1. Fourier coefficients of common signals.
- The Fourier coefficient of the periodic square wave signal is calculated as follows.
Assuming that the period of the square wave signal x(t) is T, the amplitude is 1, the pulse width is τ, and the duty cycle is 1/2, we can get T=2τ. The image is shown below.
The derivation process of c0 is shown in the figure below.
In the above figure, in the integration interval [-τ/2, τ/2], x(t)=1, and c0 can be obtained by bringing it in.
The derivation process of ck is as follows.
From: ω0=2π/T, we get: ω0 T=2π, and because: T=2τ, so: ω0 2τ=2π, we get: ω0*τ=π, the square wave signal ck is the formula in the figure below .
C0 can also be obtained directly from the general formula of ck. When k->0, sin(kπ/2)/(kπ/2) -> 1, and since sin(kπ/2)/(kπ/2) is an elementary function, the function value is equal to the limit value, thus It can be seen that c0 is equal to 0.5 in this example. - The Fourier coefficient of the periodic rectangular signal is as follows.
The periodic square wave signal actually belongs to the periodic rectangular signal. The duty ratio of the periodic square wave signal is equal to 2, and the duty ratio of the periodic rectangular signal is n. Therefore, it can be extended to the periodic rectangular signal through the periodic square wave signal.
Assuming that the amplitude of the periodic rectangular signal is 1, the pulse width is τ, the period is T, and the duty cycle is 1/n, it is brought into the following formula by T=nτ, ω0 nτ=2π, ω0 τ=2π/n Zizhong.
After bringing it in, the general term ck of the Fourier coefficient of the periodic rectangular signal can be obtained by calculation. As shown below.
It can be seen from the formula in the above figure that when the duty cycle is 1/2, that is, when n=2, the Fourier coefficient of the square wave signal with amplitude 1 is substituted.
2. Discrete spectrum of common periodic signals.
- The discrete spectrum of the cosine function is shown in the figure below.
Assume that the cosine signal is as shown in the figure below.
The formula in the above figure is derived from Euler's formula by eliminating the term containing the imaginary number j. It can be seen that the decomposition of the complex exponential signal of the cosine function is obtained by Euler's formula.
The three-dimensional frequency spectrum of the cosine function in the example is shown in the figure below.
The three-bit spectrum in the above figure uses the angular velocity w as the horizontal axis, and the frequency f can also be used as the horizontal axis. The xy plane is used to represent ck, because ck is a complex number.
The amplitude spectrum of the cosine function is shown in the figure below.
The amplitude of the vertical axis in the figure above represents the modulus of ck.
The phase spectrum of the cosine function is shown in the figure below.
The phase of the vertical axis in the figure above represents the phase of ck, because ck is a complex number. Since the ck of the cosine function at -w0 and w0 are both 0.5, corresponding to the positive direction of the real axis in the complex plane Cartesian coordinate system, the argument is 0, so the phase is 0 at these two points. - The frequency spectrum of a sinusoidal signal.
Suppose the sinusoidal signal is as shown in the figure below.
Like the cosine signal, the decomposition of the sine signal into a complex exponential signal can be obtained through the elimination of Euler's formula, without the need for approximation through the Fourier series.
The three-dimensional spectrum of the sinusoidal signal is shown in the figure below.
The amplitude spectrum of the sinusoidal signal is shown in the figure below.
The vertical axis represents the modulus of ck.
The phase spectrum of the sinusoidal signal is shown in the figure below.
In the above figure, when w is equal to -w0, ck is equal to 0.5j, which corresponds to the positive direction of the imaginary axis, so the argument is π/2. - The frequency spectrum of a periodic square wave signal.
Assume that the period of the periodic square wave signal: T=1, the pulse width: τ=0.5, and the duty cycle: 1/n=τ/T=1/2. As shown below.
According to the general term of the Fourier coefficient ck of the periodic rectangular signal, the general term ck of the periodic square wave signal in this example can be obtained.
The general term of the Fourier coefficient ck of the periodic rectangular signal is shown in the figure below.
The general term of the Fourier coefficient of the periodic square wave signal is shown in the figure below.
The three-dimensional frequency spectrum of the periodic square wave signal is shown in the figure below.
The bottom horizontal axis represents k, k=0, ±1, ±2,..., which means that the unit of the horizontal axis on the horizontal axis of k is w0 or f0, that is, the distance between discrete points is w0 or f0, The point on the horizontal axis is k w0 or k f0, and different units correspond to different independent variables. The vertical axis represents ck. - The frequency spectrum of a periodic rectangular signal.
