Signal Representation/The Convolution Theorem and Operation - Revision history

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@@ Line 29: / Line 29: @@
 :$$X_1 ( f ) \cdot X_2 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}{ {x} }_{\rm{1} } ( t ) * { {x} }_{\rm{2} } (t ).$$
-*The convolution is&nbsp; &raquo;$\text{commutative}$&laquo;  &nbsp; ⇒  &nbsp;The order of the operands can be changed:
+*The convolution is&nbsp; &raquo;commutative&laquo;  &nbsp; ⇒  &nbsp;The order of the operands can be changed:
 :$$x_1(t) * x_2(t) =x_2(t) * x_1(t).$$
-*The&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation#Proof_of_the_Convolution_Theorem|&raquo;$\text{Proof&laquo;}$]]&nbsp; will be shown at the end of this chapter.}}
+*The&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation#Proof_of_the_Convolution_Theorem|&raquo;Proof&laquo;]]&nbsp; will be shown at the end of this chapter.}}

@@ Line 203: / Line 203: @@
 {{BlaueBox|TEXT=
 $\text{Definition: }$&nbsp;
-The following relation of time functions&nbsp; $x_1(t)$&nbsp; and&nbsp; $x_2(t)$&nbsp; is called the&nbsp; $\text{convolution}$&nbsp; and represents this functional relation with a star:
+The following relation of time functions&nbsp; $x_1(t)$&nbsp; and&nbsp; $x_2(t)$&nbsp; is called the&nbsp; &raquo;'''convolution'''&laquo;&nbsp; and represents this functional relation with a star:
 :$$x_{\rm{1} } (t) * x_{\rm{2} } (t) = \int_{ - \infty }^{ + \infty } {x_1 ( \tau  ) }  \cdot x_2 ( {t - \tau } ) \hspace{0.1cm}{\rm d}\tau.$$
-This results in the following&nbsp; $\text{Fourier correspondence}$:
+*This results in the following&nbsp; &raquo;'''Fourier correspondence'''&laquo;:
 :$$X_1 ( f ) \cdot X_2 ( f )\hspace{0.1cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.1cm}{ {x} }_{\rm{1} } ( t ) * { {x} }_{\rm{2} } (t ).$$}}

@@ Line 170: / Line 170: @@
 ==Descriptive interpretation of convolution==
 <br>
-We assume an impulse response&nbsp; $h(t)$&nbsp; which is first constant for one millisecond and then decreases linearly to zero until &nbsp; $t = 3 \,\text{ms}$&nbsp;.
+We assume an impulse response&nbsp; $h(t)$&nbsp; which is first constant for one millisecond and then decreases linearly to zero until &nbsp; $t = 3 \,\text{ms}$.
-*If a Dirac delta&nbsp; $K_0 \cdot \delta(t)$&nbsp; is applied to the input of this low-pass filter, the output signal&nbsp; $y(t)$&nbsp; has the same shape as the impulse response&nbsp; $h(t)$.&nbsp; <br>The situation is shown in red in the figure.
+[[File:P_ID524__Sig_T_3_4_S5_rah.png|right|frame|An interpretation of&nbsp; "Convolution"]]
-*A&nbsp; $T= 1 \,\text{ms}$&nbsp; shifted  Dirac delta with weight&nbsp; $K_1 > K_0$&nbsp; results in the output signal&nbsp; $y_1(t)$&nbsp; which is delayed with respect to the red signal and increased in amplitude.
+*If&nbsp; $K_0 \cdot \delta(t)$&nbsp; is applied to the input of this low-pass filter,&nbsp; the output signal&nbsp; $y(t)$&nbsp; has the same shape as the impulse response&nbsp; $h(t)$.&nbsp; The situation is shown in red in the figure.
-[[File:P_ID524__Sig_T_3_4_S5_rah.png|right|frame|An interpretation of&nbsp; "Convolution"]]
+*The&nbsp; $($blue$)$&nbsp; $T= 1 \,\text{ms}$&nbsp; shifted  Dirac delta with weight&nbsp; $K_1 > K_0$&nbsp; results in the output signal&nbsp; $y_1(t)$&nbsp; which is delayed with respect to the red signal and increased in amplitude.
