Signal Representation/The Fourier Transform and its Inverse - Revision history

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@@ Line 51: / Line 51: @@
 ==Closer examination of the Fourier coefficients==
 <br>
-We assume a periodic signal&nbsp; $x_{\rm P}(t)$&nbsp; with period duration&nbsp; $T_0$&nbsp; which corresponds to the explanations in section&nbsp; [[Signal_Representation/Fourier_Series#Complex_Fourier_series|"Complex Fourier series"]].&nbsp;
+We assume a periodic signal&nbsp; $x_{\rm P}(t)$&nbsp; with period duration&nbsp; $T_0$&nbsp; which corresponds to the explanations in section&nbsp; [[Signal_Representation/Fourier_Series#Complex_Fourier_series|&raquo;Complex Fourier series&laquo;]].&nbsp;
 [[File:P_ID538__Sig_T_3_1_S2b_rah.png|right|frame|Periodic signal&nbsp; $x_{\rm P}(t)$&nbsp; and&nbsp; $x_{\rm P}\hspace{0.01cm}'(t)$&nbsp; and its line spectra]]
 *This signal can be described as follows:

@@ Line 225: / Line 225: @@
 This signal is shown in blue color in the left sketch.
-Since the signal&nbsp; $x(t)$&nbsp; extends to infinity on both sides, we must add a suitable convergence factor&nbsp; $\text{e}^{-\varepsilon \hspace{0.05cm} \cdot \hspace{0.05cm}\vert \hspace{0.05cm} t \hspace{0.05cm} \vert}$&nbsp; is valid&nbsp; $($&nbsp; $\varepsilon > 0)$&nbsp;  in order to calculate the Fourier transform for both sections.&nbsp; The resulting time function is then
+Since the signal&nbsp; $x(t)$&nbsp; extends to infinity on both sides, we must add a suitable convergence factor&nbsp; $\text{e}^{-\varepsilon \hspace{0.05cm} \cdot \hspace{0.05cm}\vert \hspace{0.05cm} t \hspace{0.05cm} \vert}$&nbsp; with&nbsp; $($&nbsp; $\varepsilon > 0)$&nbsp;  in order to calculate the Fourier transform for both sections.&nbsp; The resulting time function is then
 :$$x_\varepsilon  (t) = \left\{ {\begin{array}{*{20}c}   { {\rm{e} }^{ - \varepsilon \hspace{0.05cm} \cdot \hspace{0.05cm}t} } & { {\rm{for} }\quad t > 0,}  \\   { {\rm{ - e} }^{\hspace{0.05cm}\varepsilon\hspace{0.05cm} \cdot \hspace{0.05cm}  t} } & { {\rm{for} }\quad t < 0.}  \\\end{array} } \right.$$
-Following a similar procedure as in the section&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|$\text{Dirac  delta function in the frequency domain}$]]&nbsp; results for the corresponding spectral function:
+Following a similar procedure as in section&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|&raquo;Dirac  delta function in the frequency domain&laquo;]]&nbsp; results for the corresponding spectral function:
 :$$X_\varepsilon  (f) = \frac{1}{ {\varepsilon  + {\rm{j} }2{\rm{\pi } }f} } - \frac{1}{ {\varepsilon  - {\rm{j} }2{\rm{\pi } }f} } = \frac{ { - {\rm{j4\pi } }f} }{ {\varepsilon ^2  + \left( {2{\rm{\pi } }f} \right)^2 } }.$$
-But actually we are interested in the spectrum of the actual jump function
+But actually we are interested in the spectrum of the&nbsp; &raquo;jump function&laquo;
 :$$x(t) = \mathop {\lim }\limits_{\varepsilon  \hspace{0.05cm}\to \hspace{0.05cm}0 } x_\varepsilon  (t).$$
-Therefore, the spectral function&nbsp; $X(f) =\text{F}\big[x(t)\big]$&nbsp; has to be determined as limit value of&nbsp; $X_\varepsilon(f)$&nbsp; for&nbsp; $\varepsilon \to 0$&nbsp;:
+Therefore,&nbsp; the spectral function&nbsp; $X(f) =\text{F}\big[x(t)\big]$&nbsp; has to be determined as limit value of&nbsp; $X_\varepsilon(f)$&nbsp; for&nbsp; $\varepsilon \to 0$:
 :$$X(f) = \mathop {\lim }\limits_{\varepsilon \hspace{0.05cm} \to \hspace{0.05cm}0 } X_\varepsilon  (f) = \frac{ { - {\rm{j} } } }{ { {\rm{\pi } }f} } = \frac{1}{ { {\rm{j\pi } }f} }.$$
@@ Line 243: / Line 243: @@
 In the right graph the imaginary spectral function&nbsp; $X(f)$&nbsp; is shown as a blue curve.&nbsp; You can see that&nbsp; $\vert X(f) \vert$&nbsp; decreases continuously with increasing frequency.
