diff options
| -rw-r--r-- | chapter04/content_ch04.tex | 4 | ||||
| -rw-r--r-- | chapter06/content_ch06.tex | 5 | ||||
| -rw-r--r-- | exercise04/exercise04.tex | 44 | ||||
| -rw-r--r-- | main/exercise04.tex | 4 |
4 files changed, 38 insertions, 19 deletions
diff --git a/chapter04/content_ch04.tex b/chapter04/content_ch04.tex index c4e04f6..a5fc3bd 100644 --- a/chapter04/content_ch04.tex +++ b/chapter04/content_ch04.tex @@ -895,7 +895,7 @@ The \ac{DTFT} is derived from the \ac{CTFT}. Therefore, all properties apply lik Analogous to the Fourier and Laplace transform, the \acf{DTFT} is a special case of the z-transform. -\begin{definition}{Discrete-time Fourier transform} +\begin{definition}{z-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} @@ -1065,7 +1065,7 @@ The $N \times N$ transformation matrix $\underline{\mat{F}}$ is the \index{DFT m \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.}. +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.}.\nomenclature[Sw]{$\underline{w}_N$}{$N$-th primitive root of unity} \begin{equation} \underline{w} = e^{j \frac{2 \pi}{N}} \end{equation} diff --git a/chapter06/content_ch06.tex b/chapter06/content_ch06.tex index 03da595..a9597f0 100644 --- a/chapter06/content_ch06.tex +++ b/chapter06/content_ch06.tex @@ -569,6 +569,7 @@ The \ac{LO} generates both a cos-signal and a \SI{90}{\degree}-phase-shifted sin \begin{itemize} \item Signals at lower sampling rates require less computation time and memory (software), or lower hardware complexity (less logic gates). The power consumption is reduced. \item The \ac{ADC} can be operated at maximum sampling rate. The signal is oversampled. Down-sampling provides processing gain and enhances the receiver performance. + %TODO \item Implementing the timing recovery \end{itemize} \subsection{Down-sampling} @@ -605,6 +606,8 @@ Down-sampling is in fact the sampling of an already sampled time-discrete signal \end{itemize} \end{itemize} +%TODO Sampling phase (timing) must be considered. + \begin{figure}[H] \centering @@ -1225,6 +1228,8 @@ The bandwidth of the interpolation filter shall be less or equal to the input si % %\todo{CIC filter} +%TODO Rational (fractional) resampling + \section{Fast Fourier Transform} In Chapter 4, we have learnt about the \ac{DFT} and how it can be used in digital systems by periodic continuation and windowing. diff --git a/exercise04/exercise04.tex b/exercise04/exercise04.tex index ea80fda..2215a23 100644 --- a/exercise04/exercise04.tex +++ b/exercise04/exercise04.tex @@ -97,9 +97,11 @@ \centering \begin{tabular}{|l|r|r|r|r|r|r|r|r|r|} \hline - $n$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$
\\ + $n$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ +\\ \hline - $t$ in \si{\micro\second} & $0.0$ & $0.125$ & $0.25$ & $0.375$ & $0.5$ & $0.625$ & $0.75$ & $0.875$ & $1.0$
