Engineering basics-Fourier analysis, Fuli Analysis

Engineering basics-Fourier analysis, Fuli Analysis

Introduction

If you are confused about why Newton believes in God in his later years, Fourier analysis will be able to feel more or less the reverence and confusion of scientific masters on natural laws, the beauty of symmetry does not help but doubt that the laws of nature are developed by the so-called creators. Otherwise, the seemingly mixed phenomena imply subtle internal laws, today, we are not very concerned about the existence of our creators, comrade Xiaoping's discussion of science and technology is the first productive force. In China, it even fades away from the "strange tricks and tricks" that have been shrouded in science and technology for a long time and the Western theology, as a means to build an ideal human society and enhance human well-being, science and technology are increasingly practical.

The Fourier analysis method belongs to the field of communication technology in the engineering field. It is used to construct the characteristic relationship between the communication signal and the communication system in the time domain and frequency domain, it can be said that without Fourier analysis, there will be no modern communication technology, because the development of communication technology is closely related to the degree of human cognition of the frequency domain or the frequency domain, similar to other field technologies that we don't have in our daily senses, the frequency domain is invisible to anyone. This also makes the theory of Fourier analysis very difficult, in particular, the conclusions derived from rigorous mathematical formulas make it difficult for people without perceptual knowledge to understand and accept and use them to guide practices, although I am a major in communication, I am deeply touched by the pain of the formula. Therefore, I want to explain the Fourier analysis I understand in a way that is fully described.

I. signal and system Classification

From the time domain point of view, the signal hasTime and amplitude coordinate axes (Time Functions)Description
From the perspective of time domain, the system hasTime and amplitude coordinate axes (Unit Impulse Response characteristics)Description

From the perspective of frequency domain, the signal hasAmplitude and frequency coordinate axes (amplitude-frequency characteristics)AndPhase and frequency coordinate axes (Phase-frequency characteristics)Description
From the perspective of frequency domain, the system hasAmplitude and frequency coordinate axes (amplitude-frequency characteristics)AndPhase and frequency coordinate axes (Phase-frequency characteristics)Description

From the time domain description, signals can be divided:

Continuous/analog signal-continuous Signal Time and amplitude
Discrete/sampled signal-time discrete signal and continuous Amplitude
Digital/quantified signal-Discrete Signal Time and amplitude

In actual engineering, discrete/sampled signals are used only as the form of analog-to-digital conversion and are not used in other signal processing processes. Therefore, the system uses the processing of the other two forms of signals as the basis for division.

From the time domain description, the system can be divided:

Continuous System-Unit Impulse Response Characteristic CurveH (t) Continuous Time, input and output analog signal
Discrete System-Unit Impulse Response Characteristic CurveH (n) discrete time, input and output Digital Signal

Note: The default discrete signal refers to a digital signal. If the discrete signal is considered to correspond to the time axis, it can be considered as a special analog signal.

The Time Domain descriptions of signals, system continuity, and discretization are divided and defined respectively. However, both theoretical analysis and engineering practice require the construction of corresponding signals and System Frequency Domain descriptions, therefore, a set of analysis methods are required to establish corresponding time-frequency connections.

Ii. Fourier Analysis

Premise conclusion:

1. the time domain description constructed in Fourier analysis is a time-range change from negative infinity to positive infinity, that is, a stable frequency domain description corresponds to a time-domain description of a stable infinity time.

2. only the system can have a stable and infinite time domain description, that is, a stable frequency domain description (spectrum). in engineering practice, the signal must have a stable time domain description within the period of symbol, that is to say, the frequency domain description of a signal changes with the change of the symbol cycle, and there are different stable descriptions in different symbol periods.

3. In both theoretical analysis and engineering practice, it is assumed that the signals in the time domain are cyclical and the system is non-cyclical. In practice, it is often assumed that the period of the Finite Long message number is extended as a periodic signal.