-The green curve in the left graph shows the signal&nbsp; $y(t)$, which differs from&nbsp; $x(t)$&nbsp; only in the negative time section.
+&rArr; &nbsp; The green curve in the left graph shows the signal&nbsp; $y(t)$,&nbsp; which differs from&nbsp; $x(t)$&nbsp; only in the negative time section.
 *In this area&nbsp; $y(t) = 0$.&nbsp; The corresponding spectral function&nbsp; $Y(f)$&nbsp; is only half as large as&nbsp; $X(f)$&nbsp; in the entire range.&nbsp; This is shown in the following calculation:
@@ Line 249: / Line 249: @@
 :$$Y(f) = \mathop {\lim }\limits_{\varepsilon  \to 0 } Y_\varepsilon  (f) = \mathop {\lim }\limits_{\varepsilon \hspace{0.05cm} \to \hspace{0.05cm}0 }\frac{1}{ {\varepsilon  + {\rm{j} }2{\rm{\pi } }f} } = \frac{1}{ { {\rm{j2\pi } }f} }.$$
-*In addition there is now a Dirac delta function at&nbsp; $f = 0$&nbsp; with weight&nbsp; $1/2$, due to the equal part.&nbsp; This is explained in detail in the example in section&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Assignment_Theorem|"Assignment Theorem"]].&nbsp; }}
+*In addition there is a Dirac delta function at&nbsp; $f = 0$&nbsp; with weight&nbsp; $1/2$, due to the equal part.&nbsp; This is explained in the example in section&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Assignment_Theorem|&raquo;Assignment Theorem&laquo;]].&nbsp; }}
 ==The second Fourier integral==
 <br>
-Up to now, it has only been shown how to calculate the corresponding spectral function&nbsp; $x(t)$&nbsp; $X(f)$&nbsp; for an aperiodic, pulse-like signal&nbsp; $x(t)$.&nbsp; Now we turn to the reverse task, namely: &nbsp; How to determine the time function&nbsp; $x(t)$&nbsp; from the spectral function&nbsp; $X(f)$?
+Up to now,&nbsp; it has only been shown how to calculate the corresponding spectral function&nbsp; $X(f)$&nbsp; for an aperiodic,&nbsp; pulse-like signal&nbsp; $x(t)$.&nbsp;
 [[File:EN_Sig_T_3_1_S7.png|right|frame|On the second Fourier integral]]
-With the same designations as in the first sections of this chapter, you can write the signal&nbsp; $x(t)$&nbsp; as a Fourier series, where now the boundary&nbsp; ${f_0}' \to 0$&nbsp; is to be considered:
+With the same designations as in the first sections of this chapter,&nbsp; you can write the signal&nbsp; $x(t)$&nbsp; as Fourier series,&nbsp; where now the limit&nbsp; ${f_0}' \to 0$&nbsp; is to be considered:
 :$$x(t)=\lim_{{f_{\rm 0}}'  \hspace{0.05cm}\to  \hspace{0.05cm}0} \sum^{+\infty}_{\nu = -\infty}{D_{\it \nu}}' \cdot \rm e^{j\hspace{0.03cm} 2  \hspace{0.03cm}\pi  \hspace{0.03cm}\it\nu \hspace{0.03cm} {f_{\rm 0}}' t}.$$
@@ Line 267: / Line 267: @@
 The limit crossing&nbsp; ${f_0}' \to 0$&nbsp; has the following effects:
-*The&nbsp; (infinite)&nbsp; sum becomes an integral, where&nbsp;  ${f_0}'$&nbsp; has to be formally replaced by the differential quantity&nbsp; $\text{d}f$&nbsp; (integration variable).
+#The&nbsp; $($infinite$)$&nbsp; sum becomes an integral,&nbsp; where&nbsp;  ${f_0}'$&nbsp; has to be formally replaced by the differential quantity&nbsp; $\text{d}f$&nbsp; $($integration variable$)$.
