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 $x_a\left(t\right)$, an overlap of spectral components may occur and $x_a\left(t\right)$ cannot be recovered from its samples $x\left[n\right]$. This overlap in frequency domain is called aliasing.

C/D conversion

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

$$\begin{array}{}X\left(e^{j\theta }\right)=X_s\left(\omega \right){|}_{\omega =\frac{\theta }{T_s}}=\frac{1}{T_s}\sum _{k=-\infty }^{\infty }{X}_a(\frac{\theta }{T_s}-k\cdot \frac{2\pi }{T_s})& \text{(1)}\end{array}$$

D/C conversion

Interpolation equations ideal LPF $H\left(\omega \right)$ $$\begin{array}{}h\left(t\right)=\frac{1}{2\pi }{\int }_{-\pi /T_s}^{\pi /T_s}{T}_s{e}^{j\omega t}\text{d}\omega =\frac{\sin \left(\frac{\pi }{T_s}t\right)}{\frac{\pi }{T_s}t}& \text{(2)}\end{array}$$ $$\begin{array}{}{x}_a\left(t\right)=x_s\left(t\right)\ast h\left(t\right)=\sum _{n=-\infty }^{\infty }x\left[n\right]\left(\frac{\sin (\frac{\pi }{T_s}\cdot (t-n{T}_s\left)\right)}{\frac{\pi }{T_s}\cdot (t-n{T}_s)}\right)& \text{(3)}\end{array}$$ $$\begin{array}{}{X}_a\left(\omega \right)=\{\begin{array}{cc}{T}_s\cdot X\left(e^{j\theta }\right){|}_{\theta =\omega T_s}& |\omega |<\frac{\pi }{T_s}\\ 0& \text{otherwise}\end{array}& \text{(4)}\end{array}$$

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

$$\begin{array}{}{h}_0\left(t\right)=\{\begin{array}{cc}1& 0\le t<T_s\\ 0& \text{otherwise}\end{array}& \circ -\circ & H_0\left(\omega \right)=\frac{\sin \left(\omega \frac{T_s}{2}\right)}{\omega \frac{T_s}{2}}\cdot e^{-j\omega \frac{T_s}{2}}& \text{(5)}\end{array}$$

Relation analog vs discrete-time filter

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

$$\begin{array}{}{H}_a\left(\omega \right)=\{\begin{array}{cc}{H}_d\left(e^{j\theta }\right){|}_{\theta =\omega \cdot T_s}& |\omega |<\frac{\pi }{T_s}\\ 0& \text{otherwise}\end{array}& \text{(6)}\end{array}$$

$$\begin{array}{}{H}_d\left(e^{j\theta }\right)=H_a\left(\omega \right){|}_{\omega =\frac{\theta }{T_s}}\text{for}|\theta |<\pi & \text{(7)}\end{array}$$

Sample Rate Conversion

SRD

$$\begin{array}{}y[n\cdot T_y]=x[n\cdot (M\cdot T_x\left)\right]& \text{(8)}\end{array}$$

$$\begin{array}{}Y\left(e^{j\theta }\right)=\frac{1}{M}\sum _{p=0}^{M-1}X\left(e^{j(\theta -p\cdot 2\pi )/M}\right)\$& \text{(9)}\end{array}$$

SRI

$$\begin{array}{}y[n\cdot T_y]=\{\begin{array}{cc}x\left[n\right(T_x/L\left)\right]& n=0,\pm L,\cdots \\ 0& \text{otherwise}\end{array}& \text{(10)}\end{array}$$

$$\begin{array}{}Y\left(e^{j\theta }\right)=X\left(e^{jnL\theta }\right)& \text{(11)}\end{array}$$