Fourier analysis transform pair:

Time Domain <-> frequency domain <-> Conversion Type
Continuous non-periodic <-> non-periodic continuous <-> Fourier Transformation
Continuous period <-> non-periodic discretization <-> Fourier Series
Fourier transformation of discrete non-periodic <-> periodic continuity <-> Sequences
Discrete period <-> periodic discretization <-> Discrete Fourier transformation/Discrete Fourier Series

The following describes the important laws of Fourier Transformation:
Both the time-frequency and time-frequency values meet the requirements.Continuous <-> non-periodicAndDiscrete <-> Period

The premise argument can extend the Fourier transform pair to the corresponding signal and system type:

Time Domain <-> frequency domain <-> Conversion Type <-> signal/system type
Continuous non-cyclic h (t) <-> non-cyclic continuous H (jw) <-> Fourier transformation <-> continuous system
Continuous period f (t) <-> non-periodic discretization <-> Fourier series <-> continuous signal
Discrete non-periodic h (n) <-> periodic continuous H (e ^ jw) <-> Fourier transformation of sequences <-> Discrete Systems
Discrete period x (n) <-> periodic discretization <-> Discrete Fourier transformation/Discrete Fourier series <-> Discrete Signal

Classic rule:

1. Signal Time-Domain sampling is the discretization of the signal time-domain. This will inevitably result in a periodic signal frequency domain, that is, the pattern of the change from a continuous signal to a discrete signal spectrum;The periodic frequency in the frequency domain is the sampling frequency..

2. the frequency domain description of a continuous system is non-periodic, that is, high and low bandwidths are for the infinite frequency domain; the frequency domain description of a discrete system is periodic, that is, high and low bandwidths are for each frequency domain period.

3. by using the low-pass filtering feature of a continuous system, only the discrete signal spectrum of a period can be retained, and thus the discrete signal can be restored to a continuous signal; you can also use the raised cosine filter to perform Pulse Forming (interpolation) on discrete signals and convert them into continuous baseband signals that resist inter-code interference.

4. assume that the finite long sequence carries a periodic extension in the time domain, and the discrete Fourier transform is used to describe the frequency domain of the periodic sequence. Because the frequency domain of the periodic sequence is described as a periodic, therefore, you only need to find the frequency domain description cycle extension in a certain period. The Discrete Fourier Transform is essentially a period of Fourier transformation for a finite long sequence.N points and other interval sampling.

5.Principles of DFT/FFT:

Limited long message number timeT-> sampling frequencyFs-> sampling periodTs = 1/fs-> Number of discrete SequencesN=T*fs=T/Ts
PairSequence execution in the N-point PeriodN pointDFT-> resolution in Frequency DomainDf = fs/N = 1/T & frequency-domain Coordinate-fs/2:df:fs/2-df

For computing efficiencyFFT Algorithm ImplementationDFT,Original calculated by FFTN-point Frequency Domain description Positive and Negative frequency axes are reversedThe N-point data is divided into two groups.(1: 1: N/2 ),(N/2 + 1: 1: N), the actual frequency is:(1: 1: N/2) <-> (0: df :( N/2-1) * df ),(N/2 + 1: 1: N) <-> (-N/2 * df:-df );Use functions in matlabFftshift implements this conversion.

The actual "spectrum leakage ":

1. the jump of each frequency component within the cycle will lead to the corresponding transitional high and low harmonic energy. The harmonic energy of this series is emitted by the signal source, used to transition the hop of each frequency component phase; if some high-low-order harmonic waves are filtered by the filter, the jump of the corresponding frequency component will be smoother.

2. the frequency components of phase hops in a cycle and the corresponding transitional high-low-order harmonic to the spectrum, that is to say, the main line representing the main frequency component is formed and the side lobe representing the corresponding transitional high and low harmonic; that is, the side lobe is used to describe the amplitude and phase hops of the corresponding frequency component within the cycle.

