diff --git a/README.html b/README.html index 31cca24..c807bcc 100644 --- a/README.html +++ b/README.html @@ -543,18 +543,18 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\

AM

CAM

-

ma=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))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 signalPc=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{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 $m(t)$)}\\ - 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}\\ P_c &=\frac{ {A_c}^2}{2}\quad\text{Carrier power}\\ P_x &=\frac{1}{4}{m_a}^2{A_c}^2\\ \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

@@ -567,17 +567,17 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\

FM/PM

-

s(t)=Accos⁡[2πfct+kpm(t)]Phase modulated (PM)s(t)=Accos⁡(θi(t))=Accos⁡[2πfct+2πkf∫−∞tm(τ)dτ]Frequency modulated (FM)s(t)=Accos⁡[2πfct+βsin⁡(2πfmt)]FM single tonefi(t)=12πddtθi(t)Instantaneous frequency from instantaneous phaseΔf=βfm=kfAmfm=max⁡t(kfm(t))−min⁡t(kfm(t))Maximum frequency deviationΔf=max⁡t(fi(t))−min⁡t(fi(t))Maximum frequency deviationβ=Δffm=kfAmModulation indexD=ΔfWmDeviation ratio, where Wm is bandwidth of m(t) (Use FT)\begin{align*} +

s(t)=Accos⁡[2πfct+kpm(t)]Phase modulated (PM)s(t)=Accos⁡(θi(t))=Accos⁡[2πfct+2πkf∫−∞tm(τ)dτ]Frequency modulated (FM)s(t)=Accos⁡[2πfct+βsin⁡(2πfmt)]FM single tonefi(t)=12πddtθi(t)=fc+kfm(t)=fc+Δfmaxm^(t)Instantaneous frequencyΔfmax=max⁡t∣fi(t)−fc∣=kfmax⁡t∣m(t)∣Maximum frequency deviationΔfmax=kfAmMaximum frequency deviation (sinusoidal)β=ΔfmaxfmModulation indexD=ΔfmaxWmDeviation ratio, where Wm is bandwidth of m(t) (Use FT)\begin{align*} s(t) &= A_c\cos\left[2\pi f_c t + k_p m(t)\right]\quad\text{Phase modulated (PM)}\\ s(t) &= A_c\cos(\theta_i(t))=A_c\cos\left[2\pi f_c t + 2 \pi k_f \int_{-\infty}^t m(\tau) d\tau\right]\quad\text{Frequency modulated (FM)}\\ s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right]\quad\text{FM single tone}\\ - f_i(t) &= \frac{1}{2\pi}\frac{d}{dt}\theta_i(t)\quad\text{Instantaneous frequency from instantaneous phase}\\ - \Delta f&=\beta f_m=k_f A_m f_m = \max_t(k_f m(t))- \min_t(k_f m(t))\quad\text{Maximum frequency deviation}\\ - \Delta f&=\max_t(f_i(t))- \min_t(f_i(t))\quad\text{Maximum frequency deviation}\\ - \beta&=\frac{\Delta f}{f_m}=k_f A_m\quad\text{Modulation index}\\ - D&=\frac{\Delta f}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)}\\ + f_i(t) &= \frac{1}{2\pi}\frac{d}{dt}\theta_i(t)=f_c+k_f m(t)=f_c+\Delta f_\text{max}\hat m(t)\quad\text{Instantaneous frequency}\\ + \Delta f_\text{max}&=\max_t|f_i(t)-f_c|=k_f \max_t |m(t)|\quad\text{Maximum frequency deviation}\\ + \Delta f_\text{max}&=k_f A_m\quad\text{Maximum frequency deviation (sinusoidal)}\\ + \beta&=\frac{\Delta f_\text{max}}{f_m}\quad\text{Modulation index}\\ + D&=\frac{\Delta f_\text{max}}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)}\\ \end{align*} -

+

Bessel form and magnitude spectrum (single tone)

s(t)=Accos⁡[2πfct+βsin⁡(2πfmt)]⇔s(t)=Ac∑n=−∞∞Jn(β)cos⁡[2π(fc+nfm)t]\begin{align*} @@ -595,17 +595,16 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\

Carson's rule to find BB (98% power bandwidth rule)

-

B=2Mfm=2(β+1)fm=2(Δf+fm)=2(kfAm+fm)=2(D+1)WmB={2(Δf+fm)FM, sinusoidal message2(Δϕ+1)fmPM, sinusoidal message\begin{align*} +

