diff --git a/README.html b/README.html index c807bcc..c71d0ab 100644 --- a/README.html +++ b/README.html @@ -231,7 +231,7 @@ along with this program. If not, see http exp⁡(−πf2)\exp(-\pi f^2) -1−∣t∣T,∣t∣<T1 - \frac{\lvert t \rvert}{T},\quad\lvert t \rvert < T +1−∣t∣T,∣t∣<Ttri(t/T)1 - \frac{\lvert t \rvert}{T},\quad\lvert t \rvert < T\quad\text{tri}(t/T) Tsinc2(fT)T \text{sinc}^2(fT) @@ -383,14 +383,15 @@ along with this program. If not, see http -

u(t)={1,t>012,t=00,t<0Unit Step Functionsgn(t)={+1,t>00,t=0−1,t<0Signum Functionsinc(2Wt)=sin⁡(2πWt)2πWtsinc Functionrect(t)=Π(t)={1,−0.5<t<0.50,∣t∣>0.5Rectangular/Gate Functiong(t)∗h(t)=(g∗h)(t)=∫∞∞g(τ)h(t−τ)dτConvolution\begin{align*} +

u(t)={1,t>012,t=00,t<0Unit Step Functionsgn(t)={+1,t>00,t=0−1,t<0Signum Functionsinc(2Wt)=sin⁡(2πWt)2πWtsinc Functionrect(t)=Π(t)={1,−0.5<t<0.50,∣t∣>0.5Rectangular/Gate Functiontri(t/T)={1−∣t∣T,∣t∣<T0,∣t∣>T=Π(t/T)∗Π(t/T)Triangle Functiong(t)∗h(t)=(g∗h)(t)=∫∞∞g(τ)h(t−τ)dτConvolution\begin{align*} u(t) &= \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}&\text{Unit Step Function}\\ \text{sgn}(t) &= \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}&\text{Signum Function}\\ \text{sinc}(2Wt) &= \frac{\sin(2\pi W t)}{2\pi W t}&\text{sinc Function}\\ \text{rect}(t) = \Pi(t) &= \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}&\text{Rectangular/Gate Function}\\ + \text{tri}(t/T) &= \begin{cases} 1 - \frac{|t|}{T}, & \lvert t\rvert < T \\ 0, & \lvert t \rvert > T \end{cases}=\Pi(t/T)*\Pi(t/T)&\text{Triangle Function}\\ g(t)*h(t)=(g*h)(t)&=\int_\infty^\infty g(\tau)h(t-\tau)d\tau&\text{Convolution}\\ \end{align*} -

+

Fourier transform of continuous time periodic signal

Required for some questions on sampling:

@@ -401,14 +402,27 @@ along with this program. If not, see http

Calculate CnC_n coefficient as follows from xp(t)x_p(t):

-

Cn=1Ts∫Tsxp(t)exp⁡(−j2πfst)dt=1TsX(nfs)(TODO: Check)x(t−nTs) is contained in the interval Ts\begin{align*} - C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\ - &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$} +

Cn=1Ts∫Tsxp(t)exp⁡(−j2πfst)dt=1TsX(nfs)(TODO: Check)x(t−nTs) is contained in the interval Ts\begin{align*} + C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$} \end{align*} -

+

-

rect\text{rect} function

-

rect

+

Shape functions

+ + + + + + + + + + + + + +
rect\text{rect} functiontri\text{tri} function
rectTODO: Add graphic.
+

Tri placeholder: For tri(t/T)=1−∥t∥/T\text{tri}(t/T)=1-\|t\|/T, Intersects xx axis at −T-T and TT and yy axis at 11.

Bessel function

∑n∈ZJn2(β)=1Jn(β)=(−1)nJ−n(β)\begin{align*} \sum_{n\in\mathbb{Z}}{J_n}^2(\beta)&=1\\ @@ -543,10 +557,10 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\

AM

CAM

-

x(t)=Accos⁡(2πfct)[1+kam(t)]=Accos⁡(2πfct)[1+mam(t)/Ac],where m(t)=Amm^(t) and m^(t) is the normalized modulating signalma=min⁡t∣kam(t)∣Acka is the amplitude sensitivity (volt−1), ma is the modulation index.ma=Amax−AminAmax+Amin (Symmetrical m(t))ma=kaAm (Symmetrical m(t))Pc=Ac22Carrier powerPx=14ma2Ac2η=Signal PowerTotal Power=PxPx+PcBT=2fm=2B\begin{align*} +

x(t)=Accos⁡(2πfct)[1+kam(t)]=Accos⁡(2πfct)[1+mam(t)/Ac],where m(t)=Amm^(t) and m^(t) is the normalized modulating signalma=∣min⁡t(kam(t))∣Acka is the amplitude sensitivity (volt−1), ma is the modulation index.ma=Amax−AminAmax+Amin (Symmetrical m(t))ma=kaAm (Symmetrical m(t))Pc=Ac22Carrier powerPx=14ma2Ac2η=Signal PowerTotal Power=PxPx+PcBT=2fm=2B\begin{align*} x(t)&=A_c\cos(2\pi f_c t)\left[1+k_a m(t)\right]=A_c\cos(2\pi f_c t)\left[1+m_a m(t)/A_c\right], \\ &\text{where $m(t)=A_m\hat m(t)$ and $\hat m(t)$ is the normalized modulating signal}\\ - m_a &= \frac{\min_t|k_a m(t)|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\ + m_a &= \frac{|\min_t(k_a m(t))|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\ m_a &= \frac{A_\text{max}-A_\text{min}}{A_\text{max}+A_\text{min}}\quad\text{ (Symmetrical $m(t)$)}\\ m_a&=k_a A_m \quad\text{ (Symmetrical $m(t)$)}\\ P_c &=\frac{ {A_c}^2}{2}\quad\text{Carrier power}\\ @@ -554,7 +568,7 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ \eta&=\frac{\text{Signal Power}}{\text{Total Power}}=\frac{P_x}{P_x+P_c}\\ B_T&=2f_m=2B \end{align*} -