The periodic square wave signal is a type of periodic rectangular signal, and the duty cycle of the periodic square wave signal is 2.
Assume that the amplitude of the periodic rectangular signal is 1, the pulse width is τ, and the duty cycle is 1/n.
The Fourier coefficients of a periodic rectangular signal are shown in the figure below.
The n in the figure above represents the duty cycle. The Fourier coefficients of the periodic rectangular signal in the figure above can be transformed as shown in the figure below.
Incorporating the formula in the figure above into the sinc function can get the formula in the figure below.
According to the formula in the above figure, the discrete spectrum ck of a periodic rectangular signal with an amplitude of 1, a pulse width of τ and a duty ratio of 1/n is sampled from the formula in the above figure, and the sampling interval is f0.
The pulse width of the periodic rectangular signal is 0.5, the periods are 1, 2, 4, and the corresponding duty ratios are 1/2, 1/4, and 1/8. The waveforms are shown in the figure below.
In the figure above, the horizontal axis represents the period, and the vertical axis represents the amplitude.
The pulse width of the periodic rectangular signal is 0.5, the periods are 1, 2, 4, and the corresponding duty ratios are 1/2, 1/4, and 1/8. The waveforms are shown in the figure below.
In the above figure, the horizontal axis represents k, and the interval unit is f0, that is, the interval between sampling points k and k+1 is f0.
3. Continuous spectrum of non-periodic signals.
- Discrete spectrum of aperiodic rectangular signal.
The Fourier coefficient expression of the periodic rectangular signal, namely the discrete spectrum, is shown in the figure below.
The waveform of the non-periodic rectangular signal is equivalent to that obtained when the period T of the periodic signal tends to infinity. If the analysis is performed according to the discrete spectrum of the periodic rectangular signal in the above figure, when T tends to infinity, n also tends to infinity, so the line interval and length of the spectrum will approach zero, as shown in the following figure.
This will cause inconvenience to the analysis of non-periodic rectangular signals. Therefore, when analyzing non-periodic signals, the Fourier coefficients of non-periodic rectangular signals are generally not analyzed directly, that is, the discrete spectrum is generally not discussed, but the continuous research is carried out after some transformation. Spectrum. - Continuous spectrum of non-periodic rectangular signal.
The non-periodic rectangular signal is generally described using ck/f0. From the expression of the Fourier coefficient of the periodic signal, the formula in the figure below can be obtained.
It can be seen from the above figure that the value of ck/f0 is the flat top sampling of τsinc(τf), and the sampling interval is f0.
Draw together the ck/f0 stepped polyline and discrete spectrum of a periodic rectangular signal with amplitude 1, pulse width τ=0.5 and periods 1, 2, and 4, as shown in the figure below.
As can be seen from the above figure, as the period increases, the stepped broken line gradually approaches the curve of τsinc(τf). Since the period of the non-periodic rectangular signal tends to infinity, it can be understood as the continuous spectrum of the non-periodic rectangular signal It is the curve of τsinc(τf), which is shown in the figure below.
4. Fourier transform of non-periodic signals.
- Rectangular pulse signal.
Assuming that the amplitude of the rectangular pulse signal is 1 and the pulse width is τ, its image and its Fourier transformed image are shown in the figure below.
The Fourier transform is equivalent to turning the signal in the time domain into a signal in the frequency domain. The image of the non-periodic rectangular signal on the left in the figure above is the image in the time domain. The horizontal axis is time, the vertical axis is amplitude, and the right one The picture is an image of the non-periodic rectangular signal in the frequency domain. According to the continuum spectrum of the non-periodic rectangular signal that was pushed before, the image is a sinc function. Therefore, the Fourier transform of the non-periodic rectangular signal is a sinc function . - sinc pulse signal.
Assuming that the function of the sinc pulse signal is τsinc(τt), its Fourier transform is a non-periodic rectangular function . As shown below.
The Fourier transform has a certain symmetry. If the A function is an even function and the Fourier of the A function is the B function, the Fourier transform of the B function is the A function (change the argument f in the B function to The t, B function can be understood as changing from the frequency domain to the time domain, and the Fourier transform becomes the A function, that is, the independent variable t in the A function is changed to f). If it is an odd function, it also has Symmetry, but not the same as even functions. - Unit impulse signal The
unit impulse signal can be obtained from the sinc pulse signal τsinc(τt), when τ->infinity, the Fourier transform can also be obtained from the Fourier transform of the sinc pulse signal, that is, the non-periodic rectangular signal is also It is also obtained when τ->infinity, as shown in the figure below.
As can be seen from the figure above, the Fourier transform of the unit impulse function is a DC signal function .