Difference between revisions of "Modulation Methods/Double-Sideband Amplitude Modulation"

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According to this equation, two models can be given for double-sideband amplitude modulation.   These are to be interpreted as follows:  
 
According to this equation, two models can be given for double-sideband amplitude modulation.   These are to be interpreted as follows:  
 
*The upper model directly describes the relationship given above,  where the carrier  $z(t) = \cos(ω_{\rm T}t + ϕ_{\rm T})$  is applied without a unit.  
 
*The upper model directly describes the relationship given above,  where the carrier  $z(t) = \cos(ω_{\rm T}t + ϕ_{\rm T})$  is applied without a unit.  
*The lower model is more in line with the physical condition  "each signal also has a unit".   If  $q(t)$  and  $z(t)$  are voltages,  the model still needs to provide a scaling with the modulator constant  $K_{\rm AM}$  (unitso that the output signal  $s(t)$  also represents a voltage waveform.  
+
*The lower model is more in line with the physical condition  "each signal also has a unit".   If  $q(t)$  and  $z(t)$  are voltages,  the model still needs to provide a scaling with the modulator constant  $K_{\rm AM}$  with unit   $\rm 1/V$,  so that the output signal  $s(t)$  also represents a voltage waveform.  
*If we set  $K_{\rm AM} = 1/A_{\rm T}$, both models are the same.  In the following, we will always assume the simpler model.
+
*If we set  $K_{\rm AM} = 1/A_{\rm T}$,  both models are the same.  In the following,  we will always assume the lower, simpler model.
 
<br clear=all>
 
<br clear=all>
[[File:P_ID977__Mod_T_2_1_S2b_neu.png|right|frame|Signal waveforms in DSB–AM without a carrier]]
+
[[File:P_ID977__Mod_T_2_1_S2b_neu.png|right|frame|Signal waveforms in DSB–AM without carrier]]
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
$\text{Example 1:}$&nbsp; The graph shows in red the transmitted signals &nbsp;$s(t)$&nbsp; for DSB–AM with two different carrier frequencies.&nbsp;  
+
$\text{Example 1:}$&nbsp; The graph shows in red the transmitted signals &nbsp;$s(t)$&nbsp; for DSB–AM with two different carrier frequencies.&nbsp;
*The source signal &nbsp;$q(t)$&nbsp; with bandwidth &nbsp;$B_{\rm NF} = 4\text{ kHz}$, which is the same in both cases, is drawn in solid blue and the signal &nbsp;$-q(t)$&nbsp is dashed.
+
*The carrier signal &nbsp;$z(t)$&nbsp; has a cosine shape in both cases.  
+
The source signal &nbsp;$q(t)$&nbsp; with bandwidth &nbsp;$B_{\rm NF} = 4\text{ kHz}$,&nbsp; which is the same in both cases,&nbsp; is drawn in solid blue and the signal &nbsp;$-q(t)$&nbsp; is dashed.
*In the upper image, the carrier frequency is &nbsp;$f_{\rm T} = 20\text{ kHz}$&nbsp; and the lower image &nbsp;$f_{\rm T} = 100\text{ kHz}$.}}  
+
 
 +
The carrier signal &nbsp;$z(t)$&nbsp; has a cosine shape in both cases.&nbsp; In the upper sketch,&nbsp; the carrier frequency&nbsp; (German:&nbsp; "Trägersignal" &nbsp; &rArr; &nbsp; subscript "T")&nbsp; is &nbsp;$f_{\rm T} = 20\text{ kHz}$&nbsp; and in the lower sketch &nbsp;$f_{\rm T} = 100\text{ kHz}$.}}  
  
 
==Ring modulator==
 
==Ring modulator==

Revision as of 14:32, 22 November 2021

# OVERVIEW OF THE SECOND MAIN CHAPTER #


$\Rightarrow \hspace{0.5cm}\text{We are just beginning the English translation of this chapter.}$


After some general explanations of modulation and demodulation,  now a detailed description of  "amplitude modulation"  and their associated  "demodulators". This chapter deals in detail with:

  • the description and realization of  »double-sideband amplitude modulation«  $\text{(DSB–AM)}$  in frequency and time domain,
  • the characteristics of a  »synchronous demodulator«  and the possible applications of an  »envelope demodulator«,
  • the similarities/differences of  »single-sideband modulation«  $\text{(SSB–AM)}$  compared to DSB-AM and   »modified AM methods«.


