bpsk signal full form A Geometrical Representation for BPSK Signals Bandwidth for BPSK ? transmitter and receiver diagram ?

**A Geometrical Representation for BPSK Signals**

** **We know that BPSK signal carries the information about two symbols. These symbols are symbol ‘1’ and symbol ‘0’. We can represent BPSK signal geometrically to show those two symbols. From equation (8.9), we know that BPSK signal is expressed as,

s(t) = b(t) cos(2f_{c}t) …(8.18)

Let us rearrange the last equation as,

s(t) = b(t) . cos(2f_{c}t) …(8.19)

Now, let (t) = cos (2f_{c}t) represents an orthonormal carrier signal. Equation (8.17) also gives equation for carrier. It is slightly different than (t) defined here. Then, we may write equation (8.19) as,

s(t) = b(t) (t) …(8.20)

The bit energy E_{b} is defined in terms of power `*P*‘ and bit duration T_{b} as,

E_{b} = PT_{b} …(8.21)

Therefore, equation (8.20) becomes,

s(t) = b(t) (t) …(8.22)

Here, b(t) is simply ± 1.

Thus, on the single axis of , (*t*), there will be two points. One point will be located at or and other point will be located at or . This has been shown in figure 8.13.

**diagram**

**FIGURE 8.13*** Geometrical representation of BPSK signal.*

At the receiver end, the point at on (t) represents symbol ‘1’ and point, at, represents symbol ‘0’. The separation between these two points represents the isolation in symbols ‘1’ and ‘0’ in BPSK signal. This separation is generally called distance ‘d’. From figure 8.13, it is obvious that the distance between the two points is,

d = – () = …(8.23)

As this distance ‘d’ increases, the isolation between the symbols in BPSK signal is more. Thus, probability of error reduces.

**8.6.5. Bandwidth for BPSK Signal**

** **As discussed earlier, the spectrum of the BPSK signal is centred around the carrier frequency f_{c}.

If fb = , then for BPSK, the maximum frequency in the baseband signal will be f* _{b}* as shown in figure 8.12. In this figure, the main lobe is centred around carrier frequency f

*and extends from f*

_{c}*– f*

_{c}*to f*

_{b}*+ f*

_{c}*.*

_{b}Therefore Bandwidth of BPSK signal will be,

BW = Highest frequency — Lowest frequency in the main lobe

BW = f

*+ f*

_{c}*—(f*

_{b}*— f*

_{c }*)*

_{b}or BW = 2

_{fb}…(8.24)

Hence, the minimum bandwidth of BPSK signal is equal to twice of the highest frequency contained in baseband signal.

**8.6.6. Salient Features of BPSK**

**(i) BPSK has a bandwidth which is lower than that of a BFSK signal.**

(ii) BPSK has the best performance of all the three digital modulation techniques in presence of noise. It yields the minimum value of probability of error.

(iii) Binary phase shift keying (BPSK) has a very good noise immunity.

**8.6.7. Drawbacks of BPSK**

Figure 8.9 shows the block diagram of BPSK receiver. To regenerate the carrier in the receiver, we start by squaring b(t) cos (2f

_{c }

*t*+ ) . If the received signal is -b (t) cos (2f

_{c }

*t*+ ), then the squared signal remains same as before. Hence, the recovered carrier is unchanged even if the input signal has changed its sign. Therefore, it is not possible to determine whether the received signal is equal to b(t) or -b(t). Infact, this results in ambiguity in the output signal.

**Remedy**

**This problem can be removed if we use differential phase shift keying (DPSK). However, differential phase shift keying (DPSK) also has some other problems. DPSK will be discussed in detail later on in this chapter. Other problems of BPSK are ISI and Interchannel interference. However, these problems can be reduced to some extent by making use of filters.**

**8.6.8. Bit Error Rate (BER) or Probability of Error**

**The expression for probability of error, Pe, of a BPSK system is given by**

P

*=*

_{e}The above expression shows that the probability of error depends only on the energy contents of the signal i.e.,

*E*. Also, as the energy increases, the value of complementary error function erfc decreases and the value of P

*reduces.*

_{c}**8.7 COHERENT BINARY FREQUENCY SHIFT KEYING (BFSK)**

**In binary frequency shift keying (BFSK), the frequency of a sinusoidal carrier is shifted according to the binary symbol. In other words, the frequency of a sinusoidal carrier is shifted between two discrete values. However, the phase of the carrier is unaffected. This means that we have two different frequency signals according to binary symbols. Let there be a frequency shift by W. Then we can write following equations.**

If b(t) = ‘1’, then s

_{H}(t) = …(8.25)

If b(t) = ‘0’, then s

_{L}(t) = …(8.26)

Hence, there is increase or decrease in frequency by W. Let us use the following conversion table to combine above two FSK equations:

**Table 8.2. Conversion table for BPSK representation**

b(t) Input | d(t) |
P_{H}(t) |
P_{L}(t) |

1 0 |
+ 1V – 1V |
+ 1V 0V |
0V +1V |

The equations (8.25) and (8.26) combinely may be written as

s(t) = …(8.27)

Hence, if symbol ‘1’ is to be transmitted, the carrier frequency will be f_{c} + and is represented by f* _{H}*. If symbol ‘0’ is to be transmitted, then the carrier frequency will be f

_{c}+ and is represented by f

_{L}.

Therefore, we have

Thus, f

_{H}= f

_{c}+ for symbol ‘1’ …(8.28)

f

_{L}= f

_{c}+ for symbol ‘0’ …(8.29)