Difference between revisions of "Modulation Methods/Linear Digital Modulation"

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{{Header
 
{{Header
|Untermenü=Digitale Modulationsverfahren
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|Untermenü=Digital Modulation Methods
|Vorherige Seite=Pulscodemodulation
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|Vorherige Seite=Pulse Code Modulation
|Nächste Seite=Quadratur–Amplitudenmodulation
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|Nächste Seite=Quadrature Amplitude Modulation
 
}}
 
}}
==Unterschiede zwischen analogen und digitalen Modulationsverfahren==
+
==Differences between analog and digital modulation methods==
Die Grafik zeigt oben ein analoges Übertragungssystem und darunter gezeichnet ein Digitalsystem.  
+
<br>
 +
The diagram shows an analog transmission system at the top and a digital system drawn below.&nbsp; The main differences are highlighted in red: 
 +
[[File:EN_Mod_T_4_2_S1.png |right|frame|Analog and digital transmission system]]
  
 +
While in the upper system the analog source signal &nbsp;$q(t)$&nbsp; is applied to the modulator input,&nbsp; in the lower digital system the modulating signal &nbsp;$q_{\rm D}(t)$&nbsp; is a digital signal,&nbsp; characterized by the amplitude coefficients &nbsp;$a_ν$,&nbsp; the basic pulse &nbsp;$g_q(t)$&nbsp; and the symbol duration &nbsp;$T$:
 +
:$$q_{\rm D}(t) = \sum_{\nu=-\infty}^{+\infty}a_\nu \cdot g_q(t - \nu \cdot T) \hspace{0.05cm}.$$
  
[[File: P_ID1687__Mod_T_4_2_S1_ganz_neu.png | Analoges und digitales Übertragungssystem]]
+
*The A/D conversion can be done for example by&nbsp; [[Modulation_Methods/Pulse_Code_Modulation|$\text{pulse code modulation}$]]&nbsp; and includes the functions sampling,&nbsp; quantization,&nbsp; binary coding and signal shaping.&nbsp;
 +
*The basic pulse &nbsp;$g_q(t)$&nbsp; is often assumed to be NRZ-rectangular with amplitude &nbsp;$s_0$&nbsp; and duration &nbsp;$T$.&nbsp; For the spectral function holds with &nbsp;${\rm sinc}(x) = \sin(\pi x)/(\pi x)$:
 +
:$$G_q(f) = s_0 · T · {\rm sinc}(f T).$$
  
 +
*The modulators can be the same for both systems.&nbsp; They change one of the three signal parameters of the carrier&nbsp; $z(t)$&nbsp; according to the modulator input.&nbsp; The digital variants of AM, PM, FM  are called
 +
#"Amplitude Shift Keying"&nbsp; $\rm (ASK)$,
 +
#"Phase Shift Keying"&nbsp; $\rm (PSK)$,&nbsp;
 +
#"Frequency Shift Keying"&nbsp; $\rm (FSK)$.
  
Die wesentlichen Unterschiede sind rot hervorgehoben:
+
*In contrast,&nbsp; the demodulator of the digital system differs fundamentally from an analog demodulator by the required decision component&nbsp; (in hardware or software).&nbsp; Like &nbsp;$q_{\rm D}(t)$,&nbsp; the demodulator  output signal &nbsp;$v_{\rm D}(t)$&nbsp; is digital and must then be D/A converted into the analog sink signal &nbsp;$v(t)$.&nbsp;
*Während beim oberen System am Modulatoreingang das analoge Quellensignal $q(t)$ anliegt, ist beim unteren Digitalsystem das modulierende Signal $q_{\rm D}(t)$ ein Digitalsignal, gekennzeichnet durch die Amplitudenkoeffizienten $a_ν$, den Grundimpuls $g_q(t)$ sowie die Symboldauer $T$:
+
 
$$q_{\rm D}(t) = \sum_{\nu=-\infty}^{+\infty}a_\nu \cdot g_q(t - \nu \cdot T) \hspace{0.05cm}.$$
+
*The decisive quality criterion for both systems is the&nbsp; [[Modulation_Methods/Quality_Criteria#Signal.E2.80.93to.E2.80.93noise_.28power.29_ratio|$\text{sink SNR}$]] &nbsp; &rArr; &nbsp; quotient of the power of source signal $q(t)$ and error signal &nbsp;$ε(t) = v(t) \ – \ q(t)$.&nbsp; In a digital system,&nbsp; one usually makes do with the quality characteristic&nbsp; [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_rate|$\text{bit error rate}$]]&nbsp; $($ $\rm BER)$,&nbsp; which refers to the digital signals &nbsp;$q_{\rm D}(t)$&nbsp; and &nbsp;$v_{\rm D}(t)$.&nbsp; This can be converted into a&nbsp; $\rm  SNR$.&nbsp; 
*Die A/D–Wandlung kann z. B. mittels Pulscodemodulation erfolgen und umfasst die Funktionen Abtastung, Quantisierung, Binärcodierung und Signalformung. Der Grundimpuls $g_q(t)$ wird im Folgenden meist als NRZ–rechteckförmig mit Amplitude $s_0$ und Dauer $T$ angenommen, so dass für die Spektralfunktion $G_q(f) = s_0 · T · {\rm si}(π f T)$ mit ${\rm si}(x) = \sin(x)/x$ gilt.  
 
*Die Modulatoren können bei beiden Systemen durchaus gleich sein. Sie verändern einen der drei Signalparameter des Trägersignals $z(t)$ entsprechend dem Modulatoreingangssignal. Die digitalen Varianten von AM, PM und FM heißen ''Amplitude Shift Keying'' (ASK), ''Phase Shift Keying'' (PSK) und ''Frequency Shift Keying'' (FSK).  
 
*Dagegen unterscheidet sich der Demodulator des Digitalsystems grundsätzlich von einem analogen Demodulator durch die erforderliche Entscheiderkomponente (in Hardware oder Software). Das Ausgangssignal $υ_{\rm D}(t)$ ist ebenso wie $q_{\rm D}(t)$ digital. Dieses Signal muss anschließend noch in das analoge Sinkensignal $υ(t)$ D/A–gewandelt werden.
 
*Das entscheidende Gütekriterium ist bei beiden Systemen das Sinken–SNR als der Quotient der Leistungen von Quellensignal $q(t)$ und Fehlersignal $ε(t) = υ(t) \ – \ q(t).$ Bei einem Digitalsystem begnügt man sich meist mit dem Qualitätsmerkmal Bitfehlerquote (engl.: ''Bit Error Rate,'' BER), das sich auf die beiden Digitalsignale $q_{\rm D}(t)$ und $υ_{\rm D}(t)$ bezieht. Diese ist in ein SNR umrechenbar.  
 
  
 
==ASK – Amplitude Shift Keying==
 
==ASK – Amplitude Shift Keying==
Die Grafik zeigt das digitale Quellensignal $q(t)$ – auf den Index „D” wird ab sofort verzichtet sowie das ASK–Sendesignal $s_{\rm ASK}(t) = q(t) · \sin(2π · f_{\rm T} · t)$, wobei hier von unipolaren Amplitudenkoeffizienten $a_ν ∈$ {0, 1} und einem sinusförmigen Träger ausgegangen wird. Dieses Verfahren wird insbesondere bei optischen Übertragungssystemen eingesetzt (da es bekanntlich keine negativen Lichtimpulse gibt) und ist auch unter der Bezeichnung On–Off–Keying bekannt.
+
<br>
 +
The diagram shows the digital source signal &nbsp;$q(t)$&nbsp; the index "D" is omitted from now on &nbsp; and the ASK transmitted signal
 +
[[File:P_ID1688__Mod_T_4_2_S2_neu.png|right|frame|Signals and power-spectral density&nbsp; $\rm (PSD)$&nbsp; at ASK]]
 +
 +
:$$s_{\rm ASK}(t) = q(t) · \sin(2π · f_{\rm T} · t),$$
 +
where unipolar amplitude coefficients &nbsp;$a_ν ∈ \{0, 1\}$&nbsp; and a sinusoidal carrier&nbsp; (German:&nbsp; "Träger" &nbsp; &rArr; &nbsp; subscript "T")&nbsp; with frequency&nbsp; $f_{\rm T}$&nbsp; are assumed here.&nbsp;
 +
 
 +
For example,&nbsp; this method is used in optical transmission systems (since there are no negative light pulses) and is also known as&nbsp; "On–Off–Keying".
 +
 
 +
In the right half of the figure,&nbsp; the corresponding power-spectral densities are shown - although not to scale.&nbsp;
 +
 
 +
With the rectangular basic pulse &nbsp;$g_q(t)$&nbsp; and equally probable unipolar amplitude coefficients holds:
 +
:$$\begin{align*}{{\it \Phi}_{q}(f)}& = \frac{{s_0}^2 \cdot T}{4} \cdot {\rm sinc}^2 (f T) + \frac{{s_0}^2 }{4} \cdot \delta (f)\hspace{0.05cm},\\ {{\it \Phi}_{s}(f)}& = \frac{1}{4} \cdot \big [ {{\it \Phi}_{q}(f- f_{\rm T})}+  {{\it \Phi}_{q}(f+ f_{\rm T})}\big]\hspace{0.05cm}.\end{align*}$$
 +
 
 +
It should be noted about these equations:
 +
#The DC component &nbsp;$m_q = s_0/2$&nbsp; of the source signal leads to a Dirac delta function at frequency &nbsp;$f = 0$&nbsp; with weight &nbsp;${s_0}^2/4$&nbsp; in the power-spectral density &nbsp;$ϕ_q(f)$.
 +
#The power-spectral density of the ASK transmitted signal&nbsp; $s(t)$&nbsp; is equal to &nbsp;$ϕ_s(f) = ϕ_q(f) ∗ ϕ_z(f)$,&nbsp; where the PSD &nbsp;$ϕ_z(f)$&nbsp; of the carrier &nbsp;$z(t)$&nbsp; is composed of two Dirac delta functions at &nbsp;$±f_{\rm T}$&nbsp; with respective weight &nbsp;$1/4$.&nbsp; The sign&nbsp; "$\star$"&nbsp; describes the convolution.&nbsp; The equation is also valid with other carrier phase.
 +
#The power-spectral density &nbsp;$ϕ_s(f)$&nbsp; is identical in shape to &nbsp;$ϕ_q(f)$&nbsp; except for the shift by&nbsp; $±f_{\rm T}$ &nbsp; &rArr; &nbsp;  the ASK belongs to the&nbsp; &raquo;'''linear digital modulation methods'''&laquo;.  
  