Description in the frequency domain


We consider the following problem:  a source signal $q(t)$,  whose spectrum  $Q(f)$  is bandlimited to the range  $\pm B_{\rm NF}$  (subscript  "NF"  from German "Niederfrequenz"   ⇒   low frequency), 

  • is to be shifted to a higher frequency range where the channel frequency response  $H_{\rm K}(f)$  has favorable characteristics, 
  • using a harmonic oscillation of frequency  $f_{\rm T}$, which we will refer to as the carrier signal  $z(t)$.


The diagram illustrates the task, with the following simplifying assumptions:

Representation of amplitude modulation in the frequency domain
  • The spectrum  $Q(f)$  drawn here is schematic.  It states that only spectral components in the range  $|f| ≤ B_{\rm NF}$  are included in  $q(t)$.  $Q(f)$  could also be a line spectrum.
  • Let the channel be ideal in a bandwidth range  $B_{\rm K}$  around frequency  $f_{\rm M}$,  that is, let  $H_{\rm K}(f) = 1$  for  $|f - f_{\rm M}| ≤ B_{\rm K}/2.$  Impairments by noise are ignored for now.
  • Let the carrier signal be cosine   $($phase  $ϕ_{\rm T} = 0)$  and have amplitude  $A_{\rm T} = 1$  (without a unit).  Let the carrier frequency  $f_{\rm T}$  be equal to the center frequency of the transmission band.
  • Thus,  the spectrum of the carrier signal  $z(t) = \cos(ω_{\rm T} · t)$  is
    (plotted in green in the graph):
$$Z(f) = {1}/{2} \cdot \delta (f + f_{\rm T})+{1}/{2} \cdot \delta (f - f_{\rm T})\hspace{0.05cm}.$$

Those familiar with the  laws of spectral transformation  and in particular with the  Convolution Theorem  can immediately give a solution for the spectrum  $S(f)$  of the modulator output signal:

$$S(f)= Z(f) \star Q(f) = 1/2 \cdot \delta (f + f_{\rm T})\star Q(f)+1/2 \cdot \delta (f - f_{\rm T})\star Q(f) = 1/2 \cdot Q (f + f_{\rm T})+ 1/2 \cdot Q(f - f_{\rm T}) \hspace{0.05cm}.$$

$\text{Please note:}$  This equation takes into account

  • that the convolution of a shifted Dirac delta function  $δ(x - x_0)$  with an arbitrary function $f(x)$  yields the   shifted function  $f(x - x_0)$.


The diagram displays the result.  One can identify the following characteristics:

Spectrum of double-sideband amplitude modulation without carrier;
other name:  "double-sideband amplitude modulation with carrier suppression"
  • Due to the system-theoretic approach with positive and negative frequencies, $S(f)$  is composed of two parts around  $\pm f_{\rm T}$,  each of which have the same shape as  $Q(f)$.
  • The factor  $1/2$  results from the carrier amplitude  $A_{\rm T} = 1$.  Thus,  $s(t = 0) = q(t = 0)$  and the integrals over their spectral functions  $S(f)$  and  $Q(f)$  must also be equal.
  • The channel bandwidth  $B_{\rm K}$  must be at least twice the signal bandwidth  $B_{\rm NF}$,  which gives the name 
    Double-Sideband Amplitude Modulation  $\text{(DSB–AM)}$.
  • It should be noted that $B_{\rm NF}$ and $B_{\rm K}$  are absolute and  non-equivalent bandwidths.  The latter are defined over rectangles of equal area and are denoted in our tutorial by  $Δf_q$  and  $Δf_{\rm K}$,  resp.
  • The spectral function  $S(f)$  does not include any Dirac-lines at the carrier frequency  $(\pm f_{\rm T})$.  Therefore, this method is also referred to as  "DSB-AM without carrier".
  • The frequency components
  • above the carrier frequency  $f_{\rm T}$  are called the  "upper sideband"  $\rm (USB)$,
  • and those below  $f_{\rm T}$  are the  "lower sideband"  $\rm (LSB)$.

