Appunti Telecomunicazioni

Appunti presi al corso di Telecomunicazioni

Appunti presi al corso di Telecomunicazioni tenuto da Leonardo Badia, A.A. 2014/2015 12 34 56 7

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.

-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: 10logA(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 powers, but amplitudes (voltages/currents).

We will define the bit (a dimensionless unit). The “byte”yt in manyny systemsys = 8 bit. Some say:ay 20logR(dB) R= 1010 1 kilobytelobyte = 2 byte = 1024 byte, wrong.

201 megabyte = 2 byte = 1048576104857 byte.

Powers have the meaning of “signal squared”2 so if P = X, A = P /P, R = X /X then:

In the SI only base 10 units are admitted!dmitted!

1 2 1 2 P X. Solution to avoid any ambiguity: 1 110log 20logA(dB) R(dB)= = =10 10. Convert all the measures in bits P X2 2.

6 one kilobit = 1000 bits, one megabit = 10 bits...8

• Lower-case k in every “kilo” (km, kg, kb).
• No symbol B for “byte” (because B = bel).
• The symbol of seconds is “s” not “sec”.
• Symbols never have dots.
• My gravitational pull is about 700 newtons, Isaac and Helmut are two famous Newtons.

Band and bandwidth

We are familiar with time-domain and frequency-domain representations of signals.

From x(t) to X(f ) (and back) via Fourier transform.

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

Duration = measure of the support.

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

≥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

Limited duration unlimited band.

Band is [f ,f ] which are the closest to center1 2. So, we can use a practical definition of band frequency for which X(f ) = X(f ) = X(f ), with α α<11 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 closest to center frequency c.

Typical values for )αX(f:

• 0.7071 (A) or 0.5 (E): 3 dB band)=X(f )=0 for which X(f1 2.
• 0.1 (A) or 0.01 (E): 20 dB band.

And so on.

Vector spaces

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 inner product <x,y> 2space 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 φ1 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.

The space of signals

A valid choice for a vector space is also the space of energy-limited signals.

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=21 2Ω.

A probability function is associated with subsets of.

A random process is the same thing, but now xis 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)xω.

A signal which at every t is the average in of x (t)ω ω.

M Statistical power of x (t) = deterministic (t)xω.

Also a signal equal to the power of x (t) for every tω (t)m x.

Autocorrelation of x (t) is a signal (t, )r τxω t t t time1 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 averagesx 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

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 metricω.

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. Ergodic stationary (but not necessarily )).

m m It should have been (t ) = (t - t ) for every tx x 0 0.

E.g.: x (t) = (stationary, but not ergodic)ωω.

Wide sense stationary (WSS) if true for mean 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 constantx x.

Autocorrelation must be a function of only (not t)τ.

m E.g.: ergodic in mean mean of x (t) =1 x10.

Power spectral density

For a WSS rp x(t), the PSD (f ) is the Fourier ω x (t, ) = [x (t) y (t - )](transform of the autocorrelation function ( ). τ τrr τ xy ω ωx.

M*Note: (f ) df = (0) = (t) x *(t)] =([xr$ x x xω ω.

Thus, (f ) df is seen as the contribution to ! WSS rps x (t) and y (t) are jointly WSS if their xf ω ω1 τ cross correlation only depends on but not tM the power term in the frequency range [f ,f ]x 1 2.

* r(f ) is the infinitesimal contribution in [f, f +df ] x y xyx.

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.

Cross correlation, cross PSD

Cross correlation of 2 rps x (t) and y (t):

* (t, ) = [x (t) y (t - )]( τ τ rxy ω ω.

If (t, ) = 0, for all t, x and y are orthogonal!τ τr∫ xy.

If (t, ) = (t) (t - ) for all t, uncorrelated!τ m m τ τrxy x y.

WSS rps x (t) and y (t) are jointly WSS if their cross correlation only depends on but not t.

Cross PSD (f ): Fourier transform of ( )! τ* r(f ) is the infinitesimal contribution in [f, f +df ] x y xyx.

Property: (f ) = (f )x y y x.

Filtering of a WSS rp

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 ) =

Physical meaning of PSD