# Ber Calculation

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BER calculation

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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