Signal Representation/Signal classification - Revision history

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← Older revision		Revision as of 12:46, 22 June 2023
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	*We refer  $T_{\rm A}$  as  »sampling time interval«  and its reciprocal  $f_{\rm A} = 1/T_{\rm A}$  as  »sampling frequency«.		*We refer  $T_{\rm A}$  as  »sampling time interval«  and its reciprocal  $f_{\rm A} = 1/T_{\rm A}$  as  »sampling frequency«.

−	*Such a signal may be created by sampling a»'''continuous-time signal'''«. }}	+	*Such a signal may be created by sampling a »'''continuous-time signal'''«. }}

← Older revision		Revision as of 12:45, 22 June 2023
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	*We refer  $T_{\rm A}$  as  »sampling time interval«  and its reciprocal  $f_{\rm A} = 1/T_{\rm A}$  as  »sampling frequency«.		*We refer  $T_{\rm A}$  as  »sampling time interval«  and its reciprocal  $f_{\rm A} = 1/T_{\rm A}$  as  »sampling frequency«.

−	*Such a signal may be created by sampling a continuous-time signal. }}	+	*Such a signal may be created by sampling a»'''continuous-time signal'''«. }}

← Older revision		Revision as of 14:08, 7 June 2023
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	*With pulse-shaped signals like the signal  $x_1(t)$  in the above example,  the energy is always limited.  Mostly, the signal values here are different from zero only for a finite time period.   In other words:  Such signals are often time-limited,  too.		*With pulse-shaped signals like the signal  $x_1(t)$  in the above example,  the energy is always limited.  Mostly, the signal values here are different from zero only for a finite time period.   In other words:  Such signals are often time-limited,  too.

−	*But even signals that are unlimited in time can have a finite energy.  In~~ [[Signal_Representation/Fourier_Transform_and_Its_Inverse\|~~ later chapters]] you will find more information about energy–limited and therefore aperiodic signals, for example the   [[Signal_Representation/Special_Cases_of_Pulses#Gaussian_pulse\|»Gaussian pulse«]]  and the  [[Aufgaben:Exercise_3.1:_Spectrum_of_the_Exponential_Pulse\|»exponential pulse«]].	+	*But even signals that are unlimited in time can have a finite energy.  In later chapters you will find more information about energy–limited and therefore aperiodic signals, for example the   [[Signal_Representation/Special_Cases_of_Pulses#Gaussian_pulse\|»Gaussian pulse«]]  and the  [[Aufgaben:Exercise_3.1:_Spectrum_of_the_Exponential_Pulse\|»exponential pulse«]].

← Older revision		Revision as of 14:05, 7 June 2023
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−	Since the time function  $x(t)$  for all times  $t$  is known and can be specified unambiguously,  there always exists a spectral function  $X(f)$  which can be calculated using the  [[Signal_Representation/Fourier_Series#Fourierreihe\|»$\text{Fourier series}$«]]  or the  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse#Das_erste_Fourierintegral~~\|»$\text{Fourier transform}$«]] .	+	Since the time function  $x(t)$  for all times  $t$  is known and can be specified unambiguously,  there always exists a spectral function  $X(f)$  which can be calculated using the  [[Signal_Representation/Fourier_Series#Fourierreihe\|»$\text{Fourier series}$«]]  or the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse\|»$\text{Fourier transform}$«]] .

