Bandwidth and cutoff frequency
https://blog.csdn.net/a419116194/article/details/103335162
http://m.elecfans.com/article/588901.html
1. Bandwidth, signal frequency concept explanation
Bandwidth: The signal bandwidth is the width of the signal spectrum. The signal wave is formed by the superposition of n sine waves. Then the signal bandwidth is the difference between the highest frequency component and the lowest frequency component of the n harmonic signals.
Signal frequency: The frequency components contained in a signal can be observed from the signal spectrogram. The difference between the highest frequency and the lowest frequency of the harmonics contained in a signal, that is, the frequency range possessed by the signal, is defined as the bandwidth of the signal. The greater the frequency range of the signal, the wider the bandwidth of the signal.
For example: a square wave signal superimposed by n sine waves, its lowest frequency component is its fundamental frequency, assuming f=2kHz, and its highest frequency component is its 7th harmonic frequency, that is, 7f =7×2=14kHz , So the signal bandwidth is 7f-f =14-2=12kHz.
The channel bandwidth defines the lower limit frequency and the upper limit frequency of the signal allowed to pass through the channel, that is, it defines a frequency passband .
For example: the allowable passband of a channel is 1.5kHz to 15kHz, and its bandwidth is 13.5kHz. Of course, all frequency components of the above square wave signal can pass through this channel. If attenuation, delay, noise and other factors are not considered, pass this The signal of the channel will be undistorted.
However, if a square wave with a fundamental frequency of 1kHz is passed through the channel, the distortion will definitely be serious; if the fundamental frequency of a square wave signal is 2kHz, but the highest harmonic frequency is 18kHz, the bandwidth exceeds the channel bandwidth, and its higher harmonics are also Will be filtered out by the channel, and the square wave received through this channel is not of good quality;
2. Cut-off frequency
Low-pass filter circuit
Low-pass filter refers to the circuit in the car power amplifier that allows low-frequency signals to pass but not medium and high-frequency signals. Its function is to filter out the mid-range and high-pitched components in the audio signal, and enhance the bass component to drive The woofer of the speaker. In addition, high-pass filters often appear in pairs with low-pass filters, no matter which one is used to send a certain sound frequency to the unit where it should go.
Features: low loss and high suppression; accurate split point; double copper tube protection; good frequency shielding and strong waterproof function.
Usage: The product has a wide range of uses and is used in many communication systems, such as CATV EOC and other systems. And it can effectively remove the signal outside the passband and the interference of redundant frequency bands and frequencies.
The following figure shows a 10MHz low-pass filter. This low-pass filter uses a high-speed current feedback integrated operational amplifier 0PA603 with a bandwidth of up to 100MHz to form a second-order Butterworth low-pass filter. In the figure, R1=R2=159Ω, C1=C2 =100pF, its cut-off frequency is fc=1/2πR1C1=10MHz, and its zero-frequency gain is G0=1+Rf/R=1.6. 
Cutoff frequency
When keeping the amplitude of the input signal unchanged, changing the frequency to reduce the output signal to 0.707 times the maximum value, that is, the frequency response characteristic is expressed as the -3dB point is the cutoff frequency, which is used to illustrate the frequency characteristic index Of a special frequency. 
The cut-off frequency calculation method
uses the modulus of the system function to represent the magnification of the circuit. Since 20lgA(ω)=-3dB, the solution is: A(ω)=10-0.15=0.707945784≈1/√2, and because of A(ω) =|H(jω)|, then |H(jω)|^2=1/2
There is a cut-off frequency at the high-frequency end and the low-frequency end, respectively called the upper cut-off frequency and the lower cut-off frequency. The frequency range between the two cutoff frequencies is called the passband.
Open-loop cut-off frequency and closed-loop cut-off frequency The
open-loop cut-off frequency is also called the shear frequency, which is the frequency at which the amplitude-frequency characteristic curve crosses the 0dB line in the open-loop amplitude-frequency characteristic, which is recorded as ωc; the closed-loop cut-off frequency is also called the bandwidth frequency, Refers to when the closed-loop amplitude-frequency characteristic drops to 3dB below the decibel value when the frequency is zero, the corresponding frequency is recorded as ωb.
The open-loop cut-off frequency and the closed-loop cut-off frequency have the same direction. The open-loop cut-off frequency and the closed-loop cut-off frequency are two different physical quantities, which are used to describe the amplitude-frequency characteristics of the open-loop system and the closed-loop system, but there is a certain correlation between them, namely: the open-close cut-off frequency and its unit The closed-loop cutoff frequency of negative feedback increases in the same direction. And it has the following relationship: ωb>ωc.
Since the closed-loop cutoff frequency can be used to characterize the transient response speed of the closed-loop system, the higher the closed-loop cutoff frequency ωb, the faster the transient response speed. Since ωc and ωb have the same direction, the transient response speed of the system can be known through the Bode diagram of the system, that is, the higher the shear frequency ωc, the faster the transient response speed.
The relationship between the
bandwidth of the low-pass filter and the cut-off frequency Bandwidth B is generally defined by the cut-off frequency f=ωH/2π, B=1/f=2π/ωH, it can also be understood that the B bandwidth is actually the difference between the upper and lower cut-off frequencies. In
practical applications For the negative frequency part of the signal, it needs to go through the Hilbert transformer to eliminate the negative frequency part, which can effectively save bandwidth after the signal is modulated.
The bandwidth of a low-pass filter is equal to the cutoff frequency, and a high-pass filter generally seldom talks about the bandwidth, only the cutoff frequency. There is a significant difference between the bandwidth of the bandpass filter and the cutoff frequency
The formula for the cutoff frequency of the second-order low-pass filter is fc=1/(2 pi (R1 R2 C1*C2)^.5)
If it is the following circuit, can a square wave be converted into a sine wave? If I want to convert a square wave signal of 1KHz~100KHz into a sine wave signal, how can I change the magnitude of R and C? 
If you want to convert a 1kHz square wave into a sine wave, you can use this circuit, as long as you set the cutoff frequency at 1kHz; if you want to convert a 100kHz square wave into a sine wave, you can also use it, but you need to re-adjust the cutoff frequency. To convert the range of 1~100kHz, this circuit can't do it, because it can pass the low pass of the 100kHz sine wave and unblock the 1kHz square wave (the square wave contains 3, 5, 7, 9 times... etc. The equal harmonic components are all within 100kHz).
To change the passband frequency, as long as C1 and C2 are changed, the capacity increases several times, and the frequency decreases several times, and vice versa. Of course, without changing the capacitance, it is also possible to change R1 and R2, and the results are similar. No engineer calculates the cut-off frequency by himself, and he uses the manual method to solve the problem; for your ready-made circuit, you can also use software simulation to measure the frequency response.
How to design the circuit to realize the waveform conversion of 1~100KHz arbitrary frequency?
1. Hardware circuit: Use the function signal chip 8038 to directly generate three waveforms: square wave, triangle wave, and sine wave. The core is a triangle wave oscillator, in which the sine wave is obtained by "breaking point shaping" of the triangle wave (it cannot be adjusted with a square wave), and the square wave is obtained by using the upper and lower double threshold comparators.
2. Software circuit: Store the sine wave function table in ROM, read it repeatedly, and output it through D/A.