Ber Calculation

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:

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:

0        1

s

d2

Pb = Q @  min A = Q

2N0

where the Q function is de ned as:

Q(x) = p1 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:

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:

Using the tight approximation of Q function for z   0: