From c13283b6e9c7c663820e33786fd3efbd9b741ca0 Mon Sep 17 00:00:00 2001 From: Philipp Le Date: Sat, 23 May 2020 23:56:36 +0200 Subject: WIP: Chapter 4 - Quantization --- chapter04/content_ch04.tex | 430 ++++++++++++++++++++++++++++++++++++++++----- 1 file changed, 386 insertions(+), 44 deletions(-) diff --git a/chapter04/content_ch04.tex b/chapter04/content_ch04.tex index b3a3f97..64a098e 100644 --- a/chapter04/content_ch04.tex +++ b/chapter04/content_ch04.tex @@ -880,41 +880,6 @@ The normalization is of minor importance for the \ac{DTFT}, but must be consider Both expressions are equivalent. \end{definition} -\begin{excursus}{z-Transform} - Analogous to the Fourier and Laplace transform, the \acf{DTFT} is a special case of the z-transform. The \index{z-transform} \textbf{z-transform} is: - \begin{equation} - \underline{X}\left(\underline{z}\right) = \mathcal{Z}\left\{\underline{x}[n]\right\} = \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot \underline{z}^{-n} - \end{equation} - $\underline{z}$ is the complex frequency variable, which can be decomposed into: - \begin{equation} - \underline{z} = A e^{j \phi} - \end{equation} - where $A$ represents the gain and $e^{j \phi}$ the frequency. - \begin{figure}[H] - \centering - \begin{tikzpicture} - \draw[->] (-2.2,0) -- (2.2,0) node[below, align=left]{$\Re\left\{\underline{z}\right\}$}; - \draw[->] (0,-2.2) -- (0,2.2) node[left, align=right]{$\Im\left\{\underline{z}\right\}$}; - %\draw (0:1) arc(0:360:1); - \draw (1,0.2) -- (1,-0.2) node[below]{$1$}; - \draw (-1,0.2) -- (-1,-0.2) node[below]{$-1$}; - \draw (0.2,1) -- (-0.2,1) node[left]{$1$}; - \draw (0.2,-1) -- (-0.2,-1) node[left]{$-1$}; - - \draw[thick, red] (0:1) arc(0:360:1); - \draw[dashed, red] (60:1) -- (45:1.5) node[right, align=left, color=red]{$e^{j \phi}$}; - \end{tikzpicture} - \caption{Complex plane of the complex frequency variable $\underline{z}$} - \label{fig:ch04:ztrafo_z_cmplx_plane} - \end{figure} - - In the \acf{DTFT}, $A = 1$ as a special case. The remainig $e^{j \phi}$ describes the unit circle in the complex plane. Like the Fourier transform, it assumes a steady-state, whereas the z-transform delivers a complete description of a time-discrete system. The z-transform is preferred for transient analysis of a time-discrete system. Its zeros $\underline{z}_0$ and poles $\underline{z}_\infty$ determine the stability of the system. - - \vspace{0.5em} - - Figure \ref{fig:ch04:ztrafo_z_cmplx_plane} makes evident the $2 \pi$-periodicity of both the \ac{DTFT} and z-transform. The frequency $e^{j \phi}$ repeats every $2 \pi$. -\end{excursus} - \subsubsection{Properties} The \ac{DTFT} is derived from the \ac{CTFT}. Therefore, all properties apply likewise. @@ -926,6 +891,51 @@ The \ac{DTFT} is derived from the \ac{CTFT}. Therefore, all properties apply lik \item Symmetry rules \end{itemize} +\subsection{z-Transform} + +Analogous to the Fourier and Laplace transform, the \acf{DTFT} is a special case of the z-transform. + +\begin{definition}{Discrete-time Fourier transform} + The \index{z-transform} \textbf{z-transform} of a time-discrete signal $\underline{x}[n]$ with the sampling period $T_S$ is: + \begin{equation} + \underline{X}\left(\underline{z}\right) = \mathcal{Z}\left\{\underline{x}[n]\right\} = \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot \underline{z}^{-n} + \end{equation} + $\underline{z}$ is the complex frequency variable. + + The \index{z-transform!inverse} \textbf{inverse z-transform} is: + \begin{equation} + \underline{x}[n] = \mathcal{Z}^{-1}\left\{\underline{x}[n]\right\} = \frac{1}{2 \pi j} \oint\limits_{C} \underline{X}\left(\underline{z}\right) \underline{z}^{n-1} \, \mathrm{d} \underline{z} + \end{equation} + $C$ is a counter-clockwise closed path enclosing the origin and the region of convergence. In the case of the \ac{DTFT}, $C$ is the unit circle, i.e., $C = [e^{-j \pi}, e^{j \pi}]$. +\end{definition} + +$\underline{z}$ can be decomposed into: +\begin{equation} + \underline{z} = A e^{j \phi} +\end{equation} +where $A$ represents the gain and $e^{j \phi}$ the frequency. +\begin{figure}[H] + \centering + \begin{tikzpicture} + \draw[->] (-2.2,0) -- (2.2,0) node[below, align=left]{$\Re\left\{\underline{z}\right\}$}; + \draw[->] (0,-2.2) -- (0,2.2) node[left, align=right]{$\Im\left\{\underline{z}\right\}$}; + %\draw (0:1) arc(0:360:1); + \draw (1,0.2) -- (1,-0.2) node[below]{$1$}; + \draw (-1,0.2) -- (-1,-0.2) node[below]{$-1$}; + \draw (0.2,1) -- (-0.2,1) node[left]{$1$}; + \draw (0.2,-1) -- (-0.2,-1) node[left]{$-1$}; + + \draw[thick, red] (0:1) arc(0:360:1); + \draw[dashed, red] (60:1) -- (45:1.5) node[right, align=left, color=red]{$e^{j \phi}$}; + \end{tikzpicture} + \caption{Complex plane of the complex frequency variable $\underline{z}$} + \label{fig:ch04:ztrafo_z_cmplx_plane} +\end{figure} + +In the \acf{DTFT}, $A = 1$ as a special case. The remainig $e^{j \phi}$ describes the unit circle in the complex plane. Like the Fourier transform, it assumes a steady-state, whereas the z-transform delivers a complete description of a time-discrete system. The z-transform is preferred for transient analysis of a time-discrete system. Its zeros $\underline{z}_0$ and poles $\underline{z}_\infty$ determine the stability of the system. + +Figure \ref{fig:ch04:ztrafo_z_cmplx_plane} makes evident the $2 \pi$-periodicity of both the \ac{DTFT} and z-transform. The frequency $e^{j \phi}$ repeats every $2 \pi$. + \subsection{Discrete Fourier Transform} \subsubsection{Periodic Sequences} @@ -1014,12 +1024,14 @@ This is the inverse \ac{DFT}. Again the summation boundaries of $[-\frac{N}{2}, \begin{definition}{Discrete Fourier transform} The \index{discrete Fourier transform} \textbf{\acf{DFT}} of a $N$-periodic sequence $\underline{x}[n]$ is: \begin{equation} - \underline{X}[k] = \sum\limits_{n \in N} \underline{x}[n] \cdot e^{-j 2\pi \frac{k}{N} n} + \underline{X}[k] = \mathcal{F}_{\text{DFT}}\left\{\underline{x}[n]\right\} = \sum\limits_{n \in N} \underline{x}[n] \cdot e^{-j 2\pi \frac{k}{N} n} + \label{eq:ch04:dft} \end{equation} The \index{inverse discrete Fourier transform} \textbf{inverse discrete Fourier transform} is: \begin{equation} - \underline{x}[n] = \frac{1}{N} \sum\limits_{k \in N} \underline{X}[k] \cdot e^{+ j 2\pi \frac{k}{N} n} + \underline{x}[n] = \mathcal{F}_{\text{DFT}}^{-1}\left\{\underline{X}[k]\right\} = \frac{1}{N} \sum\limits_{k \in N} \underline{X}[k] \cdot e^{+ j 2\pi \frac{k}{N} n} + \label{eq:ch04:idft} \end{equation} Both $\underline{X}[k]$ and $\underline{x}[n]$ are $N$-periodic. The