Description in the time domain


Adapting the notation and nomenclature to this problem,  the convolution theorem reads:

$$S(f) = Z(f) \star Q(f)\hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm} s(t) = q(t) \cdot z(t) = q(t) \cdot \cos(\omega_{\rm T}\cdot t + \phi_{\rm T})\hspace{0.05cm}.$$
Models of DSB–AM without carrier

This result is still true if the restrictions made on the last page are removed:

  1. real-valued spectrum  $Q(f)$, 
  2. carrier phase  $ϕ_{\rm T} = 0$ 

In general,  this results in a complex-valued spectrum  $S(f)$.

According to this equation, two models can be given for double-sideband amplitude modulation.  These are to be interpreted as follows:

  • The upper model directly describes the relationship given above,  where the carrier  $z(t) = \cos(ω_{\rm T}t + ϕ_{\rm T})$  is applied without a unit.
  • The lower model is more in line with the physical condition  "each signal also has a unit".  If  $q(t)$  and  $z(t)$  are voltages,  the model still needs to provide a scaling with the modulator constant  $K_{\rm AM}$  with unit   $\rm 1/V$,  so that the output signal  $s(t)$  also represents a voltage waveform.
  • If we set  $K_{\rm AM} = 1/A_{\rm T}$,  both models are the same.  In the following,  we will always assume the lower, simpler model.


Signal waveforms in DSB–AM without carrier

$\text{Example 1:}$  The graph shows in red the transmitted signals  $s(t)$  for DSB–AM with two different carrier frequencies. 

The source signal  $q(t)$  with bandwidth  $B_{\rm NF} = 4\text{ kHz}$,  which is the same in both cases,  is drawn in solid blue and the signal  $-q(t)$  is dashed.

The carrier signal  $z(t)$  has a cosine shape in both cases.  In the upper sketch,  the carrier frequency  (German:  "Trägersignal"   ⇒   subscript "T")  is  $f_{\rm T} = 20\text{ kHz}$  and in the lower sketch  $f_{\rm T} = 100\text{ kHz}$.

Ring modulator


One possibility to realize "double sideband amplitude modulation with carrier suppression" is offered by a so-called ring modulator, also known as double push-pull diode modulator. Below you can see the circuit on the left and a simple functional diagram on the right.

Ring modulator to realize DSB–AM without a carrier

Without claiming to be complete, the principle can be stated as follows:

  • Let the amplitude of the harmonic carrier oscillation  $z(t)$  be much larger than the maximum value $q_{\rm max}$  of the message signal $q(t)$.  Thus, all diodes are operated as switches.
  • When the half-wave of the carrier is positive  $(z(t) > 0)$  the two magenta diodes conduct while the light green ones block. Thus, without considering losses, $s(t) = q(t)$ holds.
  • For a negative half-wave, the light green diodes conduct and the diodes in the longitudinal branches block. As can be seen from the right image,;$s(t) = \ – q(t)$ holds for this lower switch position.
  • Due to the operation of this switch, the harmonic oscillation  $z(t)$  can also be replaced by a periodic square wave signal with identical period duration:
$$z_{\rm R}(t) = \left\{ \begin{array}{c} +1 \\ -1 \\ \end{array} \right.\quad \begin{array}{*{10}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c} {z(t) >0,} \\ {z(t) <0.} \\ \end{array}$$
  • The modulated signal $s(t)$  is then obtained as the product of the message signal  $q(t)$  and this square wave signal  $z_{\rm R}(t)$, whereas in ideal DSB-AM one multiplies by a cosine signal.
  • The carrier  $z(t)$  is not itself included in the signal $s(t)$.  Since this is supplied via the center taps of the transformers, the induced voltages cancel out ("DSB-AM without a carrier").


