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Autor:   •  April 20, 2017  •  Term Paper  •  1,568 Words (7 Pages)  •  90 Views

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BER calculation

Vahid Meghdadi

reference: Wireless Communications by Andrea Goldsmith

January 2008

  • SER and BER over Gaussian channel
  1. BER for BPSK modulation

In a BPSK system the received signal can be written as:

y = x + n

(1)

where x 2 f A; Ag, n CN (0; 2) and 2 = N0. The real part of the above equation is yre = x + nre where nre N (0; 2=2) = N (0; N0=2). In BPSK constellation dmin = 2A and b is de ned as Eb=N0 and sometimes it is called SNR per bit. With this de nition we have:

b :=

Eb

=

A2

=

dmin2

(2)

4N0

N0

N0

So the bit error probability is:

Pb = P fn > Ag = Z

1

1

e

x2

(3)

2 2=2

A

2  2=2

This equation can be simpli ed using Q-

function as:

p

0        1

s

d2

Pb = Q @  min A = Q

2N0

where the Q function is de ned as:

Q(x) = p1 2

 

dmin

= Q  p

2 b

p

(4)

2N0

Zx1

e

x2

dx

(5)

2

  1. BER for QPSK

QPSK modulation consists of two BPSK modulation on in-phase and quadrature components of the signal. The corresponding constellation is presented on gure 1. The BER of each branch is the same as BPSK:

 

p

Pb = Q   2 b        (6)

1


[pic 1]

Figure 1: QPSK constellation

The symbol probability of error (SER) is the probability of either branch has a bit error:

 

p

Ps = 1  [1  Q   2 b  ]2        (7)

Since the symbol energy is split between the two in-phase and quadrature com-ponents, s = 2 b and we have:

Ps = 1  [1  Q (p

)]2

(8)

s

We can use the union bound to give an upper bound for SER of QPSK. Regard-ing gure 1, condition that the symbol zero is sent, the probability of error is bounded by the sum of probabilities of 0 ! 1, 0 ! 2 and 0 ! 3. We can write:

Ps

Q(d01=  2N0) + Q(d02=

2N0) + Q(d03=

2N0)

(9)

=

2Q(A= p

) + Q(p2A=p

)

p

(10)

N0

2N0

p

p

Since  s = 2 b = A2=N0, we can write:

Ps   2Q(p

) + Q(p

)   3Q(p

)

2 s

(11)

s

s

Using the tight approximation of Q function for z   0:

...

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