From b1a22abfe12b001c1832911da178be87684f62d4 Mon Sep 17 00:00:00 2001 From: Philipp Le Date: Thu, 4 Jun 2020 01:22:08 +0200 Subject: WIP Chapter 5: Complex Mixing --- chapter05/content_ch05.tex | 717 ++++++++++++++++++++++++++++++++++++++++++++- common/acronym.tex | 2 + 2 files changed, 714 insertions(+), 5 deletions(-) diff --git a/chapter05/content_ch05.tex b/chapter05/content_ch05.tex index 9350d57..3579757 100644 --- a/chapter05/content_ch05.tex +++ b/chapter05/content_ch05.tex @@ -650,6 +650,7 @@ The information-carrying signal and the carrier are usually at different frequen } \caption{A selection of heterodyne architectures} + \label{fig:ch05:trx_if_arch} \end{figure} Definitions: @@ -875,19 +876,723 @@ So, the input signal's spectrum consists of a positive and a negative part: \subsection{Technical Realization of Mixers} -\todo{Non-linear component} +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=1} + \begin{circuitikz} + \node[adder](Adder){}; + \node[block,draw,right=of Adder](NonLin){Non-linear\\ component}; + \node[oscillator,below=of Adder](LO){}; + + \draw (LO.south) node[below,align=center,yshift=-5mm]{\acs{LO}}; + + \draw[latex-o] (Adder.west) -- ++(-2cm,0) node[left,align=right]{Input\\ signal $x_i(t)$}; + \draw[-latex] (Adder.east) -- (NonLin.west); + \draw[-latex] (NonLin.east) -- ++(2cm,0) node[right,align=left]{Output\\ signal $x_o(t)$}; + \draw[-latex] (LO.north) -- node[midway,right,align=left]{$u_{LO}(t)$} (Mix.south); + \draw[latex-o] (Adder.north) -- (0,1.5cm) node[above,align=center]{\acs{DC} bias\\ (optional)}; + \end{circuitikz} + \end{adjustbox} + \caption[Basic principle of a mixer]{Basic principle of a mixer. The input signals and optionally a \acs{DC} bias are combined. A non-linearity implements the mixing process.} +\end{figure} + +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=1} + \begin{circuitikz} + \draw (0,2) node[left,align=right]{Input\\ signal} to[short,o-] (1,2) to[R,l=$R$] (3,2); + \draw (0,0) node[left,align=right]{\acs{LO}\\ signal} to[short,o-] (1,0) to[R,l=$R$] (3,0) to[short,-*] (3,2); + \draw (3,2) to[R,l=$R$] (5,2) to[C,l=$C$] (7,2) to[empty diode] (9,2) to[short,-o] (10,2) node[right,align=left]{Output\\ signal}; + \draw (7,4) node[above,align=center]{\acs{DC} bias} to[L,l=$L$,o-*] (7,2); + + \draw[dashed] (1,3) -- (5,3) -- (5,-1) -- (1,-1) node[below right,align=left]{Power combiner} -- cycle; + \end{circuitikz} + \end{adjustbox} + \caption[Passive unbalanced mixer]{Passive unbalanced mixer. The input signals are added. Then a \acs{DC} bias is injected. The diode is the non-linearity where the mixing happens. The carrier is not suppressed.} + \label{fig:ch05:pass_unbal_mixer} +\end{figure} -\todo{IP3} +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.8} + \begin{circuitikz} + % current source + \draw(0,0) to[I=$2 I_T$,*-] ++(0,-2) + node[rground]{}; + + % driver stage + \draw(-3,1) node[npn](Q5){} node[right]{Q5}; + \draw(3,1) node[npn,xscale=-1](Q6){} node[left]{Q6}; + \draw(0,0) to[R,l^=$Z_e$] ++(-3,0) + |- (Q5.E); + \draw(0,0) to[R,l_=$Z_e$] ++(3,0) + |- (Q6.E); + \draw(Q5.B) to[short,-o] ++(-1,0) + node[left]{Differential input (+)}; + \draw(Q6.B) to[short,-o] ++(1,0) + node[right]{Differential input (-)}; + + % switching quad + \draw(Q5) ++(-1,1.5) node[npn](Q7){} node[right]{Q7}; + \draw(Q5) ++(1,1.5) node[npn,xscale=-1](Q8){} node[left]{Q8}; + \draw(Q6) ++(-1,1.5) node[npn](Q9){} node[right]{Q9}; + \draw(Q6) ++(1,1.5) node[npn,xscale=-1](Q10){} node[left]{Q10}; + \draw(Q5.C) to[short,*-] ++(-1,0) + |- (Q7.E); + \draw(Q5.C) to[short] ++(1,0) + |- (Q8.E); + \draw(Q6.C) to[short,*-] ++(-1,0) + |- (Q9.E); + \draw(Q6.C) to[short] ++(1,0) + |- (Q10.E); + \draw(Q7.B) to[short,-o] ++(-1,0) + node[left]{Differential \acs{LO} (-)}; + \draw(Q10.B) to[short,-o] ++(1,0) + node[right]{Differential \acs{LO} (-)}; + \draw(Q8.B) to[short] (Q9.B); + \draw(0,|-Q8.B) to[short,*-o] ++(0,-0.5) + node[below]{Differential \acs{LO} (+)}; + \draw(Q7.C) to[short] ++(0,0.5) + to[R,l_=$R$] ++(0,2) + node[vcc]{}; + \draw(Q9.C) to[short] (Q7.C) + to[short,*-o] ++(-2,0) + node[left]{Differential output (-)}; + \draw(Q10.C) to[short] ++(0,0.5) node(AboveQ10){} + to[R,l_=$R$] ++(0,2) + node[vcc]{}; + \draw(Q8.C) to[short] ++(0,0.5) + to[short] (AboveQ10) + to[short,*-o] ++(2,0) + node[right]{Differential output (+)}; + \end{circuitikz} + \end{adjustbox} + \caption[Active double-balanced mixer with differential inputs and output]{Active double-balanced mixer with differential inputs and output. The switching stages (balanced transistor pairs) implement the non-linearity. The balanced switching mode suppresses the carrier.