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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.

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

An infinitely tall and thin waveform
*with unit area* :

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

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

Notice from the sifting property that

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

Thus, Euler's equations

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

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

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

Often we draw this as

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

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

(from course webpage).

Notice that
`plottf.m`

only plots frequencies
$f\in [-\frac{1}{2{T}_{s}},\frac{1}{2{T}_{s}})$ .

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

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

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

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
*causal* linear-phase filters with group delay
$=\frac{\mathtt{Lf}}{2}$ samples.
Thus, the filtered output

`firls,firpm,fir2`

generate
`y`

will be delayed by
$\frac{\mathtt{Lf}}{2}$ samples relative to
`x`

.-
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Source:
OpenStax, Introduction to analog and digital communications. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10968/1.2

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