \dfrac{k^2 E_x}{|a|}\)
rectangular pulse
\(energy=A^2\times(duration)\)
triangular pulse
\(energy=\frac{A^2}{3}\times(duration)\)
convolution
\(\displaystyle h(t)*x(t) = \int_{-\infty}^{\infty}x(\tau)\cdot h(t-\tau)d\tau\)
time delay
\(x_1(t-t_1)*x_2(t-t_2)=y(t-t_1-t_2)\)
time scaling
\(x_1(at)*x_2(at)=\dfrac{1}{|a|}y(at)\)
\(x(t)*u(t) = \int _{-\infty} ^t x(\tau) d\tau\)
\(x(t)*\delta(t-t_1) = x(t-t_1)\)
area
\(x_1(t)*x_2(t) =y(t)\)
\(A_{x_1(t)} \times A_{x_2(t)} = A_{y(t)}\)
sampling
\(M_s(\omega) = f_s \sum \limits_{n=-\infty}^{\infty} M(\omega-n\omega_s)\)
\(M_s(f) = f_s \sum \limits_{n=-\infty}^{\infty} M(f-nf_s)\)
no aliasing
\(f_s \ge 2f_m\)
filter cutoff
\(f_m \le f_c \le f_s-f_m\)
nyquist
\(NR=f_{NY} = 2 f_{max} = (f_s) _{min}\)
\(NI =T_{NY}=\dfrac{1}{f_{NY}}\)
rectangular
\(M_s(f) = \sum \limits_{n=-\infty}^{\infty}\frac{2A}{a}sinc(\frac{2n}{a}) M(f-nf_s)\)
bandpass sampling theorem
\(f_s \ge \dfrac{2f_H}{k}\)
\(k=\Big[\dfrac{f_H}{f_H-f_L}\Big]\) \([\cdot]\) → GIF
fourier series¶
dirichlet conditions
signal must have finite number of maxima and minima over the range of time period.
signal must have a finite number of discontinuities over the range of time period.
signal must be absolutely integrable over a period.
\(\color{lightgreen} x(t)=a_o+\sum_{n=1} ^{\infty}a_n cos(n\omega_ot)+\sum_{n=1} ^{\infty}b_n sin(n\omega_ot)\)
\(\displaystyle a_o=\frac{1}{T_o}\int_{T_o} x(t) dt\)
\(\displaystyle a_n=\frac{2}{T_o}\int_{T_o} x(t) cos(n\omega_ot) dt\)
\(\displaystyle b_n=\frac{2}{T_o}\int_{T_o} x(t) sin(n\omega_ot) dt\)
\(x(t)=\sum\limits_{n=-\infty}^{\infty} C_n e^{jn\omega_ot}\)
\(\displaystyle C_n=\frac{1}{T_o} \int _{T_o} x(t) e^{-jn\omega_o t}dt\)
\(\displaystyle C^{*}_{-n}=\frac{1}{T_o} \int _{T_o} x^*(t) e^{-jn\omega_o t}dt\)
properties
conjugation
\(x^*(t) \longleftrightarrow C_{-n} ^*\)
time reversal
\(x(-t) \longleftrightarrow C_{-n}\)
time scaling
\(x(at) \longleftrightarrow C_{n}\) but period = \(\dfrac{T_o}{a}\)
time shifting
\(x(t+t_o) \longleftrightarrow e^{+jn\omega_o t_o}C_{n}\)
frequency shifting
\(x(t+t_o)e^{\pm jm\omega_o t_o} \longleftrightarrow C_{n\mp m}\)
convolution in time
\(x_1(t)*x_2(t) \longleftrightarrow T_o(C_{1n}\cdot C_{2n} )\)
multiplication in time
\(x_1(t)\cdot x_2(t) \longleftrightarrow C_{1n}* C_{2n}\)
differentiation in time
\(\dfrac{d^kx(t)}{dt^k} \longleftrightarrow (jn\omega_o)^kC_{n}\)
