Fourier Theory Background:- The mathematical analysis of the modulation and multiplexing methods used in communication systems assumes sine wave carriers and information signals. This simplifies the analysis and makes operation predictable. However, in the real world, not all information signals are sinusoidal. Information signals are typically more complex voice and video signals that are essentially composites of sine waves of many frequencies and amplitudes. Information signals can take on an infinite number of shapes, including rectangular waves (i.e., digital pulses), triangular waves, sawtooth waves, and other nonsinusoidal forms. Such signals require that a non–sine wave approach be taken to determine the characteristics and performance of any communication circuit or system. One of the methods used to do this is Fourier analysis, which provides a means of accurately analyzing the content of most complex nonsinusoidal signals. Although Fourier analysis requires the use of calculus and advanced mathematical techniques beyond the scope of this text, its practical applications to communication electronics are relatively straightforward.
Fig. 2-55(a) shows a basic sine wave with its most important dimensions and the equation expressing it. A basic cosine wave is illustrated in Fig. 2-55(b). Note that the cosine wave has the same shape as a sine wave but leads the sine wave by 90°. A harmonic is a sine wave whose frequency is some integer multiple of a fundamental sine wave. For example, the third harmonic of a 2-kHz sine wave is a sine wave of 6 kHz. Fig. 2-56 shows the first four harmonics of a fundamental sine wave. What the Fourier theory tells us is that we can take a nonsinusoidal waveform and break it down into individual harmonically related sine wave or cosine wave components. The classic example of this is a square wave, which is a rectangular signal with equal duration positive and negative alternations. In the ac square wave in Fig. 2-57, this means that t1 is equal to t2. Another way of saying this is that the square wave has a 50 percent duty cycle D, the ratio of the duration of the positive alteration t1 to the period T expressed as a percentage:
Fourier analysis tells us that a square wave is made up of a sine wave at the fundamental frequency of the square wave plus an infinite number of odd harmonics. For example, if the fundamental frequency of the square wave is 1 kHz, the square wave can be synthesized by adding the 1-kHz sine wave and harmonic sine waves of 3 kHz, 5 kHz, 7 kHz, 9 kHz, etc. Fig. 2-58 shows how this is done. The sine waves must be of the correct amplitude and phase relationship to one another. The fundamental sine wave in this case has a value of 20 V peak to peak (a 10-V peak). When the sine wave values are added instantaneously, the result approaches a square wave. In Fig. 2-58(a), the fundamental and third harmonics are added.
Note the shape of the composite wave with the third and fifth harmonics added, as in Fig. 2-58(b). The higher harmonics that are added, the more the composite wave looks like a perfect square wave. Fig. 2-59 shows how the composite wave would look with 20 odd harmonics added to the fundamentals. The results very closely approximate a square wave.
The implication of this is that a square wave should be analyzed as a collection of harmonically related sine waves rather than a single square wave entity. This is confirmed by performing a Fourier mathematical analysis on the square wave. The result is the following equation, which expresses voltage as a function of time:
where the factor 4V/π is a multiplier for all sine terms and V is the square wave peak voltage. The first term is the fundamental sine wave, and the succeeding terms are the third, fifth, seventh, etc., harmonics. Note that the terms also have an amplitude factor. In this case, the amplitude is also a function of the harmonic. For example, the third harmonic has an amplitude that is one-third of the fundamental amplitude, and so on. The expression could be rewritten with f 5 1/T. If the square wave is direct current rather than alternating current, as shown in Fig. 2-57(b), the Fourier expression has a dc component:
In this equation, V/2 is the dc component, the average value of the square wave. It is also the baseline upon which the fundamental and harmonic sine waves ride. A general formula for the Fourier equation of a waveform is
where n is odd. The dc component, if one is present in the waveform, is V/2. By using calculus and other mathematical techniques, the waveform is defined, analyzed, and expressed as a summation of sine and/or cosine terms, as illustrated by the expression for the square wave above. Fig. 2-60 gives the Fourier expressions for some of the most common nonsinusoidal waveforms
Example 2-26 An ac square wave has a peak voltage of 3 V and a frequency of 48 kHz. Find (a) the frequency of the fifth harmonic and (b) the RMS value of the fifth harmonic. Use the formula in Fig. 2-60(a).
a. 5×48 kHz =240 kHz
b. Isolate the expression for the fifth harmonic in the formula, which is 1/5 sin 2π(5/T )t. Multiply by the amplitude factor 4V/π. The peak value of the fifth harmonic VP is
The triangular wave in Fig. 2-60(b) exhibits the fundamental and odd harmonics, but it is made up of cosine waves rather than sine waves. The sawtooth wave in Fig. 2-60(c) contains the fundamental plus all odd and even harmonics. Fig. 2-60(d) and (e) shows half sine pulses like those seen at the output of half and full-wave rectifiers. Both have an average dc component, as would be expected. The half wave signal is made up of even harmonics only, whereas the full wave signal has both odd and even harmonics. Fig. 2-60( f ) shows the Fourier expression for a dc square wave where the average dc component is Vt0/T.
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