summation boundaries can be chosen to any sequence of length $N$. @@ -1036,8 +1048,88 @@ The \ac{DFT} is derived from the \ac{DTFT} and \ac{CTFT}. Therefore, all propert \item Symmetry rules \end{itemize} +\subsection{Orthogonality of the \acs{DFT} Frequency Vectors} + +Both the time-domain sequence $\underline{x}[n]$ and frequency-domain sequence $\underline{X}[k]$ can be interpreted as vectors: +\begin{itemize} + \item $\cmplxvect{x} = \left[\underline{x}[0], \underline{x}[1], \dots, \underline{x}[N-1]\right]^{\mathrm{T}}$ + \item $\cmplxvect{X} = \left[\underline{X}[0], \underline{X}[1], \dots, \underline{X}[N-1]\right]^{\mathrm{T}}$ +\end{itemize} + +The \ac{DFT} \eqref{eq:ch04:dft} can be expressed as a linear system of equation: +\begin{equation} + \cmplxvect{X} = \underline{\mat{F}} \cdot \cmplxvect{x} +\end{equation} + +The $N \times N$ transformation matrix $\underline{\mat{F}}$ is the \index{DFTmatrx} \textbf{ac{DFT} matrix} with the elements: +\begin{equation} + \underline{F}_{pq} = \underline{w}^{p \cdot q} +\end{equation} +where $\underline{w}$ is the $N$-th \index{primitive root of unity} \textbf{primitive root of unity}\footnote{The primitive root of unity divide the unit circle $e^{j \phi}$ into equally sized segments.}. +\begin{equation} + \underline{w} = e^{j \frac{2 \pi}{N}} +\end{equation} +So +\begin{equation} + \underline{\mat{F}} = \left[ + \begin{matrix} + 1 & 1 & 1 & 1 & \ldots & 1 \\ + 1 & \underline{w} & \underline{w}^2 & \underline{w}^3 & \ldots & \underline{w}^{N-1} \\ + 1 & \underline{w}^2 & \underline{w}^4 & \underline{w}^6 & \ldots & \underline{w}^{2\left(N-1\right)} \\ + 1 & \underline{w}^3 & \underline{w}^6 & \underline{w}^9 & \ldots & \underline{w}^{3\left(N-1\right)} \\ + \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ + 1 & \underline{w}^{N-1} & \underline{w}^{2\left(N-1\right)} & \underline{w}^{3\left(N-1\right)} & \ldots & \underline{w}^{\left(N-1\right)\left(N-1\right)} \\ + \end{matrix} + \right] +\end{equation} + +The inverse \ac{DFT} is, using the conjugate complex $\overline{\underline{\mat{F}}}$: +\begin{equation} + \cmplxvect{x} = \frac{1}{N} \overline{\underline{\mat{F}}} \cdot \cmplxvect{X} +\end{equation} + +Each row and column of $\underline{\mat{F}}$ is a vector of powers of the $N$-th primitive root of unity $\underline{w}$. A row with the index $k$ is $\cmplxvect{u}_k$. +\begin{equation} + \begin{split} + \cmplxvect{u}_k &= \left[\left.\underline{w}^{k \cdot q}\right| q = 0, 1, \dots, N-1\right]^{\mathrm{T}} \\ + &= \left[1, \underline{w}^{k}, \underline{w}^{2 k}, \underline{w}^{3 k}, \dots, \underline{w}^{k \left(N-1\right)} \right]^{\mathrm{T}} \\ + &= \left[1, e^{j \frac{2 \pi}{N} k}, e^{j \frac{2 \pi}{N} 2 k}, e^{j \frac{2 \pi}{N} 3 k}, \dots, e^{j \frac{2 \pi}{N} k \left(N-1\right)} \right]^{\mathrm{T}} \\ + \end{split} +\end{equation} +Each vector $\cmplxvect{u}_k$ is the basis for the associated frequency sample $\underline{X}[k]$. + +It can be shown that the vectors $\cmplxvect{u}_k$ are orthogonal. They form an \index{orthogonal basis} \textbf{orthogonal basis}. This can be proven by their inner product: +\begin{equation} + \begin{split} + \langle \cmplxvect{u}_p, \overline{\cmplxvect{u}_q} \rangle &= \sum\limits_{n=0}^{N-1} \left(e^{j \frac{2 \pi}{N} p