Difference between revisions of "Aufgaben:Exercise 2.3: DSB-AM Realization"

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[[File:EN_Mod_A_2_3.png|right|frame|Nonlinear characteristic curve <br>forr AM&ndash;realization]]
+
[[File:EN_Mod_A_2_3.png|right|frame|Nonlinear characteristic curve <br>for DSB-AM realization]]
In order to realize the so-called "DSB-AM with carrier", an amplifier with the characteristic curve
+
In order to realize the so-called&nbsp; "Double-Sideband Amplitude Modulation&nbsp; $\text{(DSB-AM)}$&nbsp; with carrier",&nbsp; an amplifier with the following characteristic curve must be used:
 
:$$y = g(x) = U \cdot \left( 1 -{\rm e} ^{-x/U}\right)$$
 
:$$y = g(x) = U \cdot \left( 1 -{\rm e} ^{-x/U}\right)$$
must be used. Here, &nbsp;$x = x(t)$&nbsp; and &nbsp;$y = y(t)$&nbsp;are time-dependent voltages at the input and output of the amplifier, respectively. &nbsp; The parameter &nbsp;$U = 3 \ \rm V$&nbsp; indicates the saturation voltage of the amplifier.
+
*Here, &nbsp;$x = x(t)$&nbsp; and &nbsp;$y = y(t)$&nbsp; are time-dependent voltages at the input and output of the amplifier,&nbsp; respectively. &nbsp;  
 +
*The parameter &nbsp;$U = 3 \ \rm V$&nbsp; indicates the saturation voltage of the amplifier.
  
This curve is operated at the operating point &nbsp;$A_0 = 2\ \rm  V$.&nbsp; This is achieved, for example, by the input signal
+
 
 +
This curve is used at the operating point &nbsp;$A_0 = 2\ \rm  V$.&nbsp; This is achieved,&nbsp; for example,&nbsp; by the input signal
 
:$$x(t) = A_0 + z(t) + q(t)\hspace{0.05cm}.$$
 
:$$x(t) = A_0 + z(t) + q(t)\hspace{0.05cm}.$$
 
Assume cosine oscillations for both the carrier and the source signal:
 
Assume cosine oscillations for both the carrier and the source signal:
 
:$$ z(t)  = A_{\rm T} \cdot \cos (2 \pi f_{\rm T} t),\hspace{0.2cm} A_{\rm T} = 1\,{\rm V},\hspace{0.2cm} f_{\rm T} = 30\,{\rm kHz},$$
 
:$$ z(t)  = A_{\rm T} \cdot \cos (2 \pi f_{\rm T} t),\hspace{0.2cm} A_{\rm T} = 1\,{\rm V},\hspace{0.2cm} f_{\rm T} = 30\,{\rm kHz},$$
 
:$$ q(t)  =  A_{\rm N} \cdot \cos (2 \pi f_{\rm N} t),\hspace{0.2cm} A_{\rm N} = 1\,{\rm V},\hspace{0.2cm} f_{\rm N} = 3\,{\rm kHz}\hspace{0.05cm}.$$
 
:$$ q(t)  =  A_{\rm N} \cdot \cos (2 \pi f_{\rm N} t),\hspace{0.2cm} A_{\rm N} = 1\,{\rm V},\hspace{0.2cm} f_{\rm N} = 3\,{\rm kHz}\hspace{0.05cm}.$$
In solving this problem, use the auxiliary quantity
+
In solving this problem,&nbsp; use the auxiliary quantity
 
:$$w(t) = x(t) - A_0 = z(t) + q(t)\hspace{0.05cm}.$$
 
:$$w(t) = x(t) - A_0 = z(t) + q(t)\hspace{0.05cm}.$$
  
The nonlinear characteristic curve can be developed according to a ''Taylor series'' around the operating point:
+
The nonlinear characteristic curve can be developed according to a&nbsp; "Taylor series"&nbsp; around the operating point:
 
