8/18/2019 Cheat Sheet Communications

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Cheat Sheet: ECE 318Prepared by Walid Abediseid

1 Modulation Schemes   (see Table below)

–   Why Modulation? Make transmission more reliable.Reduce noise and interference. To facilitate channel as-signment and multiplexing.

–   For DSB-LC:   µ   =   |max f (t)|Ac represents the modula-tion index. In this case, one may rewrite the DSB-LC as:   s(t) =   Ac(1 +  µf (t)) cos(ωc  +  φ), conditionedon   |max f (t)|   = 1. The modulation efficiency is de-

fined by  η   =  P f /2P f /2+A2c/2

, or   η   =  µ2P f /2µ2P f /2+1/2

  =  µ2P f 1+µ2P f 

(for the other representation). A special case, is whenf (t) = cos(ωf t) or sin(ωf t). In this case, we have

P f  = 1/2, and  η  =  µ2

2+µ2 .

–  For SSB:  f̂ (t) =   1πt∗f (t) is the Hilbert Transform of f (t)(hard to find in time-domain). In frequency domain:

F̂ (ω) =  − jsgn(ω)F (ω). Hilbert Transform acts as a90◦ phase delay. Example   cos(ωt) = cos(ωt −  π/2) =sin(ωt), in this case no need to do Fourier Transform.

–  All the above modulation schemes can be demodulated(coherently) using the following:

s(t)   ×

cos(ωct + φ)

LPF   Acf (t)

However, for VSB modulation scheme, the filter at the

transmitter side   hVSB(t) must satisfy the condition:H (ω−ωc) + H (ω + ωc) = constant, so that one can usethe above demodulator to recover the actual messagesignal f (t).

–   An exception to the above demodulator, is the DSB-LC. If the modulation index  µ  such that the envelopeof  s(t) is greater than zero, i.e., (Ac + µf (t)) >  0, thena simple RC circuit can be used to recover f (t).

2 Noise Filtering in LTI Systems

A random noise signal,   n(t), is called white if it has a

power spectral density function,   S n(ω), that is flat for

all frequencies, i.e.,   S n(ω) =  K 2   where   K   is a consta

The average power of a signal   x(t) with power spect

density   S x(ω) is given by:   P x   =  12π

∞ 

−∞

S x(ω)   dω   a

since   S x(ω) is a non-negative function, one can write above equation as   P x   =

  12π   ×   [Total Area under S x(ω

Signal-to-Noise Ratio (SNR) is defined by: SNR Average Power of Transmitted Signal

Average Power of Noise  . The following pr

erty of LTI systems is very important:

r(t)

S r(ω)

LTI

H (ω)g(t) =  r(t) ∗ h(t)

S g(ω) =  S r(ω)|H (ω)|2

Example 1.  Suppose we have  r(t) =  x(t) + n(t)  is the into the above LTI system, where   n(t)   is a white noise wS n(t) =  K/2, and 

ω

S x(ω)

−2πB

1

2πBω

H (ω)

−πB

1/2

πB

Find the SNR at the output? 

Solution:   Let  y(t) =  x(t) ∗ h(t), and  n(t) =  n(t) ∗ h(t).order to find the average power of   y(t)   and   n(t), we nto find their corresponding power spectral density functi

S y(ω),  S n(ω):

ω

S y(ω)

−πB

1/4

1/8

πBω

S n(ω)

−πB

K/8

πB

In this case, the average power of   y(t)   can be eily evaluated using:   P y   =

  12π [Area under S y(ω)]

12π [Area of rectangle +Area of triangle] =

  316

B. Also, P n1

2π [2πB  × K 

8 ] =  BK 

8   . Therefore, the SNR=

  3B/16

BK/8   =  3

2K .

Scheme Format Power P s   Band-Width  Bs

DSB-LC   s(t) = (Ac + f (t)) cos(ωct + φ)  A2c

2   +  µ2P f 

2   2Bf 

DSB-SC   s(t) =  Acf (t) cos(ωct + φ)  A2cP f 

2   2Bf 

USSB   s(t) =  Acf (t) cos(ωct + φ) − Ac f̂ (t)sin(ωct + φ)  A2cP f 

4  Bf 

LSSB   s(t) =  Acf (t) cos(ωct + φ) + Ac f̂ (t)sin(ωct + φ)  A2cP f 

4  Bf 

VSB   s(t) =  sDSB−SC(t) ∗ hVSB(t) depend on filter depend on filter

savals Thm:  ∞−∞ x(t)y(t)

∗ dt   =   12π  ∞−∞ X (ω)Y (ω)

∗ dω. Averag power of   x(t),   P x   = limT →∞1T 

  T/2−T/2 |x(t)|

2 dt. Power[C

=1 C k cos(ωkt + φk)] =  C 20  +

 nk=1

C 2k2   , where  ωi  =  ωj   for   i  =  j . If the input to an LIT system with Frequency Response H (

he complex exponential  A exp( jω0t) =⇒   the output is simply given by  A|H (ω0)| exp( j[ω0t + ∠H (ω0)]). Autocorrelation of  xt) =  F −1{S x(ω)}. Average power also equal to  Rx(0).