n}\right) \overline{\left(e^{j \frac{2 \pi}{N} q n}\right)} \\ + &= \sum\limits_{n=0}^{N-1} e^{j \frac{2 \pi}{N} \left(p - q\right) n} \\ + &= N \delta_{pq} + \end{split} +\end{equation} + +\textit{Remark:} $\delta_{pq}$ is the Kronecker delta here. + +\begin{itemize} + \item The vectors $\cmplxvect{u}_p$ and $\cmplxvect{u}_q$ are orthogonal ($\delta_{pq} = 0$) for $p \neq q$. + \item $\delta_{pq}$ is non-zero only if $p = q$. +\end{itemize} + +\begin{fact} + The basis of the frequency samples of a \ac{DFT} are orthogonal. +\end{fact} + +\subsection{Windowing Non-Periodic Signals} + +\todo{Windowing and Periodic continuation} + +\todo{Window Filters} + +\todo{Spectral Leakage} + \section{Analogies Of Time-Continuous and Time-Discrete Signals and Systems} +All relations shown here are analogous to the \ac{CTFT}. Their deduction is analogous to Chapters 2 and 3. + \subsection{Transforms} \begin{table}[H] @@ -1061,7 +1153,7 @@ The \ac{DFT} is derived from the \ac{DTFT} and \ac{CTFT}. Therefore, all propert \vspace{0.5em} - Fourier transform: + \acf{CTFT}: \begin{equation*} \underline{X}(j \omega) = \int\limits_{t = -\infty}^{\infty} \underline{x}(t) \cdot e^{-j \omega t} \, \mathrm{d} t \end{equation*} @@ -1082,7 +1174,7 @@ The \ac{DFT} is derived from the \ac{DTFT} and \ac{CTFT}. Therefore, all propert \vspace{0.5em} - Discrete-time Fourier transform: + \acf{DTFT}: \begin{equation*} \underline{X}_{2\pi}(e^{j \phi}) = \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot e^{- j \phi n} \end{equation*} @@ -1126,7 +1218,7 @@ The \ac{DFT} is derived from the \ac{DTFT} and \ac{CTFT}. Therefore, all propert \vspace{0.5em} - Discrete Fourier transform: + \acf{DFT}: \begin{equation*} \underline{X}[k] = \sum\limits_{n = 0}^{N - 1} \underline{x}[n] \cdot e^{- j \frac{2 \pi}{N} k n} \end{equation*} @@ -1142,21 +1234,271 @@ The \ac{DFT} is derived from the \ac{DTFT} and \ac{CTFT}. Therefore, all propert \end{itemize} \end{minipage} +\subsubsection{Properties of the \acs{DFT}} + +\begin{itemize} + \item Linearity: + \begin{equation} + \mathcal{F}_{\text{DFT}}\left\{a \underline{x}[n] + b \underline{y}[n]\right\} = a \mathcal{F}_{\text{DFT}}\left\{\underline{x}[n]\right\} + b \mathcal{F}_{\text{DFT}}\left\{\underline{y}[n]\right\} + \end{equation} + \item Time shift: + \begin{equation} + \mathcal{F}_{\text{DFT}}\left\{\underline{x}[n - m]\right\}[k] = \underline{X}[k] \cdot e^{-j 2 \pi \frac{k}{N} m} + \end{equation} + \item Frequency shift: + \begin{equation} + \mathcal{F}_{\text{DFT}}\left\{\underline{x}[n] \cdot e^{-j 2 \pi \frac{n}{N} m}\right\}[k] = \underline{X}[k-m] + \end{equation} + \item Multiplication in the time-domain becomes convolution in the frequency domain: + \begin{equation} + \mathcal{F}_{\text{DFT}}\left\{\underline{x}[n] \cdot \underline{y}[n]\right\}[k] = \underline{X}[k] * \underline{Y}[k] = \sum_{l=0}^{N} \underline{X}[l] \underline{Y}[(k - l) \mod N] + \end{equation} + \item Convolution in the time-domain becomes multiplication in the frequency domain: + \begin{equation} + \mathcal{F}_{\text{DFT}}\left\{\underline{x}[n] * \underline{y}[n]\right\}[k] = \mathcal{F}_{\text{DFT}}\left\{\sum_{l=0}^{N} \underline{x}[l] \underline{y}[(n - l) \mod N]\right\}[k] = \underline{X}[k] \cdot \underline{Y}[k] + \end{equation} + \item