NYQUIST'S CRITERION FOR DISTORTIONLESS BASEBAND BINARY TRANSMISSION

NYQUIST’S CRITERION FOR DISTORTIONLESS BASBAND BINARY TRANSMISSION                    (U.P. Tech, Sem. Exam. 2004-05) (10 marks)

1. Basic concept

In the previous section, we have observed that in absence of the ISI, we have
y(t_i) = µa_i                                                                           …(6.105)
This expression shows that under these conditions, the i^th transmitted bit can be decoded correctly. In order to minimize the effects of ISI, we have to design the transmitting and receiving filters properly. The transfer function of the channel and the shape of transmitted pulse are generally specified. Therefore, it becomes the first step towards design of filters. From this information we have to determine the transfer functions of the transmitting and receiving filters, to reconstruct the transmitted data sequence {b_k}. This is achieved by first extracting and then decoding the corresponding sequence of weights from the output y(t).
As expressed by following equation
y(t) =   p(t – kT_b)                                   …(6.106)
 
This shows that output y(t) is dependent on ak, the received pulse p(t) and the sealing factor .

2. Extraction

Extraction is basically the process of sampling. The signal y(t) is sampled at t = iT_b.

3. Decoding

The decoding should be such that the contribution of the weighted pulse i.e., a_k p(iT_b — kT_b) for i = k be free from ISI. This can be stated mathematically as under:
EQUATION
where p(0) = 1 due to normalizing.
If p(t) i.e., received pulse satisfies the above expression, then the receiver output given by equation (6.107) reduces to
y(t_i) = ai                                                        …(6.108)
which indicates zero ISI in the absence of noise.

4. Frequency Domain Representation

To obtain the transfer function, we have to transform the condition stated in equation (6.107) into frequency domain. Let us consider that the sequence of samples is represented by p(nT_b) where n = 0, ± 1, ± 2… after the process of extraction. The Fourier transform of such a sequence of samples can be obtained, as we did for the sampling process of low pass signal. Therefore, the frequency domain representation of p(nT_b) will be as under:
EQUATION                                      …(6.109)
where R_b = 1/T_b i.e., bit rate.
But, (f) also represents the Fourier transform of an infinite periodic sequence of unit impulses whose strengths are weighted by the respective sample values of p(t). Hence, (f) can be written as under:
Equation                                      …(6.110)
where (EQUATION) o(mT_b)  (t – mT_b)  represents the sequence of unit impulses weighted by the respective sample values.
Now, let m = i — k, therefore, if i = k then m = 0 and if i  k then m 0.
Hence, let us apply condition of equation (6.107) to equation (6.110) to get, for m = 0.
EQUATION
Using the sifting property of delta function, we have
(f) = p(0)
But, p(0) = 1 due to normalization, hence (f) = 1.
Substituting this into equation (6.109), we obtain
EQUATION
Therefore, we have
EQUATION
This expression is called as the Nyquist criterion for distortionless baseband transmission in the absence of noise.
6.28     IDEAL SOLUTION

1. Basic concept

Note that the L.H.S. of equation (6.111) represents a series of shifted spectrums. For n = 0, the LHS corresponds to P(f) and it represents a frequency function with the narrowest band which satisfies equation (6.111). The range of frequencies for P(f) will extend from – B₀ to B₀ where B₀ corresponds to half the bit rate.
Hence,                                     B₀ = R_b/2                                                         …(6.112)
Therefore, P(f) can be specified in the following form:
P(f) =                                               …(6.113)
and it has been shown graphically in figure 6.34(a). This is the spectrum of a signal which produces zero ISI. Hence, the signal that produces zero ISI can be obtained by taking the IFT of P(f).
This means that we have
EQUATION
Figure 6.48(b) shows the plot of this function. This function p(t) can be regarded as the impulse response of an ideal low pass filter (LPF) with bandwidth B₀.
Thus the shape of a pulse should be a sine pulse rather than being a rectangular one to eliminate the ISI.
DIAGRAM
FIGURE 6.48

2. Advantages of Using the Sinc Pulse

            (i)         Bandwidth requirement (of the channel) is reduced.
(ii)        ISI is reduced to zero.

3. Possible Difficulties

(i)         It is necessary that the amplitude characteristics of P(f) should be flat from -B₀ to B₀ and zero outside this band. But, abrupt transition at ± B₀ is not physically realizable.
(ii)        Due to discontinuity of P(f) at ± Bo, there is practically no margin of error in sampling times at the receiver end.
6.29 RAISED COSINE SPECTRUM

1. Concept and Mathematical Expression

The two difficulties experienced by the ideal Nyquist channel can be overcome by increasing the bandwidth from its minimum value B₀ = R_b/2 to an adjustable value between B₀ and 2 B₀. A condition is put on the overall frequency response PO to satisfy the given condition.
As per equation (6.111), we have
EQUATION
Expanding the summation sign, we get
…P(f  + R_b) + P(f) + P(f – R_b) + P (f – 2 R_b) … = T_b
But                              B₀ =        ⸫  R_b = 2B₀
Thus, we write
…P(f  + 2B₀) + P(f) + P(f – 2B₀) + P (f – 4B₀) +  … =
We retain only the three terms or LHS which correspond to n = – 1, n = 0 and n = 1 and restrict the frequency band of interest to (- B₀, B₀) to get,
P(f + 2B₀) + P(f) + P(f – 2B₀) =   and given – B₀ ≤ f ≤ B₀
It is possible to devise several bandlimited functions which will satisfy above equation, one of them is called as the raised cosine spectrum. This spectrum consists of a flat portion and a roll off portion. The raised cosine spectrum is expressed mathematically as under:
EQUATION
The relation between frequency parameter f₁ and the bandwidth B₀ are related as under :
a = 1 –
where α is called as the roll off factor. It indicates the excess bandwidth over the ideal solution B₀.
            The transmission bandwidth B_T. is defined as under :
B_T= 2B₀– f₁ = B₀(1 + α)
The normalized frequency response of raised cosine function is obtained by multiplying P(f) by 2B₀ and it is plotted in figure 6.49(α), for different values of α. The corresponding time response p(t) is shown in figure 6.49(b).
DIAGRAM
FIGURE 6.49 Responses for different roll-off factors, a.

2. Observations

(i)         For α = 0.5 and 1, the characteristics of P(f) changes gradually with respect to frequency. Hence it is easier to realize this characteristics practically.
(ii)        The time response has a sinc shape and all the sinc functions pass through zero at t = ± T_b, ± 2 T_b,…..
(iii)       The amplitude of side lobes increases with reduction in the value of α.
(iv)       With α = 0, the bandwidth requirement is maximum equal to 2B₀.