POWER SPECTRAL DENSITY (PSD) OF NRZ POLAR FORMAT , POWER SPECTRAL DENSITY (PSD)

POWER SPECTRAL DENSITY (PSD) OF NRZ BIPOLAR FORMAT , POWER SPECTRAL DENSITY (PSD) OF NRZ POLAR FORMAT Here, let us follow a similar procedure as followed in the previous section to obtain the psd of NRZ polar format.
For the NRZ polar format, we have
EQUATION                                      …(6.31)
We know the basic pulse p(t) is same for all the PAM formats, therefore, its Fourier transform will also remain same.
Hence,             P(f) = T_b sinc (fT_b)
Thus, the power spectral density (psd) is given by,
S(f) = A²T_b sinc² (fT_b)                                     …(6.32)
The normalized form of this expression has been plotted in figure 6.13 as curve b.*
diagram
FIGURE 6.13 Power spectra of different line codes.
* Curve h of figure 6.13 shows that most of the power of polar NRZ format lies inside the main lobe of the sine shaped curve extending upto the bit rate 1/T_b.
Similarly, it is possible to obtain the expressions for the power spectral densities (psds) of other PAM formats.
6.16 POWER SPECTRAL DENSITY (PSD) OF NRZ BIPOLAR FORMAT
We know that the bipolar NRZ format has three levels A, 0 and – A. If we assume that the 1s and 0s in the binary input data occur with equal probability then the probabilities of occurrence of these levels will be given by
P(a_k = A) = 1/4
P(a_k = 0) = 1/2
P(a_k = – A) = 1/4
Therefore, for n = 0, we can have
E[] = a²P(a_k = A) + (0)² P (a_k =0) + (-A)² P(a_k = -A) = A²/2

Similarly, for n = 1, the input sequence can assume four possible formats as (0, 0), (0, 1), (1, 0) and (1, 1). The corresponding values of the product a_k a_k-1 are 0, 0, 0 and 1 respectively.
Now, assuming that the 0s and 1s in the binary sequence have equal probability, we can write
E[a_k a_k-1] = 3(0) (1/4) + (- A²) (1/4) = -A²/4
Further, for n > 1 we find that
E[a_ka_k-1] = 0
Therefore, for the bipolar NRZ format, we have
EQUATION
Also, the basic pulse p(t) has the Fourier transform which is given by
P(f) = T_b sinc (fT_b)
Hence, putting equations (6.33) and (6.34) into the equation (6.27), we have
EQUATION
Hence, we get the power spectral density (psd) of NRZ bipolar format as under:
EQUATION
or                                                           EQUATION
or                                                           S(f) = sin²c(fT_b) [1-cos (2fT_b)]
or                                                           S(f) = A²T_b sin²c (fT_b) sin² (T_b)
The normalized form of this equation is shown plotted in ‘c’ in figure
POWER SPECTRAL DENSITY (PSD) OF THE manchester FORMAT
Here, let us assume that the input binary data consists of 0s and 1s of equal probability (i.e., ). Then the autocorrelation function of this format is same as that for the NRZ polar format.
First, let us express the machester signal x(t) mathematcially in time domain.
The basic pulse p(t) of the Machester signal is as shown in figure 6.14. We  can express the Machester signal in time domain as under:
DIAGRAM
FIGURE 6.14 Manchester signal.
EQUATION
The Fourier transform of p(t) is given by
EQUATION                                      …(6.36)
or                                                            EQUATION
or                                                            EQUATION
or                                                            EQUATION
Now, let us separately solve Term 1 and Term 2 as under:
Term 1:
EQUATION
EQUATION
EQUATION
Term 2:
EQUATION
EQUATION
EQUATION
EQUATION
Now, adding both the terms, we obtain
EQUATION
or                                                          EQUATION
or                                                          EQUATION
or                                                          EQUATION
Further, multiplying and dividing by T_b/2, we obtain

 EQUATION
Rearranging terms, we obtain
EQUATION
This is the required Fourier transform.
Now, let us obtain the expression for S(f).
We know that the power spectral density (psd) S(f) is given by
EQUATION
Substituting the expression for P(f), and R(n), we obtain
EQUATION
or                                                          EQUATION                                                   …(6.38)
This is the required expression for the power spectral density (psd). The normalized form of this power spectral density (psd) is plotted as curve “d” in figure 6.13. Most of the power lies inside the bandwidth equal to 2/T_b which is twice as large as that of unipolar, polar and bipolar formats of NRZ type.
6.17.1 Salient Features of the psd of a Machester Format.
(i)         The main lobes extend from 0 to 2f_b rather than 0 to f_b (as NRZ).
(ii)        At f = 0, S(f) as shown in figure 6.15.
(iii)       if the Manchester signal is transmitted through an ideal low pass filter with a cut-off frequency of 2f_b then about 95% power will be transmitted but if the cut-off frequency is reduced to f_b then only about 70% power is passed.
(iv)       The bandwidth required is 2f_b which is double the bandwidth of the NRZ signal.
diagram
FIGURE 6.15 Power spectra of different line codes.
6.18     COMPARISON OF VARIOUS DISCRETE PAM FORMATS   ON THE BASIS OF POWER SPECTRA
As a matter of fact, power spectra is a graph power spectral density (psd) plotted against frequency. It gives the distribution of power of different frequency components. The power spectra for four important line codes have been shown in figure 6.15.
(i) Power Spectra of NRZ Unipolar
            Curve a of figure 6.15 shows a normalized graph. Both the power spectral density (psd) and frequency are normalized. Most of the power of the NRZ unipolar format is concentrated between dc and the bit rate of input data.
(ii)        Power Spectra of NRZ Polar Format
            Curve b in figure 6.15 shows a normalized power spectra of NRZ polar format. Here, also like unipolar NRZ format, most of the power lies in main lobe of the sinc shaped curve. This main lobe extends upto a bit rate of 1/T_b.
(iii)       NRZ Bipolar (AMI) Format
            Curve c in figure 6.15 shows the normalized power spectra for the AMI format. It can be seen that almost all the power lies inside the bandwidth equal to 1/T_b. Similar to the two NRZ formats discussed earlier, however the spectral content of the NRZ bipolar format is relatively small around the zero frequency.
(iii) Manchester Format
Curve d in figure 6.15 shows the normalized power spectra of Manchester format. Here most of power lies inside the bandwidth equal to 2/T_b which is twice the bandwidth of unipolar, polar and bipolar NRZ formats.
Table 6.3. Comparison of Various Line Codes