What are the methods of linear modulation?

There are 4 methods of linear modulation, namely: 1. Conventional double-sideband amplitude modulation "DSB-AM"; 2. Double-sideband amplitude modulation "DSB"; 3. Single-sideband modulation "SSB"; 4 , vestigial sideband modulation "VSB".

# Modulation method According to the transmission characteristics, the modulation method can be divided into linear modulation and non-linear modulation. Generalized linear modulation refers to the modulation process in which the modulated parameters in the modulated wave change linearly with the modulating signal. Linear modulation in a narrow sense refers to the modulation process of moving the spectrum of the modulated signal to both sides of the carrier frequency to form upper and lower sidebands.

Overview of linear modulation theory

Continuous wave modulation CWM (Sine wave): It is a modulation method in which sine wave is the carrier wave. There are two major categories:

Linear modulation: Z out = ∑ ki Zin ( f -f oi )

Nonlinear modulation: There is no above-mentioned linear relationship.

Analog linear modulation

1. Conventional double sideband amplitude modulation (DSB-AM)

2. Modulation of double sideband amplitude modulation (DSB)

3. Single sideband modulation (SSB)

4. Vestigial sideband modulation (VSB)

Conventional double sideband amplitude modulation (DSB-AM)

S ​​AM (t ) = [ A0 f (t )] cos(ωc t θc )

A Where: 0 is the external DC; f (t ) is the modulation signal; ωc is the angular frequency of the carrier signal ; θc is the starting phase of the carrier signal. This is a simple and intuitive modulation method, and the original modulated signal can be easily recovered using envelope detection. [

The prerequisite for detection without distortion is: A0 f (t )] ≥ 0; otherwise, over-amplitude modulation will occur. Here is an example.

①The modulation signal is a single frequency cosine, so f (t ) = Am cos(Ω mt θ m ) S AM (t ) = [ A0 Am cos(Ω mt θ m )] cos(ωc t θ c ) = A0 [1 β AM cos(Ω mt θ m )] cos(ωc t θ c )β where: AM Am = ; is the amplitude modulation index, and its value should be ≤1. A0

②The modulated signal spectrum when the modulated signal is a deterministic signal Let S AM (t ) = [ A0 f (t )] cos(ωc t θ c ) 1 = [ A0 f (t )] [e j (ωct θ c ) e ? j (ωct θ c ) ] 2If the spectrum of f(t) is F(ω), by Fourier transform F [ A0 ] = 2πA0δ (ω )F [ f (t )e ± jωct ] = F (ω m ωc ) can be obtained1 S AM (ω ) = [2πA0δ (ω ? ωc ) F (ω ? ωc )]e jθ c 2 1 [2πA0δ (ω ωc ) F (ω ωc )] e ? jθ c 2 For simplicity, let θ=0, then 1 S AM (ω ) = πA0δ (ω ? ωc ) F (ω ? ωc ) 2 1 πA0δ (ω ωc ) F (ω ωc ) 2If Convolution representation, let θ=0, then S AM (t ) = [ A0 f (t )] cos(ωc t ) = m(t ) ? c(t ) 1 S AM (ω ) = [m(ω ) ? c(ω )] 2π

Where: m(t ) = A0 f (t ), c(t ) = cos ωc t M (ω ) = F [m(t )] = 2πA0δ ( ω ) F (ω ) C (ω ) = F [cos ωc t ] = π [δ (ω ? ωc ) δ (ω ωc )]This result is exactly the same as the above result.

③Power distribution (average power) 2 S AM = S AM (t ) = [ A0 f (t )]2 cos 2 ωc t Since f (t ) = 0, cos 2ωc t = 0 S AM A02 f 2 (t ) = = Sc S f 2 2 Sc═ Carrier power Sf ═ Sideband power The result of average power includes carrier power and sideband power. It can be known from the definition that only sideband power is related to the modulated signal. So we can define the modulation efficiency as η AM = Sf S AM = f 2 (t ) A02 f 2 (t ) 2 When the modulation signal is a single-frequency cosine, f (t ) 2 = Am / 2, at this time η AM 2 2 Am β AM = = 2 2 2 2 A0 Am 2 β AM When at the critical point, βAM=1, the maximum modulation efficiency is etaAM=1/3. The modulation signal with the maximum modulation efficiency is a square wave with amplitude A0, etaAM= 0.5

Conclusion: The carrier component C does not carry information, but it occupies a large amount of power. This part of the power is wasted. If the carrier component can be suppressed, this part of the power can be saved, so another evolution occurs. One modulation method: suppressed carrier double sideband modulation.

④ When the modulated signal is a random signal, the power spectral density of the modulated signal is known. The power spectral density can be obtained through the autocorrelation function of the signal to study the modulation efficiency and characteristics. For ergodic stationary random processes/generalized stationary random processes, the relationship between the power spectral density and the autocorrelation function is a pair of Fourier transforms. Autocorrelation characteristics of signal waveform → autocorrelation function; power spectral density → average power → modulation efficiency.