integration in time
\(\displaystyle \int_{-\infty}^{t} x(\tau) d\tau \longleftrightarrow \dfrac{C_{n}}{jn\omega_o}\)
transforms¶
laplace transform
bilateral
\(\displaystyle F(s)=\int_{-\infty}^{+\infty}f(t)e^{-st}dt\)
unilateral
\(\displaystyle F(s)=\int_{0}^{\infty}f(t)e^{-st}dt\)
properties
conjugation
\(f^*(t)\leftrightarrow F^*(s^*)\)
time reversal
\(f(-t)\leftrightarrow F(-s)\)
time scaling
\(f(at)\leftrightarrow\frac{1}{\mid a \mid}F(\frac{s}{a})\)
time shifting
\(f(t\pm t_0)\leftrightarrow F(s)e^{\pm st_0}\)
frequency shifting
\(e^{\pm s_0 t}f(t) \leftrightarrow F(s\mp s_0)\)
convolution in time
\(f_1(t)*f_2(t)\leftrightarrow F_1(s)\times F_2(s)\)
convolution in frequency
\(f_1(t)\times f_2(t)\leftrightarrow \frac{1}{2\pi j}\lbrack F_1(s)*F_2(s) \rbrack\)
differentiation in time
bilateral
\(\dfrac{d^n}{dt^n}f(t) \leftrightarrow s^nF(s)\)
unilateral
\(\dfrac{d^n}{dt^n}f(t) \leftrightarrow s^nF(s)- s^{n-1}f(0^-)-s^{n-2}f'(0^-)-...\)
integration in time
bilateral
\(\displaystyle \int_{-\infty}^tf(\tau)d\tau \leftrightarrow \frac{F(s)}{s}\)
unilateral
\(\displaystyle\int_{-\infty}^tf(\tau)d\tau \leftrightarrow \frac{F(s)}{s} + \frac{\int_{-\infty}^{0^-}f(\tau)d\tau }{s}\)
differentiation in frequency
\(\displaystyle t^nf(t) \leftrightarrow (-1)^{n}\frac{d^n}{ds^n}F(s)\)
integration in frequency
\(\displaystyle\frac{f(t)}{t} \leftrightarrow \int_{s}^\infty F(s)d\tau\)
initial value theorem
\(f(0^+)=\underset{s\rightarrow\infty}{lim}\space sF(s)\)
note : initial value theorem is applied only on remainder function
final value theorem
\(f(\infty)=\underset{s\rightarrow0}{lim}\space sF(s)\)
standard results
\(\delta(t)\)
\(\delta \leftrightarrow 1\)
\(u(t)\)
\(u(t)\leftrightarrow \frac{1} {s}\)
\(t^nu(t)\)
\(t^nu(t) \longleftrightarrow \dfrac{n!}{s^{n+1}}\)
\(cos(\omega_o t)u(t)\)
\(cos(\omega_o t)u(t)\leftrightarrow \dfrac{s} {s^2+\omega_o^2}\)
\(sin(\omega_o t)u(t)\)
\(sin(\omega_o t)u(t)\leftrightarrow \dfrac{\omega_o} {s^2+\omega_o^2}\)
fourier transform
\(\rm{X\left(j\omega\right)=\displaystyle \int \limits_{ - \infty }^\infty {x(t)e^{-j\omega t}}dt}\) (CTFT)
\(\rm{x\left(t\right)=\dfrac{1}{2\pi}\displaystyle \int \limits_{ - \infty }^\infty {X(j\omega)e^{j\omega t}}d\omega}\) (inverse CTFT)
\(\rm{X\left(e^{j\omega}\right)=\displaystyle \sum \limits_{ - \infty }^\infty {x[n]e^{-j\omega n}}}\) (DTFT)
