Amplitude Modulation vs. Angle Modulation

---

foreword

Record the learning notes related to modulation and demodulation in communication principles.

---

1. Introduction to Modulation

1. Modulation definition

Metaphor - cargo transportation: loading cargo into a certain position of the plane/ship

Modulation : Put the message signal on a certain parameter of the carrier to form a modulated signal.

Carrier : Some kind of high-frequency periodic oscillating signal, such as a sine wave.
   The modulated carrier is called a modulated signal , which contains message signal characteristics.

Demodulation : The reverse process of modulation,recovering the information signal from the modulated signal .

2. Modulation purpose

• Match channel characteristics, reduce antenna size, and improve radiation efficiency
• Spectrum relocation to realize channel multiplexing and improve channel utilization
• Extend signal bandwidth and improve system anti-interference ability
• To achieve the exchange of bandwidth and signal-to-noise ratio (efficiency and reliability)

通过提高发射信号的频率进而减小制作天线的尺寸长度。

3. Classification of Modulation

①. Modulation process

②. The following figure shows the modulation methods:

2. Amplitude modulation (linear modulation)

1. General model of amplitude modulation

Amplitude modulation - the message signal controls the amplitude of the sinusoidal carrier

Time domain: $$S_m(t)=[m(t)\cos {oh}_{c}t]\ast h\left(t\right)$$
interval:$$S_m(t)=\frac{1}{2}\left[M\right(o+{oh}_c)+M(o-{oh}_c)]H(\omega )$$
$m(t)\leftrightarrow M(\omega )$；$h(t)\leftrightarrow H\left(\omega \right)$

2. Conventional double sideband AM

①, AM signal generation

Condition: Let the message signal m ( t ) m(t)The mean of$m$$($$t$$)$$\overline{m(t)}=0$，且 $\mid m\left(t\right){|}_{max}\le A_0$, plus DC bias $A_0$, and then multiplied by the carrier to get the AM signal.

②, AM modulator model

③, AM waveform and spectrum

④, AM signal characteristics

• $|m(t){|}_{max}\le A_0$When , the envelope of the AM wave is proportional to the frequency modulation signal $m\left(t\right)$ , so envelope detection can be used.
• The AM spectrum consists of carrier frequency components , upper and lower sidebands .
• The AM transmission bandwidth is twice the modulation signal bandwidth, $B_{AM}=2f_H$。
• The advantage of AM is that the receiver is simple, and it is widely used in short and medium AM broadcasting .
• The disadvantage of AM is mainly low modulation efficiency (power utilization) . ( ${the}_{AM}\le 50$%）

⑤, AM envelope detection

条件：$\mid m(t){|}_{max}\le A_0$
Features: Simple; no coherent carrier (non-coherent demodulation); however, there is a threshold effect when the signal-to-noise ratio is small.

⑥, AM coefficient

——reflecting the amplitude modulation signal $The\; extent\; to\; which\; m\left(t\right)$ changes the carrier amplitude
- concerns AM'spower allocationandmodulation efficiency

3. Suppress carrier double sideband DSB

① How to improve modulation efficiency?

——Suppress the carrier component in the modulated signal (equivalent to removing the DC bias A 0 A_0 in the baseband signal$A_0$), this method is called double sideband modulation with suppressed carrier, referred to as double sideband modulation DSB

②, DSB waveform and spectrum

③. Characteristics of DSB signal

• The envelope is no longer related to $m\left(t\right)$ is proportional; when$When\; m\left(t\right)$ changes symbols, the carrier phase is reversed, so envelope detection cannot be used, and coherent demodulation is required.
• Five carrier frequency components, only upper and lower sidebands.
• The bandwidth is the same as for AM: $B_{DSB}=B_{AM}=2f_H$。
• The modulation efficiency is 100%, that is, the power utilization rate is high.
• It is mainly used as the technical basis of SSB and VSB, the difference signal modulation in FM stereo, etc.

