# 0.8 Discrete-time implementation of digital communication

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In this module, we derive the discrete-time implementation of a (baseband-equivalent) digital communication system, which is useful for Matlab simulation and error-rate analysis. To do this, we use the fundamental DSP tool known as "sinc reconstruction", which says that a bandlimited waveform can be represented exactly by a sum of shifted sinc pulses. Sinc reconstruction is separately applied to the transmission pulse, the reception pulse, and the baseband equivalent channel impulse response, after which those three steps can be represented by discrete-time filtering operations. For this, we show that it is essential to sample the filters at least twice as fast as the symbol rate.

Digital implementation of transmitter pulse-shaping and receiver filtering is much more practical than analog. First, recall the analog implementation of our digital communication system: from the module Digital Communication .

To proceed further, we need an important DSP concept called “sinc reconstruction”:

$\text{If}\phantom{\rule{3pt}{0ex}}\text{waveform}\phantom{\rule{3pt}{0ex}}x\left(t\right)\phantom{\rule{3pt}{0ex}}\text{is}\phantom{\rule{3pt}{0ex}}\text{bandlimited}\phantom{\rule{3pt}{0ex}}\text{to}\phantom{\rule{3pt}{0ex}}\frac{1}{2{T}_{s}}\text{Hz,}\phantom{\rule{3pt}{0ex}}\text{then}$
$x\left(t\right)=\sum _{n=-\infty }^{\infty }x\left[n\right]sinc\left(\frac{1}{{T}_{s}},\left(t\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}n{T}_{s}\right)\right)\phantom{\rule{5pt}{0ex}}\text{for}\phantom{\rule{5pt}{0ex}}x\left[n\right]=x\left(n{T}_{s}\right).$

In other words, a bandlimited waveform can be reconstructed from its samples via sinc pulse shaping.

## Discrete-time pulse-shaping

Applying $\frac{T}{P}$ -sampling and reconstruction to $g\left(t\right)$ (where the SRRC pulse bandwidth $\frac{1+\alpha }{2T}$ requires the use of $P\ge 2$ ),

$\begin{array}{ccc}\hfill g\left(\tau \right)& =& {\sum }_{l}g\left[l\right]sinc\left(\frac{P}{T}\left(\tau -l\frac{T}{P}\right)\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}g\left[l\right]=g\left(l\frac{T}{P}\right)\hfill \\ \hfill \stackrel{˜}{m}\left(t\right)& =& {\sum }_{n}a\left[n\right]g\left(t-nT\right)\hfill \\ & =& {\sum }_{n}a\left[n\right]{\sum }_{l}g\left[l\right]sinc\left(\frac{P}{T}\left(t-nT-l\frac{T}{P}\right)\right)\hfill \\ & =& {\sum }_{n}a\left[n\right]{\sum }_{k}g\left[k-nP\right]sinc\left(\frac{P}{T}\left(t-k\frac{T}{P}\right)\right)\phantom{\rule{4.pt}{0ex}}\text{via}\phantom{\rule{4.pt}{0ex}}k=nP-l\hfill \\ & =& {\sum }_{k}\underset{:=\stackrel{˜}{m}\left[k\right]}{\underbrace{{\sum }_{n}a\left[n\right]g\left[k-nP\right]}}sinc\left(\frac{P}{T}\left(t-k\frac{T}{P}\right)\right)\hfill \end{array}$

The sequence $\stackrel{˜}{m}\left[k\right]$ , a weighted sum of P -shifted pulses $g\left[k\right]$ , can be generated by P -upsampling $a\left[n\right]$ (i.e., inserting $P\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}1$ zeros between every pair of samples) and filtering with $g\left[k\right]$ :

sinc-pulse shaping $\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}$ digital-to-analog conversion (DAC).

Applying $\frac{T}{P}$ -sampling and reconstruction to bandlimited $q\left(\tau \right)$ :

$\begin{array}{ccc}\hfill q\left(\tau \right)& =& \sum _{l=-\infty }^{\infty }q\left[l\right]sinc\left(\frac{P}{T},\left(\tau -l\frac{T}{P}\right)\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}q\left[l\right]=q\left(l\frac{T}{P}\right)\hfill \end{array}$

