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| -rw-r--r-- | DigSig1.tex | 76 |
1 files changed, 55 insertions, 21 deletions
diff --git a/DigSig1.tex b/DigSig1.tex index 2ca6207..10379ae 100644 --- a/DigSig1.tex +++ b/DigSig1.tex @@ -8,13 +8,19 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Packages -\usepackage{tex/hsrstud} -\usepackage{tex/docmacros} - %% Font configuration \usepackage{fontspec} -% \usepackage{gfsbaskerville} -\setmainfont[Ligatures = TeX]{TeX Gyre Pagella} +\usepackage{fouriernc} + +%% Own packages +% \usepackage{tex/hsrstud} +\usepackage{tex/docmacros} + +%% Mathematics +\usepackage{amssymb} + +%% Frames +\usepackage{framed} %% Language configuration \usepackage{polyglossia} @@ -37,7 +43,7 @@ \authoremail{naoki.pross@ost.ch} \author{\textsl{Naoki Pross} -- \texttt{\theauthoremail}} -\title{\texttt{\themodule} Lecture Notes} +\title{Digital Signal Processing Lecture Notes} \date{\thesemester} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -66,7 +72,7 @@ The \emph{distribution function} of a RV is a function \(F_x : \mathbb{R} \to [0 \[ F_x(\alpha) = \Pr{x \leq \alpha}. \] -The probability density function (PDF) is +The \emph{probability density function} (PDF) of a RV is \[ f_x(\alpha) = \frac{dF_x}{d\alpha}. \] @@ -90,7 +96,7 @@ for example \end{align*} The \emph{variance} of a RV is \[ - \sigma^2 = \Var{x} = \E{(x - \E{x})^2} = \E{x^2} - \E{x}^2, + \Var{x} = \sigma^2 = \E{(x - \E{x})^2} = \E{x^2} - \E{x}^2, \] where \(\sigma\) is called the \emph{standard deviation}. The variance is sometimes also called the \emph{second moment} of a RV, the \emph{\(n\)-th moment} of a RV is \(\E{x^n}\). @@ -99,9 +105,10 @@ The variance is sometimes also called the \emph{second moment} of a RV, the \emp \section{Analog signals} -\paragraph{Notation} \(\Omega = 2\pi f\) is used for physical analog frequencies (in radians / second), whereas \(\omega\) is for digital frequencies (in radians / sample). +In this document we will use the notation \(\Omega = 2\pi f\) for physical analog frequencies (in radians / second), and \(\omega\) for digital frequencies (in radians / sample). -\paragraph{Transformations} Recall the three important operations for the analysis of analog signals. +\subsection{Transformations} +Recall the three important operations for the analysis of analog signals. \begin{flalign*} \textit{Fourier Transform} && X(\Omega) &= \int_\mathbb{R} x(t) e^{j\Omega t} \,dt \\ @@ -114,7 +121,7 @@ The variance is sometimes also called the \emph{second moment} of a RV, the \emp \end{flalign*} The Laplace transform reduces to the Fourier transform under the substitution \(s = j\Omega\). -\paragraph{Linear Systems} +\subsection{Linear Systems} Recall that superposition holds. Thus the system is characterized completely by the impulse response function \(h(t)\). The output in the time domain \(y(t)\) is given by the convolution product @@ -128,16 +135,43 @@ and in the frequency domain \(Y(\Omega) = H(\Omega) X(\Omega)\), where \(H(\Omeg \section{Sampling and reconstruction} -Sampling theorem: \(f_s = 2 f_\text{max}\) is called Nyquist rate. In other words you need at least 2 samples/cycle to reconstruct a signal. -%% TODO: ideal sampler -Nyquist intervals are bounded by Nyquist frequencies, i.e. \(\left[-f_s / 2, f_2 / 2\right]\) - -Alias frequency \(f_a = f \pmod f_s\). - -Anti-aliasing: analog LP prefilter cutoff \@ \(f_s/2\) - -Processing: Upper limit on sampling frequency given by processing time \(T_\text{proc}\). Thus \(2f_\text{max} \leq f_s \leq f_\text{proc}\). - +To sample a signal \(x(t)\) it means to measure (take) the value at a periodic interval every \(T\) seconds. \(T\) is thus called the \emph{sample interval} and \(f_s =1/T\) is the \emph{sampling frequency}. We will introduce the notation +\[ + x[n] = x(nT) +\] +to indicate that a signal is a set of discrete samples. + +\subsection{Sampling theorem} + +To represent a signal \(x(t)\) by its samples \(x[n]\) two conditions must be met: +\begin{enumerate} + \item \(x(t)\) must be \emph{bandlimited}, i.e. there must be a frequency \(f_\text{max}\) after which the spectrum of \(x(t)\) is always zero. + \item The sampling rate \(f_s\) must be chosen so that + \[ + f_s \geq 2 f_\text{max}. + \] +\end{enumerate} +In other words you need at least 2 samples / period to reconstruct a signal. +When \(f_s = 2 f_\text{max}\), the edge case, the sampling rate is called \emph{Nyquist rate}. +The interval \(\left[-f_s / 2, f_2 / 2\right]\), and its multiples are called \emph{Nyquist intervals}, as they are bounded by the Nyquist frequencies. +It would be good to have an arbitrarily high sampling frequency but in reality there is upper limit given by processing time \(T_\text{proc}\). Thus \(2f_\text{max} \leq f_s \leq f_\text{proc}\). + +\subsection{Discrete-Time Fourier Transform} + +Mathematically speaking, to sample a signal is equivalent multiplying a function with the \emph{impulse train distribution}\footnote{Sometimes it is also called the Dirac comb.} +\[ + \Comb_T (t) = \sum_{n = -\infty}^{\infty} \delta(t - nT), +\] +so \(x[n] = \Comb_T(t)\, x(t)\). +Interestingly the impulse train is periodic, and has thus a Fourier series with all coefficients equal to \(1/T\). +So the Fourier transform of a comb is also a comb, i.e. +\[ + \Comb_T(t) \leftrightarrow \Comb_{1/T}(\Omega). +\] + +\subsection{Spectrum replication and aliasing} +% Alias frequency \(f_a = f \pmod{f_s}\). +% Anti-aliasing: analog LP prefilter cutoff \@ \(f_s/2\) |