④, coherent demodulation (synchronous detection)

Demodulate DSB signals by coherent demodulation

Applicable: AM, DSB, SSB, VSB
Requirements: carrier synchronization $c(t)=\cos {oh}_{c}t$ - local carrier (coherent carrier-----与调制载波同频同向)

若接收端提供的本地载波不相干（如同频不同向），那么对解调带来衰减甚至失真问题

4. Single sideband modulation SSB

①, SSB signal generation

(1) Filtering method

Principle: firstly form DSB signal, then sideband filtering can get upper or lower sideband signal
Requirement: filter $H_{SSB}\left(\omega \right)$ has a steep cut-off characteristic at the carrier frequency. ——Oneof the technical difficulties

(2) Phase shift method

$$S_{SSB}(t)=\frac{1}{2}m(t)\cos {oh}_{c}t\pm \frac{1}{2}\widehat{m\left(t\right)}\sin {oh}_{c}t-SSBexpression$$

②. Characteristics of SSB signal

• One of the advantages is high frequency band utilization. Its transmission bandwidth is only half of AM/DSB: $B_{SSB}=B_{AM}/2=f_H$, Therefore, it has been widely used in communication occasions with crowded spectrum, especially in short-wave communication and multi-carrier telephone.
• The second advantage is the low power consumption characteristic, because there is no need to transmit the carrier and another sideband and save power. This is especially important for mobile communication systems.
• The disadvantage is that the equipment is more complicated and there are technical difficulties. Coherent demodulation is also required.

5. VSB modulation with vestigial sideband

- A compromise between SSB and DSB

**The problem with SSB is that it is difficult to achieve steep sideband filtering characteristics. The solution is to cut gradually, and the roll will be smooth .

①, VSB filter characteristics H ( ω ) H(\omega)What conditions should$H$$($$\omega$$)\; satisfy$

To restore m ( t ) m(t) without distortion$m(t)$，$H\left(\omega \right)$ should satisfy the following conditions:
$$\left[H\right(o+{oh}_c)+H(o-{oh}_c)]=constant,|\omega |\le {oh}_H$$

Meaning: $H\left(\omega \right)$ must satisfy:Complementary symmetriccharacteristicsatthe carrier frequency

②. Geometric explanation of VSB filter

is to translate the properties with complementary symmetry left and right ${oh}_c$The result of the superposition is equal to the constant
$$\left[H\right(o+{oh}_c)+H(o-{oh}_c)]=constant,|\omega |\le {oh}_H$$

③. Characteristics of VSB signal

• There is only a small increase in bandwidth required by SSB, but in exchange for simplicity of the circuit. $f_H<B_V<2f_H$。
• Applications: Video signal transmission in commercial TV broadcasting , etc.

6. AM/DSB/SSB/VSB relationship

3. Angle modulation (non-linear modulation)

Since frequency and phase are in the relationship of calculus , no matter frequency modulation or phase modulation, the angle of the carrier will change, so frequency modulation and phase modulation are collectively referred to as angle modulation .

1. The basic concept of frequency modulation and phase modulation

Angle modulation signal: $S_m(t)=A\cos [o_{c}t+\varphi (t)]$ , where the amplitude$A$ is constant,$\varphi \left(t\right)$ as$m\left(t\right)$ change

<1>, phase modulation ( PM ): phase offset
If the phase offset $\varphi \left(t\right)$ is proportional to the message signal$m\left(t\right)$ , the modulated signal is a phase modulated signal, where$K_p$is the phase shift constant (rad/V).

$\varphi \left(t\right)=K_{p}m(t)$

$S_{PM}=A\cos [o_c(t)+K_{p}m(t)]$

<2>, frequency modulation ( FM ): angular frequency deviation
if angular frequency deviation $\frac{d\varphi (t)}{dt}$proportional to the message signal $m\left(t\right)$ , and the phase offset$\varphi \left(t\right)$ is proportional to the message signal$The\; integral\; of\; m\left(t\right)$ , then the modulated signal is a frequency modulated signal, where$K_f$It is the frequency deviation constant (Hz/V), indicating the frequency deviation generated by the unit modulation voltage, also known as frequency modulation sensitivity.