\(\rm{x\left[n\right]=\dfrac{1}{2\pi}\displaystyle \int \limits_{ - \pi }^\pi {X(j\omega)e^{j\omega n}}d\omega}\)
\(\text{z-transform} \overset{z=e^{j\omega}} \longleftrightarrow \text{DTFT}\)
duality
\(x(t) \leftrightarrow X(\omega)\)
\(\boxed{X(t) \leftrightarrow 2\pi x(-\omega)}\)
\(x(t) \leftrightarrow X(f)\)
\(\boxed{X(t) \leftrightarrow x(-f)}\)
modulation
\(x(t)\cdot cos(\omega_ot) \leftrightarrow \dfrac{2\pi}{2} [X(\omega-\omega_o)+X(\omega+\omega_o)]\)
\(\boxed{x(t)\cdot cos(\omega_ot) \leftrightarrow \frac{1}{2} [X(f-f_o)+X(f+f_o)]}\)
\(x(t)\cdot sin(\omega_ot) \leftrightarrow \dfrac{2\pi }{2j} [X(\omega-\omega_o)-X(\omega+\omega_o)]\)
\(x(t)\cdot sin(\omega_ot) \leftrightarrow \dfrac{1}{2j} [X(f-f_o)-X(f+f_o)]\)
sinc
\(Sa(bt)=\dfrac{sin(bt)}{bt}\)
\(\dfrac{sin(at)}{bt}=\dfrac{a}{b}sinc(\dfrac{at}{\pi})\)
\(\boxed{A\cdot rect(\frac{t}{T}) \longleftrightarrow AT\cdot Sa(\frac{\omega T}{2})}\)
\(\boxed{A\cdot tri(\frac{t}{T}) \longleftrightarrow AT\cdot Sa^2(\frac{\omega T}{2})}\)
properties
time shifting
\(x(t\pm t_0)\leftrightarrow X(j\omega)e^{\pm j\omega t_0}\)
\(x[n\pm n_0]\leftrightarrow X(e^{j\omega})e^{\pm j\omega n_0}\)
frequency shifting
\(e^{\pm j\omega_0 t}x(t)\leftrightarrow X[j(\omega\mp\omega)]\)
\(e^{\pm j\omega_0 t}x[n]\leftrightarrow X(e^{j (\omega\mp\omega)})\)
conjugation
\(x^*[n]\leftrightarrow X^*(e^{-j\omega})\)
time reversal
\(x[-n] \leftrightarrow X(e^{-j\omega})\)
area under \(x(t)\)
\(X(0)=\displaystyle\int_{-\infty}^{\infty}x(t)dt\)
area under \(X(j\omega)\)
\(2\pi\cdot x(0)=\displaystyle\int_{-\infty}^{\infty}X(j\omega)d\omega\)
differentiation in frequency
\(t^nx(t) \leftrightarrow j\frac{d^n}{d\omega^n}X(j\omega)\)
differentiation in time
\(\frac{d^n}{dt^n}x(t) \leftrightarrow (j \omega)^nX(j\omega)\)
\(x[n]-x[n-1] \leftrightarrow (1-e^{-j\omega}) X(e^{j\omega})\)
integration/accumulation in time
\(\int _{-\infty} ^{t} x(\tau)d\tau \leftrightarrow \frac{X(j\omega)}{j\omega} + \pi X(0) \delta (\omega)\)
\(\sum_{k=-\infty} ^n x[k] \leftrightarrow \dfrac{X[e^{j\omega}]}{1-e^{-j\omega}} + \pi X(e^{j0}) \sum_{k=-\infty} ^\infty\delta (\omega-2\pi k )\)
\(x(t) \leftrightarrow X(\omega) \space pairs\)
\(\begin{array}{c:c} \color{orange} {x(t)} & \color{orange} {X(\omega)} \\ \hdashline R & CS \\ CS & R \\ I & CAS \\ CAS & I \\ R+E & R+E \\ I+E & I+E \\ R+O & I+O \\ I+O & R+O & \end{array}\)