$\text{Example 2:}$  Now we will explain the mode of operation of a ring modulator using typical signal characteristics.  Let the carrier frequency be  $f_{\rm T} = 10\text{ kHz}$.

Signals to illustrate a ring modulator
  • The top graph shows the signals  $q(t)$  and  $-q(t)$  as magenta and light green waveforms respectively.


  • The bipolar square wave signal $z_{\rm R}(t)$  is shown in blue dashes, and takes the values  $±1$ .


  • The middle chart shows the modulated signal from the ring modulator:
$$s_{\rm RM}(t) = q(t) · z_{\rm R}(t).$$


  • For comparison, the conventional DSB-AM signal is shown in the bottom graph:
$$s(t) = q(t) · \cos(ω_{\rm T} · t).$$


One can see significant differences, but these can be compensated for in a simple way:

  • The Fourier series representation of the periodic square wave signal $z_{\rm R}(t)$  is:
$$z_{\rm R}(t) = \frac{4}{\pi} \cdot \cos(\omega_{\rm T}\cdot t)-\frac{4}{3\pi} \cdot \cos(3\omega_{\rm T}\cdot t) +\frac{4}{5\pi} \cdot \cos(5\omega_{\rm T}\cdot t)- \text{ ...}$$
  • The associated spectral function consists of Dirac lines at  $±f_{\rm T}, ±3f_{\rm T}, ±5f_{\rm T}$, etc.  Convolution with  $Q(f)$  leads to the spectral function (the subscript stands for "ring modulator"):
$$S_{\rm RM}(f) = \frac{2}{\pi} \cdot Q (f \pm f_{\rm T})-\frac{2}{3\pi} \cdot Q (f \pm 3f_{\rm T})+\frac{2}{5\pi} \cdot Q (f \pm 5f_{\rm T}) -\text{ ...} \hspace{0.05cm}$$
  • From this, it can be seen that by appropriately band-limiting $($for example, to  $±2f_{\rm T})$ and attenuating with  $π/4 ≈ 0.785$  the familiar DSB-AM spectrum can be obtained:
$$S(f) = {1}/{2} \cdot Q (f \pm f_{\rm T})\hspace{0.05cm}.$$

Here it must be taken into account that in the above reasoning,  $B_{\rm NF} \ll f_{\rm T}$  can always be assumed.

AM signals and spectra with a harmonic input signal


Now we will consider a special case, which is important for testing purposes, where not only the carrier signal  $z(t)$ is a harmonic oscillation, but also the message signal to be modulated  $q(t)$ :

$$\begin{align*}q(t) & = A_{\rm N} \cdot \cos(\omega_{\rm N} \cdot t + \phi_{\rm N})\hspace{0.05cm}, \\ \\ z(t) & = \hspace{0.15cm}1 \hspace{0.13cm} \cdot \hspace{0.1cm}\cos(\omega_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.\end{align*}$$

Please note:   since we are describing modulation processes, the phase term is used with a plus sign in the above equations.

  • Thus,  $ϕ_{\rm N} = - 90^\circ$  represents a sinusoidal input signal  $q(t)$  and  $ϕ_{\rm T} = - 90^\circ$  denotes a sinusoidal carrier signal $z(t)$.
  • Therefore, the equation for the modulated signal is:
$$s(t) = q(t) \cdot z(t) = A_{\rm N} \cdot \cos(\omega_{\rm N} t + \phi_{\rm N})\cdot \cos(\omega_{\rm T} t + \phi_{\rm T})\hspace{0.05cm}.$$


This equation can be transformed using the trigonometric addition theorem:

$$s(t) = A_{\rm N}/{2} \cdot \cos \big [(\omega_{\rm T} +\omega_{\rm N})\cdot t + \phi_{\rm T}+ \phi_{\rm N} \big ] + A_{\rm N}/{2} \cdot \cos \big [(\omega_{\rm T} -\omega_{\rm N})\cdot t + \phi_{\rm T}- \phi_{\rm N} \big ]\hspace{0.05cm}.$$
  • For cosinusoidal signals  $(ϕ_{\rm T} = ϕ_{\rm N} = 0)$ , this equation simplifies to
$$s(t) = {A_{\rm N}}/{2} \cdot \cos\big[(\omega_{\rm T}+\omega_{\rm N})\cdot t\big] + {A_{\rm N}}/{2} \cdot \cos\big[(\omega_{\rm T} -\omega_{\rm N})\cdot t \big]\hspace{0.05cm}.$$
  • Using a Fourier transform we arrive at the spectral function:
$$S(f) = {A_{\rm N}}/{4} \cdot \big[\delta ( f - f_{\rm T} - f_{\rm N})+\delta ( f + f_{\rm T} + f_{\rm N})\big)] + {A_{\rm N}}/{4} \cdot \big[ \delta ( f - f_{\rm T}+ f_{\rm N})+\delta ( f+ f_{\rm T} - f_{\rm N} ) \big]\hspace{0.05cm}.$$


This result, which would also have been arrived at via convolution, states:

  • The spectrum consists of four Dirac lines at frequencies  $±(f_{\rm T} + f_{\rm N})$  and  $±(f_{\rm T} - f_{\rm N})$, where in both bracket expressions the first Dirac function indicates the one for positive frequencies.
  • The weights of all Dirac functions are equal and each is  $A_{\rm N}/4$.  The sum of these weights - that is, the integral over $S(f)$ – is equal to the signal value  $s(t = 0) = A_{\rm N}$.
  • The Dirac lines remain for  $ϕ_{\rm T} ≠ 0$  and/or  $ϕ_{\rm N} ≠ 0$  at the same frequencies.  However, complex rotation factors must then be added to the weights  $A_{\rm N}/4$ .


$\text{Example 3:}$  The following diagram shows the spectral functions  $S(f)$  for different values of  $ϕ_{\rm T}$  and  $ϕ_{\rm N}$, respectively.  The other parameters are assumed to be  $f_{\rm T} = 50\text{ kHz}$,  $f_{\rm N} = 10\text{ kHz}$  and  $A_{\rm N} = 4\text{ V}$ . Thus, the magnitudes of all Dirac lines are $A_{\rm N}/4 = 1\text{ V}$.

Typical spectra for DSB-AM


  • The upper left image shows the case just discussed: both the carrier and the message signal are cosine. Thus, the amplitude modulated signal  $s(t)$  s composed of two cosine oscillations with  $ω_{60} = 2 π · 60\text{ kHz}$  and  $ω_{40} = 2 π · 40\text{ kHz}$ .


  • For the other three constellations, at least one of the signals  $q(t)$  or  $z(t)$  sinusförmig, is sinusoidal, so that  $s(0) = 0$  always holds.  Thus, for these spectra, the sum of the four impulse weights each add up to zero.


  • The bottom right image depicts  $s(t) = A_{\rm N} · \sin(ω_{\rm N} t) · \sin(ω_{\rm T}t)$.  Multiplying two odd functions yields the even function  $s(t)$  and thus a real spectrum  $S(f)$.  In contrast, the other two constellations each result in imaginary spectral functions.

Double-Sideband Amplitude Modulation with carrier


The following diagram shows how to get from "DSB-AM without a carrier" to the better known variant "DSB-AM with carrier". This has the advantage that the demodulator can be realized much more easily and cheaply by a simple manipulation at the transmitter.

Models of DSB–AM with carrier

The diagram is to be interpreted as follows:

  • The top plot shows the physical model of the "DSB-AM with carrier", with changes from the "DSB-AM without a carrier" highlighted in red.
  • The carrier signal  $z(t) = A_{\rm T} · \cos(ω_{\rm T} · t)$  is added to the signal  $s(t)$  , which causes two additional Dirac functions in the spectrum  $±f_{\rm T}$  bewirkt,  ach with impulse weight  $A_{\rm T}/2$.
  • Adding the DC signal  $A_{\rm T}$  to the source signal and then multiplying by the dimensionless carrier  $z(t)$  as shown in the lower sketch results in the same signal  $s(t)$  and spectrum  $S(f)$  as above.
  • Thus, the second representation is equivalent to the upper model. In both cases the carrier phase is only set to  $ϕ_{\rm T} = 0$  for the sake of simplified presentation.


