Sampling, reconstruction and multirate signal processing

Introduction

Sounds that reach our ears are signals in the form of air pressure. This air pressure is a continuous signal, i.e. it is defined for all time instances and can have any value on a reasonable domain. These kind of signals cannot be stored directly on a computer, which uses numbers to perform calculations. The signal first needs to be approximated in two ways in order to allow for computations on a computer and to limit the required memory.

Module overview

The concepts covered in this module are:

1. Mathematical description sampling process [⯈] - The conversion of a sample frequency can give rise to several undesired artifacts. The correct filtering of the signal is therefore of crucial importance.
2. Basic building blocks of multirate signal processing [⯈] - In order to perform the sample conversion, several common filter architectures can be utilized, which will be discussed in this section.

Summary

If the sampling frequency $f_s$ is lower than two times the highest frequency $f_{max}$ of the analog signal $x_a(t)$, an overlap of spectral components may occur and $x_a(t)$ cannot be recovered from its samples $x[n]$. This overlap in frequency domain is called aliasing.

C/D conversion

Note: In practice analog LPF needed before input $x_a(t)$ to prevent frequencies $\omega > \frac{\pi}{T_s}$.

\begin{equation} X(e^{j\theta})= X_s(\omega)|_{\omega=\frac{\theta}{T_s}}= \frac{1}{T_s} \sum_{k=- \infty}^{\infty} X_a(\frac{\theta}{T_s} - k \cdot \frac{2 \pi}{T_s}) \end{equation}

D/C conversion

Interpolation equations ideal LPF $H(\omega)$ \begin{equation} h(t) = \frac{1}{2 \pi} \int_{-\pi/ T_s}^{\pi/ T_s} T_s e^{j\omega t} \mbox{d} \omega = \frac{\sin(\frac{\pi}{T_s} t)}{\frac{\pi}{T_s} t} \end{equation} \begin{equation} x_a(t) = x_s(t) \ast h(t) = \sum_{n=-\infty}^{\infty} x[n] \left ( \frac{\sin (\frac{\pi}{T_s} \cdot (t- n T_s))}{\frac{\pi}{T_s} \cdot (t - n T_s)} \right ) \end{equation} \begin{equation} X_a(\omega) = \begin{cases} T_s \cdot X(e^{j\theta})|_{\theta = \omega T_s} & |\omega| < \frac{\pi}{T_s} \newline 0 & \mbox{ otherwise} \end{cases} \end{equation}

Practice D/C conversion: Ideal LPF $\Rightarrow$ Zero Order Hold filter:

\begin{eqnarray} h_0(t) = \begin{cases} 1 & 0 \leq t < T_s \newline 0 & \mbox{otherwise} \end{cases} & \circ \hspace{-4px} - \hspace{-4px} \circ & H_0(\omega) = \frac{\sin (\omega \frac{T_s}{2})}{\omega \frac{T_s}{2}} \cdot e^{-j\omega \frac{T_s}{2}} \end{eqnarray}

Relation analog vs discrete-time filter

Bandlimited $x(t)$ and LPF in D/C

\begin{equation} H_a(\omega) = \begin{cases} H_d(e^{j\theta})|_{\theta= \omega \cdot T_s} & |\omega| < \frac{\pi}{T_s} \newline 0 & \mbox{otherwise} \end{cases} \end{equation}

\begin{equation} H_d(e^{j\theta}) = H_a(\omega)|_{\omega= \frac{\theta}{T_s}} \mbox{ for } |\theta| < \pi \end{equation}

Sample Rate Conversion

SRD

\begin{equation} y[n \cdot T_y] = x[ n \cdot (M \cdot T_x)] \end{equation}

\begin{equation} Y(e^{j\theta})= \frac{1}{M} \sum_{p=0}^{M-1} X(e^{j(\theta - p \cdot 2 \pi)/M})$ \end{equation}

SRI

\begin{equation} y[n \cdot T_y] = \begin{cases} x[ n (T_x/L)] & n=0,\pm L, \cdots \newline 0 & \mbox{otherwise} \end{cases} \end{equation}

\begin{equation} Y(e^{j\theta}) = X(e^{jn L \theta}) \end{equation}