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=

@@ Line 176: / Line 176: @@
-[[File:P_ID831_Sig_T_1_2_S4_b_neu.png|left|frame|FSK-Signal &ndash; continuous&ndash;in&ndash;value and  binary&ndash;in&ndash;frequency]]
+[[File:P_ID831_Sig_T_1_2_S4_b_neu.png|left|frame|FSK signal &nbsp; &rArr; &nbsp; continuous&ndash;valued,&nbsp;  binary&ndash;in&ndash;frequency]]
@@ Line 192: / Line 192: @@
 ==Continuous-time and discrete-time signals==
 <br>
-For the signals considered so far, the signal parameter was defined at any given time.&nbsp; Such a signal is called&nbsp; "continuous in time".
+For the signals considered so far,&nbsp; the signal parameter was defined at any given time.&nbsp; Such a signal is called&nbsp; "continuous in time".
 {{BlaueBox|TEXT=
-$\text{Definitions:}$&nbsp;
+$\text{Definition:}$&nbsp;
-*With a&nbsp; &raquo;'''discrete-time signal'''&laquo;&nbsp; on the contrary, the signal parameter is defined only at the discrete points&nbsp; $t_\nu$.&nbsp;
-*These points in time are usually chosen equidistant: &nbsp;
+With a&nbsp; &raquo;'''discrete-time signal'''&laquo;&nbsp; on the contrary,&nbsp; the signal parameter is defined only at the discrete points&nbsp; $t_\nu$.&nbsp; These time points are usually chosen equidistant: &nbsp;
 :$$t_\nu = \nu \cdot T_{\rm A}.$$
-*We refer&nbsp; $T_{\rm A}$&nbsp; as&nbsp; "sampling time interval"&nbsp; and its reciprocal&nbsp; $f_{\rm A} = 1/T_{\rm A}$&nbsp; as&nbsp; "sampling frequency",&nbsp; because such a signal may be created by sampling a continuous-time signal. }}
+*We refer&nbsp; $T_{\rm A}$&nbsp; as&nbsp; &raquo;sampling time interval&laquo;&nbsp; and its reciprocal&nbsp; $f_{\rm A} = 1/T_{\rm A}$&nbsp; as&nbsp; &raquo;sampling frequency&laquo;.&nbsp;
-[[File:P_ID355_Sig_T_1_2_S5_neu.png|right|frame|Continuous-time and discrete-time signal]]
+*The discrete-time signal&nbsp; $x_{\rm A}(t)$&nbsp; is obtained after sampling the continuous-time and continuous-value signal&nbsp; $x(t)$&nbsp; with a uniform sampling period &nbsp; $(T_{\rm A})$.<br><br>
-{{GraueBox|TEXT=
+*The time plot&nbsp; $x_{\rm R}(t)$&nbsp; outlined below differs from the real discrete-time representation&nbsp; $x_{\rm A}(t)$&nbsp; in that the infinitely narrow samples&nbsp; $($mathematically describable with Dirac deltas$)$&nbsp; are replaced by rectangular pulses of duration&nbsp; $T_{\rm A}$.<br><br>
-$\text{Example 5:}$&nbsp;
+*Such a signal can also be called&nbsp; "discrete-time"&nbsp; according to the above definition.<br><br>
-*The time plot&nbsp; $x_{\rm R}(t)$&nbsp; outlined below differs from the real discrete-time representation&nbsp; $x_{\rm A}(t)$&nbsp; in that the infinitely narrow samples&nbsp; (mathematically describable with Dirac deltas)&nbsp; are replaced by rectangular pulses of duration&nbsp; $T_{\rm A}$&nbsp;.
+*Furthermore applies:
-*Such a signal can also be called&nbsp; "discrete-time"&nbsp; according to the above definition.