3. The original cycle waveform can be obtained by accumulating the jump waveform of each frequency component in the cycle.

Iii. Actual if Spectrum Analysis

The frequency of continuous signals only has the value of theoretical analysis. In practice, discrete Fourier transformation is applied.

Limited lengthThe continuous signal of T is assumed to be the cycle duration.T continuous signal sampling frequencyFs obtains the discrete sequence within a period, and the period can be obtained through discrete Fourier transformation.Description in the fs, based on the known sampling frequencyFs and signal cycle durationT can be used to obtain the periodic discrete spectrum of a given periodic discrete signal. It is also known that the non-cyclic frequency domain will cause the time domain to be continuous, therefore, the single-cycle frequency spectrum near zero frequency corresponds to a periodic continuous signal.

clear allclose allT = 2ts = 0.001t = 0:ts:T-tsfs = 1/tsdf = fs/length(t)f = -fs/2:df:fs/2 -dfy1 = sin(2* pi * 5* t )y2 = sin(2 * pi * 80 *t )ytmp = y1 + 2 * y2ytmp1 = [y1(1:length(t)/2) y2(length(t)/2+1:end)]y11 = y1 + 2*y2y22 = 2 * y1 + y2ytmp2 = [y11(1:length(t)/2) y22(length(t)/2+1:end)]subplot(3,1,1)plot(t, y1)subplot(3,1,2)plot(t, ytmp1)subplot(3,1,3)plot(t, ytmp2) pam1 = fftshift(fft(y1)/fs)%pam2 = fftshift(fft(ytmp1)/fs)pam2_1 = fft(ytmp1)/fspam2_1(15:end-15) = 0 pam2 = fftshift(pam2_1)ytest201 = ifft(pam2_1)*fspam2_1_1 = fft(ytmp1)/fsytest1 = ifft(pam2_1_1)*fs% pam3 = fftshift(fft(ytmp2)/fs)pam3_1 = fft(ytmp2)/fspam3_1(15:end-15) = 0 pam3 = fftshift(pam3_1)ytest301 = ifft(pam3_1)*fspam3_1_1 = fft(ytmp2)/fsytest2 = ifft(pam3_1_1)*fsfiguresubplot(2,1,1)plot(real(ytest201))subplot(2,1,2)stem(f,abs(pam2))figuresubplot(2,1,1)plot(ytest1)subplot(2,1,2)stem(f,abs(fftshift(pam2_1_1)))figuresubplot(2,1,1)plot(real(ytest301))subplot(2,1,2)stem(f,abs(pam3))figuresubplot(2,1,1)plot(ytest2)subplot(2,1,2)stem(f,abs(fftshift(pam3_1_1)))% real_pam1 = real(pam1)% image_pam1 = imag(pam1)% for count = 1:length(pam1)% if abs(real_pam1(count)) < 0.1% real_pam1(count) = 0;% end% if abs(image_pam1(count)) < 0.1% image_pam1(count) = 0;% end% pam1_f(count) = real_pam1(count) + image_pam1(count)*j;% end% % real_pam2 = real(pam2)% image_pam2 = imag(pam2)% for count = 1:length(pam2)% if abs(real_pam2(count)) < 0.1% real_pam2(count) = 0;% end% if abs(image_pam2(count)) < 0.1% image_pam2(count) = 0;% end% pam2_f(count) = real_pam2(count) + image_pam2(count)*j;% end% % real_pam = real(pam)% image_pam = imag(pam)% for count = 1:length(pam1)% if abs(real_pam(count)) < 0.1% real_pam(count) = 0;% end% if abs(image_pam(count)) < 0.1% image_pam(count) = 0;% end% pam_f(count) = real_pam(count) + image_pam(count)*j;% end% % figure% stem(f, angle(pam1_f)/pi)% grid on% figure% stem(f, angle(pam2_f)/pi)% grid on% figure% stem(f, angle(pam_f)/pi)% grid on