} +\end{figure} + +The core component of a mixer is a non-linear device. + +Figure \ref{fig:ch05:pass_unbal_mixer} depicts a simple mixer with a diode as the non-linear device. The characteristic curve of a diode is an exponential function -- the \emph{Schockley diode equation}. +\begin{equation} + I = I_S \left(e^{\frac{V_D}{n V_T}} - 1\right) +\end{equation} +The equation describes the relation of the diode current $I$ and its forward voltage $V_D$. The equation is non-linear. + +A mathematical model is depicted in Figure \ref{fig:ch05:math_model_mixer}. +\begin{figure} + \centering + \begin{circuitikz} + \draw(0,0) node[mixer](Mix){}; + \draw(-3,|-Mix) node[adder](Add){}; + \draw(Mix.4) node[above]{$M(x)$}; + \draw(Add.3) -- (Mix.1) node[inputarrow]{}; + \draw(Add.1) +(-1,0) node[above]{$x_{i}$} -- (Add.1) node[inputarrow]{}; + \draw(Add.4) +(0,1) node[right]{$a$} -- (Add.4) node[inputarrow,rotate=-90]{}; + \draw(Add.2) +(0,-1) node[right]{$x_{LO}$} -- (Add.2) node[inputarrow,rotate=90]{}; + \draw(Mix.3) -- +(1,0) node[inputarrow]{$x_{o}$}; + \end{circuitikz} + \caption[Mathematical model of the mixer]{Mathematical model of the mixer with a signal combiner ($x_{i} + x_{LO} + a$) and a non-linear device with the characteristic $M(x)$. $x_{i}$ is the input, $x_{LO}$ is the \acs{LO}, $x_{o}$ is the output and $a$ is the \acs{DC} bias defining the operating point.} + \label{fig:ch05:math_model_mixer} +\end{figure} + +The non-linearity $M(x)$ of the diode or any other non-linear devices can be expressed as a \emph{Taylor series}. The Taylor series is developed around a operating point of non-linear device which is defined by the bias $a$. +\begin{equation} + \begin{split} + x_{o} &= M(x_{i} + x_{LO} + a) = \sum\limits_{n=0}^{\infty} \frac{1}{n!} \left.\frac{\mathrm{d}^n M(x)}{\mathrm{d} x^n}\right|_{x=a} \left(x_{i} + x_{LO} + a - a\right)^n \\ + &= M(a) + \underbrace{M^{(1)}(a) \left(x_{i} + x_{LO}\right)}_{\text{Linear term}} + \underbrace{\frac{M^{(2)}(a)}{2} \left(x_{i} + x_{LO}\right)^2}_{\text{Quadratic term}} + \underbrace{\frac{M^{(3)}(a)}{6} \left(x_{i} + x_{LO}\right)^2}_{\text{Qubic term}} + \dots + \end{split} +\end{equation} + +\begin{itemize} + \item The \underline{linear term} is used in electronics for the small signal analysis of a circuit. + \item Mixers are driven with relatively strong signals. Therefore, the \underline{quadratic term} comes into play. + \item The contribution of high-order polynomials decreases with their order due to the coefficient $\frac{1}{n!}$. Because of that, polynomials of order three or higher are neglected. +\end{itemize} + +The quadratic term is the important part in the mixing process. +\begin{equation} + \left(x_{i} + x_{LO}\right)^2 = x_{i}^2 + 2 \underbrace{x_{i} x_{LO}}_{\text{Mixing}} + x_{LO}^2 +\end{equation} + +The quadratic term devolves, amongst others, into a multiplication of the input signals. This is where the mixing process happens. + +\begin{excursus}{Spurious components in the output signal} + The equation + \begin{equation*} + \left(x_{i} + x_{LO}\right)^2 = x_{i}^2 + 2 x_{i} x_{LO} + x_{LO}^2 + \end{equation*} + points out that there are more signals than the desired signals. + \begin{itemize} + \item The term $x_{i} x_{LO}$ yields the desired components in the output signal. + \begin{itemize} + \item A signal at a frequency of $\omega_i - \omega_{LO}$ + \item Another signal at the mirror frequency of $\omega_i + \omega_{LO}$ + \end{itemize} + \item The two other terms produce spurious signals + \begin{itemize} + \item at the double frequency of the \ac{LO} $2 \omega_{LO}$ and + \item at the double frequency of the input $2 \omega_{i}$. + \end{itemize} + \end{itemize} +\end{excursus} + +The spurious signals distort the output signal, decrease the \ac{SNR} and are therefore unwanted. \textbf{The output of the mixer must always be filtered, to remove spurious components.} + +\begin{excursus}{Intermodulation} + Other spurious signals are created by higher order polynomials. + \begin{itemize} + \item For weak inputs, their contribution is low (due to the coefficient $\frac{1}{n!