 +
==Coherent demodulation of ASK signals==
 +
<br>
 +
The diagram shows the block diagram of an ASK system including the receiver components.&nbsp;
 +
[[File:EN_Mod_T_4_2_S3a_neu.png|right|frame|Block diagram of an ASK system including receiver components]]
 +
*Let the source signal &nbsp;$q(t)$&nbsp; be NRZ-rectangular and unipolar:
 +
:$$a_ν ∈ \{0, \ 1\}.$$
 +
* Let the channel be initially ideal, characterized by &nbsp;
 +
:$$H_{\rm K}(f) = 1,\hspace{0.25cm} n(t) = 0 \hspace{0.5cm}  \Rightarrow  \hspace{0.5cm} r(t) = s(t).$$
  
[[File:P_ID1688__Mod_T_4_2_S2_neu.png | Signale und Leistungsdichtespektren bei ASK]]
+
The demodulation is performed coherently by means of a&nbsp; [[Modulation_Methods/Synchrondemodulation|$\text{synchronous demodulator}$]],&nbsp; the operation of which has already been described for the analog modulation methods AM and PM.
  
 +
To summarize the block diagram:
 +
*At the receiver the same carrier signal is added as at the transmitter,&nbsp; but with double amplitude.&nbsp;
 +
*$z(t)$&nbsp; denotes the carrier at the transmitter &nbsp; &rArr; &nbsp; the carrier at the receiver&nbsp; (German:&nbsp; "Empfänger" &nbsp; &rArr; &nbsp; subscript "E")&nbsp; is&nbsp; $z_{\rm E}(t) = 2 · z(t) $.
 +
*Multiplication is followed by a suitably sized low-pass filter with frequency response &nbsp;$H_{\rm E}(f)$,&nbsp; which removes the higher frequency components of the signal &nbsp;$b(t)$.
 +
*The detection signal &nbsp;$d(t)$&nbsp; is sampled at instants &nbsp;$ν · T$&nbsp; and decided using a threshold decision with threshold &nbsp;$E = {s_0}/2$.&nbsp;
 +
*The sink signal &nbsp;$v(t)$&nbsp; at the decision output is rectangular and in the noise-free case equal to the source signal &nbsp;$q(t)$&nbsp; except for the transit time &nbsp;$T/2$.
  
In der rechten Bildhälfte sind – allerdings nicht maßstäblich – die dazugehörigen Leistungsdichtespektren (abgekürzt: LDS) dargestellt. Bei rechteckförmigem Grundimpuls $g_q(t)$ und gleichwahrscheinlichen unipolaren Amplitudenkoeffizienten gilt:
 
$$\begin{align*}{{\it \Phi}_{q}(f)}& = \frac{{s_0}^2 \cdot T}{4} \cdot {\rm si}^2 (\pi f T) + \frac{{s_0}^2 }{4} \cdot \delta (f)\hspace{0.05cm},\\ {{\it \Phi}_{s}(f)}& = \frac{1}{4} \cdot \left [ {{\it \Phi}_{q}(f- f_{\rm T})}+  {{\it \Phi}_{q}(f+ f_{\rm T})}\right]\hspace{0.05cm}.\end{align*}$$
 
  
Zu diesen Gleichungen ist zu bemerken:  
+
{{BlaueBox|TEXT=
*Der Gleichanteil $m_q = s_0/2$ des Quellensignals führt im Leistungsdichtespektrum $ϕ_q(f)$ zu einer Diracfunktion bei der Frequenz $f = 0$ mit dem Gewicht ${s_0}^2/4$.  
+
$\text{To be noted:}$&nbsp;
*Das Leistungsdichtespektrum des ASK–Sendesignals ist gleich $ϕ_s(f) = ϕ_q(f) ∗ ϕ_z(f)$, wobei sich das LDS $ϕ_z(f)$ des Trägersignals $Z(t)$ aus zwei Diracfunktionen bei $±f_{\rm T}$ mit jeweiligem Gewicht 1/4 zusammensetzt. Die Gleichung gilt auch bei anderer Trägerphase, „∗” beschreibt die Faltung.
+
*&raquo;'''Coherent demodulation'''&laquo;&nbsp; requires that the carrier frequency &nbsp;$f_{\rm T}$&nbsp; and the carrier phase &nbsp;$ϕ_{\rm T}$&nbsp; are exactly known to the receiver.
*Das Leistungsdichtespektrum $ϕ_s(f)$ ist bis auf die Verschiebung um $±f_{\rm T}$ formgleich mit $ϕ_q(f)$. Deshalb gehört ASK zu den linearen digitalen Modulationsverfahren.  
+
*The receiver&nbsp; '''must extract these two quantities from the received signal &nbsp;$r(t)$''',&nbsp; which can be quite costly in the presence of strong channel distortion and large noise&nbsp; $n(t)$. Such realization aspects are dealt with, for example, in [[Aufgaben:Exercise_4.9:_Costas_Rule_Loop|"Exercise 4.9"]] to this chapter.
 +
*If the carrier phase &nbsp;$ϕ_{\rm T}$&nbsp; is not known to the receiver,&nbsp; this is called&nbsp; &raquo;'''incoherent demodulation'''&laquo;,&nbsp; even if the carrier frequency &nbsp;$f_{\rm T}$&nbsp; is known.}} 
  
==Kohärente Demodulation von ASK–Signalen (1)==
 
Die Grafik zeigt das Blockschaltbild eines ASK–Systems inklusive der Empfängerkomponenten. Das Quellensignal $q(t)$ sei NRZ–rechteckförmig und unipolar, das heißt, es gilt $a_ν ∈$ {0, 1}. Der Kanal sei zunächst ideal, gekennzeichnet durch $H_{\rm K}(f) =$ 1 und $n(t) =$ 0  ⇒  $r(t) = s(t)$.
 
  
 +
{{GraueBox|TEXT=
 +
$\text{Example 1:}$&nbsp; The diagram shows the signals mentioned in the ASK block diagram with ideal channel: &nbsp; $H_{\rm K}(f) = 1, \ \ n(t) = 0.$
 +
[[File:EN_Mod_T_4_2_S3b.png |right|frame| Signals with ASK modulation and coherent demodulation]]
 +
 +
The individual signal characteristics can be interpreted as follows:
 +
*The transmitted signal &nbsp;$s(t)$&nbsp; is the product of the unipolar source signal &nbsp;$q(t)$&nbsp; and the carrier signal&nbsp; $z(t) = \sin(2π\hspace{0.05cm}f_{\rm T}\hspace{0.05cm}t)$,&nbsp; where in the example &nbsp;$f_{\rm T} = 4/T$&nbsp; holds&nbsp; (only four oscillations per symbol duration).
 +
*The received signal &nbsp;$r(t) = s(t)$&nbsp; is first multiplied by the receiver-side carrier &nbsp;$z_{\rm E}(t) = 2 · \sin(2π\hspace{0.05cm}f_{\rm T}\hspace{0.05cm}t)$  &nbsp; ⇒ &nbsp;  twice the amplitude compared to &nbsp;$z(t)$,&nbsp; no frequency and phase offset.&nbsp; This gives:
 +
:$$b(t) = 2 \cdot z(t)\cdot r(t)= 2 \cdot z^2(t)\cdot q(t) $$
 +
:$$\Rightarrow \hspace{0.35cm}b(t) =  q(t) \cdot \big [ 1 - \cos(4\pi\hspace{0.05cm} f_{\rm T}\hspace{0.05cm} t)\big]
 +
\hspace{0.05cm}.$$
 +
*The low-pass filter with frequency response &nbsp;$H_{\rm E}(f) = {\rm sinc}(f_{\rm T}\hspace{0.05cm} T)$&nbsp; and corresponding rectangular impulse response &nbsp;$h_{\rm E}(t)$&nbsp; forms from the signal&nbsp; $b(t)$&nbsp; the detection signal &nbsp;$d(t) = b(t) \star h_{\rm E}(t)$&nbsp; .
 +
* $h_{\rm E}(t)$&nbsp; is matched to the rectangular basic pulse &nbsp;$g_q(t)$.&nbsp; This is called the &nbsp; [[Theory_of_Stochastic_Signals/Matched_Filter|$\text{matched filter configuration}$]]  &nbsp;  ⇒ &nbsp; best possible compromise between equalization and noise power limitation.
 +
*In the absence of noise &nbsp; &rArr; &nbsp; $d(νT) = q(νT) ∈ \{0, \ s_0\}$. &nbsp; In the presence of&nbsp; (moderate)&nbsp; noise,&nbsp; it is very likely that &nbsp;$d(νT) > s_0/2,$ if &nbsp;$a_ν = +1$,&nbsp; and it will be &nbsp;$d(νT) < s_0/2$&nbsp; for &nbsp;$a_ν = 0$.
 +
*The decision obtains from the comparison of the samples &nbsp;$d(νT)$&nbsp; with the threshold &nbsp;$E = s_0/2$&nbsp; the sink signal &nbsp;$v(t)$,&nbsp; which is equal to &nbsp;$q(t)$&nbsp; if the decision is error-free except for the transit time &nbsp;$T/2$.&nbsp;}}
  
[[File:P_ID1689__Mod_T_4_2_S3a_neu.png | Blockschaltbild eines ASK–Systems einschließlich Empfängerkomponente]]
+
==Incoherent demodulation of ASK signals==
 +
<br>
 +
We further assume ASK modulation as well as an ideal transmission channel, i.e.
 +
[[File:EN_Mod_T_4_2_S4a_neu.png|right|frame|Block diagram of an incoherent ASK demodulator]]
 +
#distortion-free,
 +
#attenuation-free and
 +
#noise-free.
  