-<br>Now we consider the input signal consisting of seven differently weighted and shifted Dirac deltas
 :$$x( t ) = \sum\limits_{n = 0}^6 {K_n  \cdot \delta ( {t - n \cdot T} ),}$$
@@ Line 181: / Line 183: @@
 which can be understood as a time&ndash;discrete approximation of a time&ndash;continuous signal.
-*The signal at the output of the linear system is the sum of the seven partial signals marked with different colors in the figure:
+*The output signal of the linear system is the sum of the seven partial signals marked with different colors in the figure:
 :$$y( t ) = \sum\limits_{n = 0}^6 {K_n  \cdot h( {t - n \cdot T} ).}$$
-*We now look at the signal value at time&nbsp; $t = 4.5T$&nbsp; (see dotted lines):
+*We now look at the signal value at time&nbsp; $t = 4.5T$&nbsp; $($see dotted line$)$:
 :$$y( {t = 4.5T} ) = K_2  \cdot h( {2.5T} ) + K_3  \cdot h(1.5 T ) + K_4  \cdot h( 0.5 T ).$$
-The output signal value is thus only determined by the input signal values&nbsp; $K_2$,&nbsp; $K_3$&nbsp; and&nbsp; $K_4$.&nbsp; The influence of
+*$y( {t = 4.5T} )$&nbsp; is thus only determined by the input signal values&nbsp; $K_2$,&nbsp; $K_3$&nbsp; and&nbsp; $K_4$.&nbsp; <u>Note:</u>
-* $K_4$&nbsp; is  the greatest due to&nbsp; $h(0.5T) = 1$,&nbsp;
-* $K_3$&nbsp; is  less  strong due to&nbsp; $h(1.5T) = 0.75$,&nbsp; and
+#The influence of&nbsp; $K_4$&nbsp; is  the greatest due to&nbsp; $h(0.5T) = 1$.&nbsp;
-* $K_2$&nbsp; is the lowest due to&nbsp; $h(2.5T) = 0.25$.&nbsp;
+#The influence of&nbsp; $K_3$&nbsp; is  less  strong due to&nbsp; $h(1.5T) = 0.75$.&nbsp;

@@ Line 114: / Line 114: @@
 ==Graphical convolution==
 <br>
 For the descriptions in this section the following convolution operation is assumed:
-[[File:P_ID2723__Sig_T_3_4_programm.png|right|frame|Screenshot of an older (German) version &nbsp;$\rm LNTwww$ applet&nbsp; "Convolution":<br>&nbsp; &nbsp; $x_1(t)$&nbsp; is denoted as&nbsp; $x(t)$&nbsp; and&nbsp; $x_2(t)$&nbsp; as&nbsp; $h(t)$]]
 :$$y(t) = x_1 (t) * x_2 (t) $$
 :$$\Rightarrow \hspace{0.3cm}y(t) = \int_{ - \infty }^{ + \infty } {x_1 ( \tau  )}  \cdot x_2 ( {t - \tau } )\hspace{0.1cm}{\rm d}\tau.$$
@@ Line 126: / Line 127: @@
 #&nbsp; &raquo;'''Shifting the mirrorred function'''&laquo; by&nbsp; $t$:&nbsp;  $x_2(-\tau) \to x_2(t-\tau)$.
 #&nbsp; &raquo;'''Multiplication'''&laquo;&nbsp; of both functions&nbsp; $x_1(\tau)$&nbsp; and&nbsp; $x_2(t-\tau)$.
-#&nbsp; &raquo;'''Integration'''&laquo;&nbsp; over the product respective&nbsp; $\tau$&nbsp; between the limits&nbsp; $-\infty$&nbsp; to&nbsp; $+\infty$.