+#The quantity &nbsp;  $\nu \cdot{f_0}'$&nbsp; in the exponent describes the physical frequency&nbsp; $f$.
-*The quantity &nbsp;  $\nu \cdot{f_0}'$&nbsp; in the exponent describes the physical frequency&nbsp; $f$.
+#The quotient&nbsp; ${D_{\nu}}'/{f_0}'$&nbsp; yields the spectral function&nbsp; $X(f)$&nbsp; at the frequency&nbsp; $f$.
-Taking these properties into account, the&nbsp; "second Fourier integral"&nbsp; is obtained.
+Taking these properties into account, the&nbsp; &raquo;second Fourier integral&laquo;&nbsp; is obtained.
 {{BlaueBox|TEXT=
-$\text{Second Fourier Integral:}$&nbsp;
+$\text{Second Fourier Integral:}$&nbsp; If the spectral function&nbsp; $X(f)$&nbsp; of an aperiodic and energy-limited signal is given,&nbsp; then the corresponding&nbsp; &raquo;'''time signal'''&laquo;&nbsp; is:
 :$$x(t) = \hspace{0.01cm}\int_{-\infty} ^{ {+}\infty} X(f) \, \cdot \, { \rm e}^{\rm j 2\pi \it ft} \,{\rm d}f.$$}}

@@ Line 147: / Line 147: @@
-Thus,&nbsp; a fundamental definition is obtained,&nbsp; which allows the calculation of the spectral function of an aperiodic time function.&nbsp; The name of this spectral transformation goes back to the French physicist&nbsp; [https://en.wikipedia.org/wiki/Joseph_Fourier $\text{Jean-Baptiste Joseph Fourier}$].
+Thus,&nbsp; a fundamental definition is obtained,&nbsp; which allows the calculation of the spectral function of an aperiodic time function.&nbsp; The name of this spectral transformation goes back to the French physicist&nbsp; [https://en.wikipedia.org/wiki/Joseph_Fourier &raquo;$\text{Jean-Baptiste Joseph Fourier}$&laquo;].
 {{BlaueBox|TEXT=
@@ Line 210: / Line 210: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-The generally valid functional relation&nbsp; $X(f) = F\big [x(t) \big ]$&nbsp; is called&nbsp; &raquo;'''Fourier Transform}'''&laquo;.&nbsp; For the short notation we use&nbsp; $($with the&nbsp; "white dot"&nbsp; for the time domain and the&nbsp; "filled dot"&nbsp; for the spectral domain$)$:
+The generally valid functional relation&nbsp; $X(f) = F\big [x(t) \big ]$&nbsp; is called&nbsp; &raquo;'''Fourier Transform'''&laquo;.&nbsp; For the short notation we use&nbsp; $($with the&nbsp; "white dot"&nbsp; for the time domain and the&nbsp; "filled dot"&nbsp; for the spectral domain$)$:
 :$$X(f)\ \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\,\ x(t).$$

@@ Line 121: / Line 121: @@
 ==The first Fourier integral==
 <br>
-Concerning the spectral function&nbsp; $X_{\rm P}(f)$&nbsp; and&nbsp; $X(f)$&nbsp; the following statements can be made:
+Concerning the spectral functions&nbsp; $X_{\rm P}(f)$&nbsp; and&nbsp; $X(f)$&nbsp; the following statements can be made:
 *The individual spectral lines now lie as close together as desired&nbsp; $({f_0}'=1/{T_0}' \to 0)$.
-*In the spectral function&nbsp; $X(f)$&nbsp; all possible&nbsp; (not only discrete)&nbsp; frequencies now occur within certain intervals &nbsp; &rArr; &nbsp; $X(f)$&nbsp; is no longer a line spectrum.
+*In the spectral function&nbsp; $X(f)$&nbsp; all possible&nbsp; $($not only discrete$)$&nbsp; frequencies now occur within certain intervals &nbsp; &rArr; &nbsp; $X(f)$&nbsp; is no longer a line spectrum.
-*The contribution of each individual frequency&nbsp; $f$&nbsp; to the signal is negligibly small&nbsp; $(k \to \infty, {D_{\nu}}' \to 0)$.