:$$y(x)  = y(A_0) + \frac{1}{1!} \cdot y\hspace{0.08cm}{\rm '}(A_0) \cdot (x - A_0)+ \frac{1}{2!} \cdot y\hspace{0.08cm}''(A_0) \cdot (x - A_0)^2+
 
:$$y(x)  = y(A_0) + \frac{1}{1!} \cdot y\hspace{0.08cm}{\rm '}(A_0) \cdot (x - A_0)+ \frac{1}{2!} \cdot y\hspace{0.08cm}''(A_0) \cdot (x - A_0)^2+
 
   \frac{1}{3!} \cdot y\hspace{0.08cm}'''(A_0) \cdot (x - A_0)^3 + \text{ ...}$$
 
   \frac{1}{3!} \cdot y\hspace{0.08cm}'''(A_0) \cdot (x - A_0)^3 + \text{ ...}$$
 
The output signal can then also be represented as depending on the auxiliary quantity &nbsp;$w(t)$&nbsp; as follows:
 
The output signal can then also be represented as depending on the auxiliary quantity &nbsp;$w(t)$&nbsp; as follows:
 
:$$y(t) = c_0 + c_1 \cdot w(t) + c_2 \cdot w^2(t)+ c_3 \cdot w^3(t) +\text{ ...}$$
 
:$$y(t) = c_0 + c_1 \cdot w(t) + c_2 \cdot w^2(t)+ c_3 \cdot w^3(t) +\text{ ...}$$
The DSB–AM signal &nbsp;$s(t)$&nbsp; is obtained by band-limiting&nbsp;$y(t)$&nbsp; to the frequency range from &nbsp;$\text{23 kHz}$&nbsp; to &nbsp;$\text{37 kHz}$.&nbsp; That is, all frequencies other than &nbsp;$f_{\rm T}$, &nbsp;$f_{\rm T}±f_{\rm N}$&nbsp; and &nbsp;$f_{\rm T}±2f_{\rm N}$&nbsp;  are removed by the bandpass.
+
*The DSB–AM signal &nbsp;$s(t)$&nbsp; is obtained by band-limiting&nbsp;$y(t)$&nbsp; to the frequency range from &nbsp;$\text{23 kHz}$&nbsp; to &nbsp;$\text{37 kHz}$.&nbsp;  
 
+
*That is,&nbsp; all frequencies other than &nbsp;$f_{\rm T}$, &nbsp;$f_{\rm T}±f_{\rm N}$&nbsp; and &nbsp;$f_{\rm T}±2f_{\rm N}$&nbsp;  are removed by the band-pass.
The graph shows the characteristic curve &nbsp;$g(x)$&nbsp; and the approximations &nbsp;$g_1(x)$, &nbsp;$g_2(x)$&nbsp; and &nbsp;$g_3(x)$, when the Taylor series is truncated after the first, second, or third term.  It can be seen that the approximation &nbsp;$g_3(x)$&nbsp;  is indistinguishable from &nbsp;$g(x)$&nbsp; in the range shown.
 
 
 
  
  
 +
The graph shows the characteristic curve &nbsp;$g(x)$&nbsp; and the approximations &nbsp;$g_1(x)$, &nbsp;$g_2(x)$&nbsp; and &nbsp;$g_3(x)$, when the Taylor series is truncated after the first, second, or third term.&nbsp;  It can be seen that the approximation &nbsp;$g_3(x)$&nbsp;  is indistinguishable from &nbsp;$g(x)$&nbsp; in the range shown.
  
  
  
  
''Hints:''
+
Hints:  
 
*This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Double-Sideband_Amplitude_Modulation|Double-Sideband Amplitude Modulation]].
 
*This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Double-Sideband_Amplitude_Modulation|Double-Sideband Amplitude Modulation]].
 
*Reference will also be made to the chapter&nbsp;  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions#Description_of_nonlinear_systems|Description of nonlinear systems]]&nbsp; in the book "Linear and Time Invariant Systems".
 
