fourier theory

The mathematical analysis of the modulation and multiplexing methods used in communication systems assumes sine wave carriers and information signals Fourier Theory. 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 Fourier Theory.

 

Basic Concepts

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°.



Figure 2-55 Sine and cosine waves.

 

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 Fourier Theory.



Figure 2-56 A sine wave and its harmonics.

 

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.



Figure 2-57 A square wave.



Figure 2-58 A square wave is made up of a fundamental sine wave and an infinite number of odd harmonics.

 

Fig. 2-58 shows how this is done Fourier Theory. 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 harmonic are added. Note the shape of the composite wave with the third and fifth harmonics added, as in Fig. 2-58(b). The more 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 fundamental. The results very closely approximate a square wave.



Figure 2-59 Square wave made up of 20 odd harmonics added to the fundamental.

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 Fourier Theory. 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=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 Fourier Theory.



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.

 

Time Domain Versus Frequency Domain

Most of the signals and waveforms that we discuss and analyze are expressed in the time domain. That is, they are variations of voltage, current, or power with respect to time. All the signals shown in the previous illustrations are examples of time-domain waveforms. Their mathematical expressions contain the variable time t, indicating that they are a time-variant quantity.

 

Fourier theory gives us a new and different way to express and illustrate complex signals Fourier Theory. Here, complex signals containing many sine and/or cosine components are expressed as sine or cosine wave amplitudes at different frequencies. In other words, a graph of a particular signal is a plot of sine and/or cosine component amplitudes with respect to frequency.

A typical frequency-domain plot of the square wave is shown in Fig. 2-61(a). Note that the straight lines represent the sine wave amplitudes of the fundamental and harmonics, and these are plotted on a horizontal frequency axis. Such a frequencydomain plot can be made directly from the Fourier expression by simply using the frequencies of the fundamentals and harmonics and their amplitudes. Frequency-domain plots for some of the other common nonsinusoidal waves are also shown in Fig. 2-61.



Figure 2-61 The frequency-domain plots of common nonsinusoidal waves. (a) Square wave. (b) Sawtooth. (c) Triangle. (d) Half cosine wave.

 

Note that the triangle wave in Fig. 2-61(c) is made up of fundamental and odd harmonics. The third harmonic is shown as a line below the axis, which indicates a 180° phase shift in the cosine wave making it up.



Fig. 2-62 shows how the time and frequency domains are related. The square wave discussed earlier is used as an example. The result is a three-axis three dimensional view Fourier Theory.

Figure 2-62 The relationship between time and frequency domains.

 

Signals and waveforms in communication applications are expressed by using both time-domain and frequency-domain plots, but in many cases the frequency-domain plot is far more useful. This is particularly true in the analysis of complex signal waveforms as well as the many modulation and multiplexing methods used in communication.

Test instruments for displaying signals in both time and frequency domains are readily available. You are already familiar with the oscilloscope, which displays the voltage amplitude of a signal with respect to a horizontal time axis.

The test instrument for producing a frequency-domain display is the spectrum analyzer. Like the oscilloscope, the spectrum analyzer uses a cathode-ray tube for display, but the horizontal sweep axis is calibrated in hertz and the vertical axis is calibrated in volts or power units or decibels.

 

The Importance of Fourier Theory

Fourier analysis allows us to determine not only the sine wave components in any complex signal but also how much bandwidth a particular signal occupies. Although a sine or cosine wave at a single frequency theoretically occupies no bandwidth, complex signals obviously take up more spectrum space. For example, a 1-MHz square wave with harmonics up to the eleventh occupies a bandwidth of 11 MHz. If this signal is to pass unattenuated and undistorted, then all harmonics must be passed.

 



Figure 2-63 Converting a square wave to sine wave by filtering out all the harmonics.

 

An example is shown in Fig. 2-63. If a 1-kHz square wave is passed through a low pass filter with a cutoff frequency just above 1 kHz, all the harmonics beyond the third harmonic are greatly attenuated or, for the most part, filtered out completely. The result is that the output of the low-pass filter is simply the fundamental sine wave at the square wave frequency.

