# 0.2 Preliminaries

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The module provides a review of the background concepts needed for the study of analog and digital communications: the Fourier transform, various definitions of bandwith, the Dirac delta, frequency-domain representation of sinusoids, frequency-domain plotting in Matlab, linear time invariant (LTI) systems, linear filtering, lowpass filters, and Matlab design of lowpass filters.

## Fourier transform (ft)

Definition:

$\begin{array}{ccc}\hfill W\left(f\right)& =& {\int }_{-\infty }^{\infty }w\left(t\right){e}^{-j2\pi ft}dt\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\mathcal{F}\left\{w\left(t\right)\right\}\hfill \\ \hfill w\left(t\right)& =& {\int }_{-\infty }^{\infty }W\left(f\right){e}^{j2\pi ft}df\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\mathcal{F}}^{-1}\left\{W\left(f\right)\right\}.\hfill \end{array}$

Properties:

• Linearity: $\mathcal{F}\left\{{c}_{1}{w}_{1}\left(t\right)+{c}_{2}{w}_{2}\left(t\right)\right\}={c}_{1}{W}_{1}\left(f\right)+{c}_{2}{W}_{2}\left(f\right)$ .
• Real-valued $w\left(t\right)\phantom{\rule{3.33333pt}{0ex}}⇒\left\{\begin{array}{c}\text{conjugate symmetric}W\left(f\right)\\ |W\left(f\right)|\text{symmetric around}f=0\end{array}$ .

## Dirac delta (or “continuous impulse”) $\delta \left(·\right)$

An infinitely tall and thin waveform with unit area :

that's often used to “kick” a system and see how it responds.

## Key properties:

1. Sifting: ${\int }_{-\infty }^{\infty }w\left(t\right)\delta \left(t-q\right)dt=w\left(q\right)$ .
2. Time-domain impulse $\delta \left(t\right)$ has a flat spectrum:
$\mathcal{F}\left\{\delta \left(t\right)\right\}={\int }_{-\infty }^{\infty }\delta \left(t\right){e}^{-j2\pi ft}dt\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1\phantom{\rule{3.33333pt}{0ex}}\text{(for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}f\text{)}.$
3. Freq-domain impulse $\delta \left(f\right)$ corresponds to a DC waveform:
${\mathcal{F}}^{-1}\left\{\delta \left(f\right)\right\}={\int }_{-\infty }^{\infty }\delta \left(f\right){e}^{j2\pi ft}df\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1\phantom{\rule{3.33333pt}{0ex}}\text{(for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}t\text{)}.$

## Frequency-domain representation of sinusoids

Notice from the sifting property that

${\mathcal{F}}^{-1}\left\{\delta \left(f-{f}_{o}\right)\right\}={\int }_{-\infty }^{\infty }\delta \left(f-{f}_{o}\right){e}^{j2\pi ft}df={e}^{j2\pi {f}_{o}t}.$

Thus, Euler's equations

$\begin{array}{ccc}\hfill cos\left(2\pi {f}_{o}t\right)& =& \frac{1}{2}{e}^{j2\pi {f}_{o}t}+\frac{1}{2}{e}^{-j2\pi {f}_{o}t}\hfill \\ \hfill sin\left(2\pi {f}_{o}t\right)& =& \frac{1}{2j}{e}^{j2\pi {f}_{o}t}-\frac{1}{2j}{e}^{-j2\pi {f}_{o}t}\hfill \end{array}$

and the Fourier transform pair ${e}^{j2\pi {f}_{o}t}↔\delta \left(f-{f}_{o}\right)$ imply that

$\begin{array}{ccc}\hfill \mathcal{F}\left\{cos\left(2\pi {f}_{o}t\right)\right\}& =& \frac{1}{2}\delta \left(f-{f}_{o}\right)+\frac{1}{2}\delta \left(f+{f}_{o}\right)\hfill \\ \hfill \mathcal{F}\left\{sin\left(2\pi {f}_{o}t\right)\right\}& =& \frac{1}{2j}\delta \left(f-{f}_{o}\right)-\frac{1}{2j}\delta \left(f+{f}_{o}\right).\hfill \end{array}$

Often we draw this as

## Frequency domain via matlab

Fourier transform requires evaluation of an integral. What do we do if we can't define/solve the integral?

1. Generate (rate- $\frac{1}{{T}_{s}}$ ) sampled signal in MATLAB.
2. Plot magnitude of Discrete Fourier Transform (DFT) using plottf.m (from course webpage).

Notice that plottf.m only plots frequencies $f\in \left[-\frac{1}{2{T}_{s}},\frac{1}{2{T}_{s}}\right)$ .

## Linear time-invariant (lti) systems

An LTI system can be described by either its “impulse response” $h\left(t\right)$ or its “frequency response” $H\left(f\right)=\mathcal{F}\left\{h\left(t\right)\right\}$ .

## Input/output relationships:

• Time-domain: Convolution with impulse response $h\left(t\right)$
• Freq-domain: Multiplication with freq response $H\left(f\right)$

## Linear filtering

Freq-domain illustration of LPF, BPF, and HPF:

## Lowpass filters

Ideal non-causal LPF (using $sinc\left(x\right):=\frac{sin\left(\pi x\right)}{\pi x}$ ):

Ideal LPF with group-delay t o :

A causal linear-phase LPF with group-delay t o :

but MATLAB can give better causal linear-phase LPFs...

In MATLAB, generate $\frac{1}{{T}_{s}}$ -sampled LPF impulse response via

h = firls(Lf, [0,fp,fs,1], [G,G,0,0])/Ts;

where...

The commands firpm and fir2 have the same interface, but yield slightly different results (often worse for our apps).

In MATLAB, perform filtering on $\frac{1}{{T}_{s}}$ -sampled signal x via

y = Ts*filter(h,1,x);   or   y = Ts*conv(h,x);

The routines firls,firpm,fir2 generate causal linear-phase filters with group delay $=\frac{\mathtt{Lf}}{2}$ samples. Thus, the filtered output y will be delayed by $\frac{\mathtt{Lf}}{2}$ samples relative to x .

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