}$) and can be neglected. + \item If the input signal is strong enough, polynomials of orders higher than 2 cannot be neglected any longer and must be considered as well. + \item Especially, the 3rd order polynomial comes into effect firstly. It amongst others contributes the following output frequencies: + \begin{itemize} + \item $2 \omega_a - \omega_b$ + \item $2 \omega_b - \omega_a$ + \end{itemize} + \item Example: + \begin{itemize} + \item The \ac{LO} is \SI{100}{MHz}. The \ac{RF} signal at \SI{110}{MHz} shall be mixed down to \SI{10}{MHz}. + \item There are two more very strong, disturbing signals -- so called \emph{blockers} -- at \SI{107}{MHz} and \SI{104}{MHz}. + \item These very strong blockers mix to: $2 \cdot \SI{107}{MHz} - \SI{104}{MHz} = \SI{110}{MHz}$. + \item The mixer product of the blockers disturbs the input signal and decreases its \ac{SNR}. + \item This effect is called \index{intermodulation} \textbf{intermodulation}. + \end{itemize} + \end{itemize} + + We will not bother with the theory behind this. However, the consequences are: + \begin{itemize} + \item The mixer input should be filtered to eliminate out-of-band disturbances. + \item The input signal power must not exceed a certain limit. This characteristic is given in mixer datasheets as the \index{interception point} \textbf{interception point of the 3rd order (IP3)}. + \item Too strong signals will cause intermodulation. + \end{itemize} +\end{excursus} \subsection{Zero-Intermediate-Frequency} -\todo{coherency} +Figure \ref{fig:ch05:trx_if_arch} considers different receiver architectures with a variation in \acf{IF} stages. +\begin{itemize} + \item Superheterodyne receivers allow the implementation of high quality filter to archive a good selectivity. + \item However, the cost of the implementation increases with an increasing number of \ac{IF} stages. + \item A common implementation in modern digital communication systems is omitting the \ac{IF} stages can convert directly from \ac{RF} to the baseband. + \item The filtering is accomplished in the digital signal processing chain. +\end{itemize} + +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.8} + \begin{circuitikz} + \node[ampshape](RFAmplifier){}; + \node[mixer, right=2cm of RFAmplifier](Mixer){}; + \node[oscillator, below=1cm of Mixer](LO){}; + \node[ampshape, right=2.5cm of Mixer](BBAmplifier){}; + \node[adcshape, right=2.5cm of BBAmplifier](ADC){}; + \node[block, draw, right=1cm of ADC](Baseband){Digital signal\\ processing}; + + \draw (LO.south) node[below,align=center,yshift=-5mm]{\acs{LO}}; + \draw (RFAmplifier.south) node[below,align=center,yshift=-5mm]{\acs{RF}\\ amplifier}; + \draw (Mixer.north) node[above,align=center,yshift=3mm]{Mixer}; + \draw (BBAmplifier.south) node[below,align=center,yshift=-5mm]{Baseband\\ amplifier}; + \draw (ADC.south) node[below,align=center,yshift=-5mm]{\acs{ADC}}; + + \draw (RFAmplifier.west) to[bandpass] ++(-2cm,0) node[rxantenna,xscale=-1]{}; + + \draw[-latex] (LO.north) -- (Mixer.south); + \draw[-latex] (RFAmplifier.east) -- node[midway,above,align=center]{\acs{RF}\\ signal} (Mixer.west); + \draw[-latex] (Mixer.east) to[lowpass] ++(2cm,0) -- (BBAmplifier.west); + \draw[-latex] (BBAmplifier.east) -- node[midway,above,align=center]{Baseband\\ signal} (ADC.west); + \draw[-latex] (ADC.east) -- (Baseband.west); + \end{circuitikz} + \end{adjustbox} + \caption{Zero-\acs{IF} receiver} +\end{figure} + +\begin{itemize} + \item The \ac{RF} is directly converted to the baseband close to a frequency of zero. The baseband can be seen as a \ac{IF} of zero. + \item The \ac{LO} is tuned directly to the \ac{RF} frequency. + \item The receiver architecture is called \index{zero-IF} \textbf{zero-\acs{IF}} or \index{direct conversion receiver} \textbf{direct conversion receiver}. +\end{itemize} + +This special design requires a careful consideration. Assume that the \ac{RF} signal is monochromatic. The baseband signal $x_B(t)$ is: +\begin{equation} + x_B(t) = \underbrace{\hat{X}_{RF} \cos\left(\omega_{RF} t + \varphi_{RF}\right)}_{= x_{RF}(t)} \cdot \underbrace{\cos\left(\omega_{RF} t\right)}_{= x_{RF}(t) \text{ with } \omega_{LO} = \omega_{RF}} +\end{equation} + +\begin{itemize} + \item If the phase shift $\varphi_{RF}$ of the \ac{RF} signal is $0$, the \ac{RF} signal can be received without