The Fourier analysis of binary pulses is especially useful in communication, for it gives a way to analyze the bandwidth needed to transmit such pulses. Although theoretically, the system must pass all the harmonics in the pulses, in reality, relatively few must be passed to preserve the shape of the pulse.

In addition, the pulse train in data communication rarely consists of square waves with a 50 percent duty cycle. Instead, the pulses are rectangular and exhibit varying duty cycles, from very low to very high. [The Fourier response of such pulses is given in Fig. 2-60( f ).] Look back at Fig. 2-60( f ). The period of the pulse train is T, and the pulse width is t0. The duty cycle is t0/T. The pulse train consists of dc pulses with an average dc value of Vt0/T. In terms of Fourier analysis, the pulse train is made up of a fundamental and all even and odd harmonics. The special case of this waveform is where the duty cycle is 50 percent; in that case, all the even harmonics drop out. But with any other duty cycle, the waveform is made up of both odd and even harmonics. Since this is a series of dc pulses, the average dc value is Vt0/T. A frequency-domain graph of harmonic amplitudes plotted with respect to frequency is shown in Fig. 2-65. The horizontal axis is frequency-plotted in increments of the pulse repetition frequency f, where f 5 1/T and T is the period. The fi rst component is the average dc component at zero frequency Vt0/T, where V is the peak voltage value of the pulse. Now, note the amplitudes of the fundamental and harmonics. Remember that each vertical line represents the peak value of the sine wave components of the pulse train. Some of the higher harmonics are negative; that simply means that their phase is reversed.

The dashed line in Fig. 2-65, the outline of the peaks of the individual components, is what is known as the envelope of the frequency spectrum. The equation for the envelope curve has the general form

width. This is known as the sinc function. In Fig. 2-65, the sinc function crosses the horizontal axis several times. These times can be computed and are indicated in the figure. Note that they are some multiple of 1/t0. The sinc function drawn on a frequency-domain curve is used in predicting the harmonic content of a pulse train and thus the bandwidth necessary to pass the wave. For example, in Fig. 2-65, as the frequency of the pulse train gets higher, the period T gets shorter and the spacing between the harmonics gets wider. This moves the curve out to the right. And as the pulse duration t0 gets shorter, which means that the duty cycle gets shorter, the first zero crossings of the envelope moves farther to the right. The practical significance of this is that higher-frequency pulses with shorter pulse durations have more harmonics with greater amplitudes, and thus a wider bandwidth is needed to pass the wave with minimum distortion. For data communication applications, it is generally assumed that a bandwidth equal to the first zero crossings of the envelope is the minimum that is sufficient to pass enough harmonics for reasonable waveshape:

**Example 2-27 **A dc pulse train like that in Fig. 2-60( f ) has a peak voltage value of 5 V, a frequency

of 4 MHz, and a duty cycle of 30 percent.

Most of the higher-amplitude harmonics and thus the most significant part of the signal power is contained within the larger area between zero frequency and the 1/t0 point on the curve.

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