B=2Mfm=2(β+1)fm=2(Δfmax+fm)=2(D+1)WmB={2(Δfmax+fm)=2(Δfmax+Wm)FM, sinusoidal message2(Δϕmax+1)fm=2(Δϕmax+1)WmPM, sinusoidal message\begin{align*} B &= 2Mf_m = 2(\beta + 1)f_m\\ - &= 2(\Delta f+f_m)\\ - &= 2(k_f A_m+f_m)\\ + &= 2(\Delta f_\text{max}+f_m)\\ &= 2(D+1)W_m\\ B &= \begin{cases} - 2(\Delta f+f_m) & \text{FM, sinusoidal message}\\ - 2(\Delta\phi + 1)f_m & \text{PM, sinusoidal message} + 2(\Delta f_\text{max}+f_m)=2(\Delta f_\text{max}+W_m) & \text{FM, sinusoidal message}\\ + 2(\Delta\phi_\text{max} + 1)f_m=2(\Delta \phi_\text{max}+1)W_m & \text{PM, sinusoidal message} \end{cases}\\ \end{align*} -

+

Complex envelope

s(t)=Accos⁡(2πfct+βsin⁡(2πfmt))⇔s~(t)=Acexp⁡(jβsin⁡(2πfmt))s(t)=Re[s~(t)exp⁡(j2πfct)]s~(t)=Ac∑n=−∞∞Jn(β)exp⁡(j2πfmt)\begin{align*} diff --git a/README.md b/README.md index a019b65..2b7e568 100644 --- a/README.md +++ b/README.md @@ -317,11 +317,11 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ ```math \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{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 $m(t)$)}\\ - 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}\\ P_c &=\frac{ {A_c}^2}{2}\quad\text{Carrier power}\\ P_x &=\frac{1}{4}{m_a}^2{A_c}^2\\ \eta&=\frac{\text{Signal Power}}{\text{Total Power}}=\frac{P_x}{P_x+P_c}\\ @@ -354,11 +354,11 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$ s(t) &= A_c\cos\left[2\pi f_c t + k_p m(t)\right]\quad\text{Phase modulated (PM)}\\ s(t) &= A_c\cos(\theta_i(t))=A_c\cos\left[2\pi f_c t + 2 \pi k_f \int_{-\infty}^t m(\tau) d\tau\right]\quad\text{Frequency modulated (FM)}\\ s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right]\quad\text{FM single tone}\\ - f_i(t) &= \frac{1}{2\pi}\frac{d}{dt}\theta_i(t)\quad\text{Instantaneous frequency from instantaneous phase}\\ - \Delta f&=\beta f_m=k_f A_m f_m = \max_t(k_f m(t))- \min_t(k_f m(t))\quad\text{Maximum frequency deviation}\\ - \Delta f&=\max_t(f_i(t))- \min_t(f_i(t))\quad\text{Maximum frequency deviation}\\ - \beta&=\frac{\Delta f}{f_m}=k_f A_m\quad\text{Modulation index}\\ - D&=\frac{\Delta f}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)}\\ + f_i(t) &= \frac{1}{2\pi}\frac{d}{dt}\theta_i(t)=f_c+k_f m(t)=f_c+\Delta f_\text{max}\hat m(t)\quad\text{Instantaneous frequency}\\ + \Delta f_\text{max}&=\max_t|f_i(t)-f_c|=k_f \max_t |m(t)|\quad\text{Maximum frequency deviation}\\ + \Delta f_\text{max}&=k_f A_m\quad\text{Maximum frequency deviation (sinusoidal)}\\ + \beta&=\frac{\Delta f_\text{max}}{f_m}\quad\text{Modulation index}\\ + D&=\frac{\Delta f_\text{max}}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)}\\ \end{align*} ``` @@ -392,12 +392,11 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$ ```math \begin{align*} B &= 2Mf_m = 2(\beta + 1)f_m\\ - &= 2(\Delta f+f_m)\\ - &= 2(k_f A_m+f_m)\\ + &= 2(\Delta f_\text{max}+f_m)\\ &= 2(D+1)W_m\\ B &= \begin{cases} - 2(\Delta f+f_m) & \text{FM, sinusoidal message}\\ - 2(\Delta\phi + 1)f_m & \text{PM, sinusoidal message} + 2(\Delta f_\text{max}+f_m)=2(\Delta f_\text{max}+W_m) & \text{FM, sinusoidal message}\\ + 2(\Delta\phi_\text{max} + 1)f_m=2(\Delta \phi_\text{max}+1)W_m & \text{PM, sinusoidal message} \end{cases}\\ \end{align*} ``` diff --git a/README.pdf b/README.pdf index ed99660..2aac667 100644 Binary files a/README.pdf and b/README.pdf differ