problems. + \item If the phase shift $\varphi_{RF}$ of the \ac{RF} signal is $\pm \frac{\pi}{2}$, the \ac{RF} signal is orthogonal to the \ac{LO} signal. The baseband signal be zero. The \ac{RF} signal cannot be received. + \item Values of $0 < \varphi_{RF} < \frac{\pi}{2}$ reduce the amplitude of the baseband signal, which decreases the \ac{SNR}. +\end{itemize} + +\begin{proof}{} + \begin{equation*} + \begin{split} + x_B(t) &= \hat{X}_{RF} \cos\left(\omega_{RF} t + \frac{\pi}{2}\right) \cdot \cos\left(\omega_{RF} t\right) \\ + & \quad \text{Fourier transform} \\ + \underline{X}_B\left(j\omega\right) &= \hat{X}_{RF} \pi^2 \left( e^{j \frac{\pi}{2}} \delta\left(\omega-\omega_{RF}\right) + e^{-j \frac{\pi}{2}} \delta\left(\omega+\omega_{RF}\right) \right) * \left( \delta\left(\omega-\omega_{RF}\right) + \delta\left(\omega+\omega_{RF}\right) \right) \\ + &= \hat{X}_{RF} \pi^2 \left( \underbrace{e^{j \frac{\pi}{2}} \delta\left(\omega\right) + e^{-j \frac{\pi}{2}} \delta\left(\omega\right)}_{\text{Baseband signal at zero \acs{IF}}} + \underbrace{e^{j \frac{\pi}{2}} \delta\left(\omega-2\omega_{RF}\right) + e^{-j \frac{\pi}{2}} \delta\left(\omega+2\omega_{RF}\right)}_{\text{Eliminated by the \ac{LPF}}} \right) \\ + &= \hat{X}_{RF} \pi^2 \left( \underbrace{j \delta\left(\omega\right) - j \delta\left(\omega\right)}_{= 0} \right) \\ + &= 0 + \end{split} + \end{equation*} + + If the \ac{RF} and \ac{LO} signals are orthogonal, no signal will be present in a band-limited (\ac{LPF}) baseband signal. +\end{proof} + +The problem is that the phase of the \ac{RF} $\varphi_{RF}$ is not known. The phase of the \ac{LO} must be aligned to the \ac{RF} signal phase to reduce $\varphi_{RF}$ to zero. + +To solve this issue, the mixer principle of the zero-\acs{IF} mixer must be adapted. +\begin{itemize} + \item The \ac{RF} signal is split and fed into two mixer branches. + \item One of the mixers multiplies the \ac{RF} signal with the \ac{LO} signal. + \item The other mixer multiplies the \ac{RF} signal with a $\frac{\pi}{2}$-phase-shifted copy of the \ac{LO} signal. + \item The \ac{LO} signals of the mixers are orthogonal. + \item Due to the orthogonality of the \ac{LO} signals, all variations of the \ac{RF} signal phase shift $\varphi_{RF}$ can be reliably received without \ac{SNR} degradation. +\end{itemize} +This mixer architecture is called \index{coherent mixer} \textbf{coherent}. + +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.8} + \begin{circuitikz} + \node[mixer](MixI) at(5cm,3cm) {}; + \node[mixer](MixQ) at(5cm,-3cm) {}; + \node[oscillator](LO) at(3cm,0cm){}; + \node[block, draw, minimum height=8cm](DSP) at(11cm,0cm){Digital\\ signal\\ processing}; + + \draw (LO.south) node[below,align=center,yshift=-3mm]{\acs{LO}\\ $\omega_{LO} = \omega_{RF}$}; + \draw (MixI.north) node[above,align=center,yshift=1cm]{In-phase (\acs{I}) branch}; + \draw (MixQ.south) node[below,align=center,yshift=-1cm]{Quadrature (\acs{Q}) branch}; + + \draw (0cm,0cm) node[left,align=right]{Input\\ signal} to[short,o-*] (1cm,0cm); + \draw (1cm,0cm) to[short] ([xshift=-4cm] MixI) to[lowpass] (MixI.west); + \draw (1cm,0cm) to[short] ([xshift=-4cm] MixQ) to[lowpass] (MixQ.west); + + \draw (LO.east) to[short,-*] (5cm,0cm); + \draw (5cm,0cm) to[short] (MixI.south); + \draw (5cm,0cm) to[phaseshifter,l=$\SI{90}{\degree}$] (MixQ.north); + + \draw (MixI.east) to[amp] ++(2cm,0cm) to[adc] ([yshift=3cm] DSP.west); + \draw (MixQ.east) to[amp] ++(2cm,0cm) to[adc] ([yshift=-3cm] DSP.west); + \end{circuitikz} + \end{adjustbox} + \caption{Coherent mixer suitable for zero-\acs{IF} receiver architectures (IQ demodulator)} + \label{fig:ch05:iq_down_circuit} +\end{figure} + +The coherent mixer, also called \index{IQ demodulator} \textbf{IQ demodulator}, (Figure \ref{fig:ch05:iq_down_circuit}) consists of two branches (also called paths): +\begin{itemize} + \item The \ac{I} path mixes the original \ac{LO} signal to the replica of the \ac{RF} signal. + \item The \ac{Q} path mixes the $\frac{\pi}{2}$-phase-shifted \ac{LO} signal to the replica of the \ac{RF} signal. +\end{itemize} +The coherent mixer is capable of receiving all variations of the \ac{RF} signal phase shift $\varphi_{RF}$, while the amplitude is kept intact. + +\begin{excursus}{Non-coherent demodulation} + \todo{Non-coherent demodulation} +\end{excursus} + +\begin{fact} + Pure \ac{AM} signals