+*The contribution of each individual frequency&nbsp; $f$&nbsp; to the signal&nbsp; $x(t)$&nbsp;is negligibly small&nbsp; $(k \to \infty,\ {D_{\nu}}' \to 0)$.
 *Because of the infinite number of frequencies there is a finite result in total.
-*Instead of calculating the Fourier coefficients&nbsp; ${D_{\nu}}'$&nbsp; a spectral density&nbsp; $X(f)$&nbsp; is now calculated.&nbsp; For the frequency&nbsp; $f=\nu\cdot {f_0}'$&nbsp; then applies:
+*Instead of calculating the Fourier coefficients&nbsp; ${D_{\nu}}'$:&nbsp; Now a spectral density&nbsp; $X(f)$&nbsp; is calculated.&nbsp; For the frequency&nbsp; $f=\nu\cdot {f_0}'$&nbsp; then applies:
 : $$X(f = {\rm \nu} {f_{\rm 0}}') = \lim_{{f_{\rm 0}}' \hspace{0.05cm}\to \hspace{0.05cm} 0} ({{D_{\rm \nu}}'}/{{f_{\rm 0}}'}) = \lim_{{T_{\rm 0}}' \to \infty} ({D_{\rm \nu}}' \cdot {T_{\rm 0}}').$$
-*The spectral function&nbsp; $X(f)$&nbsp; of the aperiodic signal&nbsp; $x(t)$&nbsp; is visible in the spectrum&nbsp; $X_{\rm P}(f)$&nbsp; of the periodic signal&nbsp; $x_{\rm P}(t)$ as envelope&nbsp; (see graphics in the last section).
+*The spectral function&nbsp; $X(f)$&nbsp; of the aperiodic signal&nbsp; $x(t)$&nbsp; is visible in the spectrum&nbsp; $X_{\rm P}(f)$&nbsp; of the periodic signal&nbsp; $x_{\rm P}(t)$ as envelope&nbsp; $($see graphics in the last section$)$.
-*In the lower graphic&nbsp; ${D_{\nu}}'$&nbsp; corresponds to the red-shaded area of the frequency interval around&nbsp; $\nu \cdot {f_0}'$&nbsp; with the width ${f_0}'$.
@@ Line 141: / Line 142: @@
 :$$X(f = {\rm \nu} \cdot {f_{\rm 0}}') = \lim_{{T_{\rm 0}'} \to \infty} \int ^{{T_{\rm 0}}'/2} _{-{T_{\rm 0}}'/2} x_{\rm P}(t) \, \cdot \, { \rm e}^{-\rm j 2\pi\nu \it {f_{\rm 0}}' t} \,{\rm d}t.$$
-Through the common limit crossing &nbsp; $({T_0}' \to \infty, \ {f_0}' \to 0)$&nbsp; the following transformations will happen
+Through the common limit crossing &nbsp; $({T_0}' \to \infty, \ {f_0}' \to 0)$&nbsp; the following transformations will happen:
-*from the periodic signal&nbsp; $x_{\rm P}(t)$&nbsp; to the aperiodic signal&nbsp; $x(t)$, and
+#From the periodic signal&nbsp; $x_{\rm P}(t)$&nbsp; to the aperiodic signal&nbsp; $x(t)$.
+Thus,&nbsp; a fundamental definition is obtained,&nbsp; which allows the calculation of the spectral function of an aperiodic time function.&nbsp; The name of this spectral transformation goes back to the French physicist&nbsp; [https://en.wikipedia.org/wiki/Joseph_Fourier $\text{Jean-Baptiste Joseph Fourier}$].
 {{BlaueBox|TEXT=
 $\text{First Fourier Integral:}$&nbsp;
-The&nbsp; $\text{spectral function}$&nbsp; (or short:&nbsp; the&nbsp; $\text{spectrum)}$&nbsp; of an aperiodic and simultaneously energy limited signal&nbsp; $x(t)$&nbsp; is to be calculated as follows
+The&nbsp; &raquo;'''spectral function'''&laquo;&nbsp; $($or short:&nbsp; the&nbsp;  &raquo;'''spectrum'''&laquo;$)$&nbsp; of an aperiodic and simultaneously energy limited signal&nbsp; $x(t)$&nbsp; is to be calculated as follows
 :$$X(f)= \hspace{0.05cm}\int_{-\infty} ^{ {+}\infty} x(t) \, \cdot \, { \rm e}^{-\rm j 2\pi \it ft} \,{\rm d}t.$$}}
-The following (German language) learning video should clarify the statements of the last sections:<br> &nbsp; &nbsp; &nbsp; &nbsp;[[Kontinuierliche_und_diskrete_Spektren_(Lernvideo)|"Kontinuierliche und diskrete Spektren"]] &nbsp; &rArr; &nbsp; "Continuous and discrete spectra".
