Application and Simulation of Fourier Analysis in Communication Circuit

This article mainly studies the application of Fourier analysis theory in wireless communication circuits. Firstly, the relevant basic theories of Fourier analysis are introduced, including the decomposition of periodic signal, the modulation characteristic of Fourier transform, and the system function; then the relevant applications of Fourier theory in communication circuits are introduced, such as high-frequency resonant power amplifier, amplitude modulation, demodulation and mixing, low-pass filter. Finally, the application of Fourier theory in communication circuits is simulated and analyzed through Multisim circuit simulation software.


Introduction
Fourier analysis is an important branch of analytical mathematics. From the history of mathematics development, as early as the beginning of the 18th century, D. Bernoulli, L. Euler and others once discussed trigonometric series in their work. However, the really important step was taken by the French mathematician J. Fourier. In 1822, he systematically used trigonometric series and trigonometric integrals to deal with heat conduction problems in his book "The Analytical Theory of Heat" [1]. After that, scientists such as Dirichlet, Riemann, Lipschitz and Jordan have been engaged in research in this field. Their research results not only made up for the deficiencies in Fourier's work, but also greatly developed the series theory and expanded the Fourier analysis. The application range of Fourier analysis theory has been developed rapidly.
Fourier analysis is not only a mathematical tool, but its vigorous development has made it widely used in many disciplines such as physics, electronic science, statistics, signal processing, cryptography, optics, medical, oceanography, structural mechanics [2]- [8], etc. It plays an extremely important role not only in the fields of communication and control, but also in the fields of earthquake, nuclear science, biomedicine, and electric power engineering. In the information age, communication is inseparable from people's lives. Communication technology has been greatly developed, and the development of communication technology is accompanied by the careful application of Fourier analysis! Such as modulation, power amplification, filters, sampling, frequency division multiplexing and so on.
This article mainly studies the application of Fourier analysis theory in wireless communication circuits, and uses Multisim circuit simulation software to simulate and analyze the specific applications of Fourier analysis in communication circuits, and then understand the application of Fourier analysis theory in communication circuits intuitively and vividly. The article is mainly divided into three parts: one is the introduction of Fourier analysis theory, the other is the application theory of Fourier analysis in communication circuits, and finally, the application and analysis of Fourier . when this signal satisfies Dirichlet condition, it can be decomposed into the following trigonometric series -referred to as Fourier series. series-called the Fourier series [10].
(2) Combining formula (1) with the frequency term can be written as: Equation (3) shows that the periodic signal f(t) can be decomposed into the form of superposition of DC and many cosine components.

2.2.Fourier transform frequency shift characteristics-modulation properties
The amplitude-frequency characteristics and phase-frequency characteristics of an ideal low-pass filter are shown in figure 1.

3.Basic composition of communication system
The wireless communication system mainly includes modulation, demodulation, high frequency resonant power amplifier, high frequency small signal amplifier, mixer, oscillator, frequency multiplier, filter. Among them, amplitude modulation, demodulation, and frequency mixing are all modulation characteristics of Fourier transform. High-frequency resonant power amplifier and frequency multiplier use the series decomposition theory of periodic signals. The filter is designed using the system function and the principle of low-pass filter. Fourier theory is fully applied in various functional circuits of wireless communication.

3.1.High frequency resonant power amplifier
High-frequency power amplifier, also known as Class C power amplifier, is an important part of the transmitting end of the wireless communication system. Its function is to amplify the signal to be transmitted to generate the required power to meet the antenna and load requirements [11] [12]. The high-frequency resonant power amplifier is an energy conversion device that can convert the DC energy supplied by the power supply into a high-frequency AC output.

3.1.1.Circuit composition
The working principle of the high-frequency power amplifier is shown in Figure 2. It mainly includes three parts: input loop, amplifier, and collector output loop. In the base loop, V BB is the base bias voltage, which is used to make the amplifier work in a Class C state, and its value is usually zero or negative. In the collector circuit, the load selects the LC resonant circuit, which is tuned to the center frequency of the input signal to achieve high-frequency amplification. Tuned to an integer multiple of the input signal frequency, it can be used as a frequency multiplier. R L is the external equivalent load resistance of the power amplifier. V CC is the collector DC power supply.   Figure 3 shows that the input is a cosine signal, and the output is a periodic cosine pulse signal. The cosine pulse is a periodic signal, which satisfies the Dirichri condition, so the Fourier series expansion is In formulas (11) Through the collector LC frequency selection network, the fundamental wave signal can be taken out, and high-order harmonics and direct current can be filtered out to obtain a distortion-free cosine signal, that is, the fundamental frequency signal. Compared with the input signal, the fundamental frequency signal has the same frequency and reverse phase, and the amplitude is amplified.