can be demodulated either coherently or non-coherently. \ac{PM} signals or mixed \ac{AM} and \ac{PM} signals require a coherent demodulation, because a non-coherent demodulation drops the signal phase. +\end{fact} \subsection{Mixing Complex-Valued Baseband Signals} -\todo{IQ Modulator} +\subsubsection{Down-Conversion} + +A coherent mixer (Figure \ref{fig:ch05:iq_down_circuit}) retains the phase of the \ac{RF} signal. The phase information is coded into the \ac{I} and \ac{Q} parts of the baseband signal. + +Let's consider the \ac{RF} signal $x_{RF}(t) \TransformHoriz \underline{X}_{RF}\left(j\omega\right)$ contains the information whose power is concentrated close to the \ac{RF} frequency $\omega_{RF}$. $\underline{X}_{RF}\left(j\omega\right)$ devolves into a positive and negative frequency part. +\begin{equation} + \underline{X}_{RF}\left(j\omega\right) = \begin{cases} + \underline{X}_{RF}^{+}\left(j\omega\right) & \quad \text{ if } \omega \geq 0 \\ + \underline{X}_{RF}^{-}\left(j\omega\right) & \quad \text{ if } \omega \leq 0 + \end{cases} +\end{equation} +$x_{RF}(t)$ is real-valued. Therefore, $\underline{X}_{RF}^{+}\left(j\omega\right) = \overline{\underline{X}_{RF}^{-}\left(-j\omega\right)}$. So, both the positive and negative part in fact carry the same information. + +Now, the \ac{RF} signal is mixed down to the baseband by a coherent mixer. +\begin{itemize} + \item The \ac{I} path of the coherent mixer (Figure \ref{fig:ch05:iq_down_circuit}): + \begin{equation} + \begin{split} + x_{B,I}(t) &= x_{RF}(t) \cdot \cos\left(\omega_{RF} t\right) \\ + &\quad \text{Fourier transform:} \\ + \underline{X}_{B,I}\left(j\omega\right) &= \underline{X}_{RF}\left(j\omega\right) * \left(\delta\left(\omega-\omega_{RF}\right) + \delta\left(\omega+\omega_{RF}\right)\right) \pi \\ + &= \pi \left(\underline{X}_{RF}\left(j\omega-j\omega_{RF}\right) + \underline{X}_{RF}\left(j\omega+j\omega_{RF}\right)\right) \\ + &= \pi \left(\underbrace{\underline{X}_{RF}^{-}\left(j\omega-j\omega_{RF}\right) + \underline{X}_{RF}^{+}\left(j\omega+j\omega_{RF}\right)}_{\text{\acs{I} component of the baseband signal}} \right. \\ &\qquad + \left. \underbrace{\underline{X}_{RF}^{+}\left(j\omega-j\omega_{RF}\right) + \underline{X}_{RF}^{-}\left(j\omega+j\omega_{RF}\right)}_{\text{Mirror frequencies close to $\pm 2 \omega_{RF}$, eliminated by \acs{LPF}}} \right) \\ + &\quad \text{After the \acs{LPF}:} \\ + &= \pi \left(\underline{X}_{RF}^{-}\left(j\omega-j\omega_{RF}\right) + \underline{X}_{RF}^{+}\left(j\omega+j\omega_{RF}\right) \right) + \end{split} + \end{equation} + \item The \ac{Q} path of the coherent mixer (Figure \ref{fig:ch05:iq_down_circuit}): + \begin{equation} + \begin{split} + x_{B,Q}(t) &= x_{RF}(t) \cdot \underbrace{\cos\left(\omega_{RF} t + \frac{\pi}{2}\right)}_{= \sin\left(\omega_{RF} t\right)} \\ + &\quad \text{Fourier transform:} \\ + \underline{X}_{B,Q}\left(j\omega\right) &= \underline{X}_{RF}\left(j\omega\right) * \left(e^{j\frac{\pi}{2}} \delta\left(\omega-\omega_{RF}\right) + e^{-j\frac{\pi}{2}} \delta\left(\omega+\omega_{RF}\right)\right) \pi \\ + &= \pi \left(e^{j\frac{\pi}{2}}\underline{X}_{RF}\left(j\omega-j\omega_{RF}\right) + e^{-j\frac{\pi}{2}}\underline{X}_{RF}\left(j\omega+j\omega_{RF}\right)\right) \\ + &= \pi \left(\underbrace{e^{j\frac{\pi}{2}} \underline{X}_{RF}^{-}\left(j\omega-j\omega_{RF}\right) + e^{-j\frac{\pi}{2}} \underline{X}_{RF}^{+}\left(j\omega+j\omega_{RF}\right)}_{\text{\acs{I} component of the baseband signal}} \right. \\ &\qquad + \left. \underbrace{e^{j\frac{\pi}{2}} \underline{X}_{RF}^{+}\left(j\omega-j\omega_{RF}\right) + e^{-j\frac{\pi}{2}} \underline{X}_{RF}^{-}\left(j\omega+j\omega_{RF}\right)}_{\text{Mirror frequencies close to $\pm 2 \omega_{RF}$, eliminated by \acs{LPF}}} \right) \\ + &\quad \text{After the \acs{LPF}:} \\ + &= \pi \left(e^{j\frac{\pi}{2}} \underline{X}_{RF}^{-}\left(j\omega-j\omega_{RF}\right) + e^{-j\frac{\pi}{2}} \underline{X}_{RF}^{+}\left(j\omega+j\omega_{RF}\right) \right) \\ + &= j \pi \left( \underline{X}_{RF}^{-}\left(j\omega-j\omega_{RF}\right) - \underline{X}_{RF}^{+}\left(j\omega+j\omega_{RF}\right) \right) + \end{split} + \end{equation} + \item Both $\underline{X}_{B,I}\left(j\omega\right)$ and $\underline{X}_{B,Q}\left(j\omega\right)$ are Hermitian and fulfil the symmetry rules. + \item Thus, both $x_{B,I}(t)$ and $x_{B,Q}(t)$ are real-valued signals. \textit{Anything else would make no sense, because the \ac{I} and \ac{Q} components exist as physical signals at the mixer outputs.