+The following&nbsp; $($German language$)$&nbsp; learning video should clarify the statements of the last sections:<br> &nbsp; &nbsp; &nbsp; &nbsp;[[Kontinuierliche_und_diskrete_Spektren_(Lernvideo)|&raquo;Kontinuierliche und diskrete Spektren&laquo;]] &nbsp; &rArr; &nbsp; "Continuous and discrete spectra".
 {{GraueBox|TEXT=
 $\text{Example 2:}$&nbsp;
-Given is the sketched time course&nbsp; $x(t)$.&nbsp; The corresponding spectrum&nbsp; $X(f)$&nbsp; is searched for.
+Given is the sketched time course&nbsp; $x(t)$.&nbsp; The corresponding spectrum&nbsp; $X(f)$&nbsp; is searched for using the first Fourier integral:
 [[File:P_ID330__Sig_T_3_1_S5_neu.png|right|frame|Rectangular pulse&nbsp; $x(t)$]]
+*From the above representation we can see,&nbsp; that for&nbsp; $\vert t \vert > T/2$&nbsp; the signal is&nbsp; $x(t) = 0$.
 *This means that the integration interval can be limited to the range&nbsp; $\pm T/2$.
@@ Line 182: / Line 180: @@
 {{BlaueBox|TEXT=
 $\text{Definitions:}$&nbsp; For abbreviation we define the following functions:
-*$\text{sinc&ndash;function}$&nbsp; (predominantly used in Anglo-American literature) &nbsp; &nbsp; &rArr; &nbsp; &nbsp; ${\rm sinc}( x ) =  {\sin  (\pi  x) }/(\pi  x ),$
+*&raquo;'''sinc&ndash;function&laquo;'''&nbsp; $($predominantly used in Anglo-American literature$)$
-*$\text{si&ndash;function}$&nbsp; or $\text{splitting function}$ &nbsp;(predominantly used in German literature) &nbsp; &nbsp; &rArr; &nbsp; &nbsp; ${\rm si}\left( x \right) = \sin \left( x \right)/x = {\rm sinc}(x/\pi ).$}}
+*&raquo;'''si&ndash;function'''&laquo;&nbsp; or&nbsp; &raquo;$\text{splitting function}$&laquo; &nbsp;$($predominantly used in German literature$)$
@@ Line 192: / Line 192: @@
 ==Fourier transform==
 <br>
-The spectrum&nbsp; $X(f)$&nbsp; of a signal&nbsp; $x(t)$&nbsp; is according to the "first Fourier integral":
+The spectrum&nbsp; $X(f)$&nbsp; of a signal&nbsp; $x(t)$&nbsp; is according to the&nbsp; &raquo;first Fourier integral&laquo;:
 :$$X(f)= \hspace{0.05cm}\int _{-\infty} ^{{+}\infty} x(t) \, \cdot \, { \rm e}^{-\rm j 2\pi \it ft} \,{\rm d}t.$$
-As shown in the last section with a simple example, this integral can be solved easily for an energy-limited signal&nbsp; $x(t)$.&nbsp; For non-energy limited signals, for example
+As shown in the last section with a simple example,&nbsp; this integral can be solved easily for an energy-limited signal&nbsp; $x(t)$.&nbsp; For non-energy limited signals,&nbsp; for example
-*a&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal|$\text{DC signal}$]], or
+*a&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal|&raquo;DC signal&laquo;]], or
-*a&nbsp; [[ Signal_Representation/Harmonic_Oscillation|$\text{harmonic oscillation}$]],
+*a&nbsp; [[ Signal_Representation/Harmonic_Oscillation|&raquo;harmonic oscillation&laquo;]],
-we observe a divergence of the Fourier integral.&nbsp; Including a bilateral declining auxiliary function&nbsp; $\varepsilon (t)$&nbsp; however, convergence can be forced:
+we observe a divergence of the Fourier integral.&nbsp; Including a bilateral declining auxiliary function&nbsp; $\varepsilon (t)$,&nbsp; however,&nbsp; convergence can be forced:
 :$$X(f) = \lim_{\varepsilon \to 0} \int _{-\infty} ^{{+}\infty} x(t) \cdot {\rm e}^{\it -\varepsilon  | \hspace{0.01cm} t \hspace{0.01cm} |} \cdot {\rm e}^{{-\rm j 2  \pi}\it  ft} \,{\rm d}t.$$
-Such non-energy limited signals lead  to so-called&nbsp; "Dirac delta functions"&nbsp; in the spectral domain, sometimes also called&nbsp; "distributions".