+#A discrete-time signal&nbsp;$x(t)$&nbsp; is completely determined by its series &nbsp;$\left \langle x_\nu \right \rangle$&nbsp; of sampled values.&nbsp;
-<br><br><br><br><br><br><br><br><br>
+#These sampled values can either be continuous or discrete.
-Furthermore applies:
+#The mathematical description of discrete-time signals is given in the chapter&nbsp;<br> [[Signal_Representation/Time_Discrete_Signal_Representation|&raquo;Discrete-Time Signal Representation&laquo;]].}}
 <br clear=all>
 ==Analog and digital signals==
@@ Line 233: / Line 233: @@
 <br clear=all>}}
-<br>In addition, the following specifications apply:
+<br>In addition,&nbsp; the following specifications apply:
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-*If a signal is both continuous in value &nbsp;<u>and</u>&nbsp; continuous in time, it is called an&nbsp; &raquo;'''analog signal'''&laquo;.&nbsp; Such signals represent a continuous process.
+If a signal is both continuous in value &nbsp; <u>and</u> &nbsp; continuous in time,&nbsp; it is called an&nbsp; &raquo;'''analog signal'''&laquo;.&nbsp;
 *Examples are speech signals, music signals and image signals.}}
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-*A&nbsp; &raquo;'''digital signal'''&laquo;,&nbsp; on the other hand,&nbsp; is discrete in value &nbsp; <u>and</u> &nbsp; discrete in time, and the message contained therein consists of symbols from a symbol set.
+A&nbsp; &raquo;'''digital signal'''&laquo;&nbsp; is discrete in value &nbsp; <u>and</u> &nbsp; discrete in time,&nbsp; and the message contained therein consists of symbols from a symbol set.
-*For example, it can be a sampled and quantized (as well as coded in any form) voice, music or image signal,
-*but also a data signal when a file is downloaded from a server on the Internet.}}
+*For example,&nbsp; it can be a voice signal,&nbsp; music signal or image signal after sampling,&nbsp; quantization,&nbsp; and encoding in any form.
-Depending on the number of levels, digital signals are also known by other names, for example
+Depending on the number of levels,&nbsp; digital signals are also known by other names,&nbsp; for example
 * with $M = 2$:  &nbsp; binary digital signal or&nbsp; &raquo;'''binary signal'''&laquo;,
@@ Line 256: / Line 261: @@
-The following  (German-language) learning video summarizes the classification features discussed in this chapter in a compact way:<br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;[[Analoge_und_digitale_Signale_(Lernvideo)|"Analoge und digitale Signale"]] &nbsp; &rArr; &nbsp; Analog and Digital Signals.
+The following&nbsp;  $($German-language$)$&nbsp; learning video summarizes the classification features discussed in this chapter in a compact way:<br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;[[Analoge_und_digitale_Signale_(Lernvideo)|&raquo;Analoge und digitale Signale&laquo;]] &nbsp; &rArr; &nbsp; "Analog and Digital Signals".