  
Die Demodulation erfolgt hier kohärent mittels Synchrondemodulator, dessen Funktionsweise bereits bei den analogen Modulationsverfahren AM und PM beschrieben wurde.  
+
Then applies: &nbsp; $r(t) = s(t) = q(t) \cdot \cos(2 \pi \cdot f_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.$
  
Zusammenfassend lässt sich sagen:
+
Further, it is assumed for this section
*Beim Empfänger wird das gleiche Trägersignal zugesetzt wie beim Sender, jedoch mit doppelter Amplitude. $z(t)$ bezeichnet den Träger beim Sender und $2 · z(t)$ den Träger beim Empfänger.
+
*that the receiver knows the carrier frequency &nbsp;$f_{\rm T}$,  
*Nach der Multiplikation folgt ein geeignet dimensionierter Tiefpass mit dem Frequenzgang $H_{\rm E}(f)$, der die höherfrequenten Anteile des Signals $b(t)$ entfernt.
+
*but not the carrier phase &nbsp;$ϕ_{\rm T}$.&nbsp;
*Schließlich wird das Detektionssignal $d(t)$ zu den Detektionszeitpunkten $ν · T$ abgetastet und mit Hilfe eines Schwellenwertentscheiders mit der Entscheiderschwelle $E = {s_0}/2$ entschieden.
 
*Das Sinkensignal $υ(t)$ am Ausgang des Entscheiders ist rechteckförmig und im rauschfreien Fall (oder bei nur kleinen Rauschstörungen) bis auf die Laufzeit $T/2$ gleich dem Quellensignal $q(t)$.
 
  
  
 +
The diagram shows such an&nbsp; &raquo;'''incoherent demodulator'''&laquo; &nbsp; &rArr; &nbsp; the&nbsp; '''result of the demodulator is independent of the carrier phase &nbsp;$ϕ_{\rm T}$''',&nbsp; which the receiver does not know.
  
{{Box}}
+
The operation of this demodulator will be given here only as an outline:&nbsp;
'''Zu beachten ist:''' Eine kohärente Demodulation erfordert, dass dem Empfänger die Trägerfrequenz $f_{\rm T}$ und die Trägerphase $ϕ_{\rm T}$ exakt bekannt sind. Der Empfänger muss diese beiden Größen aus dem Empfangssignal $r(t)$ extrahieren, was bei starken Kanalverzerrungen und großen Rauschstörungen durchaus aufwändig sein kann. Solche Realisierungsaspekte werden zum Beispiel in der Aufgabe A4.8 zu diesem Kapitel behandelt.  
+
*The signals &nbsp;$d_1(t)$&nbsp; and &nbsp;$d_2(t)$&nbsp; after the two &nbsp;[[Theory_of_Stochastic_Signals/Matched_Filter|$\text{matched filters}$]]&nbsp; with respective frequency response &nbsp;$H_{\rm E}(f)$&nbsp; are identical in shape to the detection signal &nbsp;$d(t)$ &nbsp; &rArr; &nbsp; $d_{\rm coh}(t)$&nbsp; according to the &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Coherent_demodulation_of_ASK_signals|$\text{previous block diagram}$]],&nbsp; but generally attenuated with respect to it due to the lack of phase matching:
{{end}}
+
:$$d_1(t) = d_{\rm coh}(t) \cdot \cos( \phi_{\rm T}), \hspace{0.5cm}d_2(t) = -d_{\rm coh}(t) \cdot \sin( \phi_{\rm T}) \hspace{0.05cm}.$$
 +
*If the amplitude coefficient is &nbsp;$a_ν = 0$,&nbsp; then in the noise-free case the two signal values are zero: &nbsp;
 +
:$$d_1(ν · T) = 0,\hspace{0.5cm} d_2(ν · T) = 0.$$
 +
*Otherwise &nbsp;$(a_ν = 1)$&nbsp; applies:
 +
:$$d_1(\nu \cdot T) = s_{\rm 0} \cdot \cos( \phi_{\rm T}),
 +
\hspace{0.5cm}d_2(\nu \cdot T) = -s_{\rm 0} \cdot \sin( \phi_{\rm
 +
T}) \hspace{0.05cm}.$$
 +
*After squaring the two partial signals we get for the sum signal:
 +
:$$d(\nu \cdot T) = \left\{ \begin{array}{c} 0 \\ {s^2_0} \end{array} \right.\quad
 +
\begin{array}{*{1}c} {\rm if}\hspace{0.15cm}a_\nu = 0,
 +
\\  {\rm if}\hspace{0.15cm}a_\nu = 1. \\ \end{array}$$
 +
*By threshold decision – sensibly with the decision threshold &nbsp;$E = {s_0}^2/2$ – the amplitude coefficients &nbsp;$a_ν$&nbsp; can be decided.&nbsp; <br>However in the noisy case,&nbsp; this results in a somewhat larger &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Error_probabilities_-_a_brief_overview|$\text{bit error probability}$]]&nbsp; than with coherent demodulation.
  
==Kohärente Demodulation von ASK–Signalen (2)==
+
==BPSK – Binary Phase Shift Keying==
Die Grafik zeigt die im ASK–Blockschaltbild  genannten Signale bei idealem Kanal: $H_{\rm K}(f) =$ 1, $n(t) =$ 0.
+
<br>
 +
For &nbsp;[[Modulation_Methods/Phasenmodulation_(PM)|$\text{analog phase modulation}$]]&nbsp; $\rm (PM)$,&nbsp; the transmitted signal is: &nbsp;$s_{\rm PM}(t) = s_0 \cdot \cos\big [2 \pi  f_{\rm T}  t + \phi_{\rm T}+
 +
K_{\rm PM} \cdot q(t)\big ]\hspace{0.05cm}.$&nbsp; With
 +
[[File:P_ID1692__Mod_T_4_2_S5_ganz_neu.png |right|frame| Signals and power-spectral densities&nbsp; $\rm (PSD)$&nbsp; for Binary Phase Shift Keying]]
 +
#bipolar source signal &nbsp; ⇒ &nbsp; $a_ν ∈ \{-1, +1\}$,&nbsp;
 +
#carrier phase &nbsp;$ϕ_{\rm T} = π \ (180^\circ)$&nbsp;
 +
#modulator constant &nbsp;$K_{\rm PM} = π/(2s_0)$,&nbsp;
  
  
[[File: P_ID1690__Mod_T_4_2_S3b_neu.png | Signale bei ASK–Modulation und kohärenter Demodulation]]
+
we obtain in the &nbsp;$ν$–th time interval:  
 +
:$$s_{\rm BPSK}(t) = \left\{ \begin{array}{c} s_0 \cdot \cos(2 \pi  f_{\rm T}  t +
 +
\pi+ \pi/2) \\ s_0 \cdot \cos(2 \pi  f_{\rm T}  t + \pi- \pi/2) \end{array} \right.\quad \begin{array}{*{1}c} {\rm if}\hspace{0.15cm}a_\nu = +1, \\ {\rm if}\hspace{0.15cm}a_\nu = -1. \\ \end{array}$$
  