*Reference will also be made to the chapter&nbsp;  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions#Description_of_nonlinear_systems|Description of nonlinear systems]]&nbsp; in the book "Linear and Time Invariant Systems".
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<quiz display=simple>
 
<quiz display=simple>
{In what range can the input signal &nbsp;$x(t)$&nbsp;vary? variieren?&nbsp;Give the minimum and maximum values of the auxiliary variable  &nbsp;$w(t) = x(t) - A_0$&nbsp;.
+
{In what range can the input signal &nbsp;$x(t)$&nbsp;vary?&nbsp;Give the minimum and maximum values of the auxiliary variable  &nbsp;$w(t) = x(t) - A_0$.
 
|type="{}"}
 
|type="{}"}
 
$w_{\rm min} \ = \ $ { -2.06--1.94 } $\ \text{V}$
 
$w_{\rm min} \ = \ $ { -2.06--1.94 } $\ \text{V}$
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$c_1 \ = \ $ { 0.513 3% }  
 
$c_1 \ = \ $ { 0.513 3% }  
  
{What are the coefficients &nbsp;$c_2$&nbsp; and &nbsp;$c_3$&nbsp of the nonlinear characteristic curve?
+
{What are the coefficients &nbsp;$c_2$&nbsp; and &nbsp;$c_3$&nbsp; of the nonlinear characteristic curve?
 
|type="{}"}
 
|type="{}"}
 
$c_2\ = \ $ { -0.088--0.084 }  $\ \rm V^{ -1 }$
 
$c_2\ = \ $ { -0.088--0.084 }  $\ \rm V^{ -1 }$
 
$c_3\ = \ $ { 0.0095 3% }  $\ \rm V^{ -2 }$
 
$c_3\ = \ $ { 0.0095 3% }  $\ \rm V^{ -2 }$
  
{Show that a "DSB-AM with carrier" constellation results when &nbsp;$c_3$&nbsp; is considered negligibly small.  What is the modulation depth &nbsp;$m$?
+
{Show that a&nbsp; "DSB-AM with carrier"&nbsp; constellation results when &nbsp;$c_3$&nbsp; is considered negligibly small.&nbsp; What is the modulation depth &nbsp;$m$?
 
|type="{}"}
 
|type="{}"}
 
$m \ = \ $ { 0.335 3% }  
 
$m \ = \ $ { 0.335 3% }  
  
{Assuming that &nbsp;$c_3$&nbsp; cannot be considered negligibly small, which of the following statements are true?
+
{Assuming that &nbsp;$c_3$&nbsp; cannot be considered negligibly small,&nbsp; which of the following statements are true?
 
|type="[]"}
 
|type="[]"}
 
- The weight of the spectral line at&nbsp;$f_{\rm T}$&nbsp; is unchanged.
 
- The weight of the spectral line at&nbsp;$f_{\rm T}$&nbsp; is unchanged.
 
+ $s(t)$&nbsp; now includes Dirac lines at&nbsp;$f_{\rm T} ± 2f_{\rm N}$.
 
+ $s(t)$&nbsp; now includes Dirac lines at&nbsp;$f_{\rm T} ± 2f_{\rm N}$.
 
+ The cubic term leads to nonlinear distortions.
 
+ The cubic term leads to nonlinear distortions.
- The cubic term leads to linear distortions..
+
- The cubic term leads to linear distortions.
 
