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:

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:

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:

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

Then the signal

also represents the periodic signal with the period

and its spectrum is equal to:

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:

Substitute expanding into Fourier series instead of the signal

in (6):

Change manipulations of integrating and summing in (7) and receive:

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:

The amplitude and phase spectrums of the real signal are equal to:

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:

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:

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

where

is the conjugate signal

Substitute Fourier series equations in the complex form in (13):

It is easy to show that the integral in equation (14) for any integers

and

is equal to:

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:

Making equal (16) and (13) we receive Parseval’s Theorem connecting the average capacity of the periodic signal in the time and frequency domains:

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

[1]
Воробьев Н.Н.
Теория рядов.
Москва, Наука, Главная редакция физико-математической литературы, 1979.

[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.