If the low-pass filter were set to cut off at a frequency above the third harmonic, then the output of the filter would consist of a fundamental sine wave and the third harmonic. Such a wave shape was shown in Fig. 2-58(a). As you can see, when the higher harmonics are not all passed, the original signal is greatly distorted. This is why it is important for communication circuits and systems to have a bandwidth wide enough to accommodate all the harmonic components within the signal waveform to be processed.

Fig. 2-64 shows an example in which a 1-kHz square wave is passed through a bandpass filter set to the third harmonic, resulting in a 3-kHz sine wave output. In this case, the filter used is sharp enough to select out the desired component.

Pulse Spectrum

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 =1/T and T is the period. The first 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.



Figure 2-65 Frequency domain of a rectangular pulse train.

 

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 (sin x)/x, where



and t₀ is the pulse 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 crossing 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 crossing of the envelope is the minimum that is sufficient to pass enough harmonics for reasonable wave shape:



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

 

The Relationship between Rise Time and Bandwidth

Because a rectangular wave such as a square wave theoretically contains an infinite number of harmonics, we can use a square wave as the basis for determining the bandwidth of a signal. If the processing circuit should pass all or an infinite number of harmonics, the rise and fall times of the square wave will be zero. As the bandwidth is decreased by rolling off or filtering out the higher frequencies, the higher harmonics are greatly attenuated. The effect this has on the square wave is that the rise and fall times of the waveform become finite and increase as more and more of the higher harmonics are filtered out. The more restricted the bandwidth, the fewer the harmonics passed and the greater the rise and fall times. The ultimate restriction

is where all the harmonics are filtered out, leaving only the fundamental sine wave (Fig. 2-63).

The concept of rise and fall times is illustrated in Fig. 2-66. The rise time t_r is the time it takes the pulse voltage to rise from its 10 percent value to its 90 percent value. The fall time t_f is the time it takes the voltage to drop from the 90 percent value to the 10 percent value. Pulse width t₀ is usually measured at the 50 percent amplitude points on the leading (rise) and trailing (fall) edges of the pulse.

A simple mathematical expression relating the rise time of a rectangular wave and the bandwidth of a circuit required to pass the wave without distortion is



This is the bandwidth of the circuit required to pass a signal containing the highest frequency component in a square wave with a rise time of t_r. In this expression, the bandwidth is really the upper 3-dB down cutoff frequency of the circuit given in megahertz. The rise time of the output square wave is given in microseconds. For example, if the square wave output of an amplifier has a rise time of 10 ns (0.01 μs), the bandwidth of the circuit must be at least BW= 0.35/0.01= 35 MHz.

Rearranging the formula, you can calculate the rise time of an output signal from the circuit whose bandwidth is given: t_r= 0.35/ BW. For example, a circuit with a 50-MHz bandwidth will pass a square wave with a minimum rise time of t_r= 0.35/50= 0.007 μs= 7 ns.

This simple relationship permits you to quickly determine the approximate bandwidth of a circuit needed to pass a rectangular waveform with a given rise time. This relationship is widely used to express the frequency response of the vertical amplifier in an oscilloscope. Oscilloscope specifications often give only a rise time figure for the vertical amplifier. An oscilloscope with a 60-MHz bandwidth would pass rectangular waveforms with rise times as short as t_r= 0.35/60= 0.00583 μs = 5.83 ns.

Similarly, an oscilloscope whose vertical amplifier is rated at 2 ns (0.002 μs) has a bandwidth or upper cutoff frequency of BW= 0.35/0.002=175 MHz. What this means is that the vertical amplifier of the oscilloscope has a bandwidth adequate to pass a sufficient number of harmonics so that the resulting rectangular wave has a rise time of 2 ns. This does not indicate the rise time of the input square wave itself. To take this into account, you use the formula



where t_ri= rise time of input square wave

t_ra=rise time of amplifier

t_r= composite rise time of amplifier output

The expression can be expanded to include the effect of additional stages of amplification by simply adding the squares of the individual rise times to the above expression before taking the square root of it.

Keep in mind that the bandwidth or upper cut-off frequency derived from the rise time formula on the previous page passes only the harmonics needed to support the rise time. There are harmonics beyond this bandwidth that also contribute to unwanted emissions and noise Fourier Theory.