</quiz>
 
</quiz>
  
 
===Solution===
 
===Solution===
 
{{ML-Kopf}}
 
{{ML-Kopf}}
'''(1)'''&nbsp; From&nbsp; $x(t) = A_0 + z(t) + q(t)$&nbsp;, t&nbsp; $A_0 = 2\ \rm  V$&nbsp; and&nbsp; $A_{\rm T} = A_{\rm N} = 1 \ \rm  V$&nbsp; we get the possible range&nbsp; $0 \ {\rm V} ≤ x(t) ≤ 4\ \rm  V$.  
+
'''(1)'''&nbsp; From&nbsp; $x(t) = A_0 + z(t) + q(t)$,&nbsp; with &nbsp; $A_0 = 2\ \rm  V$&nbsp; and &nbsp; $A_{\rm T} = A_{\rm N} = 1 \ \rm  V$,&nbsp; we get the possible range&nbsp; $0 \ {\rm V} ≤ x(t) ≤ 4\ \rm  V$.  
*Thus, the auxiliary quantity&nbsp; $w(t)$&nbsp; can take values between&nbsp; $w_{\rm min}\hspace{0.15cm}\underline{ = -2 \ \rm V}$&nbsp; and&nbsp; $w_{\rm max}\hspace{0.15cm}\underline{ = +2 \ \rm V}$&nbsp;.
+
*Thus,&nbsp; the auxiliary quantity&nbsp; $w(t)$&nbsp; can take values between&nbsp; $w_{\rm min}\hspace{0.15cm}\underline{ = -2 \ \rm V}$&nbsp; and&nbsp; $w_{\rm max}\hspace{0.15cm}\underline{ = +2 \ \rm V}$.
  
  
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'''(2)'''&nbsp; The coefficient&nbsp; $c_0$&nbsp; is equal to the characteristic value at the operating point.&nbsp; Using&nbsp; $A_0 = 2 \ \rm V$&nbsp; and&nbsp; $U = 3 \ \rm V$&nbsp; we obtain:
 
'''(2)'''&nbsp; The coefficient&nbsp; $c_0$&nbsp; is equal to the characteristic value at the operating point.&nbsp; Using&nbsp; $A_0 = 2 \ \rm V$&nbsp; and&nbsp; $U = 3 \ \rm V$&nbsp; we obtain:
 
:$$c_0 = y(A_0) = U \cdot \left( 1 -{\rm e} ^{-A_0/U}\right) \hspace{0.15cm}\underline {= 1.460\,{\rm V}}\hspace{0.05cm}.$$
 
:$$c_0 = y(A_0) = U \cdot \left( 1 -{\rm e} ^{-A_0/U}\right) \hspace{0.15cm}\underline {= 1.460\,{\rm V}}\hspace{0.05cm}.$$
*Accordingly, for the Taylor coefficient&nbsp; $c_1$:
+
*Accordingly,&nbsp; for the Taylor coefficient&nbsp; $c_1$:
 
:$$c_1 = y\hspace{0.06cm}'(A_0)= {\rm e} ^{-A_0/U}\hspace{0.15cm}\underline { = 0.513}\hspace{0.05cm}.$$
 
:$$c_1 = y\hspace{0.06cm}'(A_0)= {\rm e} ^{-A_0/U}\hspace{0.15cm}\underline { = 0.513}\hspace{0.05cm}.$$
  
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'''(4)'''&nbsp; Setting&nbsp; $c_3 = 0$, the output signal of the amplifier is:
+
'''(4)'''&nbsp; Setting&nbsp; $c_3 = 0$,&nbsp; the output signal of the amplifier is:
 
:$$y(t) = c_0 + c_1 \cdot (z(t) + q(t)) + c_2 \cdot (z^2(t) + q^2(t) + 2 \cdot z(t) \cdot q(t))\hspace{0.05cm}.$$
 
:$$y(t) = c_0 + c_1 \cdot (z(t) + q(t)) + c_2 \cdot (z^2(t) + q^2(t) + 2 \cdot z(t) \cdot q(t))\hspace{0.05cm}.$$
*Thus, after the bandpass, the following signal components remain:
+
*Thus,&nbsp; after the band-pass,&nbsp; the following signal components remain:
 
:$$s(t)  =  c_1 \cdot z(t) + 2 \cdot c_2 \cdot z(t) \cdot q(t)  
 
:$$s(t)  =  c_1 \cdot z(t) + 2 \cdot c_2 \cdot z(t) \cdot q(t)  
 
   =  \left[c_1 \cdot A_{\rm T} + 2 \cdot c_2 \cdot A_{\rm T} \cdot A_{\rm N} \cdot \cos(\omega_{\rm N} t)\right] \cdot \cos(\omega_{\rm T} t)\hspace{0.05cm}.$$
 