+#&nbsp; &raquo;'''Integration'''&laquo;&nbsp; over the product resp.&nbsp; $\tau$&nbsp; between the limits&nbsp; $-\infty$&nbsp; to&nbsp; $+\infty$.
-Since the convolution is commutative,&nbsp; $x_1(\tau)$&nbsp; can also be mirrored instead of&nbsp; $x_2(\tau)$.
+<u>Note:</u>
-The topic is illustrated by the (newer) HTML5 applet&nbsp; [[Applets:Graphical_Convolution|"Graphical Convolution"]].
+*The topic is also illustrated by the HTML5 applet&nbsp; [[Applets:Graphical_Convolution|&raquo;Graphical Convolution&laquo;]].
 <br clear=all>
-[[File:P_ID582__Sig_T_3_4_S4_neu.png|right|frame|Jump function convoluted with the exponential function]]
+{{GraueBox|TEXT=
 The procedure for the graphical convolution is now explained with a detailed example:
-*At the input of a filter there is a jump function&nbsp; $x(t) = \gamma(t)$.
+#At the input of a filter there is a jump function&nbsp; $x(t) = \gamma(t)$.
-*The impulse response of the first order low-pass filter is&nbsp; $h( t ) = {1}/{T} \cdot {\rm{e} }^{ - t/T}.$
+#The impulse response of the first order low-pass filter is&nbsp; $h( t ) = {1}/{T} \cdot {\rm{e} }^{ - t/T}.$
-*The time axis is already renamed to&nbsp; $\tau$.
+#The time axis is already renamed to&nbsp; $\tau$.
 The graphic shows
-*the (red colored) input signal&nbsp; $x(\tau)$,
+#the&nbsp; $($red colored$)$&nbsp;  input signal&nbsp; $x(\tau)$,
-*the (blue colored) impulse response&nbsp; $h(\tau)$,&nbsp;
+#the&nbsp; $($blue colored$)$&nbsp; impulse response&nbsp; $h(\tau)$,&nbsp;
-*the (grey colored) output signal&nbsp; $y(\tau)$.
+#the&nbsp; $($grey colored$)$&nbsp; output signal&nbsp; $y(\tau)$.
@@ Line 154: / Line 157: @@
 Some more remarks:
-*The output value at&nbsp; $t = 0$&nbsp; is obtained by mirroring the input signal&nbsp; $x(\tau)$.&nbsp; This mirrored signal&nbsp; $x(-\tau)$&nbsp; is multiplied by the impulse response&nbsp; $h(\tau)$&nbsp; and integrated above it.
+*The output value at&nbsp; $t = 0$&nbsp; is obtained by mirroring the input signal&nbsp; $x(\tau)$.&nbsp; This mirrored signal&nbsp; $x(-\tau)$&nbsp; is multiplied by the impulse response&nbsp; $h(\tau)$&nbsp; and integrated over it.
-*As there is no time interval, where both the blue curve&nbsp; $h(\tau)$&nbsp; and at the same time also the red dashed mirroring&nbsp; $x(-\tau)$&nbsp; is not equal to zero, the result is&nbsp; $y(t=0)=0$.
-*For any other time&nbsp; $t$&nbsp; the input signal must be shifted &nbsp; ⇒ &nbsp; $x(t-\tau)$, for example according to the green dotted curve for&nbsp; $t=T$.
+*As there is no time interval,&nbsp; where both the blue curve&nbsp; $h(\tau)$&nbsp; and at the same time also the red dashed mirrored curve&nbsp; $x(-\tau)$&nbsp; is not equal to zero,&nbsp; the result is&nbsp; $y(t=0)=0$.