 +
This equation for&nbsp; &raquo;'''Binary Phase Shift Keying'''&laquo;&nbsp; $($ $\rm BPSK)$&nbsp; can be transformed as follows:
 +
:$$s_{\rm BPSK}(t) =
 +
a_\nu \cdot s_0 \cdot \sin(2 \pi  f_{\rm T}  t ) $$
 +
:$$\Rightarrow \hspace{0.3cm}s_{\rm BPSK}(t)  = \left\{ \begin{array}{c} s_0 \cdot \sin(2 \pi  f_{\rm T}  t ) \\
 +
-s_0 \cdot \sin(2 \pi  f_{\rm T}  t ) \end{array} \right.\quad \begin{array}{*{1}c} {\rm if}\hspace{0.15cm}a_\nu = +1, \\  {\rm if}\hspace{0.15cm}a_\nu = -1. \\ \end{array}$$
 +
<br clear=all>
 +
In the diagram the signals and the corresponding power-spectral densities are sketched.&nbsp; It can be seen:
 +
*Like the ASK signal, the BPSK signal can be represented as the product of the source signal &nbsp;$q(t)$&nbsp; and the carrier signal &nbsp;$z(t)$.&nbsp; The only difference lies in the bipolar amplitude coefficients &nbsp;$a_ν ∈ \{-1, +1\}$&nbsp; compared to the unipolar coefficients&nbsp; $(0$&nbsp; or&nbsp; $1)$&nbsp; with ASK.
 +
*In contrast to ASK,&nbsp; with BPSK&nbsp; $($as with any form of phase modulation$)$&nbsp; the envelope is constant.&nbsp; The information is transmitted here by the phase jumps within the transmitted signal &nbsp;$s(t)$ &nbsp; &rArr; &nbsp; gray backgrounds in the diagram.
 +
*The power-spectral densities with BPSK differ from those with ASK only by the missing Dirac delta functions&nbsp; $($since now &nbsp;$q(t)$&nbsp; does not contain a DC component$)$&nbsp; and by the factor &nbsp;$4$&nbsp; with respect to the continuous PSD components.
 +
*It further follows that binary phase modulation&nbsp; $\rm (BPSK)$&nbsp; can be counted as a linear modulation method.&nbsp; In general, (analog) phase modulation&nbsp; $\rm (PM)$&nbsp; is,&nbsp;  with a few exceptions,&nbsp; non-linear with respect to the source signal.
 +
*For the diagrams,&nbsp; different carrier phases were chosen for display reasons in the section &nbsp;[[Modulation_Methods/Lineare_digitale_Modulation#ASK_.E2.80.93_Amplitude_Shift_Keying|"ASK"]] &nbsp; &rArr; &nbsp; "sine"&nbsp; and here for&nbsp; "BPSK"&nbsp; &rArr; &nbsp; "minus cosine".&nbsp; However,&nbsp; this arbitrary determination is not a restriction.&nbsp; Both methods work in the same way for other carrier phases.
 +
 +
==Demodulation and detection of BPSK signals==
 +
<br>
 +
Due to the constant envelope of the BPSK signal,&nbsp; the demodulation must always be coherent here.&nbsp; The &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Coherent_demodulation_of_ASK_signals|$\text{same block diagram}$]]&nbsp; can be assumed as for coherent ASK demodulation.
 +
 +
[[File:EN_Mod_T_4_2_S4b.png|right|frame|Signals with BPSK modulation and coherent demodulation]]
 +
{{GraueBox|TEXT=
 +
$\text{Example 2:}$&nbsp; The diagram shows from top to bottom
 +
#the bipolar source signal &nbsp;$q(t)$,
 +
#the received signal &nbsp;$r(t) = s(t)$&nbsp; with ideal channel,
 +
#$b(t)$&nbsp; after multiplication by the receiver-side carrier &nbsp;$z_{\rm E}(t) = 2 \cdot z(t)$,
 +
#the detection signal &nbsp;$d(t)$&nbsp; after&nbsp; "integration"&nbsp; by the matched filter,
 +
#the sink signal &nbsp;$v(t)$.
  
Die einzelnen Signalverläufe können wie folgt interpretiert werden:
 
*Das Sendesignal $s(t)$ ist das Produkt aus dem unipolaren Quellensignal $q(t)$ und dem Trägersignal $z(t) = \sin(2πf_{\rm T}t)$, wobei hier $f_{\rm T} = 4/T$ gilt (nur jeweils vier Schwingungen pro Symboldauer).
 
*Das Empfangssignal $r(t) = s(t)$ wird zunächst mit dem Träger $z_{\rm E}(t) = 2 · \sin(2πf_{\rm T}t)$  ⇒  doppelte Amplitude gegenüber $z(t)$, kein Frequenz– und Phasenversatz – multipliziert. Damit ergibt sich:
 
$$b(t) = 2 \cdot z(t)\cdot r(t)= 2 \cdot z^2(t)\cdot q(t) = q(t) \cdot \left [ 1 - \cos(4\pi f_{\rm T} t)\right]
 
\hspace{0.05cm}.$$
 
*Das TP–Filter mit dem Frequenzgang $H_{\rm E}(f) = {\rm si}(πf_{\rm T}T)$ und dementsprechend rechteckförmiger Impulsantwort $h_{\rm E}(t)$ formt aus dem Signal $b(t)$ das Detektionssignal $d(t) = b(t) \star h_{\rm E}(t)$.
 
* $h_{\rm E}(t)$ ist an den rechteckförmigen Grundimpuls $g_q(t)$ angepasst; man spricht vom sog. Matched–Filter  ⇒  bestmöglicher Kompromiss zwischen Entzerrung und Rauschleistungsbegrenzung.
 
*Ohne Rauschen gilt $d(νT) = q(νT) ∈$ {0, $s_0$}. Bei (moderaten) Rauschstörungen ist mit großer Wahrscheinlichkeit $d(νT) > s_0/2,$ falls $a_ν = +1$, und es wird $d(νT) < s_0/2$ für $a_ν = 0$ gelten.
 
*Der Schwellenwertentscheider gewinnt aus dem Vergleich der Detektionsabtastwerte $d(νT)$ mit der Entscheiderschwelle $E = s_0/2$ das Sinkensignal $υ(t)$, das bei fehlerfreier Entscheidung bis auf die Laufzeit $T/2$ identisch mit $q(t)$ ist.
 
  
==Inkohärente Demodulation von ASK–Signalen==
+
A comparison with the corresponding&nbsp; [[Modulation_Methods/Linear_Digital_Modulation#Coherent_demodulation_of_ASK_signals|$\text{signals}$]]&nbsp; in the coherent demodulation of the ASK shows:
Wir gehen weiter von ASK–Modulation sowie dem idealen (das heißt: verzerrungs–, dämpfungs– und rauschfreien) Übertragungskanal aus, so dass gilt:  
+
*The rectangular signals &nbsp;$q(t)$&nbsp; and &nbsp;$v(t)$&nbsp; are now bipolar.
$$r(t) = s(t) = q(t) \cdot \cos(2 \pi \cdot f_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.$$
 
Weiter wird für diesen Abschnitt  vorausgesetzt, dass dem Empfänger zwar die Trägerfrequenz $f_{\rm T}$, nicht jedoch die Trägerphase $ϕ_{\rm T}$ bekannt ist. Üblich ist, auch diesen Demodulator als inkohärent zu bezeichnen.  
 
  
  
[[File:P_ID1691__Mod_T_4_2_S4a_neu.png | Inkohärenter ASK–Demodulator]]
+
*For the detection signal at BPSK,&nbsp; compared to ASK,&nbsp; the following applies:
 +
:$$d_{\rm BPSK}(t) =2  \cdot d_{\rm ASK}(t)-s_0.$$
  
 +
*In the considered attenuation-free, distortion-free and noise-free case,&nbsp; all detection samples are &nbsp;$d(ν · T) = ±s_0$.&nbsp; Therefore,&nbsp; the decision threshold &nbsp;$E = 0$&nbsp; should be used here.
  
Die Grafik zeigt einen solchen inkohärenten Demodulator, dessen Funktionsweise hier nur stichpunktartig angegeben werden soll. Das Demodulationsergebnisse ist unabhängig von der Trägerphase $ϕ_{\rm T}$, die der Empfänger nicht kennt.
+
*Die Signale $d_1(t)$ und $d_2(t)$ nach den beiden Matched–Filtern mit jeweiligen Frequenzgang $H_{\rm E}(f)$ sind formgleich mit dem Detektionssignal $d_{\rm koh}(t)$ entsprechend der Grafik des letzten Abschnitts, aber gegenüber diesem im Allgemeinen wegen der fehlenden Phasenanpassung gedämpft:
+
*One can see the double distance of the BPSK detection samples&nbsp; (circle marks)&nbsp; from the threshold,&nbsp; which decisively improves &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Error_probabilities_-_a_brief_overview|$\text{error probability}$]].&nbsp; }}
$$d_1(t) = d_{\rm koh}(t) \cdot \cos( \phi_{\rm T}), \hspace{0.2cm}d_2(t) = -d_{\rm koh}(t) \cdot \sin( \phi_{\rm T}) \hspace{0.05cm}.$$
 
*Ist der Amplitudenkoeffizient $a_ν =$ 0, so sind im rauschfreien Fall die beiden Signalwerte $d_1(ν · T)$ und $d_2(ν · T)$ jeweils 0. Andernfalls $(a_ν = 1)$ gilt für den Zeitpunkt $ν · T$:
 
$$d_1(\nu \cdot T) = s_{\rm 0} \cdot \cos( \phi_{\rm T}),
 
\hspace{0.2cm}d_2(\nu \cdot T) = -s_{\rm 0} \cdot \sin( \phi_{\rm
 
T}) \hspace{0.05cm}.$$
 
*Nach Quadrierung der zwei Teilsignale erhält man für das Summensignal:
 
$$d(\nu \cdot T) = \left\{ \begin{array}{c} 0 \\ {s_0}^2 \end{array} \right.\quad
 
\begin{array}{*{1}c} {\rm falls}\hspace{0.15cm}a_\nu = 0,
 
\\  {\rm falls}\hspace{0.15cm}a_\nu = 1. \\ \end{array}$$
 
*Durch die Schwellenwertentscheidung – sinnvollerweise mit der Entscheiderschwelle $E = {s_0}^2/4$ – können die Amplitudenkoeffizienten $a_ν$ entschieden werden. Allerdings ergibt sich eine etwas größere Bitfehlerwahrscheinlichkeit als bei kohärenter Demodulation.
 