\(\begin{array}{c:c} \color{orange} {x(t)} & \color{orange} {X(\omega)} \\ \hdashline continuous & NP \\ NP & continuous \\ discreate & P \\ P & discreate \\ continuous +P & discreate +NP \\ continuous + NP & continuous + NP \\ discreate+P & discreate+P \\ discreate+NP & continuous+P \end{array}\)
standard results
\(e^{-at}u(t)\)
\(e^{-at}u(t) \longleftrightarrow \dfrac{1}{a+j\omega}\)
\(e^{-a|t|}\)
\(e^{-a|t|} \longleftrightarrow \dfrac{2a}{a^2+\omega^2}\)
\(e^{-a t^2}\)
\(e^{-a t^2} \longleftrightarrow \sqrt{\dfrac{\pi}{a}}e^{-\frac{\omega^2}{4a}}\)
\(u(t)\)
\(u(t) \longleftrightarrow \dfrac{1}{j\omega} + \pi\delta(\omega)\)
\(u[n]\)
\(\boxed{a^nu[n] \longleftrightarrow \dfrac{1}{1-ae^{-j\omega}} }\) , |a| < 1
\(u[n] \longleftrightarrow \dfrac{1}{1-e^{-j\omega}} + \pi \sum\limits_{k=-\infty} ^\infty\delta (\omega-2\pi k )\)
z-transform
\(X[z] \longleftrightarrow \sum \limits_{ n=- \infty }^\infty x[n] z^{-n}\)
properties
time shifting
\(x[n-n_o]\leftrightarrow z^{-n_o} X[z]\)
time scaling
\(x[\frac{n}{m}]\leftrightarrow X[z^m]\) ROC : \(R^{\frac{1}{m}}\)
time reversal
\(x[-n] \leftrightarrow X[z^{-1}]\) ROC : \(R^{-1}\)
scaling of z (exponential sequence property)
\(a^n x[n] \leftrightarrow X[\frac{z}{a}]\) ROC : \(|a| R\)
conjugation
\(x^*[n] \leftrightarrow X^* [z^*]\) ROC : R
accumulation
\(\displaystyle\sum_{k=-\infty} ^n x[k] \leftrightarrow \dfrac{X[z]}{1-z^{-1}}\) ROC \(> R \cap |z|>1\)
convolution in time
\(x_1[n]*x_2[n] \leftrightarrow X_1[z]\cdot X_2[z]\)
multiplication in time
\(x_1[n]\cdot x_2[n] \leftrightarrow \dfrac{1}{2\pi j} \{ X_1[z] *X_2[z] \}\)
differentiation in time
\(x[n]-x[n-1] \leftrightarrow (1-z^{-1}) X[z]\)
differentiation in z-domain
\(n\cdot x[n] \leftrightarrow -z \dfrac{d X[z]}{dz}\)
initial value theorem
\(x[n] _{n=0} = \lim \limits_{z\rightarrow \infty} X[z]\)
condition : \(x[n]=0, n<0\)
final value theorem
\(\lim \limits_{n\rightarrow \infty} x[n] \leftrightarrow \lim \limits_{z\rightarrow 1} \{ (1-z^{-1})X[z]\}\)
condition 1 : \(x[n]=0, n<0\)
condition 2 : \((1-z^{-1})X[z]\) should have poles inside unit circle in z-plane
standard results
\(u[n]\)
\(\boxed{a^nu[n] \longleftrightarrow \dfrac{1}{1-az^{-1}} }\) , |a| < 1
\(\boxed{-a^nu[-n-1] \longleftrightarrow \dfrac{1}{1-az^{-1}} }\) , |a| > 1
\(u[n] \longleftrightarrow \dfrac{1}{1-z^{-1}}\) |z|>1
\(-u[-n-1] \longleftrightarrow \dfrac{1}{1-z^{-1}}\) |z|<1