} + \item Now, the \ac{I} and \ac{Q} components are composed to a complex-valued signal: + \begin{equation} + \underline{x}_B(t) = x_{B,I}(t) + j \cdot x_{B,Q}(t) + \end{equation} + \item The composition can be thought of interpreting the \ac{I} component $x_{B,I}(t)$ as the real part of $\underline{x}_B(t)$ and the \ac{Q} component $x_{B,Q}(t)$ as the imaginary part of $\underline{x}_B(t)$. This interpretation usually happens in the digital signal processing. + \item The effect of this re-interpretation becomes clear in the frequency-domain: + \begin{equation} + \begin{split} + \underline{X}_{B}\left(j\omega\right) &= \mathcal{F}\left\{\underline{x}_B(t)\right\} = \mathcal{F}\left\{x_{B,I}(t) + j \cdot x_{B,Q}(t)\right\} \\ + &= \mathcal{F}\left\{x_{B,I}(t)\right\} + j \cdot \mathcal{F}\left\{\cdot x_{B,Q}(t)\right\} \\ + &= \pi \left(\underline{X}_{RF}^{-}\left(j\omega-j\omega_{RF}\right) + \underline{X}_{RF}^{+}\left(j\omega+j\omega_{RF}\right) \right) \\ & \qquad + \underbrace{j^2}_{= -1} \pi \left( \underline{X}_{RF}^{-}\left(j\omega-j\omega_{RF}\right) - \underline{X}_{RF}^{+}\left(j\omega+j\omega_{RF}\right) \right) \\ + &= \pi \left( \underline{X}_{RF}^{+}\left(j\omega+j\omega_{RF}\right) + \underline{X}_{RF}^{+}\left(j\omega+j\omega_{RF}\right) \right. \\ & \qquad + \left. \underbrace{\underline{X}_{RF}^{-}\left(j\omega-j\omega_{RF}\right) - \underline{X}_{RF}^{-}\left(j\omega-j\omega_{RF}\right)}_{= 0} \right) \\ + &= 2 \pi \underline{X}_{RF}^{+}\left(j\omega+j\omega_{RF}\right) + \end{split} + \end{equation} + \item $\underline{X}_{B}\left(j\omega\right)$ is not Hermitian. $\underline{x}_B(t)$ is complex-valued. + \item In $\underline{X}_{B}\left(j\omega\right)$ the negative part of the baseband signal $\underline{X}_{RF}^{-}\left(j\omega-j\omega_{RF}\right)$ is completely eliminated. + \item In contrast to that, both $\underline{X}_{B,I}\left(j\omega\right)$ and $\underline{X}_{B,Q}\left(j\omega\right)$ contained the two equivalent parts $\underline{X}_{RF}^{+}\left(j\omega+j\omega_{RF}\right)$ and $\underline{X}_{RF}^{-}\left(j\omega-j\omega_{RF}\right)$. +\end{itemize} + +\textbf{In fact, the band around the \ac{RF} frequency $\omega_{RF}$ is shifted down to zero-\acs{IF} as a whole without losing the information contained. This is the reason, why the phase information of the \ac{RF} signal is retained.} + + +\begin{figure}[H] + \centering + + \subfloat[The \ac{RF} signal and \acs{LO} signal in the frequency-domain] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.1\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{X}_{RF}|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-4.6, + xmax=4.6, + ymin=0, + ymax=1.2, + xtick={-1.9, 0, 1.9}, + xticklabels={$-\omega_{RF}$, $0$, $\omega_{RF}$}, + ytick={0}, + ] + \draw[red, thick] (axis cs:-1.8,0) -- (axis cs:-2,0.7) node[above left,align=right]{$\underline{X}_{RF}^{-}$} -- (axis cs:-2.5,0); + \draw[red, thick] (axis cs:1.8,0) -- (axis cs:2,0.7) node[above right,align=left]{$\underline{X}_{RF}^{+}$} -- (axis cs:2.5,0); + + \draw[-latex, blue, very thick] (axis cs:-1.9,0) -- (axis cs:-1.9,1) node[above,align=center]{\acs{LO}}; + \draw[-latex, blue, very thick] (axis cs:1.9,0) -- (axis cs:1.9,1) node[above,align=center]{\acs{LO}}; + \end{axis} + \end{tikzpicture} + } + + \subfloat[\acs{I} component of the baseband signal in the frequency-domain] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.12\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$\Re\left\{\underline{X}_{B,I}\right\}$ (real)}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-4.6, + xmax=4.6, + ymin=0, + ymax=1.2, + xtick={-3.8, -1.9, 0, 1.9, 3.8}, + xticklabels={$-2\omega_{RF}$, $-\omega_{RF}$, $0$, $\omega_{RF}$, $2\omega_{RF}$}, + ytick={0}, + ] + % X- + \draw[red, dashed, thick] (axis cs:-3.7,0) -- (axis cs:-3.9,0.7) node[above right,align=left]{Eliminated by \acs{LPF}} -- (axis cs:-4.4,0); + \draw[red, thick] (axis cs:0.1,0) -- (axis cs:-0.1,0.7) -- (axis cs:-0.6,0); + + % X+ + \draw[red, dashed, thick] (axis cs:3.7,0) -- (axis cs:3.9,0.7) node[above left,align=right]{Eliminated by \acs{LPF}} -- (axis cs:4.4,0); + \draw[red, thick] (axis cs:-0.1,0) -- (axis cs:0.1,0.7) -- (axis