@@ Line 150: / Line 150: @@
 *All power&ndash;limited signals are also infinitely extended in time.&nbsp; Examples are the&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal|&raquo;DC signal&raquo;]]&nbsp; and&nbsp; [[Signal_Representation/Harmonic_Oscillation|&raquo;harmonic oscillations&laquo;]]&nbsp; such as the cosine signal&nbsp; $x_2(t)$&nbsp; in&nbsp; $\text{Example 3}$,&nbsp; which are described in detail in chapter&nbsp; [[Signal_Representation/General_Description|&raquo;Periodic Signals&laquo;]].
-*Even most of the stochastic signals are power&ndash;limited - see the book&nbsp; [[Theory_of_Stochastic_Signals|&raquo;Theory of Stochastic Signals&laquo;]].
+*Even most of the stochastic signals are power&ndash;limited &nbsp; &rArr; &nbsp;  see the book&nbsp; [[Theory_of_Stochastic_Signals|&raquo;Theory of Stochastic Signals&laquo;]].
@@ Line 157: / Line 157: @@
 {{BlaueBox|TEXT=
 $\text{Definitions:}$&nbsp;
-*A signal is &nbsp; &raquo;'''continuous in value'''&laquo; or&nbsp; &raquo;'''continuous-valued'''&laquo;, if the decisive signal parameter - for example the instantaneous value - can take all values of a continuum&nbsp; (for example of an interval)&nbsp;.
+*A signal is&nbsp; &raquo;'''continuous in value'''&laquo; or&nbsp; &raquo;'''continuous-valued'''&laquo;,&nbsp; if the decisive signal parameter &ndash; for example the instantaneous value &ndash; can take all values of a continuum&nbsp; $($e.g. of an interval$)$.
-*In contrast, if only countable many different values are possible for the signal parameter, then the signal is&nbsp; &raquo;'''discrete in value'''&laquo; or&nbsp; &raquo;'''discrete-valued'''&laquo;. The number&nbsp; $M$&nbsp; of possible values is called the&nbsp; "level number"&nbsp; or the&nbsp;"symbol set size".}}
+*In contrast,&nbsp; if only countable many different values are possible for the signal parameter,&nbsp; then the signal is&nbsp; &raquo;'''discrete in value'''&laquo; or&nbsp; &raquo;'''discrete-valued'''&laquo;.&nbsp; The number&nbsp; $M$&nbsp; of possible values is called the&nbsp; &raquo;level number&laquo;&nbsp; or the&nbsp; &raquo;symbol set size&laquo;.}}
 *Analog transmission systems always work with continuous-valued signals.
 *For digital systems, on the other hand, most but not all signals are discrete-valued.
 {{GraueBox|TEXT=
 $\text{Example 4:}$&nbsp;
 The upper diagram shows in blue a section of a continuous-valued signal&nbsp; $x(t)$, which can take values between&nbsp; $\pm 8\ \rm V$&nbsp;.
-*In red you can see the signal&nbsp; $x_{\rm Q}(t)$&nbsp; discretized on &nbsp; $M = 8$&nbsp; quantization levels  with the possible signal values&nbsp; $\pm 1\ \rm V$,&nbsp; $\pm 3\ \rm V$,&nbsp; $\pm 5\ \rm V$&nbsp; and&nbsp; $\pm 7\ \rm V$.
+[[File:P_ID358_Sig_T_1_2_S4_a_neu.png|right|frame|Continuous-valued and discrete-valued signal]]
-*For this signal&nbsp; $x_{\rm Q}(t)$&nbsp; the instantaneous value was considered the decisive signal parameter.
+*In red you can see the signal&nbsp; $x_{\rm Q}(t)$&nbsp; discretized on &nbsp; $M = 8$&nbsp; quantization levels  with the possible signal values&nbsp; $\pm 1\ \rm V$,&nbsp; $\pm 3\ \rm V$,&nbsp; $\pm 5\ \rm V$&nbsp; and&nbsp; $\pm 7\ \rm V$.
-−
@@ Line 182: / Line 183: @@
-Therefore the signal&nbsp; $s_{\rm FSK}(t)$&nbsp; shown below is also called discrete-valued with level number&nbsp; $M = 2$&nbsp; and  possible frequencies&nbsp; $1 \ \ \rm kHz$&nbsp; and&nbsp; $5 \ \ \rm kHz$, although the instantaneous value is continuous.}}
+Therefore the signal&nbsp; $s_{\rm FSK}(t)$&nbsp; shown below is also called&nbsp; <u>discrete-valued</u>&nbsp; with level number&nbsp; $M = 2$&nbsp; and  possible frequencies&nbsp; $1 \ \ \rm kHz$&nbsp; and&nbsp; $5 \ \ \rm kHz$, although the instantaneous value is continuous.}}