  
==BPSK – Binary Phase Shift Keying==
 
Bei analoger Phasenmodulation lautet das Sendesignal entsprechend Kapitel 3.1 allgemein:
 
$$s_{\rm PM}(t) = s_0 \cdot \cos(2 \pi  f_{\rm T}  t + \phi_{\rm T}+
 
K_{\rm PM} \cdot q(t))\hspace{0.05cm}.$$
 
Bei bipolarem Quellensignal ⇒ $a_ν ∈$ {–1, +1}, der angenommenen Trägerphase $ϕ_{\rm T} = π$ (180°) und mit der geeignet dimensionierten Modulatorkonstanten $K_{\rm PM} = π/(2s_0)$ ergibt sich im $ν$–ten Zeitintervall:
 
$$s_{\rm BPSK}(t) = \left\{ \begin{array}{c} s_0 \cdot \cos(2 \pi  f_{\rm T}  t +
 
\pi+ \pi/2) \\ s_0 \cdot \cos(2 \pi  f_{\rm T}  t + \pi- \pi/2) \end{array} \right.\quad \begin{array}{*{1}c} {\rm falls}\hspace{0.15cm}a_\nu = +1, \\  {\rm falls}\hspace{0.15cm}a_\nu = -1. \\ \end{array}$$
 
Diese Gleichung für die binäre Phasenmodulation (BPSK) lässt sich wie folgt umformen:
 
$$s_{\rm BPSK}(t) =
 
a_\nu \cdot s_0 \cdot \sin(2 \pi  f_{\rm T}  t ) = \left\{ \begin{array}{c} s_0 \cdot \sin(2 \pi  f_{\rm T}  t ) \\
 
-s_0 \cdot \sin(2 \pi  f_{\rm T}  t ) \end{array} \right.\quad \begin{array}{*{1}c} {\rm falls}\hspace{0.15cm}a_\nu = +1, \\  {\rm falls}\hspace{0.15cm}a_\nu = -1. \\ \end{array}$$
 
  
 +
==DPSK – Differential Phase Shift Keying==
 +
<br>
 +
[[File:EN_Mod_T_4_2_S7.png|right|frame| DPSK transmitter]]
 +
The diagram on the right shows the block diagram of the modulator for &nbsp;"Differential Phase Shift Keying"&nbsp; $\rm (DPSK)$.
 +
*The bipolar source signal &nbsp;$q(t)$&nbsp; with the amplitude coefficients &nbsp;$q_ν ∈ \{-1, +1\}$&nbsp; is transformed according to this mapping into the signal &nbsp;$m(t)$&nbsp; with the amplitude coefficients
 +
:$$m_{\nu} = m_{\nu -1} \cdot q_{\nu} \in \{ -1, +1\}$$
  
 +
:before it is fed to the BPSK modulator.
  
[[File:P_ID1692__Mod_T_4_2_S5_ganz_neu.png | Signale und Leistungsdichtespektren bei BPSK]]
 
  
 +
*If &nbsp;$q_ν = m_{ν-1}$,&nbsp; the mapped amplitude coefficient is &nbsp;$m_ν = +1$.&nbsp; In contrast, &nbsp;$m_ν = -1$&nbsp; indicates that the amplitude coefficients &nbsp;$q_ν$&nbsp; and &nbsp;$m_{ν-1}$&nbsp; are different.
  
In der Grafik sind die Signale und die dazugehörigen Leistungsdichtespektren skizziert. Man erkennt:
 
*Das BPSK–Signal lässt sich wie das ASK–Signal als Produkt von Quellensignal $q(t)$ und Trägersignal $z(t)$ darstellen. Der einzige Unterschied liegt in den bipolaren Amplitudenkoeffizienten $a_ν ∈$ {–1, +1} gegenüber den unipolaren Koeffizienten (0 oder 1) bei ASK.
 
*Im Gegensatz zur ASK ist bei der BPSK – wie bei jeder Form von Phasenmodulation – die Hüllkurve konstant. Die Information wird hier durch die Phasensprünge innerhalb des Sendesignals $s(t)$ übermittelt (graue Hinterlegungen in der Grafik).
 
*Die Leistungsdichtespektren bei BPSK unterscheiden sich von denen bei ASK lediglich durch die fehlenden Diracfunktionen (da nun $q(t)$ keinen Gleichanteil beinhaltet) sowie durch den Faktor 4 bezüglich der kontinuierlichen LDS–Anteile.
 
*Daraus folgt weiter, dass die binäre Phasenmodulation zu den linearen Modulationsverfahren gezählt werden kann. Im Allgemeinen ist nämlich die (analoge) Phasenmodulation bis auf wenige Ausnahmen hinsichtlich des Quellensignals nichtlinear.
 
*Für die Grafiken wurden bei ASK (Sinus) und BPSK (Minus–Cosinus) aus Darstellungsgründen verschiedene Trägerphasen gewählt. Diese willkürliche Festlegung ist jedoch keine Einschränkung. Beide Verfahren funktionieren bei anderen Trägerphasen in gleicher Weise.
 
  
==Demodulation und Detektion von BPSK–Signalen==
+
*An essential advantage of differential binary phase modulation is that the resulting signal &nbsp;$s(t)$&nbsp; can be demodulated even without knowledge of the carrier phase &nbsp;$ϕ_{\rm T}$,&nbsp; see next section.&nbsp;  Although the exact carrier frequency &nbsp;$f_{\rm T}$&nbsp; must be known to the receiver,&nbsp; it is still referred to as an&nbsp; "incoherent PSK demodulator",&nbsp; and sometimes as a&nbsp; "differential coherent PSK demodulator".
Aufgrund der konstanten Hüllkurve des BPSK–Signals muss hier die Demodulation kohärent erfolgen. Es kann dabei vom gleichen Blockschaltbild wie bei der kohärenten ASK–Demodulation ausgegangen werden. Die Grafik zeigt die Signale $q(t), r(t) = s(t), b(t), d(t)$ und $υ(t)$.
+
<br clear=all>
 +
[[File:EN_Mod_T_4_2_S7b.png|right|frame|Signals at the DPSK transmitter]]
 +
{{GraueBox|TEXT=
 +
$\text{Example 3:}$&nbsp;
 +
The diagram on the right shows
 +
#the bipolar source signal &nbsp;$q(t)$,
 +
#the mapping signal &nbsp;$m(t)$,&nbsp; and
 +
#the DPSK transmitted signal &nbsp;$s(t)$.  
  
  
[[File: P_ID1693__Mod_T_4_2_S6_ganz_neu.png | Signale bei BPSK–Modulation und kohärenter Demodulation]]
+
In the following, &nbsp;$v_ν$&nbsp; denotes the coefficients after the decision,&nbsp; which should coincide with the transmission-side amplitude coefficients &nbsp;$q_ν$.&nbsp;
  
 +
The following principle can be seen:
 +
*Whenever the receiver detects a phase jump,&nbsp; it decides&nbsp; $v_ν = -1$:
 +
:$$v_3 = v_5 =v_6 =-1.$$
 +
*If no phase jump is recognizable, &nbsp;$v_ν = +1$&nbsp; is set:
 +
:$$v_1 = v_2 =v_4 =+1.$$
  
Ein Vergleich mit den entsprechenden ASK–Signalen  zeigt:  
+
Below the phase values &nbsp;$ϕ_{\rm S}$&nbsp; are given for the first six symbols:
*Die Rechtecksignale $q(t)$ und $υ(t)$ sind nun bipolar und für das Detektionssignal bei BPSK gilt:
+
*Additional phase rotation on the channel,&nbsp; e.g. by &nbsp;$70.3π$,&nbsp; does change the absolute phase values to &nbsp;$69.8π$, &nbsp;$69.8π$, &nbsp;$70.$, &nbsp;$70.8π$, &nbsp;$69.8π$&nbsp; and &nbsp;$70.$.  
$$d_{\rm BPSK}(t) =
+
*However,&nbsp; the phase difference of adjacent symbols is preserved, so the differential coherent demodulation still works.&nbsp;
2  \cdot d_{\rm ASK}(t)-s_0.$$
+
*A corresponding demodulator is presented in the following section. }}
*Im betrachteten dämpfungs–, verzerrungs– und rauschfreien Fall sind alle Detektionsabtastwerte $d(ν · T) = ±s_0$. Deshalb muss hier die Entscheiderschwelle $E =$ 0 verwendet werden.  
 
*Man erkennt den doppelten Abstand der BPSK–Detektionsabtastwerte (Kreismarkierungen) von der Schwelle, was sich bezüglich der Fehlerwahrscheinlichkeit entscheidend auswirkt.  
 
