Appunti Telecomunicazioni

Appunti presi al corso di Telecomunicazioni tenuto da Leonardo Badia, A.A. 2014/2015 1

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Units of measurement Prefixes and suffixes

!

! Often, the basic unit is too “small” or too “big”

Use the correct units from the SI .

! All units are lower-case when written in full ! -4

Scientific notation: 0.0004 = 4 10 . Or, use

! . .

even when their symbol is a capital letter -3 3

milli (symbol m) = 10 kilo (symbol k) = 10

. .

! So, the unit of: -6 6

micro (symbol µ / u) = 10 mega (symbol M) = 10

! current intensity is the ampere /'amˌper/ (symbol: A) . .

-9 9

10 giga (symbol G) = 10

nano (symbol n) =

∆

! of electrical potential is the volt /'volt/ (symbol: V) . .

-12 12

pico (symbol p) = 10 tera (symbol T) = 10

! electrical energy is the joule /'dʒuːl/ (symbol: J)

! electrical power is the watt /'wɒt/ (symbol: W) ! Or further: femto, atto (small) or peta, exa (big)

! frequency (1/t) is the hertz /'hɜː(ɹ)ts/ (symbol: Hz)

The decibel (dB) The decibel (dB)

! !

Another way to express very large/small ratios to write “dB:” a lower-case “d” and a capital “B”

! where B stands for “bel” and means the log-scale

! If A is a ratio of (something related to) powers ! It is a logarithmic scale, thus:

10log

A(dB) A

= 10 ! only positive values can be represented

! if A = P /P where P is a power, you can write: ! a sum in dB = a product in linear scale

1 2 k (see: 10 dB + 20 dB = 1000)

instead of: A = 10 A = 10 dB

! ! Special values of the dB scale:

A = 100, 1000... A = 20 dB, 30 dB...

! ! 0 dB means that the ratio is equal to 1

A = 0.1, 0.01... A = -10 dB, -20 dB...

! ≈

! 3 dB means a ratio of ~2 (as log 2 0.30103)

10

The decibel (dB) Bits and bytes

! !

Different definition if ratio R does not involve We will define the bit (a dimensionless unit)

powers, but amplitudes (voltages/currents) ! The “byte”

yt in many

ny systems

ys = 8 bit. Some say:

ay

20log

R(dB) R

= ! 10

10 1 kilobyte

lobyte = 2 byte = 1024 byte, wrong!

! 20

1 megabyte = 2 byte = 1048576

104857 byte

! Powers have the meaning of “signal squared”

2

so if P = X , A = P /P , R = X /X then: ! In the

he SI only base 10 units are admitted!

dmitted!

1 2 1 2

P X ! Solution to avoid any ambiguity:

1 1

10log 20log

A(dB) R(dB)

= = =

10 10 ! convert all the measures in bits

P X

2 2 ! 6

one kilobit = 1000 bits, one megabit = 10 bits...

8 To avoid mistakes Band and bandwidth

! !

Lower-case k in every “kilo” (km, kg, kb) We are familiar with time-domain and

frequency-domain representations of signals

! No symbol B for “byte” (because B = bel) ! From x(t) to X(f ) (and back) via Fourier transform

! The symbol of seconds is “s” not “sec” ≠

! support = set of time values where x(t) 0

! duration = measure of the support

! Symbols never have dots ≠

! !

full band = set of frequencies where X(f ) 0

! My gravitational pull is about 700 newtons,

Isaac and Helmut are two famous Newtons ≥0 ≠

! !

band = set of frequencies where X(f ) 0

× -

! !

! real x(t) X(f ) even full band = 2 band ( )

! !

! !

bandwidth B = measure of the band B =| |

Band in practice Band in practice

! !

limited duration unlimited band Amplitude

! band is [f ,f ] which are the closest to center

1 2

! So, we can use a practical definition of band frequency for which X(f ) = X(f ) = X(f ), with

α α<1

1 2 c

! X(f ) will be like this – band around center frequency ! Energy X(f )

! c

First zero ! same but with energy criterion

! band is [f ,f ] which are the α:

1 2 ! Typical values for )

αX(f

closest to center frequency c

! 0.7071 (A) or 0.5 (E): 3 dB band

)=X(f )=0

for which X(f

1 2 ! 0.1 (A) or 0.01 (E): 20 dB band

! and so on

Vector spaces The space of signals

! For any M vectors, find an orthonormal basis of

! " #

A linear (or vector) space = quadruple ( , ,+,)

n for their span: Gram-Schmidt procedure

! " $ # $

The common example is = , = ≤

! %

The result will have size M

! Certain vector spaces also have ! A valid choice for a vector space is also the

! inner product <x,y>

2

space of energy-limited signals, = ( )

! " $

norm ||x|| norm induced by the i.p.: ||x||=

! <x,x>

!

