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It may be proved mathematically by Fourier analysis Graphical synthesis may be used. The preceding statement may be verified in any one of three different ways. As a general rule, it may be added that the higher the harmonic, the lower its energy level, so that in bandwidth calculations the highest harmonics are often ignored. For some waveforms only the even (or perhaps only the odd) harmonics will be present. Some non-sine wave recurring at a rate of 200 times per second will consist of a 200-Hz fundamental sine wave, and harmonics at 400, 600 and 800 Hz, and so on. There are an infinite number of such harmonics. The frequency of the lowest-frequency, or fundamental, sine wave is equal to the repetition rate of the nonsinusoidal waveform, and all others are harmonics of the fundamental. It may be shown that any nonsinusoidal, single-valued repetitive waveform consists of sine waves and/or cosine waves. Bandwidth Requirement in Communication System will therefore he considerably greater than might have been expected if only the repetition rate of such a wave had been taken into account. If any nonsinusoidal waves, such as square waves, are to be transmitted by a communications system, then it is important to realize that each such wave may be broken down into its component sine waves.
#BANDWIDTH IN COMMUNICATION SERIES#
The first four terms of this series for the rectangular waveform are:įrequency Spectra of Non Sinusoidal Wave: The Fourier coefficients for the rectangular waveform in Figure 1-4 are: If we substitute ω 0 for 2π∕T(ω 0=2πf 0=2π∕T) in Equation (1-4), we can rewrite the Fourier series in radian terms:Įquation (1-4) supports the statement The makeup of a square or rectangular wave is the sum of ( harmonics) the sine wave components at various amplitudes.
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The expression will become clearer when the first four terms are illustrated: The form for the Fourier series is as follows:Įach term is a simple mathematical symbol and shall be explained as follows: This simplified review of the Fourier series is meant to reacquaint the student with the basics. Figure 1-4 is an example of a rectangular wave, where A designates amplitude, T represents time, and τ indicates pulse width. Some examples of waveforms are sine, square, rectangular, triangular, and sawtooth. To expand upon the topic of Bandwidth Requirement in Communication System, we will define the terms of the expressions and provide examples so that these topics can be clearly understood.Ī periodic waveform has amplitude and repeats itself during a specific time period T. Next we will review the Fourier series, which is used to express periodic time functions in the frequency domain, and the Fourier transform, which is used to express non periodic time domain functions in the frequency domain. The symbol f in Equation (1-1) represents the frequency of the sine wave signal. If the voltage waveform described by this expression were applied to the vertical input of an oscilloscope, a sine wave would be displayed on the CRT screen. Described mathematically in the time domain and in the frequency domain, this signal may be represented as follows: It is very important in communications to have a basic understanding of a sine wave signal. Sine Wave and Fourier Series Formula Review: Since such nonsinusoidal waves occur very frequently as modulating signals in communications. However, if the modulating signals are non sinusoidal, a much more complex situation results.
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If this consists of sinusoidal signals, there is no problem, and the occupied bandwidth will simply be the frequency range between the lowest and the highest sine-wave signal. Before trying to estimate the Bandwidth Requirement in Communication System of a modulated transmission, it is essential to know the bandwidth occupied by the modulating signal itself.