cs:0.6,0); + \end{axis} + \end{tikzpicture} + } + + \subfloat[\acs{Q} component of the baseband signal in the frequency-domain] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.2\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$\Im\left\{\underline{X}_{B,Q}\right\}$ (imaginary)}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-4.6, + xmax=4.6, + ymin=-1.2, + ymax=1.2, + xtick={-3.8, -1.9, 0, 1.9, 3.8}, + xticklabels={$-2\omega_{RF}$, $-\omega_{RF}$, $0$, $\omega_{RF}$, $2\omega_{RF}$}, + ytick={0}, + ] + % X- + \draw[red, dashed, thick] (axis cs:-3.7,0) -- (axis cs:-3.9,-0.7) node[below right,align=left]{Eliminated by \acs{LPF}} -- (axis cs:-4.4,0); + \draw[red, thick] (axis cs:0.1,0) -- (axis cs:-0.1,0.7) -- (axis cs:-0.6,0); + + % X+ + \draw[red, dashed, thick] (axis cs:3.7,0) -- (axis cs:3.9,0.7) node[above left,align=right]{Eliminated by \acs{LPF}} -- (axis cs:4.4,0); + \draw[red, thick] (axis cs:-0.1,0) -- (axis cs:0.1,-0.7) -- (axis cs:0.6,0); + \end{axis} + \end{tikzpicture} + } + + \subfloat[Complex-valued baseband signal in the frequency-domain] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.1\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{X}_{B}|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-4.6, + xmax=4.6, + ymin=0, + ymax=1.2, + xtick={-3.8, -1.9, 0, 1.9, 3.8}, + xticklabels={$-2\omega_{RF}$, $-\omega_{RF}$, $0$, $\omega_{RF}$, $2\omega_{RF}$}, + ytick={0}, + ] + \draw[red, thick] (axis cs:-0.1,0) -- (axis cs:0.1,0.7) -- (axis cs:0.6,0); + \end{axis} + \end{tikzpicture} + } + + \caption{Coherent down-conversion} + \label{fig:ch05:iq_down_freqdomain} +\end{figure} + +\subsubsection{Up-Conversion} + +The process can be reversed. +\begin{itemize} + \item A complex-valued baseband signal can be mixed to a real-valued \ac{RF} signal. + \item The spectrum of the baseband signal is shifted up to $\omega_{RF}$ as a whole without losing any information. + \item The device mixing the complex baseband up to the \ac{RF} band is called \index{IQ modulator} \textbf{IQ modulator}. + \item The \emph{IQ modulator} is the counterpart of the \emph{IQ demodulator} (\emph{coherent mixer}). +\end{itemize} + +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.8} + \begin{circuitikz} + \node[block, draw, minimum height=8cm](DSP) at(0cm,0cm){Digital\\ signal\\ processing}; + \node[mixer](MixI) at([shift={(6cm,3cm)}] DSP.east) {}; + \node[mixer](MixQ) at([shift={(6cm,-3cm)}] DSP.east) {}; + \node[oscillator](LO) at([shift={(4cm,0cm)}] DSP.east){}; + \node[adder](Add) at([shift={(10cm,0cm)}] DSP.east){}; + + \draw (LO.south) node[below,align=center,yshift=-3mm]{\acs{LO}\\ $\omega_{LO} = \omega_{RF}$}; + \draw (MixI.north) node[above,align=center,yshift=1cm]{In-phase (\acs{I}) branch}; + \draw (MixQ.south) node[below,align=center,yshift=-1cm]{Quadrature (\acs{Q}) branch}; + + \draw (LO.east) to[short,-*] ([shift={(6cm,0cm)}] DSP.east); + \draw ([shift={(6cm,0cm)}] DSP.east) to[short] (MixI.south); + \draw ([shift={(6cm,0cm)}] DSP.east) to[phaseshifter,l=$\SI{90}{\degree}$] (MixQ.north); + + \draw ([shift={(0cm,3cm)}] DSP.east) to[dac] ++(2cm,0cm) to[lowpass] ++(2cm,0cm) to[amp] (MixI.west); + \draw ([shift={(0cm,-3cm)}] DSP.east) to[dac] ++(2cm,0cm) to[lowpass] ++(2cm,0cm) to[amp] (MixQ.west); + + \draw[-latex] (MixI.east) to[lowpass] ++(2cm,0cm) -| (Add.north); + \draw[-latex] (MixQ.east) to[lowpass] ++(2cm,0cm) -| (Add.south); + \draw[-latex] (Add.east) -- ++(1cm,0cm) node[right,align=left]{\acs{RF}\\ signal}; + \end{circuitikz} + \end{adjustbox} + \caption{IQ modulator mixing up a complex-valued zero-\acs{IF} baseband} + \label{fig:ch05:iq_up_circuit} +\end{figure} +\begin{figure}[H] + \centering + + \subfloat[Complex-valued baseband signal and \acs{LO} signal in the frequency-domain] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.1\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{X}_{B}|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-3, + xmax=3, + ymin=0, + ymax=1.2, + xtick={-1.9, 0, 1.9}, + xticklabels={$-\omega_{RF}$, $0$, $\omega_{RF}$}, + ytick={0}, + ] + \draw[red, thick] (axis cs:-0.1,0) -- (axis cs:0.1,0.7) -- (axis cs:0.6,0); + + \draw[-latex, blue, very thick] (axis cs:-1.9,0) -- (axis cs:-1.9,1) node[above,align=center]{\acs{LO}}; + \draw[-latex, blue, very thick] (axis cs:1.9,0) -- (axis cs:1.9,1) node[above,align=center]{\acs{LO}}; + \end{axis} + \end{tikzpicture} + } + + \subfloat[The \ac{RF} signal in the frequency-domain] { + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.1\textheight}, + width=0.9\linewidth, + scale only axis, + xlabel={$\omega$}, + ylabel={$|\underline{X}_{RF}|$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + grid=none, + legend pos=north east, + axis y line=middle, + axis x line=middle, + every axis x label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=north, + }, + every axis y label/.style={ + at={(ticklabel* cs:1.05)}, + anchor=east, + }, + xmin=-3, + xmax=3, + ymin=0, + ymax=1.2, + xtick={-1.9, 0, 1.9}, + xticklabels={$-\omega_{RF}$, $0$, $\omega_{RF}$}, + ytick={0}, + ] + \draw[red, thick] (axis cs:-1.8,0) -- (axis cs:-2,0.7) -- (axis cs:-2.5,0); + \draw[red, thick] (axis cs:1.8,0) -- (axis cs:2,0.7) -- (axis cs:2.5,0); + \end{axis} + \end{tikzpicture} + } + + \caption{Up-conversion of a complex-valued baseband signal} + \label{fig:ch05:iq_up_freqdomain} +\end{figure} + +The \ac{RF} signal is always real-valued and contains the basebased shifted as a whole (including its non-symmetric positive and negative parts) to the \ac{RF} frequency $\omega_{RF}$. + +\begin{proof}{} + The complex-valued baseband signal $\underline{x}_{B}(t)$ can be decomposed into its real and imaginary values, the \ac{I} and \ac{Q} components. + \begin{equation} + \underline{x}_{B}(t) = x_{B,I}(t) + j \cdot x_{B,Q}(t) + \end{equation} + In the frequency-domain: + \begin{equation} + \underline{X}_{B}\left(j\omega\right) = \underline{X}_{B,I}\left(j\omega\right) + j \cdot \underline{X}_{B,Q}\left(j\omega\right) + \label{eq:ch05:baseband_tx_freqdom} + \end{equation} + + Both $\underline{X}_{B,I}\left(j\omega\right)$ and $\underline{X}_{B,Q}\left(j\omega\right)$ must be Hermitian (real-valued in time-domain) and can be decomposed into: + \begin{subequations} + \begin{align} + \underline{X}_{B,I}\left(j\omega\right) &= \begin{cases} + \underline{X}_{B,I}^{+}\left(j\omega\right) & \quad \text{ if } \omega \geq 0 \\ + \underline{X}_{B,I}^{-}\left(j\omega\right) & \quad \text{ if } \omega \leq 0 + \end{cases} \\ + \underline{X}_{B,Q}\left(j\omega\right) &= \begin{cases} + \underline{X}_{B,Q}^{+}\left(j\omega\right) & \quad \text{ if } \omega \geq 0 \\ + \underline{X}_{B,Q}^{-}\left(j\omega\right) & \quad \text{ if } \omega \leq 0 + \end{cases} + \end{align} + \end{subequations} + + Following conditions must be true to fulfil \eqref{eq:ch05:baseband_tx_freqdom} and the symmetry rules of $\underline{X}_{B,I}\left(j\omega\right)$ and $\underline{X}_{B,Q}\left(j\omega\right)$: + \begin{subequations} + \begin{align} + \underline{X}_{B,I}^{+}\left(j\omega\right) = -j \underline{X}_{B,Q}^{+}\left(j\omega\right) &= \begin{cases} + \frac{1}{2} \underline{X}_{B}\left(j\omega\right) & \quad \text{ if } \omega \geq 0 \\ + 0 & \quad \text{ if } \omega \leq 0 + \end{cases} \\ + \underline{X}_{B,I}^{-}\left(j\omega\right) = j \underline{X}_{B,Q}^{-}\left(j\omega\right) &= \begin{cases} + 0 & \quad \text{ if } \omega \geq 0 \\ + \frac{1}{2} \overline{\underline{X}_{B}\left(j\omega\right)} & \quad \text{ if } \omega \leq 0 + \end{cases} + \end{align} + \end{subequations} +\end{proof} + \section{Digital Modulation Techniques} \subsection{Amplitude-Shift Keying} @@ -905,10 +1610,12 @@ So, the input signal's spectrum consists of a positive and a negative part: \todo{signal chain: S/P -> constellation diagram -> iFFT -> IQ} -\subsection{Coherent and Non-Coherent Demodulation} +%\subsection{Coherent and Non-Coherent Demodulation} \subsection{Inter-Symbol Interference} +\todo{Cyclic Prefixes? No -> OFDM} + \subsection{Synchronization 2: Carrier Recovery} \todo{Frequency and phase offset} diff --git a/common/acronym.tex b/common/acronym.tex index 06d52d2..c605c5a 100644 --- a/common/acronym.tex +++ b/common/acronym.tex @@ -69,6 +69,7 @@ \acro{HF}{high frequency} \acro{HPF}{high pass filter} \acro{HTTP}{Hypertext Transfer Protocol} + \acro{I}{in-phase} \acro{IC}{integrated circuit} \acro{ID}{identification} \acro{IEEE}{Institute of Electrical and Electronics Engineers} @@ -125,6 +126,7 @@ \acro{PPDU}{physical layer protocol data unit} \acro{PSDU}{physical layer service data unit} \acro{PSK}{phase-shift keying} + \acro{Q}{quadrature} \acro{QAM}{quadrature amplitude modulation} \acro{QED}{quod erat demonstrandum} \acro{QOS}{quality of service} -- cgit v1.1