P31Q DIGITAL COMMUNICATIONS

LECTURE 9

BPSK II

BPSK RECIVER CONT'D

Suppose we transmit an all-zero sequence only. The shaded area shown below (purple) is the probability with which the coordinate is received in the range from 0 to +infinity.

Probability distribution of the cordinate x1

The shaded area probability is the transition probability P(1|0) of a BSC, because the BPSK receiver will decide that a binary digit '1' must have been sent if

is greater than or equal to 0.

Referring to the DCS shown below, recall that the transistion probablity P(1|0) represent the probability with a binary digit 1 is received if a binary digit 0 is transmitted. If the DCS uses the BPSK modulation scheme, then we can determine an expression for P(1|0) by finding an expression for the shaded area. This is the subject of this lecture.

Digital Communications System model in which BPSK is used to transfer binary digits across the channel

BPSK FORWARD TRANSITION PROBABILITY P(1|0)

The mathematical expression for the Gaussian distribution curve which corresponds to sending an all-zero sequence is given by

To gain a better understanding of this equation, click here.

The shaded area = Transistion probability P(1|0) is given by

To integrate, we shall use the method of substitution.

The reason for this will become clear shortly

Note that Eb is a constant.

Always be sure to change the integral limits

Therefore The expression for f(z) is very special !

where eq3a.gif (1779 bytes) is a zero mean, unit variance Gaussian distribution as shown below.

Zero mean, variance = 1, Gaussian distribution

The integral of f(z) cannot be evaluated. Instead, its value is tabulated and expressed as a special function known as the Q-function

Q-FUNCTION

The Q-function is defined as

Notice that the Q-function is simply the area to the right of the coordinate u , on a unit variance zero mean Guassian distribution, as shown below.

Unit Variance, zero mean Gaussian distribution.

For large values of u, we can approximate the Q-function as
The values of Q(u) are normally tabulated in the appendix of a text book on Digital Communications.

Hence Notice how the integral is now written in terms of Q(.)

By symmetry, for a BSC P(1|0) = P(0|1). Therefore, the probability of an error in a binary digit

We know this already, that the BSC crossover probability = Pe

[Student Excercise] (a) Show that , (b) Prove the above equation for the probability of an error in a binary digit, but now assuming that an all-one sequence was transmitted.

Part (a) Part (b)

HARD AND SOFT DECISION

The output of the demidulator need not be a binary digit '0' or '1'. The range of can be split into more than two regions as shown below.

fig1.gif (4908 bytes)

Hard-Decision Soft-Decision

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APLICACIÓN DE UN TÉCNICA DE MUESTREO ENFATIZADO COMPUESTO A LA SIMULACIÓN DE UN SISTEMA DE COMUNICACIONES W-CDMA

: Gracía Martínez, Adolfo
: Ingeniería Telecomunicación

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BPSK II

Send mail to janak with questions or comments.copy , yearLast modified: Wednesday November 22, 2000

APLICACIÓN DE UN TÉCNICA DE MUESTREO ENFATIZADO COMPUESTO A LA SIMULACIÓN DE UN SISTEMA DE COMUNICACIONES W-CDMA

: Gracía Martínez, Adolfo : Ingeniería Telecomunicación

Send mail to janak with questions or comments.
copy , year
Last modified: Wednesday November 22, 2000

: Gracía Martínez, Adolfo
: Ingeniería Telecomunicación