FourierTrans_06

FourierTrans_06 - Fourier Transform Fourier series analysis...

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Fourier Transform Fourier series analysis of LTI systems is very useful when the input signals are periodic. Unfortunately, it is of limited use, because periodic signals do not exist in the real world. All physical signals are not periodic. The term "aperiodic" is frequently used to describe signals that are not periodic. If Fourier analysis is to be more useful it must be extended to include aperiodic signals. Investigating the Properties of a Pulse Train Consider the Fourier coefficients of the pulse train illustrated below, The Fourier coefficients are easily obtained () 00 0 2 2 0 2 2 22 0 0 0 0 11 2 sin 2 sin 2 2 sin 2 jk t jk t jk t k T jk jk A X x t e dt Ae dt e TT j k T AA ee k jk T k T k A kS a k kT T T k τ ωω ω ττ −− −+ == =  =− = =   = ∫∫ Now consider what happens to these Fourier coefficients as the values of A , τ , and T change. First it can be seen that changing the value of A increases or decreases the value of all of the coefficients by the same amount. The effect of changing T or τ is more complex and more interesting. Substituting into the above expression for , 0 2 T ωπ = it can be written k A XS a k π =
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2 In this expression only appear as a ratio . This implies that the Fourier and T τ T coefficients for all pulses that occupy half of the period will have the same Fourier coefficients, regardless of the value of the period. Thus a pulse train that repeats every microsecond with a pulse width of half a microsecond, will have the same Fourier Coefficients as a pulse train that repeats every second with pulses of duration of half a second. The shape of their spectrums will be exactly the same. Of course the location on the frequency axis of these coefficients will be different. In the first case the signal that repeats every microsecond will appear at multiples of 1MHz, while the signal that repeats every second will have spectral lines located at multiples of 1Hz. This is equivalent to scaling the frequency axis. A lot can be learned by examining how the Fourier coefficients are changed as the ratio of the pulse width to the period of the pulse train is changed. Changing the Width of the Pulse Consider two different pulse trains having the same period. The only difference between the two pulse trains is that the pulse width of one (blue) is half the width of the other (red). Their spectrums are shown in the next figure. Here the red spectrum corresponds to the red pulse train and the blue spectrum to the blue pulse train (the blue pulse is half the width of the red pulse).
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3 Three important features: 1. The spectral lines have the same separation. This is a result of the fact that they have the same period, i.e., . 0 2 T π ω = 2. The first zero in the spectrum of the pulse train, containing the wide pulse (red), appears at half the frequency of the pulse train containing the narrow pulse (blue).
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FourierTrans_06 - Fourier Transform Fourier series analysis...

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