 *As in this example&nbsp; $x(t-\tau)$&nbsp; is only&nbsp; $0$&nbsp; or&nbsp; $1$&nbsp; the integration&nbsp; $($general from&nbsp; $\tau_1$&nbsp; to&nbsp; $\tau_2)$&nbsp; is very simple and you get here with&nbsp; $\tau_1 = 0$&nbsp;  and&nbsp; $\tau_2 = t$&nbsp;:
 :$$y( t) = \int_0^{\hspace{0.05cm} t} {h( \tau)}\hspace{0.1cm} {\rm d}\tau = \frac{1}{T}\cdot\int_0^{\hspace{0.05cm} t} {{\rm{e}}^{ - \tau /T } }\hspace{0.1cm} {\rm d}\tau = 1 - {{\rm{e}}^{ - t /T } }.$$
-*This sketch is valid for &nbsp; $t=T$&nbsp; and results in the output value&nbsp; $y(t=T) = 1 – 1/\text{e} \approx 0.632$.}}
+*This sketch is valid for &nbsp; $t=T$&nbsp; and results in the output value&nbsp; $y(t=T) = 1 - 1/\text{e} \approx 0.632$.}}

@@ Line 8: / Line 8: @@
 ==Convolution in the time domain==
 <br>
-The&nbsp; "Convolution Theorem"&nbsp; is one of the most important laws of the Fourier transform, to which an own subchapter is dedicated in this tutorial.
+The&nbsp; &raquo;Convolution Theorem&laquo;&nbsp; is one of the most important laws of the Fourier transform,&nbsp; to which an own subchapter is dedicated in this tutorial.&nbsp; We will first consider the convolution theorem in the time domain and assume that the spectra of two time functions&nbsp; $x_1(t)$&nbsp; and&nbsp; $x_2(t)$&nbsp; are known:
 :$$X_1 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}x_1( t ),\quad X_2 ( f )\hspace{0.1cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.1cm}x_2 ( t ).$$
-Then for the time function of the product&nbsp; $X_1(f) \cdot X_2(f)$ applies:
+Then for the time function of the product&nbsp; $X_1(f) \cdot X_2(f)$&nbsp; applies:
 :$$X_1 ( f ) \cdot X_2 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}\int_{ - \infty }^{ + \infty } {x_1 ( \tau  )}  \cdot x_2 ( {t - \tau } )\hspace{0.1cm}{\rm d}\tau.$$
-Here&nbsp; $\tau$&nbsp; is a formal integration variable with the dimension of a time.
+$\tau$&nbsp; is a formal integration variable with the dimension of a time.
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The above connection of the time function&nbsp; $x_1(t)$&nbsp; and&nbsp; $x_2(t)$&nbsp; is called&nbsp; $\text{convolution}$&nbsp; and represents this functional connection with a star:
+$\text{Definition:}$&nbsp;
 :$$x_{\rm{1} } (t) * x_{\rm{2} } (t) = \int_{ - \infty }^{ + \infty } {x_1 ( \tau  ) }  \cdot x_2 ( {t - \tau } ) \hspace{0.1cm}{\rm d}\tau =  x_{\rm{2} } (t) * x_{\rm{1} } (t) .$$
@@ Line 29: / Line 29: @@
 :$$X_1 ( f ) \cdot X_2 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}{ {x} }_{\rm{1} } ( t ) * { {x} }_{\rm{2} } (t ).$$
-*The convolution is&nbsp; $\text{"commutative"}$  &nbsp; ⇒  &nbsp;The order of the operands can be changed:
+*The convolution is&nbsp; &raquo;$\text{commutative}$&laquo;  &nbsp; ⇒  &nbsp;The order of the operands can be changed:
 :$$x_1(t) * x_2(t) =x_2(t) * x_1(t).$$
-*The&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation#Proof_of_the_Convolution_Theorem|$\text{Proof}$]]&nbsp; will be shown at the end of the chapter.}}
+*The&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation#Proof_of_the_Convolution_Theorem|&raquo;$\text{Proof&laquo;}$]]&nbsp; will be shown at the end of this chapter.}}
-[[File:EN_Sig_T_3_4_S1.png|right|frame|On calculation of signal and spectrum of  the LTI output]]
+{{GraueBox|TEXT=
-{{GraueBox|TEXT=
+[[File:EN_Sig_T_3_4_S1.png|right|frame|On calculation of signal and spectrum of  the LTI output]]
 $\text{Example 1:}$&nbsp; Every linear time-invariant&nbsp; $\rm (LTI)$&nbsp; system can be described both,
-*by the frequency response&nbsp; $H(f)$,
+#by the frequency response&nbsp; $H(f)$,
-*by the impulse response&nbsp; $h(t)$.