+

BTB_T: Signal bandwidth BB: Bandwidth of modulating wave

diff --git a/README.md b/README.md index 2b7e568..c919c91 100644 --- a/README.md +++ b/README.md @@ -81,26 +81,26 @@ along with this program. If not, see . ## Fourier transform identities and properties -| Time domain $x(t)$ | Frequency domain $X(f)$ | -| --------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- | -| $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$ | $T \text{sinc}(fT)$ | -| $\text{sinc}(2Wt)$ | $\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)$ | -| $\exp(-at)u(t),\quad a>0$ | $\frac{1}{a + j2\pi f}$ | -| $\exp(-a\lvert t \rvert),\quad a>0$ | $\frac{2a}{a^2 + (2\pi f)^2}$ | -| $\exp(-\pi t^2)$ | $\exp(-\pi f^2)$ | -| $1 - \frac{\lvert t \rvert}{T},\quad\lvert t \rvert < T$ | $T \text{sinc}^2(fT)$ | -| $\delta(t)$ | $1$ | -| $1$ | $\delta(f)$ | -| $\delta(t - t_0)$ | $\exp(-j2\pi f t_0)$ | -| $\exp(j2\pi f_c t)$ | $\delta(f - f_c)$ | -| $\cos(2\pi f_c t)$ | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$ | -| $\cos(2\pi f_c t+\theta)$ | $\frac{1}{2}[\delta(f - f_c)\exp(j\theta) + \delta(f + f_c)\exp(-j\theta)]\quad\text{Use for coherent recv.}$ | -| $\sin(2\pi f_c t)$ | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$ | -| $\sin(2\pi f_c t+\theta)$ | $\frac{1}{2j} [\delta(f - f_c)\exp(j\theta) - \delta(f + f_c)\exp(-j\theta)]$ | -| $\text{sgn}(t)$ | $\frac{1}{j\pi f}$ | -| $\frac{1}{\pi t}$ | $-j \text{sgn}(f)$ | -| $u(t)$ | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$ | -| $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$ | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ | +| Time domain $x(t)$ | Frequency domain $X(f)$ | +| ---------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- | +| $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$ | $T \text{sinc}(fT)$ | +| $\text{sinc}(2Wt)$ | $\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)$ | +| $\exp(-at)u(t),\quad a>0$ | $\frac{1}{a + j2\pi f}$ | +| $\exp(-a\lvert t \rvert),\quad a>0$ | $\frac{2a}{a^2 + (2\pi f)^2}$ | +| $\exp(-\pi t^2)$ | $\exp(-\pi f^2)$ | +| $1 - \frac{\lvert t \rvert}{T},\quad\lvert t \rvert < T\quad\text{tri}(t/T)$ | $T \text{sinc}^2(fT)$ | +| $\delta(t)$ | $1$ | +| $1$ | $\delta(f)$ | +| $\delta(t - t_0)$ | $\exp(-j2\pi f t_0)$ | +| $\exp(j2\pi f_c t)$ | $\delta(f - f_c)$ | +| $\cos(2\pi f_c t)$ | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$ | +| $\cos(2\pi f_c t+\theta)$ | $\frac{1}{2}[\delta(f - f_c)\exp(j\theta) + \delta(f + f_c)\exp(-j\theta)]\quad\text{Use for coherent recv.}$ | +| $\sin(2\pi f_c t)$ | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$ | +| $\sin(2\pi f_c t+\theta)$ | $\frac{1}{2j} [\delta(f - f_c)\exp(j\theta) - \delta(f + f_c)\exp(-j\theta)]$ | +| $\text{sgn}(t)$ | $\frac{1}{j\pi f}$ | +| $\frac{1}{\pi t}$ | $-j \text{sgn}(f)$ | +| $u(t)$ | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$ | +| $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$ | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ | | Time domain $x(t)$ | Frequency domain $X(f)$ | Property | | ---------------------------------------- | ------------------------------------------------ | ----------------------------- | @@ -130,6 +130,7 @@ along with this program. If not, see . \text{sgn}(t) &= \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}&\text{Signum Function}\\ \text{sinc}(2Wt) &= \frac{\sin(2\pi W t)}{2\pi W t}&\text{sinc Function}\\ \text{rect}(t) = \Pi(t) &= \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}&\text{Rectangular/Gate Function}\\ + \text{tri}(t/T) &= \begin{cases} 1 - \frac{|t|}{T}, & \lvert t\rvert < T \\ 0, & \lvert t \rvert > T \end{cases}=\Pi(t/T)*\Pi(t/T)&\text{Triangle Function}\\ g(t)*h(t)=(g*h)(t)&=\int_\infty^\infty g(\tau)h(t-\tau)d\tau&\text{Convolution}\\ \end{align*} ``` @@ -156,16 +157,19 @@ Calculate $C_n$ coefficient as follows from $x_p(t)$: ```math \begin{align*} - C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\ - &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$} + C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$} \end{align*} ``` -### $\text{rect}$ function +### Shape functions -![rect](images/rect.drawio.svg) +| $\text{rect}$ function | $\text{tri}$ function | +| ------------------------------- | --------------------- | +| ![rect](images/rect.drawio.svg) | TODO: Add graphic. | + +Tri placeholder: For $\text{tri}(t/T)=1-\|t\|/T$, Intersects $x$ axis at $-T$ and $T$ and $y$ axis at $1$. ### Bessel function @@ -319,7 +323,7 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ \begin{align*} x(t)&=A_c\cos(2\pi f_c t)\left[1+k_a m(t)\right]=A_c\cos(2\pi f_c t)\left[1+m_a m(t)/A_c\right], \\ &\text{where $m(t)=A_m\hat m(t)$ and $\hat m(t)$ is the normalized modulating signal}\\ - m_a &= \frac{\min_t|k_a m(t)|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\ + m_a &= \frac{|\min_t(k_a m(t))|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\ m_a &= \frac{A_\text{max}-A_\text{min}}{A_\text{max}+A_\text{min}}\quad\text{ (Symmetrical $m(t)$)}\\ m_a&=k_a A_m \quad\text{ (Symmetrical $m(t)$)}\\ P_c &=\frac{ {A_c}^2}{2}\quad\text{Carrier power}\\ diff --git a/README.pdf b/README.pdf index 2aac667..984b2ed 100644 Binary files a/README.pdf and b/README.pdf differ