$\text{Example 4:}$  The  Double-sideband with carrier  still finds its main application in radio transmissions for

  • long wave     $($frequency range  $\text{30 kHz}$ ... $\text{300 kHz})$,
  • medium wave  $($frequency range  $\text{300 kHz}$ ... $\text{3 MHz})$,
  • short wave    $($frequency range  $\text{3 MHz}$ ... $\text{30 MHz})$.


However, these frequencies are increasingly being made available for digital applications, for example  Digital Video Broadcast  (DVB).

An application of   Double-sideband amplitude modulation without a carrier  exists for example in FM stereo broadcasting:

  • Here, the differential signal between the two stereo channels is amplitude modulated without a carrier at  $\text{39 kHz}$ .
  • Then the sum signal of the two channels  $($each in the range  $\text{30 Hz}$ ... $\text{15 kHz})$ is combined with an auxiliary carrier at $\text{19 kHz}$  and the dofferential signal and frequency modulated.


$\text{Example 5:}$  The following signal waveforms are intended to further clarify the principle of the "DSB-AM with carrier".

Signal waveforms for for DSB–AM with carrier


  • Above is a section of the source signal  $q(t)$ in the frequency range  $\vert f \vert \le 4\text{ kHz}$ .


  • $s(t)$  is obtained by adding the DC component nbsp;$A_{\rm T}$  is added to  $q(t)$  and multiplying this summed signal by the carrier signal  $z(t)$  of frequency  $f_{\rm T} = 100\text{ kHz}$ .


  • Below is the transmitted signal of the "DSB-AM without carrier" for comparison.


A comparison of these signal waveforms shows:

  • By adding the DC component  $A_{\rm T}$  the message signal  $q(t)$  can now be seen in the envelope of  $s(t)$ .
  • Thus,  envelope demodulation  can be applied, which is easier and cheaper to implement than coherent  synchronous demodulation.
  • However, a prerequisite for the application of the envelope demodulator is a modulation depth  $m <1$.  This is defined as follows:
$$m = \frac{q_{\rm max} }{A_{\rm T} } \hspace{0.3cm}{\rm mit}\hspace{0.3cm} q_{\rm max} = \max_{t} \hspace{0.05cm} \vert q(t) \vert\hspace{0.05cm}.$$
  • The advantage of a simpler demodulator is traded off with a much higher transmission power, since the power contribution of the carrier cannot be used for demodulation.
  • Furthermore, care must be taken to ensure that the source signal does not contain a DC component, since this would be masked by the carrier. For speech and music signals, however, this is not a major restriction.

Describing DSB-AM using the analytical signal


Spectrum of the analytical signal in two different forms of representation.

In the further course of this chapter, for the sake of simplifying the graphs, the spectrum  $S_+(f)$  of the  analytical signal  is usually given instead of the actual, physical spectrum  $S(f)$ . 

As an example, let us consider a "DSB-AM with carrier" and the following signals:

$$\begin{align*}s(t) & = \left(q(t) + A_{\rm T}\right) \cdot \cos(\omega_{\rm T}\cdot t + \phi_{\rm T})\hspace{0.05cm}, \\ \\ q(t) & = A_{\rm N} \cdot \cos(\omega_{\rm N}\cdot t + \phi_{\rm N})\hspace{0.05cm}.\end{align*}$$

Then the corresponding analytical signal is:

$$s_+(t) = A_{\rm T} \cdot {\rm e}^{{\rm j}\hspace{0.03cm} \cdot \hspace{0.01cm}(\omega_{\rm T}\cdot \hspace{0.02cm}t \hspace{0.05cm}+ \hspace{0.05cm}\phi_{\rm T})}+ \frac{A_{\rm N}}{2} \cdot {\rm e}^{{\rm j}\hspace{0.03cm} \cdot \hspace{0.01cm}((\omega_{\rm T} \hspace{0.05cm}+ \hspace{0.05cm} \omega_{\rm N} )\hspace{0.02cm}\cdot \hspace{0.02cm}t \hspace{0.05cm}+ \hspace{0.05cm} \phi_{\rm T}+ \phi_{\rm N})} + \frac{A_{\rm N}}{2} \cdot {\rm e}^{{\rm j}\hspace{0.03cm} \cdot \hspace{0.01cm}((\omega_{\rm T} \hspace{0.05cm}- \hspace{0.05cm} \omega_{\rm N} )\hspace{0.02cm}\cdot \hspace{0.02cm}t \hspace{0.05cm}+ \hspace{0.05cm} \phi_{\rm T}- \phi_{\rm N})} \hspace{0.05cm}.$$