$\frac{d\varphi (t)}{dt}=2\pi K_{f}m(t)$

$\varphi \left(t\right)=2\pi K_f\int m\left(\tau \right)d\tau$

$S_{FM}=A\cos [o_c(t)+2\pi K_f\int m(\tau )d\tau \_$

2. The relationship between the two

Since the frequency and phase are differential and integral relations:

frequency ~ phase: differential integral phase
modulation ~ frequency modulation: mutual conversion

①, if the modulation signal $m\left(t\right)$ is first differentiated and then frequency modulated, then the phase modulation signal can be obtained. The phase modulation

wave PM is the frequency proportional to the modulation signal$m\left(t\right)$ differentiated equal-amplitude wave, the following two cases of the message signal is a sine wave and a rectangular pulse

②, if the modulation signal$m\left(t\right)$ is first integrated and then phase modulated, then the frequency modulation signal can be obtained. Frequency

modulation wave FM is the frequency proportional to the modulation signal$m\left(t\right)$ equal-amplitude wave, the density of its waveform follows the message signalm ( t ) m(t)The size change of$m$$($$t$$)$$The\; larger\; m\left(t\right)$ -> the higher the frequency -> the denser the waveform

3. FM and Bandwidth

①, FM parameters and maximum frequency deviation

Starting with a monotone signal, monotone $m(t)=A_m(t)\cos {oh}_{m}t$，${oh}_m=2\pi f_m$

Range : $f\_=K_f{A}_m$

FM index (maximum phase shift): $m_f=\frac{f}{f_m}=\frac{K_f{A}_m}{f_m}$, note: $f_m$is the modulation frequency

调频指数是调频波的一个重要参量，涉及到 FM 的传输带宽、功率分配以及抗噪性能

②, FM spectrum

The frequency spectrum of the FM signal is distributed in the carrier frequency component ${oh}_c$Countless pairs of side frequencies on both sides ${oh}_c\pm n{o}_m$composition, whose magnitude depends on $m_f$(FM index)

③, FM bandwidth

Theoretically speaking, the bandwidth of the FM signal is infinite, but from the perspective of engineering application, 98% of the signal power is concentrated in the following frequency band

Carson's formula is commonly used to calculate the bandwidth of FM waves:
$$B_{FM}=2(m_f+1)f_m=2\Delta f+2f_m$$

• $B_{FM}\approx 2f_m，m_f<<When\; 1$ - Narrowband FM (NBFM), this time the bandwidth depends on the modulation frequency$f_m$
• $B_{FM}\approx 2\Delta f，m_f>>1$ o’clock—Wideband Frequency Modulation (WBFM), the bandwidth at this time depends on the maximum frequency deviation$f\_$
  

Generalization: For multitone or arbitrary band-limited modulated signals, FM bandwidth:
$B_{FM}=2(m_f+1)f_m$， $m_f=\frac{f}{f_m}=\frac{K_f{A}_m}{f_m}$，$f_m$is the modulation signal m ( t ) m(t)Highest frequency of$m$$($$t$$)\; -\; bandwidth$

4. Generation and demodulation of FM signal

①, FM signal generation

1) Direct method

• Principle : modulation voltage $m\left(t\right)$ directly controls the frequency of the oscillator,${oh}_i(t)={oh}_0+K_{f}m(t)$
• Advantages : simple circuit, large frequency deviation can be obtained
• Disadvantages : frequency stability is not high
• Improvement : Using PLL tuner
  

2) Indirect method

• Principle : integration -> phase modulation -> n times frequency multiplication -> WBFM
• Advantages : good frequency stability
• Disadvantages : Multiple frequency multiplication and frequency mixing are required , and the circuit is more complicated

②, FM signal demodulation (frequency discrimination)

5. FM features and applications

①, FM features

• Envelope constant (constant amplitude)
• Nonlinear modulation (frequency offset proportional to $m\left(t\right)$ , the phase deviation is proportional tom ( t ) m(t)integral of$m$$($$t$$)\; )$
• Bandwidth is larger than AM $(m_f+1)$ 倍：$B_{FM}=2(m_f+1)f_m$
• Advantages: strong noise immunity
• Cost: Occupies a large channel bandwidth and low spectrum utilization

②, FM application

High quality or high channel noise occasions, such as FM radio, TV audio, satellite communication, mobile communication, microwave communication and cellular phone...

---

My qq: 2442391036, welcome to communicate!

---