  
==DPSK Differential Phase Shift Keying==
+
==Differential coherent demodulation of the DPSK signal==
[[File:P_ID1694__Mod_T_4_2_S7_neu.png | Realisierung des DPSK–Senders | rechts]]
+
<br>
Die nebenstehende Grafik zeigt das Blockschaltbild des DPSK–Modulators (''Differential Phase Shift Keying''). Das (bipolare) Quellensignal $q(t)$ mit den Amplitudenkoeffizienten $q_ν $ {–1, +1} wird entsprechend dieses Mappings in das Signal $m(t)$ mit den Amplitudenkoeffizienten
+
The graphic shows the block diagram of a transmission system with DPSK modulation&nbsp; ("Differential Phase Shift Keying")&nbsp; and differential coherent demodulation.&nbsp; In bullet point form,&nbsp; the given system can be described as follows:
$$m_{\nu} = m_{\nu -1} \cdot q_{\nu} \in \{ -1, +1\}$$
+
[[File:EN_Mod_T_4_2_S7c_neu.png|right|frame| DPSK modulation and differential coherent demodulation]]  
 +
*Ignoring the modulation with the carrier signals &nbsp;$z(t)$&nbsp; and &nbsp;$2 · z(t)$&nbsp; resp.,&nbsp; the symbol &nbsp;$m_ν = m_{ν-1} · q_ν$&nbsp; is present in the interval &nbsp;$ν$&nbsp; at input&nbsp; '''(1)'''&nbsp; of the multiplier highlighted in yellow,&nbsp; and the symbol&nbsp; $m_{ν-1}$&nbsp; is present at input &nbsp; '''(2)'''.
 +
*The multiplication of&nbsp; '''(1)'''&nbsp; and&nbsp; '''(2)'''&nbsp; gives the desired result:
 +
:$$v_ν = m_{ν-1} · q_ν · m_{ν-1} = q_ν.$$
 +
:It is taken into account here that &nbsp;$m_{ν-1} \{+1, –1\}$&nbsp; holds.
 +
*The matched filter with frequency response &nbsp;$H_{\rm E}(f)$&nbsp; eliminates the unwanted components around the doubled carrier frequency &nbsp;$(\pm f_{\rm T})$&nbsp; which result from the twofold multiplication by transmitter-side carrier &nbsp;$z(t)$&nbsp; and receiver-side carrier &nbsp;$2 · z(t)$.&nbsp; With  rectangular pulse&nbsp; $g_q(t)$ &nbsp; &rArr; &nbsp; $H_{\rm E}(f)$&nbsp; can be realized by an integrator.
  
abgebildet, bevor es dem BPSK–Modulator zugeführt wird.
 
  
Ist $q_ν = m_{ν–1},$ so ergibt sich der gemappte Amplitudenkoeffizient $m_ν =$ +1. Dagegen weist $m_ν =$ –1 darauf hin, dass sich die Amplitudenkoeffizienten $q_ν$ und $m_{ν–1}$ unterscheiden.  
+
We now assume that the channel causes a phase rotation by &nbsp;$ϕ$,&nbsp; which the receiver does not know&nbsp; (red block):&nbsp;
 +
#For example,&nbsp; assuming the transmitter-side carrier &nbsp;$z(t) = \cos (2π · f_{\rm T} · t)$,&nbsp; the received signal &nbsp;$r(t)$&nbsp; contains a multiplicative component with &nbsp;$\cos (2π · f_{\rm T} · t + ϕ)$.&nbsp; Thus,&nbsp; the addition of the receiver-side carrier &nbsp;$2 · z(t)$&nbsp; is not phase synchronous.
 +
#The signal &nbsp;$r(t – T)$&nbsp; delayed  by a symbol duration &nbsp;$T$&nbsp; has the same phase &nbsp;$ϕ$.&nbsp; The correlation between&nbsp; $2 · r(t) · z(t)$ &nbsp; and &nbsp; $2 · r(t – T) · z(t – T )$&nbsp; makes the decision result independent of the random phase &nbsp;$ϕ$.&nbsp; This type of demodulation is called&nbsp; "differential-coherent".
  
Wesentlicher Vorteil der differentiellen binären Phasenmodulation ist, dass das so entstehende Signal $s(t)$ auch ohne Kenntnis der Trägerphase $ϕ_{\rm T}$ demoduliert werden kann, siehe nächsten Abschnitt.  Obwohl dem Empfänger die genaue Trägerfrequenz $f_{\rm T}$ bekannt sein muss, spricht man trotzdem von einem inkohärenten, manchmal auch vom differentiell–kohärenten PSK–Demodulator.
 
  
 +
==Error probabilities - a brief overview==
 +
<br>
 +
The error probabilities of the discussed digital modulation methods&nbsp; (ASK,&nbsp; BPSK,&nbsp; DPSK)&nbsp; are calculated in chapter&nbsp; [[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation|"Linear Digital Modulation - Coherent Demodulation"]]&nbsp; of the book&nbsp; "Digital Signal Transmission"&nbsp; under different boundary conditions.&nbsp; Here only some results are given without proof,&nbsp; valid for
 +
#a transmitted signal with the average energy &nbsp;$E_{\rm B}$&nbsp; per bit,
 +
#AWGN noise with the&nbsp; (one-sided)&nbsp; noise power density &nbsp;$N_0$,&nbsp; and
 +
#best possible receiver realization using the matched filter principle.
  
{{Beispiel}}
 
Die nachfolgende Grafik zeigt die Signale $q(t)$ und $m(t)$ sowie das DPSK–Sendesignal $s(t)$. Immer dann, wenn der Empfänger einen Phasensprung erkennt, entscheidet er sich für sich $υ_ν =$ –1 (Zeitpunkte $ν =$ 3, 5 und 6), andernfalls für +1. $υ_ν$ bezeichnet die Amplitudenkoeffizienten nach der Entscheidung, die möglichst mit den sendeseitigen Amplitudenkoeffizienten $q_ν$ übereinstimmen sollten.
 
  
 +
{{BlaueBox|TEXT=
 +
$\text{Here without proof:}$&nbsp;
 +
Let us first consider the&nbsp; &raquo;'''bit error probability'''&laquo;&nbsp; of &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Demodulation_and_detection_of_BPSK_signals|$\text{Binary Phase Shift Keying}$]] &nbsp; $\rm (BPSK)$&nbsp; assuming a coherent receiver:
 +
::$$p_{\text{B, BPSK, coh} } = {\rm Q}\left ( \sqrt{ {2 \cdot E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right
 +
) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{ {E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right ).$$
 +
*[[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|"Two variants of the complementary Gaussian error function"]]&nbsp; were used in the formulas:
 +
::$${\rm Q} ({\it x}) = \frac{\rm 1}{\sqrt{\rm 2\pi} }\int_{\it
 +
x}^{+\infty}{\rm e}^{ {\it -u}^{\rm 2}/\rm 2}\,{\rm d} {\it u}
 +
\hspace{0.05cm},\hspace{0.3cm} {\rm erfc} ( {\it x} ) = \frac{\rm
 +
2}{\sqrt{\rm \pi} }\int_{\it x}^{+\infty}{\rm e}^{ {\it -u}^{\rm
 +
2} }\,{\rm d} {\it u} \hspace{0.05cm}.$$
 +
*In contrast, for &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Coherent_demodulation_of_ASK_signals|$\text{Amplitude Shift Keying}$]]&nbsp; $\rm (ASK)$&nbsp; with coherent demodulation:
 +
::$$p_{\text{B, ASK, coh} } = {\rm Q}\left ( \sqrt{{E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right
 +
) ={1}/{2}\cdot {\rm erfc}\left ( \sqrt{ {E_{\rm B}}/{(2 \cdot N_0) } } \hspace{0.1cm}\right ).$$
 +
::If we plot the bit error probability over the quotient &nbsp;$E_{\rm B}/N_0$&nbsp; in double logarithmic scale,&nbsp; the ASK curve is always &nbsp;$3 \ \rm dB$&nbsp; to the right of the BPSK curve.&nbsp; This degradation is one of the reasons why ASK is rarely used in practice.
 +
*The decisive advantage of &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Differential_coherent_demodulation_of_the_DPSK_signal|$\text{Differential Phase Shift Keying}$]]&nbsp; $\rm (DPSK)$&nbsp; is that it&nbsp; '''can be demodulated without knowledge of the carrier phase'''.&nbsp;  This simple realization is bought by an increased error probability compared to coherent BPSK:
 +
::$$p_{\text{B, DPSK, incoh} } = {1}/{2}\cdot {\rm e}^{ - {E_{\rm B} }/{N_0 } } .$$
 +
*Incoherent demodulation of a BPSK signal,&nbsp; on the other hand,&nbsp; is not possible.&nbsp; For the &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Incoherent_demodulation_of_ASK_signals|$\text{incoherent ASK demodulation}$]]&nbsp; one obtains:
 +
::$$p_{\text{B, ASK, incoh} } = {1}/{2}\cdot {\rm e}^{- {E_{\rm B} }/{(2N_0) } } .$$}}
  
[[File:P_ID1695__Mod_T_4_2_S7b_neu.png | Signale beim DPSK–Sender]]
 
  
 +
The equations given here are to be evaluated in&nbsp; [[Aufgaben:Exercise_4.8:_Different_Error_Probabilities|"Exercise 4.8"]].&nbsp;
  
Unterhalb des Sendesignals sind für die ersten sechs Symbole die Phasenwerte $ϕ_{\rm S}$ angegeben. Durch eine zusätzliche Phasendrehung auf dem Kanal, zum Beispiel um 70., ändern sich zwar die absoluten Phasenwerte auf 69., 69., 70.8π, 69.8π, 70.8π, und 69.8π, die Phasendifferenz benachbarter Symbole bleibt jedoch erhalten, so dass die differentiell–kohärente Demodulation trotzdem funktioniert. Ein entsprechender Demodulator wird im nächsten Abschnitt vorgestellt.  
+
{{GraueBox|TEXT=
{{end}}
+
$\text{Example 4:}$&nbsp;
 +
*For&nbsp; $\rm BPSK$&nbsp; one needs  &nbsp;$10 · \lg  \ E_{\rm B}/N_0 ≈ 8.4 \ \rm dB$,&nbsp; to achieve the bit error probability &nbsp;$p_{\rm B} = \rm 10^{–4}$.&nbsp; However,&nbsp; this always requires coherent demodulation.
 +
*With&nbsp; $($differential coherent$)$&nbsp; $\rm DPSK$,&nbsp; about&nbsp; $9.3 \ \rm dB$&nbsp; are necessary for this, i.e. almost one decibel more.
 +
*With&nbsp; $\rm ASK$&nbsp; one needs even &nbsp;$11.4 \ \rm dB$&nbsp; $($with coherent demodulation$)$&nbsp;  or &nbsp;$12.3 \ \rm dB$&nbsp; $($with incoherent demodulation$)$.}}  
  
==Differentiell-kohärente Demodulation des DPSK-Signals==
 
Die nachfolgende Grafik zeigt das Blockschaltbild eines Übertragungssystems mit DPSK–Modulation (''Differential Phase Shift Keying'') und differentiell–kohärenter Demodulation.
 
