EEE 332 COMMUNICATION Fourier Series Text book: Louis E. Frenzel. Jr. Principles of Electronic Communication Systems, Third Ed. Mc Graw Hill.

2-4: Fourier Theory 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. Practically, 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 non-sinusoidal 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. 2

2-4: Fourier Theory 3 Figure 2-57: A sine wave and its first four harmonics. 2-4: Fourier Theory A harmonic is a sine wave whose frequency is some integer multiple of a fundamental sine wave. Ex: The 3 rd harmonic of a 2 kHz sine wave is a sine wave of 6 kHz.

2-4: Fourier Theory The Fourier theory states that a nonsinusoidal waveform can be broken down into individual harmonically related sine wave or cosine wave components. A square wave is one classic example of this phenomenon, which is a rectangular signal with equal duration positive and negative alternations. 4 sine wavecosine wave

2-4: Fourier Theory Basic Concepts Fourier analysis states 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. Ex: 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. Fourier analysis allows us to determine not only sine-wave components in a complex signal but also a signal’s bandwidth. 5

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 p-p (or 10 V p ). When the sine wave values are added instantaneously, the results approaches a square wave. 6

The more higher harmonics that are added, the more the composite wave looks like a perfect square wave. Figure below shows how the composite wave would look with more than 20 odd harmonics added to the fundamental. The results very closely approximate a square wave. 7

The implication of this is that a square wave should be analyzed as a collection of harmonically related sine waves rather than a single 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/  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 3 rd, 5 th, and 7 th harmonics. Note that the terms also have an amplitude factor. In this case, the amplitude is also a function of the harmonic. The 3 rd harmonic has an amplitude that is 1/3 of the fundamental amplitude. 8

If the square wave is direct current rather than alternating current, as shown in (b), the Fourier expression has a dc component. In this equation, V/2 is the dc component, the average value of the square. 9

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 10

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The triangular wave exhibits the fundamental and odd harmonics, but it is made up of cosine waves rather than sine waves. 12

The sawtooth wave contains the fundamental plus all odd and even harmonics. 13

Figures (d) and (e) show half cosine pulses like those seen at the output of half and full wave rectifiers, respectively. Both have an average dc component. The half wave signal is made up of even harmonics only, whereas the full wave signal has both odd and even harmonics. 14

The figure below shows the Fourier expression for a dc square wave where the average dc component is. 15

2-4: Fourier Theory Time Domain Versus Frequency Domain –Analysis of variations of voltage, current, or power with respect to time are expressed in the time domain. Most of the signals and waveforms that had been seen so far are in this domain. Their mathematical expressions contain the variable time t, indicating that they are a time-variant quantity. –A frequency domain plots sine and/or cosine component amplitude variations with respect to frequency. –Fourier theory gives us a new and different way to express and illustrate complex signals, that is, with respect to frequency. The complex signals containing many sine and/or cosine components are expressed as sine or cosine wave amplitudes at different frequencies. 16

2-4: Fourier Theory 17 Figure 2-63: The relationship between time and frequency domains.

2-4: Fourier Theory Time Domain Versus Frequency Domain –Signals and waveforms in communication applications are expressed by using both time-domain and frequency domain plots. 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. –An oscilloscope displays the voltage amplitude of a signal wrt a horizontal time axis. On the other hand, a spectrum analyzer, is the instrument used to produce a frequency-domain display. The horizontal sweep axis of the spectrum analyzer is calibrated in Hz and the vertical axis is calibrated in volts or power units or decibels. –It is the key test instrument in designing, analyzing, and troubleshooting communication equipment. 18