# Digital Data Transmission Notes

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1 1 Digital Communication Systems The input to a digital communication system is a sequence of digits. The input could be sourced by a data set, computer, or quantized and digitized analog signal. Such digital signals are coded using what is known as a line coder. Figure 1 illustrates some examples of line codes. Figure 1: Line Code Examples On-off Return to Zero (RZ): Figure 1.a is an on-off (RZ) code. It is on-off because a 1 is transmitted by a positive one and a 0 is represented as a zero. It is return to zero because the bit value (either 1 or 0) is not held for the entire duration. Instead it returns to zero before the next bit is transmitted. Polar (RZ): Figure 1.b is a polar (RZ) code. The code is polar because a 1 is transmitted by a positive one and a 0 is represented as a negative one. It is return to Class: Digital Communications Dentato 2012 Page 1

3 2 PSD of Various Line Codes Digital Data Transmission Notes We will directly apply the relationship between the PSD and the autocorrelation function to find a general expression for the PSD of a baseband modulation (line coding) output signal as shown in Figure 2. Figure 2: Random Pulse-Amplitude-Modulated Signal We will consider the pulse whose corresponding Fourier transform is. We can denote the line code symbol at time as. When the transmission rate is pulses per second, the line code generated a pulse train constructed from the basic pulse with amplitude starting at time ; in other words the kth symbol is transmitted as which can be seen in Figure 2.b. This signal consists of a succession of symbol transmissions seconds apart. The baseband signal is a pulse train of the form Class: Digital Communications Dentato 2012 Page 3

4 Note: The line coder determines the symbol pulse. as the amplitude of the The values of are random in nature and is a PAM signal. The on-off, polar, and bipolar line codes are all special cases of the pulse train where takes on the values 0, 1, or -1 randomly. We can use the PSD to analyze many line codes, but unfortunately the PSD of depends on and. This means we would have to solve for the PSD all over again if we change the shape of the pulse. To overcome this we will find a more general expression using a pulse train that consists of impulse functions. Let s look back at Figure 2.d. We can see that for an input of where we will have. Back in chapter 3 we learned that the ESD of a function (in this case ) is And also that We can rewrite this as We now have a general expression where all we need to find is the PSD of, where is an impulse train as seen in Figure 2.c, and then simply multiplication will yield the PSD for the pulse shape. Class: Digital Communications Dentato 2012 Page 4

5 Figure 3: Derivation of PSD for random PAM signal To find the PSD of the impulse train we will consider the limited for of the rectangle function. Each pulse has a width, and a pulse height of Class: Digital Communications Dentato 2012 Page 5

6 This guarantees the strength of the impulse to be or Let s designate this pulse train as then the autocorrelation function is is an even function so we only consider positive values of. Consider the case of. In this case the integral is the area under the signal multiplied by delayed by where. This area can be seen in Figure 3.b. Figure 4: Autocorrelation Again, because the autocorrelation function is an even we have as seen in Figure 4 The area associated with the pulse is for positive, and thus ( ) Class: Digital Communications Dentato 2012 Page 6

7 ( ) Where During the averaging interval, there are pulses where, thus This says that for the period there is pulse of pulse width and so is the time average of the square of the pulse amplitude we can express as. Using the time average notation Once again, we are dealing with an even function so we can express the function as ( ) We find that for we have a triangle function of height with a width of centered at. Similarly to we have And for we have To find we let in. As, the width of the triangle pulse will also and the height will in such a way that the area will remain finite. In this way the triangle function becomes an impulse as seen in Figure 3.e. We end up with ( ) thus Class: Digital Communications Dentato 2012 Page 7

8 The PSD is the Fourier transform of. Therefore Recognizing that because is even we have ( ) For a pulse shape we have ( ) In summary the PSD of a line code is fully characterized by its function. 2.1 PSD of Polar Signaling and pulse shaping Class: Digital Communications Dentato 2012 Page 8

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