The corresponding spectral function  $S_+(f)$  consists of three Dirac lines, each with complex weights corresponding to the graph:

  • The left sketch shows the magnitude  $|S_+(f)|$, where  $A_{\rm T}$  indicates the weight of the carrier and  $A_{\rm N}/2$  indicates the weights of   USB  (upper sideband) and   LSB  (lower sideband).
  • The values normalized to  $A_{\rm T}$  are given in parentheses. Since  $q_{\rm max} = A_{\rm N}$  holds here, the modulation depth  $m = A_{\rm N}/A_{\rm T}$  gives  $m/2$ as the normalized weights of both the upper and lower sideband.
  • The right sketch depicts the direction of the frequency axis and shows the phase angles of carrier  $(ϕ_{\rm T})$, LSB  $(ϕ_{\rm T} – ϕ_{\rm N})$  and USB  $(ϕ_{\rm T} + ϕ_{\rm N})$.


Amplitude modulation with a quadratic characteristic curve


Nonlinearities are usually undesirable and troublesome in communications engineering. As explained in the chapter  Nonlinear Distortions  of the book "Linear and Time Invariant Systems”, they lead to the facts that:

  • the superposition principle is no longer applicable,
  • the transmission behavior depends on the magnitude of the input signal, and
  • the distortions are of nonlinear nature and thus irreversible.


A nonlinearity of the general form

$$y(t) = c_0 + c_1 \cdot x(t) + c_2 \cdot x^2(t)+ c_3 \cdot x^3(t) + \text{...}$$

can also be used to implement DSB-AM. Provided that

  • only the coefficients  $c_1$  and  $c_2$  are present, and
  • the input signal  $x(t) = q(t) + z(t)$  is applied,


we obtain the nonlinearity's output signal:

$$y(t) = c_1 \cdot q(t) + c_1 \cdot z(t) + c_2 \cdot q^2(t)+ 2 \cdot c_2 \cdot q(t)\cdot z(t)+ c_2 \cdot z^2(t)\hspace{0.05cm}.$$

The first, third, and last components are – in terms of spectra – at  $| f | ≤ 2 · B_{\rm NF}$  and  $| f | = 2 · f_{\rm T}$, respectively.

Removing these signal components by a bandpass and considering  $z(t) = A_{\rm T} · \cos(ω_{\rm T} · t)$, we obtain the equation typical for "DSB-AM with carrier"
(only the second and fourth terms remain):

$$s(t) = c_1 \cdot A_{\rm T} \cdot \cos(\omega_{\rm T} \cdot t ) + 2 \cdot c_2 \cdot A_{\rm T} \cdot q(t)\cdot \cos(\omega_{\rm T} \cdot t )\hspace{0.05cm}.$$

The degree of modulation in this realization form is variable by the coefficients  $c_1$  and  $c_2$ :

$$m = \frac{2 \cdot c_2 \cdot q_{\rm max}}{c_1} \hspace{0.05cm}.$$

A diode and a field-effect transistor both approximate such a quadratic characteristic curve and can thus be used to realize DSB-AM. However, cubic components  $(c_3 ≠ 0)$  and higher order nonlinearities lead to (large) nonlinear distortions.


Exercises for the chapter


Exercise 2.1: DSB-AM with Cosine? Or with Sine?

Exercise 2.1Z: DSB-AM without/with Carrier

Exercise 2.2: Modulation depth

Exercise 2.2Z: Power Consideration

Exercise 2.3: DSB–AM Realization

Exercise 2.3Z: DSB-AM due to Nonlinearity