   =  \left[c_1 \cdot A_{\rm T} + 2 \cdot c_2 \cdot A_{\rm T} \cdot A_{\rm N} \cdot \cos(\omega_{\rm N} t)\right] \cdot \cos(\omega_{\rm T} t)\hspace{0.05cm}.$$
*The modulation depth is then determined as the quotient of the "amplitude" of the message oscillation over the "amplitude" of the carrier:
+
*The modulation depth is then determined as the quotient of the&nbsp; "amplitude of the message oscillation"&nbsp; over the "amplitude of the carrier":
 
:$$m = \frac{2 \cdot |c_2| \cdot A_{\rm T} \cdot A_{\rm N}}{|c_1| \cdot A_{\rm T}} = \frac{2 \cdot |c_2| \cdot A_{\rm N}}{|c_1| }= \frac{2 \cdot 0.086 \cdot 1\,{\rm V}}{0.513 }\hspace{0.15cm}\underline { = 0.335}\hspace{0.05cm}.$$
 
:$$m = \frac{2 \cdot |c_2| \cdot A_{\rm T} \cdot A_{\rm N}}{|c_1| \cdot A_{\rm T}} = \frac{2 \cdot |c_2| \cdot A_{\rm N}}{|c_1| }= \frac{2 \cdot 0.086 \cdot 1\,{\rm V}}{0.513 }\hspace{0.15cm}\underline { = 0.335}\hspace{0.05cm}.$$
  
  
  
'''(5)'''&nbsp; <u>Answers 2 and 3</u>are correct:
+
'''(5)'''&nbsp; <u>Answers 2 and 3</u>&nbsp; are correct:
 
*Considering the cubic part,&nbsp; $y(t)$&nbsp; includes the following other components:
 
*Considering the cubic part,&nbsp; $y(t)$&nbsp; includes the following other components:
 
:$$y_3(t)  =  c_3 \cdot (z(t) + q(t))^3
 
:$$y_3(t)  =  c_3 \cdot (z(t) + q(t))^3
 
=  c_3 \cdot z^3(t) + 3 \cdot c_3 \cdot z^2(t) \cdot q(t)+ 3 \cdot c_3 \cdot z(t) \cdot q^2(t) + c_3 \cdot q^3(t) \hspace{0.05cm}.$$
 
=  c_3 \cdot z^3(t) + 3 \cdot c_3 \cdot z^2(t) \cdot q(t)+ 3 \cdot c_3 \cdot z(t) \cdot q^2(t) + c_3 \cdot q^3(t) \hspace{0.05cm}.$$
*The first term results in components at&nbsp; $f_{\rm T}$&nbsp; and&nbsp; $3f_{\rm T}$, and the last term results in components at&nbsp; $f_{\rm N}$&nbsp; and&nbsp; $3f_{\rm N}$.&nbsp; The second term gives a component at &nbsp; $f_{\rm N}$&nbsp; and others at&nbsp; $2f_{\rm T} ± f_{\rm N}$:
+
*The first term results in components at&nbsp; $f_{\rm T}$&nbsp; and&nbsp; $3f_{\rm T}$, and the last term results in components at&nbsp; $f_{\rm N}$&nbsp; and&nbsp; $3f_{\rm N}$.&nbsp;  
 +
*The second term gives a component at &nbsp; $f_{\rm N}$&nbsp; and others at&nbsp; $2f_{\rm T} ± f_{\rm N}$:
 
:$$3 \cdot c_3 \cdot z^2(t) \cdot q(t)= {3}/{2 } \cdot A_{\rm T}^2 \cdot A_{\rm N} \cdot \left[ \cos(\omega_{\rm N} t) + \cos(2\omega_{\rm T} t) \cdot \cos(\omega_{\rm N} t)\right] \hspace{0.05cm}.$$
 
:$$3 \cdot c_3 \cdot z^2(t) \cdot q(t)= {3}/{2 } \cdot A_{\rm T}^2 \cdot A_{\rm N} \cdot \left[ \cos(\omega_{\rm N} t) + \cos(2\omega_{\rm T} t) \cdot \cos(\omega_{\rm N} t)\right] \hspace{0.05cm}.$$
 