\\ + $t$ in \si{\micro\second} & $0.0$ & $0.125$ & $0.25$ & $0.375$ & $0.5$ & $0.625$ & $0.75$ & $0.875$ & $1.0$ +\\ \hline \hline $u[n]$ in \si{V} & $1.0$ & $-1.73$ & $-1.0$ & $1.73$ & $1.0$ & $-1.73$ & $-1.0$ & $1.73$ & $1.0$ \\ @@ -148,7 +150,8 @@ \centering \begin{tabular}{|l|r|r|r|r|} \hline - $k$ & $0$ & $1$ & $2$ & $3$
\\ + $k$ & $0$ & $1$ & $2$ & $3$ +\\ \hline $k$ (alternate) & $0$ & $1$ & $-2$ & $-1$ \\ \hline @@ -376,7 +379,7 @@ $n$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \hline - $x[n]$ & 0.5 & 1 & 0 & 0.5 & -0.5 & -1 & -0.5 & -0.75 \\ + $x[n]$ & 0.25 & 1 & -0.5 & 0.5 & -0.5 & -1 & -0.5 & -0.75 \\ \hline \end{tabular} \end{table} @@ -524,13 +527,20 @@ From & To & Maps to & PCM data \\ \hline \hline - $-\infty$ & $\SI{-1.71}{V}$ & $\SI{-2.0}{V}$ & 0
\\ - $\SI{-1.71}{V}$ & $\SI{-1.14}{V}$ & $\SI{-1.43}{V}$ & 1
\\ - $\SI{-1.14}{V}$ & $\SI{-0.57}{V}$ & $\SI{-0.86}{V}$ & 2
\\ - $\SI{-0.57}{V}$ & $\SI{-0.0}{V}$ & $\SI{-0.29}{V}$ & 3
\\ - $\SI{0.0}{V}$ & $\SI{0.57}{V}$ & $\SI{0.29}{V}$ & 4
\\ - $\SI{0.57}{V}$ & $\SI{1.14}{V}$ & $\SI{0.86}{V}$ & 5
\\ - $\SI{1.14}{V}$ & $\SI{1.71}{V}$ & $\SI{1.43}{V}$ & 6
\\ + $-\infty$ & $\SI{-1.71}{V}$ & $\SI{-2.0}{V}$ & 0 +\\ + $\SI{-1.71}{V}$ & $\SI{-1.14}{V}$ & $\SI{-1.43}{V}$ & 1 +\\ + $\SI{-1.14}{V}$ & $\SI{-0.57}{V}$ & $\SI{-0.86}{V}$ & 2 +\\ + $\SI{-0.57}{V}$ & $\SI{-0.0}{V}$ & $\SI{-0.29}{V}$ & 3 +\\ + $\SI{0.0}{V}$ & $\SI{0.57}{V}$ & $\SI{0.29}{V}$ & 4 +\\ + $\SI{0.57}{V}$ & $\SI{1.14}{V}$ & $\SI{0.86}{V}$ & 5 +\\ + $\SI{1.14}{V}$ & $\SI{1.71}{V}$ & $\SI{1.43}{V}$ & 6 +\\ $\SI{1.71}{V}$ & $\infty$ & $\SI{2.0}{V}$ & 7 \\ \hline \end{tabular} @@ -547,7 +557,8 @@ \centering \begin{tabular}{|l|r|r|r|r|r|} \hline - $n$ & $0$ & $1$ & $2$ & $3$ & $4$
\\ + $n$ & $0$ & $1$ & $2$ & $3$ & $4$ +\\ \hline \hline $u[n]$ in \si{V} & $1.0$ & $-1.73$ & $-1.0$ & $1.73$ & $1.0$ \\ @@ -557,7 +568,8 @@ PCM data & $\left(101\right)_2$ & $\left(000\right)_2$ & $\left(010\right)_2$ & $\left(111\right)_2$ & $\left(101\right)_2$ \\ \hline \hline - $n$ & $5$ & $6$ & $7$ & $8$ &
\\ + $n$ & $5$ & $6$ & $7$ & $8$ & +\\ \hline \hline $u[n]$ in \si{V} & $-1.73$ & $-1.0$ & $1.73$ & $1.0$ & \\ @@ -574,7 +586,8 @@ \centering \begin{tabular}{|l|r|r|r|r|r|} \hline - $n$ & $0$ & $1$ & $2$ & $3$ & $4$
\\ + $n$ & $0$ & $1$ & $2$ & $3$ & $4$ +\\ \hline \hline $u[n]$ in \si{V} & $1.0$ & $-1.73$ & $-1.0$ & $1.73$ & $1.0$ \\ @@ -584,7 +597,8 @@ Quantization error in \si{V} & $0.14$ & $0.27$ & $0.14$ & $0.27$ & $0.14$ \\ \hline \hline - $n$ & $5$ & $6$ & $7$ & $8$ &
\\ + $n$ & $5$ & $6$ & $7$ & $8$ & +\\ \hline \hline $u[n]$ in \si{V} & $-1.73$ & $-1.0$ & $1.73$ & $1.0$ & \\ diff --git a/main/exercise04.tex b/main/exercise04.tex index 788ee0a..614c750 100644 --- a/main/exercise04.tex +++ b/main/exercise04.tex @@ -13,8 +13,8 @@ % Configuration \def\thekindofdocument{Exercise} \def\thesubtitle{Chapter 4: Sampling and Time-Discrete Signals and Systems} -\def\therevision{1} -\def\therevisiondate{2020-05-30} +\def\therevision{2} +\def\therevisiondate{2020-06-29} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Header |