Some Properties of Expanding Periodic Signals into Fourier Series

Introduction
In this section we will consider some properties of periodic signal spectrums. As we will see later, Fourier transform of non-periodic signals and discrete Fourier transform also have similar properties.
Linearity
Let’s suppose that there are two periodic signals and with equal repetition periods and both signals meet Dirichlet conditions and can be presented by Fourier series with expanding coefficients and where Here and elsewhere in this section we will accept signals and as periodic ones with equal repetition periods and both signals meet Dirichlet conditions. Then the signal is also a periodic signal with the period and it can be also presented by Fourier series with coefficients:
(1)
(1)
Thus, the sum spectrum of periodic signals is equal to the sum of their spectrums.
The property of multiplying by the constant is a consequence of linearity. The signal spectrum , is equal to:
(2)
(2)
Cyclic Time Shift Property
Consider the signal as a result of the cyclic time signal shift for the value . The cyclic time shift is shown in Figure 1 for positive and negative values

Figure 1. Cyclic Time Shift of the Periodic Signal
The spectrum of the signal with the cyclic time shift is equal to:
(3)
(3)
Thus, the time shift of the periodic signal for the value leads to multiplying the spectrum by the phase multiplier .
Spectrum of Cyclic Signal Folding
Let the signal represent periodic convolution of signals and
(4)
(4)
Then the signal also represents the periodic signal with the period and its spectrum is equal to:
(5)
(5)
Equation (5) uses the cyclic time shift property considered above.
Thus, the spectrum of the periodic signal represents cyclic convolution of periodic signals and is equal to the spectrum product of these signals multiplied by the repetition period It is one of the most important properties of the spectral analysis which allows to analyze systems of processing signals in the frequency domain replacing cumbersome calculating convolution of signals with the product of their spectrums.
Signal Product Spectrum
Let the signal represent the product of signals and The signal also represents the periodic signal with the period and its spectrum is equal to:
(6)
(6)
Substitute expanding into Fourier series instead of the signal in (6):
(7)
(7)
Change manipulations of integrating and summing in (7) and receive:
(8)
(8)
Thus, the spectrum of two signals product is equal to linear convolution of these signal spectrums.
Symmetry of Real Signal Spectrum
Let represent a real periodic signal. Consider its spectrum in more detail:
(9)
(9)
The amplitude and phase spectrums of the real signal are equal to:
(10)
(10)
Analyzing equation (10) it is possible to pay attention that …, i.e. the amplitude spectrum of the real periodic signal is always symmetric as regard to the zero frequency, and the phase spectrum is antisymmetric.
If the periodic signal is complex, then the signal spectrum symmetry is broken that will be shown in the following paragraph.
Frequency Shift Property
Let the signal represent the product of signals and the complex exponent with the frequency where is the arbitrary integer. The frequency choice provides periodicity of the signal as one period includes the integer of complex exponent turns. Thus, the signal meets Dirichlet conditions, and its spectrum is equal to:
(11)
(11)
Multiplying the signal by the complex exponent transfers the signal spectrum to the frequency At the same time the signal becomes complex, and its spectrum becomes asymmetrical as regard to the zero frequency.
In Figure 2 the example of the frequency signal shift upon multiplying by the complex exponent when rad/s (5 Hz) is shown.

Figure 2. Example of the Frequency Shift of the Periodic Rectangular Pulse Signal upon Multiplying by the Complex Exponent.
Figure 2 shows that the spectrum of the displaced frequency signal is the displaced copy spectrum for the frequency At the same time it is important to note that the signal has become complex (real and imaginary signal parts are shown separately in the diagram), and at that its amplitude spectrum has stopped being symmetric, and the phase one antisymmetric as regard to the zero frequency.
Now consider multiplying the signal not by the complex exponent, but by the simple harmonic signal where is the arbitrary integer, is the arbitrary initial phase. In this case we also keep the signal periodicity and its spectrum is equal to:
(12)
(12)
Thus, multiplying the signal by the simple harmonic signal leads to displacing the spectrum for the frequencies both in the positive and negative domains of frequencies, reducing amplitude in the positive and negative domains twice and adding a phase multiplier. At that we will note that the real signal remains real with the symmetric amplitude spectrum.
In Figure 3 the example of the frequency signal shift upon multiplying by when rad/s (5 Hz), and rad is shown.

Figure 3. Example of the Frequency Shift of the Periodic Rectangular Pulse Signal.
Figure 3 shows that the spectrum of the displaced for the frequency signal is the sum of the displaced spectrums of the half amplitude for the frequencies .
Parseval’s Theorem
Let’s suppose that there is a periodic signal which represents the current or voltage time-varying value. Consider the average capacity released within 1 Ohm resistance by the signal
(13)
(13)
where is the conjugate signal Substitute Fourier series equations in the complex form in (13):
(14)
(14)
It is easy to show that the integral in equation (14) for any integers and is equal to:
(15)
(15)
Taking into account (15), only components when are excluded from the double sum in equation (14), and (14) will be converted to the form of:
(16)
(16)
Making equal (16) and (13) we receive Parseval’s Theorem connecting the average capacity of the periodic signal in the time and frequency domains:
(17)
(17)
From (17) it follows that the average capacity released within 1 Ohm resistance during one signal repetition period is equal to the sum of squares of spectrum components modules of this signal.
Conclusions
In this section we have considered some properties of periodic signal spectrums: linearity, properties of time and frequency shifts, convolution spectrum and product of signals. We have also analyzed the spectrum symmetry property of the real signal and learned that the amplitude spectrum of the periodic signal is symmetric, and the phase spectrum is antisymmetric as regard to the zero frequency. And at last we have considered Parseval’s Theorem which sets a ratio of the average signal capacity in the time and frequency domains.
In the following section we will analyze expanding non-periodic signals in accordance with the system of complex exponents and we will receive the continuous Fourier transform.
References
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[2] Баскаков С.И. Радиотехнические цепи и сигналы Москва, Высшая школа, 2000.

[3] Гоноровский И.С. Радиотехнические цепи и сигналы Москва, Сов. радио, 1977.

[4] Folland G.B. Fourier Analysis and its Applications Belmont, Wadsworth& Brooks, 1992.