@@ Line 38: / Line 38: @@
 *Test signals for the design of communication systems,
 *carrier signals for frequency multiplex systems,&nbsp; and
-*a&nbsp;  "Dirac delta train"&nbsp; for sampling an analog signal or for time regeneration of a digital signal.
+*a&nbsp;  &raquo;Dirac delta comb&laquo;&nbsp; for sampling an analog signal or for time regeneration of a digital signal.

@@ Line 11: / Line 11: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-A&nbsp; $\text{deterministic signal}$&nbsp; exists,&nbsp; if its time functions&nbsp; $x(t)$&nbsp; can be described completely in analytical form.
+A&nbsp; &raquo;'''deterministic signal'''&laquo;&nbsp; exists,&nbsp; if its time functions&nbsp; $x(t)$&nbsp; can be described completely in analytical form.
 }}
@@ Line 19: / Line 19: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-One refers to a&nbsp; $\text{stochastic signal}$ &nbsp; or to a&nbsp; $\text{random signal}$,&nbsp; if the signal course&nbsp; $x(t)$&nbsp; is not - or at least not completely &ndash; describable in mathematical form.&nbsp; Such a signal cannot be predicted exactly for the future.}}
+One refers to a&nbsp; &raquo;'''stochastic signal'''&laquo;&nbsp; or to a&nbsp; &raquo;'''random signal}'''&laquo;,&nbsp; if the signal course&nbsp; $x(t)$&nbsp; is not - or at least not completely &ndash; describable in mathematical form.&nbsp; Such a signal cannot be predicted exactly for the future.}}
@@ Line 49: / Line 49: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-*A signal&nbsp; $x(t)$&nbsp; is called&nbsp; $\text{causal}$,&nbsp; if it does not exist for all times&nbsp; $t < 0$&nbsp; or is identical zero.
+*A signal&nbsp; $x(t)$&nbsp; is called&nbsp; &laquo;'''causal'''&laquo;,&nbsp; if it does not exist for all times&nbsp; $t < 0$&nbsp; or is identical zero.
-*If this condition is not fulfilled, then one speaks of a&nbsp; $\text{non-causal}$&nbsp; signal (or system).}}
+*If this condition is not fulfilled, then one speaks of a&nbsp; &laquo;'''non-causal'''&laquo;&nbsp; signal (or system).}}
@@ Line 81: / Line 81: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The&nbsp; $\text{energy}$&nbsp; of the signal&nbsp; $x(t)$&nbsp; is to calculate as follows:
+$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''energy'''&laquo;&nbsp; of the signal&nbsp; $x(t)$&nbsp; is to calculate as follows:
 :$$E_x=\lim_{T_{\rm M}\to\infty} \int^{T_{\rm M}/2} _{-T_{\rm M}/2} x^2(t)\,{\rm d}t.$$}}
@@ Line 87: / Line 87: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp;To calculate the (mean)&nbsp; $\text{power}$,&nbsp; $T_{\rm M}$&nbsp; must be divided by the time before the boundary crossing:
+$\text{Definition:}$&nbsp;To calculate the (mean)&nbsp; &raquo;'''power'''&laquo;,&nbsp; $T_{\rm M}$&nbsp; must be divided by the time before the boundary crossing:
 :$$P_x = \lim_{T_{\rm M}	\to \infty} \frac{1}{T_{\rm M} } \cdot \int^{T_{\rm M}/2} _{-T_{\rm M}/2} x^2(t)\,{\rm d}t.$$
@@ Line 125: / Line 125: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-A signal&nbsp; $x(t)$&nbsp; with finite energy&nbsp; $E_x$&nbsp; and infinitely small power&nbsp; $(P_x = 0)$&nbsp; is called&nbsp; $\text{energy&ndash;limited}$. }}
+A signal&nbsp; $x(t)$&nbsp; with finite energy&nbsp; $E_x$&nbsp; and infinitely small power&nbsp; $(P_x = 0)$&nbsp; is called&nbsp; &raquo;'''energy&ndash;limited'''&laquo;. }}
 *With pulse-shaped signals like the signal&nbsp; $x_1(t)$&nbsp; in the above example,&nbsp; the energy is always limited.&nbsp; Mostly, the signal values here are different from zero only for a finite period of time.&nbsp; In other words:&nbsp; Such signals are often time-limited, too.
 *But even signals that are unlimited in time can have a finite energy.&nbsp; In later&nbsp;[[Signal_Representation/Fourier_Transform_and_Its_Inverse|"chapters"]]&nbsp;you will find more information about energy&ndash;limited and therefore aperiodic signals, for example the &nbsp;&nbsp;[[Signal_Representation/Special_Cases_of_Pulses#Gaussian_pulse|$\text{Gaussian pulse}$]]&nbsp; and the&nbsp; [[Aufgaben:Exercise_3.1:_Spectrum_of_the_Exponential_Pulse|$\text{exponential pulse}$]].