  
  
[[File:P_ID1700__Mod_T_4_2_S7c_neu.png | DPSK–Modulation und differentiell–kohärente Demodulation]]
 
  
 +
==Exercises for the chapter==
 +
<br>
 +
[[Aufgaben:Exercise_4.7:_Spectra_of_ASK_and_BPSK|Exercise 4.7: Spectra of ASK and BPSK]]
  
Stichpunktartig lässt sich die Funktionsweise wie folgt beschreiben:  
+
[[Aufgaben:Aufgabe_4.7Z:_Signalformen_bei_ASK,_BPSK_und_DPSK|Exercise 4.7Z: Signal Shapes for ASK, BPSK and DPSK]]
*Ohne Berücksichtigung der Modulation mit den Trägersignalen $z(t)$ bzw. $2 · z(t)$ liegt im Intervall $ν$ am Eingang (1) des gelb hervorgehobenen Multiplizierers das Symbol $m_ν = m_{ν–1} · q_ν$ an und am Eingang (2) das Symbol $m_{ν–1}$. Die Multiplikation von (1) und (2) ergibt das gewünschte Ergebnis, nämlich $υ_ν = m_{ν–1} · q_ν · m_{ν–1} = q_ν$, da $m_{ν–1} ∈$ {+1, –1} gilt.  
 
*Das Matched–Filter mit dem Frequenzgang $H_{\rm E}(f)$ eliminiert die unerwünschten Anteile um die doppelte Trägerfrequenz, die durch die zweifache Multiplikation mit $z(t)$ bzw. $2 · z(t)$ entstehen. Bei rechteckförmigem Grundimpuls $g_q(t)$ lässt sich der Frequenzgang $H_{\rm E}(f)$ auch sehr einfach durch einen Integrator realisieren.
 
*Wir nehmen an, dass der Kanal eine Phasendrehung um $ϕ$ bewirkt, die der Empfänger nicht kennt (rote Markierung). Geht man beispielsweise vom sendeseitigen Trägersignal $z(t) = \cos (2π · f_{\rm T} · t)$ aus, so beinhaltet das Empfangssignal $r(t)$ einen multiplikativen Anteil mit $\cos (2π · f_{\rm T} · t + ϕ)$. Die Zusetzung des empfangsseitigen Trägers $2 · z(t)$ erfolgt also nicht phasensynchron.
 
*Aber auch das um eine Symboldauer $T$ verzögerte Signal $r(t – T)$ weist die gleiche Phase $ϕ$ auf. Durch die Korrelation zwischen $2 · r(t) · z(t)$ und $2 · r(t – T) · z(t – T)$ wird erreicht, dass das Entscheiderergebnis unabhängig von der zufälligen Phase $ϕ$ ist. Man bezeichnet diese Art der Demodulation als differentiell–kohärent.
 
  
 +
[[Aufgaben:Aufgabe_4.8:_Fehlerwahrscheinlichkeiten|Exercise 4.8: Different Error Probabilities]]
  
 +
[[Aufgaben:Aufgabe_4.8Z:_BPSK–Fehlerwahrscheinlichkeit|Exercise 4.8Z: BPSK Error Probability]]
  
 +
[[Aufgaben:Exercise_4.9:_Costas_Rule_Loop|Exercise 4.9: Costas Rule Loop]]
  
  
 
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Latest revision as of 14:47, 13 January 2023

Differences between analog and digital modulation methods


The diagram shows an analog transmission system at the top and a digital system drawn below.  The main differences are highlighted in red:

Analog and digital transmission system

While in the upper system the analog source signal  $q(t)$  is applied to the modulator input,  in the lower digital system the modulating signal  $q_{\rm D}(t)$  is a digital signal,  characterized by the amplitude coefficients  $a_ν$,  the basic pulse  $g_q(t)$  and the symbol duration  $T$:

$$q_{\rm D}(t) = \sum_{\nu=-\infty}^{+\infty}a_\nu \cdot g_q(t - \nu \cdot T) \hspace{0.05cm}.$$
  • The A/D conversion can be done for example by  $\text{pulse code modulation}$  and includes the functions sampling,  quantization,  binary coding and signal shaping. 
  • The basic pulse  $g_q(t)$  is often assumed to be NRZ-rectangular with amplitude  $s_0$  and duration  $T$.  For the spectral function holds with  ${\rm sinc}(x) = \sin(\pi x)/(\pi x)$:
$$G_q(f) = s_0 · T · {\rm sinc}(f T).$$
  • The modulators can be the same for both systems.  They change one of the three signal parameters of the carrier  $z(t)$  according to the modulator input.  The digital variants of AM, PM, FM are called
  1. "Amplitude Shift Keying"  $\rm (ASK)$,
  2. "Phase Shift Keying"  $\rm (PSK)$, 
  3. "Frequency Shift Keying"  $\rm (FSK)$.
  • In contrast,  the demodulator of the digital system differs fundamentally from an analog demodulator by the required decision component  (in hardware or software).  Like  $q_{\rm D}(t)$,  the demodulator output signal  $v_{\rm D}(t)$  is digital and must then be D/A converted into the analog sink signal  $v(t)$. 
  • The decisive quality criterion for both systems is the  $\text{sink SNR}$   ⇒   quotient of the power of source signal $q(t)$ and error signal  $ε(t) = v(t) \ – \ q(t)$.  In a digital system,  one usually makes do with the quality characteristic  $\text{bit error rate}$  $($ $\rm BER)$,  which refers to the digital signals  $q_{\rm D}(t)$  and  $v_{\rm D}(t)$.  This can be converted into a  $\rm SNR$. 

ASK – Amplitude Shift Keying


The diagram shows the digital source signal  $q(t)$  – the index "D" is omitted from now on –  and the ASK transmitted signal

Signals and power-spectral density  $\rm (PSD)$  at ASK
$$s_{\rm ASK}(t) = q(t) · \sin(2π · f_{\rm T} · t),$$

where unipolar amplitude coefficients  $a_ν ∈ \{0, \ 1\}$  and a sinusoidal carrier  (German:  "Träger"   ⇒   subscript "T")  with frequency  $f_{\rm T}$  are assumed here. 

For example,  this method is used in optical transmission systems (since there are no negative light pulses) and is also known as  "On–Off–Keying".

In the right half of the figure,  the corresponding power-spectral densities are shown - although not to scale. 

With the rectangular basic pulse  $g_q(t)$  and equally probable unipolar amplitude coefficients holds:

$$\begin{align*}{{\it \Phi}_{q}(f)}& = \frac{{s_0}^2 \cdot T}{4} \cdot {\rm sinc}^2 (f T) + \frac{{s_0}^2 }{4} \cdot \delta (f)\hspace{0.05cm},\\ {{\it \Phi}_{s}(f)}& = \frac{1}{4} \cdot \big [ {{\it \Phi}_{q}(f- f_{\rm T})}+ {{\it \Phi}_{q}(f+ f_{\rm T})}\big]\hspace{0.05cm}.\end{align*}$$

It should be noted about these equations:

  1. The DC component  $m_q = s_0/2$  of the source signal leads to a Dirac delta function at frequency  $f = 0$  with weight  ${s_0}^2/4$  in the power-spectral density  $ϕ_q(f)$.
  2. The power-spectral density of the ASK transmitted signal  $s(t)$  is equal to  $ϕ_s(f) = ϕ_q(f) ∗ ϕ_z(f)$,  where the PSD  $ϕ_z(f)$  of the carrier  $z(t)$  is composed of two Dirac delta functions at  $±f_{\rm T}$  with respective weight  $1/4$.  The sign  "$\star$"  describes the convolution.  The equation is also valid with other carrier phase.
  3. The power-spectral density  $ϕ_s(f)$  is identical in shape to  $ϕ_q(f)$  except for the shift by  $±f_{\rm T}$   ⇒   the ASK belongs to the  »linear digital modulation methods«.

Coherent demodulation of ASK signals


The diagram shows the block diagram of an ASK system including the receiver components. 

Block diagram of an ASK system including receiver components
  • Let the source signal  $q(t)$  be NRZ-rectangular and unipolar:
$$a_ν ∈ \{0, \ 1\}.$$
  • Let the channel be initially ideal, characterized by  
$$H_{\rm K}(f) = 1,\hspace{0.25cm} n(t) = 0 \hspace{0.5cm} \Rightarrow \hspace{0.5cm} r(t) = s(t).$$

The demodulation is performed coherently by means of a  $\text{synchronous demodulator}$,  the operation of which has already been described for the analog modulation methods AM and PM.