*Accordingly, the third summand in the above equation leads to
 
*Accordingly, the third summand in the above equation leads to
 
:$$3 \cdot c_3 \cdot z(t) \cdot q^2(t)= {3}/{2 }  \cdot A_{\rm T} \cdot A_{\rm N}^2 \cdot \left[ \cos(\omega_{\rm T} t) + \cos(\omega_{\rm T} t)\cdot \cos(2 \omega_{\rm N} t)\right] \hspace{0.05cm}.$$
 
:$$3 \cdot c_3 \cdot z(t) \cdot q^2(t)= {3}/{2 }  \cdot A_{\rm T} \cdot A_{\rm N}^2 \cdot \left[ \cos(\omega_{\rm T} t) + \cos(\omega_{\rm T} t)\cdot \cos(2 \omega_{\rm N} t)\right] \hspace{0.05cm}.$$
*Thus, within the frequency range from&nbsp; $\text{23 kHz}$&nbsp; to&nbsp; $\text{37 kHz}$&nbsp;, there is indeed a change in the spectral line at&nbsp; $f_{\rm T}$&nbsp; and new Dirac lines are formed at &nbsp; $f_{\rm T} ± 2f_{\rm N}$, i.e., at &nbsp; $\text{24 kHz}$&nbsp; and&nbsp; $\text{36 kHz}$.  
+
*Thus,&nbsp; within the frequency range from&nbsp; $\text{23 kHz}$&nbsp; to&nbsp; $\text{37 kHz}$,&nbsp; there is indeed a change in the spectral line at&nbsp; $f_{\rm T}$&nbsp; <br>and new Dirac lines are formed at &nbsp; $f_{\rm T} ± 2f_{\rm N}$,&nbsp; i.e.,&nbsp; at &nbsp; $\text{24 kHz}$&nbsp; and&nbsp; $\text{36 kHz}$.  
 
*The resulting distortions are thus nonlinear&nbsp; &rArr; &nbsp; Answer 3 ist correct and Answer 4 is wrong.
 
*The resulting distortions are thus nonlinear&nbsp; &rArr; &nbsp; Answer 3 ist correct and Answer 4 is wrong.
  

Revision as of 16:41, 24 March 2022

Nonlinear characteristic curve
for DSB-AM realization

In order to realize the so-called  "Double-Sideband Amplitude Modulation  $\text{(DSB-AM)}$  with carrier",  an amplifier with the following characteristic curve must be used:

$$y = g(x) = U \cdot \left( 1 -{\rm e} ^{-x/U}\right)$$
  • Here,  $x = x(t)$  and  $y = y(t)$  are time-dependent voltages at the input and output of the amplifier,  respectively.  
  • The parameter  $U = 3 \ \rm V$  indicates the saturation voltage of the amplifier.


This curve is used at the operating point  $A_0 = 2\ \rm V$.  This is achieved,  for example,  by the input signal

$$x(t) = A_0 + z(t) + q(t)\hspace{0.05cm}.$$

Assume cosine oscillations for both the carrier and the source signal:

$$ z(t) = A_{\rm T} \cdot \cos (2 \pi f_{\rm T} t),\hspace{0.2cm} A_{\rm T} = 1\,{\rm V},\hspace{0.2cm} f_{\rm T} = 30\,{\rm kHz},$$
$$ q(t) = A_{\rm N} \cdot \cos (2 \pi f_{\rm N} t),\hspace{0.2cm} A_{\rm N} = 1\,{\rm V},\hspace{0.2cm} f_{\rm N} = 3\,{\rm kHz}\hspace{0.05cm}.$$