Suppressed Carrier Double Sideband Modulation (DSB-SC)

If you want to suppress the carrier, as long as there is no additional DC component A0, you can get the double sideband amplitude modulation of the suppressed carrier. Its time expression is S DSB (t) = f (t) cos ωc t when f (t) is When the signal is known, the spectrum of the modulated signal is 1 S DSB (ω) = [ F (ω ? ωc ) F (ω ωc )] 2 Comparison of conventional double-sideband amplitude modulation and suppressed carrier conventional double-sideband amplitude modulation When A0 = 0, This is the conventional double-sideband amplitude modulation that suppresses the carrier; when A0 ≠ 0, this is the conventional double-sideband amplitude modulation. For details on the modulator, see balanced modulator and ring modulator. This type of demodulator can only use the coherent demodulation method. For example, after inserting a strong carrier at the demodulation end, the envelope detection method can be used. For example, when sending and receiving multiple signals, a strong carrier can be inserted at the signal sending end. The balanced modulator can be seen from the above figure. The input of the nonlinear unit is: x1 = f (t) cos ωc t. The output of the nonlinear unit is: x2 = ? f (t) cos ωc t y1 = a1[ f (t) cos ωc t ] a2 [ f (t ) cos ωc t ]2 y2 = a1 [? f (t ) cos ωc t ] a2 [? f (t ) cos ωc t ]2 Therefore, after band-pass filtering, the second expression of the following formula is filtered out The term can be y = y1 ? y2 = 2a1 f (t ) 4a2 f (t ) cos ωc t If the ring modulator wants to suppress the carrier, as long as the DC component A0 is not added, the double sideband amplitude modulation of suppressing the carrier can be obtained, and its time expression is When f (t ) is a known signal, the spectrum of the modulated signal is For S (t ) = C (t ) f (t ) 4 ∞ (?1) n ?1 = ∑ cos[2πf c t (2n ? 1)] f (t ) π n =1 2n ? 1 Working principle: D1D2 /D3D are turned on respectively.

Single sideband modulation (SSB)

Single sideband modulation only transmits one sideband of the double sideband modulation signal, which is the best way to save frequency bands.

1. Intuitive method: The characteristics of H SSB (ω ) formed by the filtering method are ?1 H SSB (ω ) = H USB (ω ) = ? ?0 ?1 = H LSB (ω ) = ? ? 0ω > ωc ω ≤ ωc ω < ωc ω ≥ ωc The spectrum formed by the single-sideband signal filtering method is as shown in the figure. Single-sideband demodulation cannot use simple envelope detection, and its signal envelope cannot reflect the single-sided waveform of the modulated signal. Demodulation with modulation should adopt coherent demodulation method: a certain sideband signal requires carrier frequency: 10MHz, bandwidth: 300~3400Hz. The upper and lower sideband interval: 600Hz is limited by the normalized value of the filter. The 600Hz transition band rises by 40dB. Only the two-stage filter can be selected. The first stage carrier frequency selection: 100kHz The second stage carrier frequency selection: 10MHz

2. Single The sideband modulation phase-shifting method forms a Hilbert transform/orthogonal pair/Hilbert filter/wideband phase-shifting network that must shift the signal broadband phase by -π/2, and the phase shift-π/2 must be stable and Accurate; ? All frequency components must be phase-shifted -π/2

3. Single-sideband modulation Weaver method forms the Weaver method. It uses the orthogonal component of the carrier frequency and only needs to phase-shift the carrier -π/ 2, without having to broadband phase shift the signal -π/2 The frequency range of the signal is the first carrier frequency is the actual carrier frequency is 1 2ωL ? ωHωa = (ω L ω H )ω c = ω a ωb 1 The filter cut-off frequency is (ω H ? ω L ) 2

Vestival sideband modulation (VSB)

Vestival sideband modulation is a method between single sideband and suppressed carrier double sideband modulation. In addition to transmitting one sideband, a part of the other sideband is also retained, that is, the transition band. Easier to implement. The residual sideband modulation can also use the phase shifting method. In fact, most of them use the filtering method. Filtering methods can be divided into: residual upper sideband method. Its spectrum characteristics are shown in the middle figure. Method for lower sidebands of the residual part and its spectral characteristics. The transfer function of the vestigial sideband filter must have complementary symmetry characteristics near the carrier frequency. In order to ensure that the coherent demodulation result is not distorted, H VSB (ω ? ωc ) H VSB (ω ωc ) = constant. Attenuation characteristics of the vestigial sideband filter: Yes Steeper→single sideband, or gentler→double sideband, whichever is appropriate. Filter attenuation roll-off characteristics: linear roll-off and cosine roll-off (TV signal). [1]

Linear modulation can be divided into two types: linear modulation in a broad sense and linear modulation in a narrow sense. Linear modulation in a narrow sense only changes the frequency of each component in the spectrum, but does not change the relative proportion of the amplitude of each component, so that the spectrum structure of the upper sideband is the same as the spectrum of the modulated signal, and the spectrum structure of the lower sideband is the mirror image of the spectrum of the modulated signal. Linear modulation in a narrow sense includes amplitude modulation (AM), double-sideband modulation that suppresses the carrier (DSB-SC), single-sideband modulation (SSB), and vestigial sideband modulation (VSB).

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