+Such non-energy limited signals lead  to so-called&nbsp; &raquo;Dirac delta functions&laquo;&nbsp; in the spectral domain,&nbsp; sometimes also called&nbsp; &raquo;distributions&laquo;.
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-The generally valid functional relation&nbsp; $X(f) = F\big [x(t) \big ]$&nbsp; is called&nbsp; $\text{Fourier Transform}$.&nbsp; For the short notation we use (with the "white" dot for the time domain and the filled dot for the spectral domain):
+The generally valid functional relation&nbsp; $X(f) = F\big [x(t) \big ]$&nbsp; is called&nbsp; &raquo;'''Fourier Transform}'''&laquo;.&nbsp; For the short notation we use&nbsp; $($with the&nbsp; "white dot"&nbsp; for the time domain and the&nbsp; "filled dot"&nbsp; for the spectral domain$)$:
 :$$X(f)\ \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\,\ x(t).$$
-With a ringing signal, however, convergence is only achieved as long as the time function increases less than exponentially. }}
+With a increasing signal,&nbsp; however,&nbsp; convergence is only achieved as long as the time function increases less than exponentially. }}

@@ Line 7: / Line 7: @@
 == # OVERVIEW OF THE THIRD MAIN CHAPTER # ==
 <br>
-In the second chapter periodic signals were described by different harmonic oscillations&nbsp; ("Fourier series").&nbsp;
+In the second chapter periodic signals were described by different harmonic oscillations&nbsp; (&raquo;Fourier series&laquo;).&nbsp;
-If one reduces &ndash; at least mentally &ndash; the repetition frequency of a periodic signal more and more,&nbsp; i.e.,&nbsp; the period duration becomes longer and longer,&nbsp; then one comes from the periodic signal to the unique&nbsp; $\text{aperiodic signal}$&nbsp; &ndash; often also called&nbsp; "$\text{pulse}$".
+If one reduces &ndash; at least mentally &ndash; the repetition frequency of a periodic signal more and more,&nbsp; i.e.,&nbsp; the period duration becomes longer and longer,&nbsp; then one comes from the periodic signal to the unique&nbsp; &raquo;aperiodic signal&laquo;&nbsp; &ndash; often also called&nbsp; &raquo;pulse&laquo;.
 In the following,&nbsp; such aperiodic and pulse&ndash;shaped signals are considered and mathematically described in the time and frequency domain.
@@ Line 16: / Line 16: @@
 # The derivation of the two&nbsp; &raquo;Fourier integrals&laquo;&nbsp; from the Fourier series,
 # the extension of the Fourier integral to the&nbsp; &raquo;Fourier transform&laquo;&nbsp; by means of distributions,
-# &raquo;some&nbsp; special cases&laquo;&nbsp; of pulse&ndash;like signals like rectangular and Gaussian pulses and the Dirac delta,
+# &raquo;some&nbsp; special cases&laquo;&nbsp; of pulses:&nbsp; &raquo;rectangular pulse&laquo;&nbsp; and&nbsp; &raquo;Gaussian pulse&laquo;,
-# the&nbsp; &raquo;laws&nbsp; of the Fourier transform&laquo;,&nbsp; and finally
+# the&nbsp; &raquo;laws&nbsp; of Fourier transform&laquo;,&nbsp; and finally
 # the meaning of the&nbsp; &raquo;convolution operation&laquo;&nbsp; and its various applications.
-Laplace transform&nbsp; and&nbsp; Hilbert transform,&nbsp; which are only applicable to causal signals or systems,&nbsp; will be treated in the next book&nbsp; "Linear Time-invariant Systems".
+Laplace transform&nbsp; and&nbsp; Hilbert transform,&nbsp; which are only applicable to causal signals or systems,&nbsp; will be treated in the second book&nbsp; &raquo;Linear Time-invariant Systems&laquo;.