In solving this problem,  use the auxiliary quantity

$$w(t) = x(t) - A_0 = z(t) + q(t)\hspace{0.05cm}.$$

The nonlinear characteristic curve can be developed according to a  "Taylor series"  around the operating point:

$$y(x) = y(A_0) + \frac{1}{1!} \cdot y\hspace{0.08cm}{\rm '}(A_0) \cdot (x - A_0)+ \frac{1}{2!} \cdot y\hspace{0.08cm}''(A_0) \cdot (x - A_0)^2+ \frac{1}{3!} \cdot y\hspace{0.08cm}'''(A_0) \cdot (x - A_0)^3 + \text{ ...}$$

The output signal can then also be represented as depending on the auxiliary quantity  $w(t)$  as follows:

$$y(t) = c_0 + c_1 \cdot w(t) + c_2 \cdot w^2(t)+ c_3 \cdot w^3(t) +\text{ ...}$$
  • The DSB–AM signal  $s(t)$  is obtained by band-limiting $y(t)$  to the frequency range from  $\text{23 kHz}$  to  $\text{37 kHz}$. 
  • That is,  all frequencies other than  $f_{\rm T}$,  $f_{\rm T}±f_{\rm N}$  and  $f_{\rm T}±2f_{\rm N}$  are removed by the band-pass.


The graph shows the characteristic curve  $g(x)$  and the approximations  $g_1(x)$,  $g_2(x)$  and  $g_3(x)$, when the Taylor series is truncated after the first, second, or third term.  It can be seen that the approximation  $g_3(x)$  is indistinguishable from  $g(x)$  in the range shown.



Hints:


Questions

1

In what range can the input signal  $x(t)$ vary? Give the minimum and maximum values of the auxiliary variable  $w(t) = x(t) - A_0$.

$w_{\rm min} \ = \ $

$\ \text{V}$
$w_{\rm max} \ = \ $

$\ \text{V}$

2

Calculate the coefficients  $c_0$  and  $c_1$  of the Taylor series.

$c_0 \ = \ $

$\ \text{V}$
$c_1 \ = \ $

3

What are the coefficients  $c_2$  and  $c_3$  of the nonlinear characteristic curve?

$c_2\ = \ $

$\ \rm V^{ -1 }$
$c_3\ = \ $

$\ \rm V^{ -2 }$

4

Show that a  "DSB-AM with carrier"  constellation results when  $c_3$  is considered negligibly small.  What is the modulation depth  $m$?

$m \ = \ $

5

Assuming that  $c_3$  cannot be considered negligibly small,  which of the following statements are true?

The weight of the spectral line at $f_{\rm T}$  is unchanged.
$s(t)$  now includes Dirac lines at $f_{\rm T} ± 2f_{\rm N}$.
The cubic term leads to nonlinear distortions.
The cubic term leads to linear distortions.


Solution

(1)  From  $x(t) = A_0 + z(t) + q(t)$,  with   $A_0 = 2\ \rm V$  and   $A_{\rm T} = A_{\rm N} = 1 \ \rm V$,  we get the possible range  $0 \ {\rm V} ≤ x(t) ≤ 4\ \rm V$.

  • Thus,  the auxiliary quantity  $w(t)$  can take values between  $w_{\rm min}\hspace{0.15cm}\underline{ = -2 \ \rm V}$  and  $w_{\rm max}\hspace{0.15cm}\underline{ = +2 \ \rm V}$.


(2)  The coefficient  $c_0$  is equal to the characteristic value at the operating point.  Using  $A_0 = 2 \ \rm V$  and  $U = 3 \ \rm V$  we obtain:

$$c_0 = y(A_0) = U \cdot \left( 1 -{\rm e} ^{-A_0/U}\right) \hspace{0.15cm}\underline {= 1.460\,{\rm V}}\hspace{0.05cm}.$$
  • Accordingly,  for the Taylor coefficient  $c_1$:
$$c_1 = y\hspace{0.06cm}'(A_0)= {\rm e} ^{-A_0/U}\hspace{0.15cm}\underline { = 0.513}\hspace{0.05cm}.$$