+#by the impulse response&nbsp; $h(t)$.
 The relation between these two system quantities is also given by the Fourier transform.
-If a signal&nbsp; $x(t)$&nbsp; with spectrum&nbsp; $X(f)$&nbsp; is applied to the input, the spectrum of the output signal is:
+*If the signal&nbsp; $x(t)$&nbsp; with spectrum&nbsp; $X(f)$&nbsp; is applied to the input,&nbsp; then the spectrum of the output signal is:
 :$$Y(f) = X(f) \cdot H(f)\hspace{0.05cm}.$$
-It is possible to calculate the output signal in the time domain with the convolution theorem:
+*It is possible to calculate the output signal in the time domain with the convolution theorem:
 :$$y( t ) = x(t) * h( t ) = \int_{ - \infty }^{ + \infty } \hspace{-0.15cm}{x( \tau  )}  \cdot h( {t - \tau } )\hspace{0.1cm}{\rm d}\tau =  \int_{ - \infty }^{ + \infty } \hspace{-0.15cm} {h( \tau  )}  \cdot x( {t - \tau } )\hspace{0.1cm}{\rm d}\tau = h(t) * x( t ).$$
-This equation shows again the&nbsp; "commutativity"&nbsp; of the convolution operation.  }}
+*This equation shows again the&nbsp; &raquo;commutativity&laquo;&nbsp; of the&nbsp; &raquo;convolution operation&laquo;.  }}
@@ Line 63: / Line 63: @@
 :$$x_1 ( t ) \cdot x_2 ( t )\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_1 (f) * X_2 (f) =  \int_{ - \infty }^{ + \infty } {X_1 ( \nu  )}  \cdot X_2 ( {f - \nu })\hspace{0.1cm}{\rm d}\nu.$$
-This result can be proved similarly to the&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation#Proof_of_the_Convolution_Theorem|$\text{convolution in the time domain}$]].&nbsp; However, the integration variable&nbsp; $\nu$&nbsp; now has the dimension of a frequency.
+This result can be proved similarly to the&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation#Proof_of_the_Convolution_Theorem|&raquo;convolution in the time domain&laquo;]].&nbsp; However,&nbsp; the integration variable&nbsp; $\nu$&nbsp; now has the dimension of a frequency.
-[[File:EN_Sig_T_3_4_S2.png|right|frame|Convolution in the frequency domain&nbsp; (e.g.&nbsp;  DSB&ndash;AM)]]
+*The time domain representation&nbsp; $\rm (blue)$&nbsp; shows the modulated signal&nbsp; $s(t)$&nbsp; as the product of the source signal&nbsp; $q(t)$&nbsp; and the&nbsp; $($normalized$)$&nbsp; carrier signal&nbsp; $z(t)$.
@@ Line 76: / Line 78: @@
 ==Convolution of a function with a Dirac delta function==
 <br>
-The convolution operation becomes very simple, if one of the two operands is a&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|$\text{Dirac delta function}$]].&nbsp; This applies equally to the convolution in the time and frequency domain.
+The convolution operation becomes very simple,&nbsp; if one of the operands is a&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|&raquo;Dirac delta function&laquo;]].&nbsp; This applies both to the convolution in  time and frequency domain.
 *We will consider the convolution of a function&nbsp; $x_1(t)$&nbsp; with the function
@@ Line 86: / Line 88: @@
 :$$Y( f ) = X_1 ( f ) \cdot X_2 ( f ) = X_1 ( f ) \cdot  \alpha  \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.05cm}\cdot\hspace{0.05cm}2\hspace{0.03cm}{\rm{\pi }}\hspace{0.05cm}\cdot\hspace{0.05cm}f\hspace{0.05cm}\cdot\hspace{0.05cm}T}.$$
-*The complex exponential function leads to a shift by&nbsp; $T$ &nbsp; &rArr; &nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|$\text{Shifting Theorem}$]], the factor&nbsp; $\alpha$&nbsp; to an attenuation&nbsp; $(\alpha < 1)$&nbsp; or amplification&nbsp; $(\alpha > 1)$.