! Inner product and norm exist:

%

A vector space has dimension it exists an

"

orthonormal basis , ,..., s.t. span( )= ∫

φ φ φ φ

1 2 j

% ! <x(t),y(t)> = x(t) y*(t) dt

$

&

! x with x = <x,φ >

"

x x = φ ! 2 (energy of x)

'

|| x(t) || =

j j j j j x 9

Random processes Random processes

! !

Random variables (continuous/discrete): Depending on what is signal x(t) changes

ω,

Ω signal

! A rv x is a function defined on a probability space

! A realization of the rv is x(ω) with ωΩ x (t)

!

ω=1 x (t)

!

ω=2

1 2

Ω

! A probability function is associated with subsets of

! A random process is the same thing, but now x

is a function of both and a time index t.

ω x (t)

!

ω=3 3

! We can call it x(t,ω) - or better: x (t )

ω

! i.e. seen as a signal x(t ) also depending on ωΩ time

! Choosing determines the shape of the signal x(t )

ω

Mean, power, autocorrelation Mean (seen graphically)

signal

first order description

! m

Statistical mean of rp x (t) = deterministic (t)

x

ω

! a signal which at every t is the average in of x (t)

ω ω

! M

Statistical power of x (t) = deterministic (t)

x

ω

! also a signal equal to the power of x (t) for every t

ω (t)

m x

! Autocorrelation of x (t) is a signal (t, )

r τ

x

ω t t t time

1 2 3

! τ r

function of t and of lag : (t, ) = (t) x *(t - )]

τ τ

([x !

x ω ω m (t) is the collection of all these averages

x

second order descriptions ≠

! statistical mean time averages of x (t), x (t), x (t)...

1 2 3

(this is a signal these are values!)

Stationary rps Ergodic rps

! !

rp x (t) is stationary (in a statistical description; stationary x (t) is ergodic (in a time metric;

ω ω

if not specified then it holds for all descriptions) if not specified then it holds for all the metrics)

if the description is invariant for all time shifts if any gives the same value of this metric

ω

! e.g.: ergodic in mean mean of x (t) = of x (t), x (t)...

1 2 3

! Previous example is not a stationary rp in mean ! ergodicstationary (but not necessarily ))

! m m

It should have been (t ) = (t - t ) for every t

x x 0 0 ! e.g.: x (t) = (stationary, but not ergodic)

ω

ω

! Wide sense stationary (WSS) if true for mean !

st nd

and autocorrelation (1 +2 order) For an ergodic rp, related statistical description

coincides with the time description.

! m m

For the mean, (t) must be a constant

x x

! Autocorrelation must be a function of only (not t)

τ

! m

E.g.: ergodic in mean mean of x (t) =

1 x

10 Power spectral density Cross correlation, cross PSD

! cross correlation of 2 rps x (t) and y (t):

! *

For a WSS rp x(t), the PSD (f ) is the Fourier ω ω

x

(t, ) = [x (t) y (t - )]

(

transform of the autocorrelation function ( ). τ τ

r

r τ xy ω ω

x ! if (t, ) = 0, for all t, x and y are orthogonal

!

τ τ

r

∫ xy

! M

*

Note: (f ) df = (0) = (t) x *(t)] =

([x

r

$ x x x

ω ω ! if (t, ) = (t) (t - ) for all t, uncorrelated

!

τ m m τ τ

r

xy x y

f

∫ 2 *

Thus, (f ) df is seen as the contribution to ! WSS rps x (t) and y (t) are jointly WSS if their

x

f ω ω

1 τ

cross correlation only depends on but not t

M

the power term in the frequency range [f ,f ]

x 1 2 *

! cross PSD (f ): Fourier transform of ( )

! τ

* r

(f ) is the infinitesimal contribution in [f, f +df ] x y xy

x

! the entire integral can be limited to the full band ! *

* *

! Property: (f ) = (f )

x y y x

! * (f ) is even: 2× the integral over

for real x(t), !

x

Filtering of a WSS rp Physical meaning of PSD

x(t) y(t) = (g x) (t)

g(t)

x(t) y(t) = (g x) (t)

g(t)

! If x(t) is a WSS rp, then y(t) is also a WSS rp. ! 2 ,

* *

As (f ) =