3.2.Modulation, demodulation, and mixing theory
Modulation is an indispensable and very important module in wireless communication. It linearly moves low-frequency signals to high-frequency places, which not only facilitates antenna transmission, but also facilitates multiplexing.
Let the modulation signal (low frequency signal) be a cosine signal with a single frequency, the  The formula (16) are spread: Demodulation is to move the frequency spectrum of the modulated signal from the high frequency to the low frequency, which is a linear movement of the frequency spectrum. So just multiply the AM signal by t c  cos to achieve demodulation. Mathematical expression: The modulated signal can be recovered by passing the signal of formula (17) through the low-pass filter.
Mixing only changes the center frequency of the modulated wave signal and does not change the spectrum structure, but only moves the modulated wave signal from the high frequency to the intermediate frequency. Modulation, demodulation, and mixing all use the frequency shift characteristics (modulation characteristics) of the Fourier transform to realize the shift of the frequency domain spectrum by multiplying the cosine signal in the time domain.

3.3.Low-pass filter
Low-pass filters are commonly used filters to remove noise. Butterworth low-pass filters have maximum flat amplitude-frequency response curves in the passband, and there is no ripple in the passband, so they are widely used in signal denoising.
The amplitude-frequency characteristics of Butterworth low-pass filter are [9]: Among them, c  is the cut-off angle frequency, uo A is the voltage amplification in the pass band, and n is the order of the filter. It can be found from equation (19) The larger the n , the faster the attenuation, and the more ideal the amplitude-frequency characteristics.
When  Figure 4 is a second-order low-pass active filter. High-order filters can be cascaded through low-order filters.  Figure 5 is the circuit diagram of the designed high-frequency resonant power amplifier. The circuit includes three parts: the base input circuit, the amplifier, and the collector circuit. The collector load adopts two kinds of loads, the switch of pure resistance load and LC resonant circuit is controlled by single-pole double-throw switch.

4.1.High frequency resonant power amplifier
Turn the switch to pure resistance 2 R and observe the collector output waveform as shown in Figure 6. It can be seen from figure 6 that the input is a cosine signal, and the collector output is a periodic cosine pulse with the same period as the input signal. This periodic pulse satisfies the Dirichri condition and can be decomposed into a Fourier series, decomposed into a form of superposition of many cosine signals. Through Multisim-analyses-Fourieranalysis, you can observe the decomposed frequency spectrum as shown in figure 8. From figure 7 and figure 8, it can be found that the input signal is a single frequency, and the output signal contains many frequency components and is an integer multiple of the input signal.
Set the switch to the LC resonant circuit, click Simulate, and observe the output waveform as shown in figure 9. From figure 9, it can be found that the output signal and the input signal have the same waveform, the same cosine signal, and the frequency is the same, both are MHz 1 , and the output signal has a larger amplitude than the input signal, achieving high-frequency amplification. The output spectrum is shown in figure 10, which is a single-frequency signal. Because the LC resonant circuit has a frequency selection function, the fundamental frequency signal is amplified without distortion, so the output and input signal frequency is the same, and the output signal amplitude becomes larger.  Figure 11 shows the diode balance modulation circuit, the modulation signal (low frequency signal) is 1kHz, and the high frequency carrier signal is 100kHz. Click Simulate to get the AM amplitude modulation signal waveform as shown in figure 12. It can be seen from the waveforms in figure 12 that the envelope of the AM amplitude modulation signal is consistent with the low frequency signal, achieving amplitude modulation. Use Multisim-analyses-Fourieranalysis to observe the frequency spectrum of the input low-frequency signal and output AM signal, as shown in figure 13 and figure 14. It can be seen from figure 13 and figure 14 that the input is a single spectrum, and the output AM signal spectrum is two ways that the input signal spectrum is linearly shifted to the high-frequency signal.  Figure 15 shows the second-order Butterworth low-pass filter with a cut-off frequency of 1kHz. Observe the transfer function and amplitude-frequency characteristic curve with a Baud meter. The  Figure 16 shows the amplitude-frequency characteristic curve of the Butterworth filter. From the amplitude-frequency diagram, it can be seen that the passband is flat without ripples, and the stopband drops slowly. The amplitude at 996Hz drops 4.018-1.016=3dB. Through Multisim AC analysis, observe the amplitude-frequency characteristic curve of the second-order Butterworth filter, as shown in figure 17. The signal amplitude in the passband is 1.5882V, and the cut-off frequency should be 1.5882*0.707=1.1228V. It can be seen from the scanning curve that it is 1.1204V at 1kHz and 15.7272mV at 10 times the stopband, which is 37dB lower than that at 1kHz. 40dB lower than the passband.

5.conclusion
Fourier analysis theory has extremely important applications in various fields. This paper studies the related theory of Fourier analysis and its application in wireless communication circuits, and simulates and analyzes the application of Fourier analysis theory in communication circuits. through Multisim circuit simulation software, and the application of Fourier theory in communication circuits is displayed intuitively and vividly.