@@ Line 132: / Line 132: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-A signal&nbsp; $x(t)$&nbsp; with finite power&nbsp; $P_x$&nbsp; and accordingly infinite energy&nbsp; $(E_x \to \infty)$&nbsp; is called&nbsp; $\text{power&ndash;limited}$.}}
+A signal&nbsp; $x(t)$&nbsp; with finite power&nbsp; $P_x$&nbsp; and accordingly infinite energy&nbsp; $(E_x \to \infty)$&nbsp; is called&nbsp; &raquo;'''power&ndash;limited'''&laquo;.}}
 *All power limited signals are also infinitely extended in time.&nbsp; Examples are the&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal|$\text{DC signal}$]]&nbsp; and&nbsp; [[Signal_Representation/Harmonic_Oscillation|$\text{harmonic oscillations}$]]&nbsp; such as the cosine signal&nbsp; $x_2(t)$&nbsp; in&nbsp; $\text{Example 3}$, which are described in detail in the chapter&nbsp; [[Signal_Representation/General_Description|"Periodic Signals"]].
@@ Line 142: / Line 142: @@
 {{BlaueBox|TEXT=
 $\text{Definitions:}$&nbsp;
-*A signal is &nbsp; $\text{continuous in value}$ or &nbsp; $\text{continuous-valued}$, if the decisive signal parameter - for example the instantaneous value - can take all values of a continuum&nbsp; (for example of an interval)&nbsp;.
+*A signal is &nbsp; &raquo;'''continuous in value'''&laquo; or&nbsp; &raquo;'''continuous-valued'''&laquo;, if the decisive signal parameter - for example the instantaneous value - can take all values of a continuum&nbsp; (for example of an interval)&nbsp;.
-*In contrast, if only countable many different values are possible for the signal parameter, then the signal is&nbsp; $\text{discrete in value}$ or &nbsp; $\text{discrete-valued}$. The number&nbsp; $M$&nbsp; of possible values is called the&nbsp; "level number"&nbsp; or the&nbsp;"symbol set size".}}
@@ Line 179: / Line 180: @@
 {{BlaueBox|TEXT=
 $\text{Definitions:}$&nbsp;
-*With a&nbsp; $\text{discrete-time signal}$&nbsp; on the contrary, the signal parameter is defined only at the discrete points&nbsp; $t_\nu$.&nbsp;
+*With a&nbsp; &raquo;'''discrete-time signal'''&laquo;&nbsp; on the contrary, the signal parameter is defined only at the discrete points&nbsp; $t_\nu$.&nbsp;
 *These points in time are usually chosen equidistant: &nbsp;
 :$$t_\nu = \nu \cdot T_{\rm A}.$$
@@ Line 220: / Line 221: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-*If a signal is both continuous in value &nbsp;<u>and</u>&nbsp; continuous in time, it is called an&nbsp; $\text{analog signal}$.&nbsp; Such signals represent a continuous process.
+*If a signal is both continuous in value &nbsp;<u>and</u>&nbsp; continuous in time, it is called an&nbsp; &raquo;'''analog signal'''&laquo;.&nbsp; Such signals represent a continuous process.
 *Examples are speech signals, music signals and image signals.}}
@@ Line 226: / Line 227: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-*A&nbsp; $\text{digital signal}$,&nbsp; on the other hand,&nbsp; is discrete in value &nbsp; <u>and</u> &nbsp; discrete in time, and the message contained therein consists of symbols from a symbol set.
+*A&nbsp; &raquo;'''digital signal'''&laquo;,&nbsp; on the other hand,&nbsp; is discrete in value &nbsp; <u>and</u> &nbsp; discrete in time, and the message contained therein consists of symbols from a symbol set.
 *For example, it can be a sampled and quantized (as well as coded in any form) voice, music or image signal,
 *but also a data signal when a file is downloaded from a server on the Internet.}}
@@ Line 232: / Line 233: @@
 Depending on the number of levels, digital signals are also known by other names, for example
-* with $M = 2$:  &nbsp; binary digital signal or&nbsp; $\text{binary signal}$,
+* with $M = 2$:  &nbsp; binary digital signal or&nbsp; &raquo;'''binary signal'''&laquo;,
-* with $M = 3$: &nbsp; ternary digital signal or &nbsp; $\text{ternary signal}$,
-* with $M = 4$: &nbsp; quaternary digital signal or&nbsp; $\text{quaternary signal}$.
+* with $M = 3$: &nbsp; ternary digital signal or &nbsp; &raquo;'''ternary signal'''&laquo;,