(3)  The further derivatives   $(n ≥ 2)$  are:

$$y^{(n)}(A_0)= \frac{(-1)^{n-1}}{U^{n-1}} \cdot {\rm e} ^{-A_0/U} \hspace{0.05cm}.$$
  • This results in the following coefficients:
$$ c_2 = \frac{1}{2!} \cdot y^{(2)}(A_0)= \frac{1}{2U} \cdot {\rm e}^{-A_0/U} \hspace{0.15cm}\underline {= -0.086\,{\rm V^{-1}}}\hspace{0.05cm},$$
$$c_3 = \frac{1}{3!} \cdot y^{(3)}(A_0)= \frac{1}{6U^2} \cdot {\rm e}^{-A_0/U}\hspace{0.15cm}\underline { = 0.0095\,{\rm V^{-2}}}\hspace{0.05cm}.$$


(4)  Setting  $c_3 = 0$,  the output signal of the amplifier is:

$$y(t) = c_0 + c_1 \cdot (z(t) + q(t)) + c_2 \cdot (z^2(t) + q^2(t) + 2 \cdot z(t) \cdot q(t))\hspace{0.05cm}.$$
  • Thus,  after the band-pass,  the following signal components remain:
$$s(t) = c_1 \cdot z(t) + 2 \cdot c_2 \cdot z(t) \cdot q(t) = \left[c_1 \cdot A_{\rm T} + 2 \cdot c_2 \cdot A_{\rm T} \cdot A_{\rm N} \cdot \cos(\omega_{\rm N} t)\right] \cdot \cos(\omega_{\rm T} t)\hspace{0.05cm}.$$
  • The modulation depth is then determined as the quotient of the  "amplitude of the message oscillation"  over the "amplitude of the carrier":
$$m = \frac{2 \cdot |c_2| \cdot A_{\rm T} \cdot A_{\rm N}}{|c_1| \cdot A_{\rm T}} = \frac{2 \cdot |c_2| \cdot A_{\rm N}}{|c_1| }= \frac{2 \cdot 0.086 \cdot 1\,{\rm V}}{0.513 }\hspace{0.15cm}\underline { = 0.335}\hspace{0.05cm}.$$


(5)  Answers 2 and 3  are correct:

  • Considering the cubic part,  $y(t)$  includes the following other components:
$$y_3(t) = c_3 \cdot (z(t) + q(t))^3 = c_3 \cdot z^3(t) + 3 \cdot c_3 \cdot z^2(t) \cdot q(t)+ 3 \cdot c_3 \cdot z(t) \cdot q^2(t) + c_3 \cdot q^3(t) \hspace{0.05cm}.$$
  • The first term results in components at  $f_{\rm T}$  and  $3f_{\rm T}$, and the last term results in components at  $f_{\rm N}$  and  $3f_{\rm N}$. 
  • The second term gives a component at   $f_{\rm N}$  and others at  $2f_{\rm T} ± f_{\rm N}$:
$$3 \cdot c_3 \cdot z^2(t) \cdot q(t)= {3}/{2 } \cdot A_{\rm T}^2 \cdot A_{\rm N} \cdot \left[ \cos(\omega_{\rm N} t) + \cos(2\omega_{\rm T} t) \cdot \cos(\omega_{\rm N} t)\right] \hspace{0.05cm}.$$
  • Accordingly, the third summand in the above equation leads to
$$3 \cdot c_3 \cdot z(t) \cdot q^2(t)= {3}/{2 } \cdot A_{\rm T} \cdot A_{\rm N}^2 \cdot \left[ \cos(\omega_{\rm T} t) + \cos(\omega_{\rm T} t)\cdot \cos(2 \omega_{\rm N} t)\right] \hspace{0.05cm}.$$
  • Thus,  within the frequency range from  $\text{23 kHz}$  to  $\text{37 kHz}$,  there is indeed a change in the spectral line at  $f_{\rm T}$ 
    and new Dirac lines are formed at   $f_{\rm T} ± 2f_{\rm N}$,  i.e.,  at   $\text{24 kHz}$  and  $\text{36 kHz}$.
  • The resulting distortions are thus nonlinear  ⇒   Answer 3 ist correct and Answer 4 is wrong.