where again we need $P\ge 2$ , yields

$\begin{array}{ccc}\hfill {y}_{↑}\left[k\right]& =& y\left(k\frac{T}{P}\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\int }_{-\infty }^{\infty }q\left(\tau \right)\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{v}\left(k\frac{T}{P}-\tau \right)d\tau \hfill \\ & =& {\int }_{-\infty }^{\infty }\sum _{l=-\infty }^{\infty }q\left[l\right]sinc\left(\frac{P}{T},\left(\tau -l\frac{T}{P}\right)\right)\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{v}\left(k,\frac{T}{P},-,\tau \right)d\tau \hfill \\ & =& \sum _{l=-\infty }^{\infty }q\left[l\right]\underset{{\left\{sinc,\left(\frac{P}{T},t\right),*,\stackrel{˜}{v},\left(t\right)\right\}}_{t=\left(k-l\right)\frac{T}{P}}=\stackrel{˜}{v}\left[k-l\right]}{\underbrace{{\int }_{-\infty }^{\infty }sinc\left(\frac{P}{T},{\tau }^{\text{'}}\right)\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{v}\left(\left(k-l\right),\frac{T}{P},-,{\tau }^{\text{'}}\right)d{\tau }^{\text{'}}}}\hfill \\ & =& \sum _{l=-\infty }^{\infty }q\left[l\right]\stackrel{˜}{v}\left[k-l\right]\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}q\left[k\right]*\stackrel{˜}{v}\left[k\right],\hfill \end{array}$

from which $y\left[m\right]$ is obtained by keeping only every ${P}^{th}$ sample:

i.e., downsampling. Here $sinc\left(\frac{P}{T}t\right)$ does anti-alias filtering.

## Discrete-time complex-baseband channel

Finally, we derive a discrete-time representation of the channel between $\stackrel{˜}{m}\left[k\right]$ and $\stackrel{˜}{v}\left[k\right]$ :

Using $\stackrel{˜}{w}\left[k\right]$ to refer to the noise component of $\stackrel{˜}{v}\left[k\right]$ , it can be seen from the block diagram that

$\begin{array}{ccc}\hfill \stackrel{˜}{w}\left[k\right]& =& {\int }_{-\infty }^{\infty }\stackrel{˜}{w}\left(\tau \right)sinc\left(\frac{P}{T}\left(k\frac{T}{P}-\tau \right)\right)d\tau \hfill \end{array}$

To model the signal component of $\stackrel{˜}{v}\left[k\right]$ , realize that $\stackrel{˜}{m}\left[k\right]$ is effectively pulse-shaped by $sinc\left(\frac{P}{T}t\right)*\stackrel{˜}{h}\left(t\right)*sinc\left(\frac{P}{T}t\right)$ . But since the frequency response of $sinc\left(\frac{P}{T}t\right)$ has a flat gain of $\frac{T}{P}$ over the signal bandwidth, and thus the bandwidth of $\stackrel{˜}{h}\left(t\right)$ ,

$sinc\left(\frac{P}{T}t\right)*\stackrel{˜}{h}\left(t\right)*sinc\left(\frac{P}{T}t\right)\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}\frac{{T}^{2}}{{P}^{2}}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{h}\left(t\right).$

So, with $\frac{T}{P}$ -sampling and reconstruction of $\frac{{T}^{2}}{{P}^{2}}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{h}\left(t\right)$ , i.e.,

$\begin{array}{ccc}\hfill \frac{{T}^{2}}{{P}^{2}}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{h}\left(t\right)& =& \sum _{i=-\infty }^{\infty }h\left[i\right]sinc\left(\frac{P}{T},\left(t-i\frac{T}{P}\right)\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}h\left[i\right]=\frac{{T}^{2}}{{P}^{2}}\phantom{\rule{0.166667em}{0ex}}h\left(i\frac{T}{P}\right)\hfill \end{array}$

we can write $\stackrel{˜}{v}\left[k\right]$ as

$\begin{array}{ccc}\hfill \stackrel{˜}{v}\left[k\right]& =& \stackrel{˜}{w}\left[k\right]+{\sum }_{l}\stackrel{˜}{m}\left[l\right]\phantom{\rule{0.166667em}{0ex}}\frac{{T}^{2}}{{P}^{2}}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{h}\left(k\frac{T}{P}-l\frac{T}{P}\right)\hfill \\ & =& \stackrel{˜}{w}\left[k\right]+{\sum }_{l}\stackrel{˜}{m}\left[l\right]{\sum }_{i}\stackrel{˜}{h}\left[i\right]\underset{\delta \left[k-l-i\right]}{\underbrace{sinc\left(k-l-i\right)}}\hfill \\ & =& \stackrel{˜}{w}\left[k\right]+{\sum }_{l}\stackrel{˜}{m}\left[l\right]\stackrel{˜}{h}\left[k-l\right]\hfill \end{array}$

yielding the discrete-time channel

Merging the discrete-time channel with the discrete-time modulator and demodulator yields

known as the “fractionally sampled” system model. This model is very convenient for MATLAB simulationand acts as a foundation for further analysis.

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