Duality: + \begin{equation} + \mathcal{F}_{\text{DFT}}\left\{\underline{X}[n]\right\}[k] = N \cdot \underline{x}[N - k] + \end{equation} + where $\underline{x} \TransformHoriz \underline{X}$. + \item Symmetry for real-valued $\underline{x}[n]$: + \begin{equation} + \underline{X}[k] = \overline{\underline{X}[N-k]} \qquad \forall \; \underline{x}[n] \in \mathbb{R} + \end{equation} +\end{itemize} + +%\subsubsection{Spectrum} + \subsection{Systems} -\subsection{Cross-Correlation and Autocorrelation} +\textit{Remark:} In contrast to signals, systems are analysed using the z-transform (general form of the \ac{DTFT}). For signals, the \ac{DFT} is preferred. + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \node[draw, block] (System) {System\\ $\underline{h}[n]$}; + \draw[<-o] (System.west) -- ++(-2cm, 0) node[above, align=center]{Input signal\\ $\underline{x}[n]$}; + \draw[->] (System.east) -- ++(2cm, 0) node[above, align=center]{Output signal\\ $\underline{y}[n]$}; + \end{tikzpicture} + \caption{A time-discrete system with input and output} +\end{figure} + +A time-discrete system is characterized by either +\begin{itemize} + \item its \index{transfer function} transfer function + \begin{equation} + \underline{H}(\underline{z}) = \frac{\underline{Y}(\underline{z})}{\underline{X}(\underline{z})} + \end{equation} + or + \item its impulse response. + \begin{equation} + \underline{h}[n] = \mathcal{Z}^{-1}\left\{\underline{H}(\underline{z})\right\} + \end{equation} +\end{itemize} + +In the time domain, the output is a convolution of the input and the impulse response. +\begin{equation} + \underline{y}[n] = \underline{h}[n] * \underline{x}[n] = \sum\limits_{l = -\infty}^{\infty} \underline{h}[l] \underline{x}[n-l] +\end{equation} + +The systems output is the impulse response $\underline{y}[n] = \underline{h}[n]$ if the input is a Kronecker delta function $\underline{x}[n] = \delta[n]$. +\begin{equation} + \underline{h}[n] * \delta[n] = \underline{h}[n] +\end{equation} +Or in the frequency domain +\begin{equation} + \underline{H}(\underline{z}) \cdot \underbrace{\mathcal{Z}\left\{\delta[n]\right\}}_{= 1} = \underline{H}(\underline{z}) +\end{equation} + +\begin{excursus}{Kronecker delta} + The \index{Kronecker delta} Kronecker delta $\delta[n]$ equivalent of the Dirac delta function $\delta(t)$ in the discrete domain. + + \begin{equation*} + \delta(t) = \begin{cases} + \infty & \quad \text{if } t = 0, \\ + 0 & \quad \text{if } t \neq 0. + \end{cases} + \end{equation*} + + \begin{equation*} + \delta[n] = \begin{cases} + 1 & \quad \text{if } n = 0, \\ + 0 & \quad \text{if } n \neq 0. + \end{cases} + \end{equation*} + + The Dirac delta function $\delta(t)$ is an indefinitely narrow pulse but indefinitely high. In contrast to that, the Kronecker delta $\delta[n]$ is of unity length and unity height. Both functions sum up to $1$. + \begin{equation} + \int\limits_{-\infty}^{\infty} \delta(t) \, \mathrm{d} t = \sum\limits_{n = -\infty}^{\infty} \delta[n] = 1 + \end{equation} +\end{excursus} \subsection{Spectral Density} -\subsection{Noise} +\subsubsection{Cross-Correlation and Autocorrelation} + +All considerations apply for ergodic or \ac{WSS} processes only: + +\begin{itemize} + \item Cross-correlation: + \begin{equation} + \underline{\mathrm{R}}_{xy}[n] = \left(\underline{x} \star \underline{y}\right)[n] = \sum\limits_{m=0}^{N-1} \underline{x}[m] \overline{\underline{y}[(m+n) \mod N]} + \end{equation} + \item Autocorrelation: + \begin{equation} + \underline{\mathrm{R}}_{xx}[n] = \left(\underline{x} \star \underline{x}\right)[n] = \sum\limits_{m=0}^{N-1} \underline{x}[m] \overline{\underline{x}[m+n]} + \end{equation} +\end{itemize} + +\subsection{Energy Spectral Density} + +%TODO + +Parseval's theorem for discrete systems: +\begin{equation} + \sum\limits_{n=0}^{N-1} \left|\underline{x}[n]\right|^2 = \frac{1}{N} \sum\limits_{k=0}^{N-1} \left|\underline{X}[k]\right|^2 +\end{equation} + +\begin{equation} + E = \sum\limits_{n=0}^{N-1} \left|\underline{x}[n]\right|^2 +\end{equation} + +\begin{equation} + \underline{\mathrm{S}}_{E,xx}[k] = \left|\underline{X}[k]\right|^2 +\end{equation} + +%\subsection{Noise} \section{Digital Signals and Systems} +Now, we are in the time-discrete domain. However, values are still continuous. + +Let's recapitulate the signal processing chain from the analogue to digital signals from Chapter 1: +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.8} + \begin{tikzpicture} + \draw node[draw, block](Continuous){Value-continuous,\\ time-continuous\\ signal}; + \draw node[draw, block, right=3cm of Continuous](Sampled){Value-continuous,\\ time-discrete\\ signal}; + \draw node[draw, block, right=3cm of Sampled](Digital){Value-discrete,\\ time-discrete\\ signal}; + + \draw [-latex] (Continuous) -- node[midway, align=center, above]{Sampling} (Sampled); + \draw [-latex] (Sampled) -- node[midway, align=center, above]{Quantization} (Digital); + + \draw[decorate, decoration={brace, amplitude=3mm, mirror}] ([yshift=-5mm] Continuous.south west) -- ([yshift=-5mm] Sampled.south east) node[midway, below, yshift=-3mm]{\textbf{Analogue}}; + \draw[decorate, decoration={brace, amplitude=3mm, mirror}] ([yshift=-5mm] Digital.south west) -- ([yshift=-5mm] Digital.south east) node[midway, below, yshift=-3mm]{\textbf{Digital}}; + \end{tikzpicture} + \end{adjustbox} + \caption{Conversion from analogue to digital signals (recap from Chapter 1)} + \label{fig:ch04:signals_sampling_recap} +\end{figure} + +The device converting an analogue signal to a digital signal is a \index{analog-to-digital converter} \textbf{\ac{ADC}}. An \ac{ADC} comprises the two processes \emph{sampling} and \emph{quantization}. + \subsection{Quantization} -\subsection{Quantization Error} +Quantization is the process of +\begin{itemize} + \item \textbf{mapping} the continuous (analogue) values of the samples to a finite set of discrete (digital) of values + \item by \textbf{rounding} and \textbf{truncating} the values. +\end{itemize} + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.25\textheight}, + width=0.6\linewidth, + scale only axis, + xlabel={$t$}, + ylabel={$x$}, + %grid style={line width=.6pt, color=lightgray}, + %grid=both, + xmajorgrids=false, + ymajorgrids=true, + grid style={color=lightgray, dashed}, + axis lines=left, + legend pos=north east, + xmin=0, + xmax=7, + ymin=0, + ymax=3, + xtick={0, 1, ..., 6}, + ytick={0, 0.5, ..., 2.5} + ] + \addplot[smooth, blue, dashed] coordinates {(0, 1.1) (1, 1.8) (2, 2.1) (3, 1.0) (4, 0.8) (5, 1.7) (6, 2.4)}; + \addplot[red, thick] coordinates {(0, 0) (0, 1.0)}; + \addplot[red, thick] coordinates {(1, 0) (1, 2.0)}; + \addplot[red, thick] coordinates {(2, 0) (2, 2.0)}; + \addplot[red, thick] coordinates {(3, 0) (3, 1.0)}; + \addplot[red, thick] coordinates {(4, 0) (4, 1.0)}; + \addplot[red, thick] coordinates {(5, 0) (5, 1.5)}; + \addplot[red, thick] coordinates {(6, 0) (6, 2.5)}; + \addplot[only marks, red, thick, mark=o] coordinates {(0, 1.0) (1, 2.0) (2, 2.0) (3, 1.0) (4, 1.0) (5, 1.5) (6, 2.5)}; + \end{axis} + \end{tikzpicture} + \caption[A digital, value-discrete, time-discrete signal]{A digital, value-discrete, time-discrete signal. Only certain time points and a limited set of values (in this case multiples of $0.5$) are valid. (Recap from Chapter 1)} + \label{fig:ch04:recap2} +\end{figure} + +The mapping is an irreversible function $\mathcal{Q}\left\{\cdot\right\}$. + +\begin{definition}{Quantization} + In this chapter, the process of quantization is denoted by $\mathcal{Q}\left\{\cdot\right\}$. The digital signal $\underline{x}_Q[n]$ can be distinguished by its index $Q$ from its analogue counterpart $\underline{x}[n]$. + \begin{equation} + \underline{x}_Q[n] = \mathcal{Q}\left\{\underline{x}[n]\right\} + \end{equation}% + \nomenclature[Fq]{$\mathcal{Q}\left\{\cdot\right\}$}{Quantization} + + Later chapters will assume digital signals, unless noted otherwise. The index $Q$ will not be used there. +\end{definition} + +The finite set of discrete numbers has the length $K$. +\begin{itemize} + \item There are $K$ possible, unique values of $\underline{x}_Q[n]$. + \item Usually, $K$ is a power of $2$. $K = 2^M$. $M$ is the number of bits. +\end{itemize} + +\subsubsection{Linear Mapping} + +The most common implementations distribute the $K$ discrete values equally between an interval of the continuous values $[\underline{\hat{X}}_L, \underline{\hat{X}}_H]$. So, the discrete values are spaced by +\begin{equation} + \Delta \underline{\hat{X}} = \frac{\underline{\hat{X}}_H - \underline{\hat{X}}_L}{K} +\end{equation} +This is called \index{linear mapping} \textbf{linear mapping}. + +\todo{step map} + +\todo{Example circuit} + +\textit{Remark:} There are other mapping like logarithmic mapping. However, this lecture only considers linear mapping. + +\subsection{Quantization Noise} + +Once values have been quantized, their original, continuous values cannot be reconstructed. + +\begin{fact} + The process of quantization is irreversible. +\end{fact} + +Furthermore, the quantized values differ from their original value due to rounding and truncation. This difference $\underline{e}[n]$ is the \index{quantization error} \textbf{quantization error}. + +\begin{definition}{Quantization error} + Each value-discrete (quantized) value $\underline{x}_Q[n]$ has an error $\underline{e}[n]$ from its original, value-continuous, analogue value $\underline{x}[n]$. + \begin{equation} + \underline{x}_Q[n] = \mathcal{Q}\left\{\underline{x}[n]\right\} = \underline{x}[n] + \underline{e}[n] + \end{equation} + + The error is bounded to + \begin{equation} + \left|\underline{e}[n]\right| \leq \frac{1}{2} \Delta \underline{\hat{X}} + \end{equation} +\end{definition} + +\todo{Quantization Noise Floor} -\subsection{Window Filters} +\todo{Dynamic Range, dBFS} \subsection{Time Recovery} -- cgit v1.1