+*The complex exponential function leads to a shift by&nbsp; $T$ &nbsp; &rArr; &nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|&raquo;Shifting Theorem&laquo;]],&nbsp; the factor&nbsp; $\alpha$&nbsp; to an attenuation&nbsp; $(\alpha < 1)$&nbsp; or amplification&nbsp; $(\alpha > 1)$.&nbsp; From this follows:
 :$$x_1 (t) * x_2 (t) = \alpha  \cdot x_1 ( {t - T} ).$$
@@ Line 94: / Line 94: @@
 {{BlaueBox|TEXT=
 $\text{In Words: }$&nbsp;
 *The convolution of any function with a Dirac delta function at&nbsp; $t = T$&nbsp; results in the function shifted to the right by&nbsp; $T$,&nbsp;
 *while the weighting of the Dirac delta function by the factor&nbsp; $\alpha$&nbsp; has to be taken into account.}}
-[[File:P_ID522__Sig_T_3_4_S3_neu.png|right|frame|Convolution of a rectangle pulse with a Dirac Dirac delta function function]]
+{{GraueBox|TEXT=
-{{GraueBox|TEXT=
+$\text{Example 3:}$&nbsp; A rectangle pulse&nbsp; $x(t)$&nbsp; is delayed by an LTI system by the delay time&nbsp; $\tau = 3\,\text{ ms}$&nbsp; and attenuated by the factor&nbsp; $\alpha = 0.5$.
-$\text{Example 3:}$&nbsp; A rectangle pulse&nbsp; $x(t)$&nbsp; is delayed by an LTI system by the delay time&nbsp; $\tau = 3\,\text{ ms}$&nbsp; and attenuated by the factor&nbsp; $\alpha = 0.5$&nbsp;.
+[[File:P_ID522__Sig_T_3_4_S3_neu.png|right|frame|Convolution of a rectangle pulse with a Dirac  delta  function]]
 Shift and attenuation can be recognised both
-*by the output signal&nbsp; $y(t)$&nbsp;
+*by the output signal&nbsp; $y(t)$,&nbsp;
-*as well as from the impulse response&nbsp; $h(t)$.
 }}

@@ Line 67: / Line 67: @@
 [[File:EN_Sig_T_3_4_S2.png|right|frame|Convolution in the frequency domain&nbsp; (e.g.&nbsp;  DSB&ndash;AM)]]
 {{GraueBox|TEXT=
-$\text{Example 2:}$&nbsp; The &nbsp; [[Modulation_Methods/Zweiseitenband-Amplitudenmodulation#Beschreibung_im_Zeitbereich|$\text{Double-Sideband Amplitude Modulation}$]]&nbsp; $\text{(DSB-AM)}$ with carrier suppression is described by the drawn model.
+$\text{Example 2:}$&nbsp; The &nbsp; [[Modulation_Methods/Zweiseitenband-Amplitudenmodulation#Beschreibung_im_Zeitbereich|"Double-Sideband Amplitude Modulation"]]&nbsp; $\text{(DSB-AM)}$ with carrier suppression is described by the drawn model.
 *The time domain representation&nbsp; $\rm (blue)$&nbsp; shows the modulated signal&nbsp; $s(t)$&nbsp; as the product of the source signal&nbsp; $q(t)$&nbsp; and the (normalized) carrier signal&nbsp; $z(t)$.

@@ Line 229: / Line 229: @@
-==Exercises for the Chapter==
+==Exercises for the chapter==
 <br>
 [[Aufgaben:Exercise_3.7:_Synchronous_Demodulator|Exercise 3.7: Synchronous Demodulator]]