P31Q DIGITAL COMMUNICATIONS

LECTURE 8

BPSK

BASIS FUNCTION

Let us define a basis function of unit energy

Basic function

[Student Excercise] (a) Show that the energy of the signal is 1 joule. (b) Express and in terms of .

BPSK SIGNALS

From the student excercise, we note that ,

Thus, as expected, the energy within each signal is simply because the basis function has unit energy. Click here for further explanation.

SIGNAL CONSTELLATION

Since we only require a single basis function to represent both signals, the signal constellation diagram is only one-dimensional. The coordinate on this signal space is given by

Co-ordinate along the one-dimentional axis

where is the received signal from the channel. This signal given by

where is either or , and n(t) is the noise signal.

If there is no noise in the channel i.e. n(t) = 0, then is either or .

For =, we find that
For =, we find that

The corresponding signal constellation diagram is shown below

fig1.gif (2273 bytes)

The two solid dots are referred to as signal points.

Under noisy conditions, the received signal will be corrupted. In this case, the coordinate for a given , will not lie on a signal point. It will in fact lie on either side of a the signal point depending on the nature of the noise within the channel.

The noise signal n(t) represents an Additive White Gaussian Noise (AWGN) channel. It will be discussed in further detail in lecture 15. For now its sufficient to know that its effect is to make the probability distribution of Gaussian. Click here for a further explanation by example.

The probability distribution of the two signal points if the BPSK signals are transmitted through an AWGN channel is shown below

Probability distributions

The red curve is the probability distribution of the signal point if only is transmitted, i.e. an all-zero sequence [00000....]
The blue curve is the probability distribution of the signal point if only is transmitted, i.e. an all-ones sequence [11111...]
Thus the total area under each curve (blue or red) must be one i.e.

eq6.gif (2271 bytes)

The shaded area represents probability --> NOT height

A probability distribution curve is very useful. For example, the shaded area shown is the probability with which

lies between the two coordinates a and b.

COHERENT BPSK RECEIVER

Based on the signal constellation diagram shown in the previous section, we can now create a BPSK receiver as shown below.

fig4.gif (3105 bytes)

The received signal is multiplied by the basis function and integrated over the duration of the signal to determine the coordinate on the signal constellation diagram i.e.

Co-ordinate along the one-dimentional axis

If is greater than or equal to 0, then the receiver decides that a binary digit '1' must have been sent. Otherwise, it decides that a binary digit '0' sent must have been sent. This form of decision making is referred to as Hard-decision.

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BPSK

Send mail to janak with questions or comments.copy , yearLast modified: Tuesday November 14, 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: Tuesday November 14, 2000

: Gracía Martínez, Adolfo
: Ingeniería Telecomunicación