To summarize the block diagram:

  • At the receiver the same carrier signal is added as at the transmitter,  but with double amplitude. 
  • $z(t)$  denotes the carrier at the transmitter   ⇒   the carrier at the receiver  (German:  "Empfänger"   ⇒   subscript "E")  is  $z_{\rm E}(t) = 2 · z(t) $.
  • Multiplication is followed by a suitably sized low-pass filter with frequency response  $H_{\rm E}(f)$,  which removes the higher frequency components of the signal  $b(t)$.
  • The detection signal  $d(t)$  is sampled at instants  $ν · T$  and decided using a threshold decision with threshold  $E = {s_0}/2$. 
  • The sink signal  $v(t)$  at the decision output is rectangular and in the noise-free case equal to the source signal  $q(t)$  except for the transit time  $T/2$.


$\text{To be noted:}$ 

  • »Coherent demodulation«  requires that the carrier frequency  $f_{\rm T}$  and the carrier phase  $ϕ_{\rm T}$  are exactly known to the receiver.
  • The receiver  must extract these two quantities from the received signal  $r(t)$,  which can be quite costly in the presence of strong channel distortion and large noise  $n(t)$. Such realization aspects are dealt with, for example, in "Exercise 4.9" to this chapter.
  • If the carrier phase  $ϕ_{\rm T}$  is not known to the receiver,  this is called  »incoherent demodulation«,  even if the carrier frequency  $f_{\rm T}$  is known.


$\text{Example 1:}$  The diagram shows the signals mentioned in the ASK block diagram with ideal channel:   $H_{\rm K}(f) = 1, \ \ n(t) = 0.$

Signals with ASK modulation and coherent demodulation

The individual signal characteristics can be interpreted as follows:

  • The transmitted signal  $s(t)$  is the product of the unipolar source signal  $q(t)$  and the carrier signal  $z(t) = \sin(2π\hspace{0.05cm}f_{\rm T}\hspace{0.05cm}t)$,  where in the example  $f_{\rm T} = 4/T$  holds  (only four oscillations per symbol duration).
  • The received signal  $r(t) = s(t)$  is first multiplied by the receiver-side carrier  $z_{\rm E}(t) = 2 · \sin(2π\hspace{0.05cm}f_{\rm T}\hspace{0.05cm}t)$   ⇒   twice the amplitude compared to  $z(t)$,  no frequency and phase offset.  This gives:
$$b(t) = 2 \cdot z(t)\cdot r(t)= 2 \cdot z^2(t)\cdot q(t) $$
$$\Rightarrow \hspace{0.35cm}b(t) = q(t) \cdot \big [ 1 - \cos(4\pi\hspace{0.05cm} f_{\rm T}\hspace{0.05cm} t)\big] \hspace{0.05cm}.$$
  • The low-pass filter with frequency response  $H_{\rm E}(f) = {\rm sinc}(f_{\rm T}\hspace{0.05cm} T)$  and corresponding rectangular impulse response  $h_{\rm E}(t)$  forms from the signal  $b(t)$  the detection signal  $d(t) = b(t) \star h_{\rm E}(t)$  .
  • $h_{\rm E}(t)$  is matched to the rectangular basic pulse  $g_q(t)$.  This is called the   $\text{matched filter configuration}$   ⇒   best possible compromise between equalization and noise power limitation.
  • In the absence of noise   ⇒   $d(νT) = q(νT) ∈ \{0, \ s_0\}$.   In the presence of  (moderate)  noise,  it is very likely that  $d(νT) > s_0/2,$ if  $a_ν = +1$,  and it will be  $d(νT) < s_0/2$  for  $a_ν = 0$.
  • The decision obtains from the comparison of the samples  $d(νT)$  with the threshold  $E = s_0/2$  the sink signal  $v(t)$,  which is equal to  $q(t)$  if the decision is error-free except for the transit time  $T/2$. 

Incoherent demodulation of ASK signals


We further assume ASK modulation as well as an ideal transmission channel, i.e.

Block diagram of an incoherent ASK demodulator
  1. distortion-free,
  2. attenuation-free and
  3. noise-free.


Then applies:   $r(t) = s(t) = q(t) \cdot \cos(2 \pi \cdot f_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.$

Further, it is assumed for this section

  • that the receiver knows the carrier frequency  $f_{\rm T}$,
  • but not the carrier phase  $ϕ_{\rm T}$. 


The diagram shows such an  »incoherent demodulator«   ⇒   the  result of the demodulator is independent of the carrier phase  $ϕ_{\rm T}$,  which the receiver does not know.

The operation of this demodulator will be given here only as an outline: 

  • The signals  $d_1(t)$  and  $d_2(t)$  after the two  $\text{matched filters}$  with respective frequency response  $H_{\rm E}(f)$  are identical in shape to the detection signal  $d(t)$   ⇒   $d_{\rm coh}(t)$  according to the  $\text{previous block diagram}$,  but generally attenuated with respect to it due to the lack of phase matching:
$$d_1(t) = d_{\rm coh}(t) \cdot \cos( \phi_{\rm T}), \hspace{0.5cm}d_2(t) = -d_{\rm coh}(t) \cdot \sin( \phi_{\rm T}) \hspace{0.05cm}.$$
  • If the amplitude coefficient is  $a_ν = 0$,  then in the noise-free case the two signal values are zero:  
$$d_1(ν · T) = 0,\hspace{0.5cm} d_2(ν · T) = 0.$$
  • Otherwise  $(a_ν = 1)$  applies:
$$d_1(\nu \cdot T) = s_{\rm 0} \cdot \cos( \phi_{\rm T}), \hspace{0.5cm}d_2(\nu \cdot T) = -s_{\rm 0} \cdot \sin( \phi_{\rm T}) \hspace{0.05cm}.$$
  • After squaring the two partial signals we get for the sum signal:
$$d(\nu \cdot T) = \left\{ \begin{array}{c} 0 \\ {s^2_0} \end{array} \right.\quad \begin{array}{*{1}c} {\rm if}\hspace{0.15cm}a_\nu = 0, \\ {\rm if}\hspace{0.15cm}a_\nu = 1. \\ \end{array}$$
  • By threshold decision – sensibly with the decision threshold  $E = {s_0}^2/2$ – the amplitude coefficients  $a_ν$  can be decided. 
    However in the noisy case,  this results in a somewhat larger  $\text{bit error probability}$  than with coherent demodulation.

BPSK – Binary Phase Shift Keying


For  $\text{analog phase modulation}$  $\rm (PM)$,  the transmitted signal is:  $s_{\rm PM}(t) = s_0 \cdot \cos\big [2 \pi f_{\rm T} t + \phi_{\rm T}+ K_{\rm PM} \cdot q(t)\big ]\hspace{0.05cm}.$  With

Signals and power-spectral densities  $\rm (PSD)$  for Binary Phase Shift Keying
  1. bipolar source signal   ⇒   $a_ν ∈ \{-1, +1\}$, 
  2. carrier phase  $ϕ_{\rm T} = π \ (180^\circ)$ 
  3. modulator constant  $K_{\rm PM} = π/(2s_0)$, 


we obtain in the  $ν$–th time interval:

$$s_{\rm BPSK}(t) = \left\{ \begin{array}{c} s_0 \cdot \cos(2 \pi f_{\rm T} t + \pi+ \pi/2) \\ s_0 \cdot \cos(2 \pi f_{\rm T} t + \pi- \pi/2) \end{array} \right.\quad \begin{array}{*{1}c} {\rm if}\hspace{0.15cm}a_\nu = +1, \\ {\rm if}\hspace{0.15cm}a_\nu = -1. \\ \end{array}$$

This equation for  »Binary Phase Shift Keying«  $($ $\rm BPSK)$  can be transformed as follows:

$$s_{\rm BPSK}(t) = a_\nu \cdot s_0 \cdot \sin(2 \pi f_{\rm T} t ) $$
$$\Rightarrow \hspace{0.3cm}s_{\rm BPSK}(t) = \left\{ \begin{array}{c} s_0 \cdot \sin(2 \pi f_{\rm T} t ) \\ -s_0 \cdot \sin(2 \pi f_{\rm T} t ) \end{array} \right.\quad \begin{array}{*{1}c} {\rm if}\hspace{0.15cm}a_\nu = +1, \\ {\rm if}\hspace{0.15cm}a_\nu = -1. \\ \end{array}$$


In the diagram the signals and the corresponding power-spectral densities are sketched.  It can be seen:

  • Like the ASK signal, the BPSK signal can be represented as the product of the source signal  $q(t)$  and the carrier signal  $z(t)$.  The only difference lies in the bipolar amplitude coefficients  $a_ν ∈ \{-1, +1\}$  compared to the unipolar coefficients  $(0$  or  $1)$  with ASK.
  • In contrast to ASK,  with BPSK  $($as with any form of phase modulation$)$  the envelope is constant.  The information is transmitted here by the phase jumps within the transmitted signal  $s(t)$   ⇒   gray backgrounds in the diagram.
  • The power-spectral densities with BPSK differ from those with ASK only by the missing Dirac delta functions  $($since now  $q(t)$  does not contain a DC component$)$  and by the factor  $4$  with respect to the continuous PSD components.
  • It further follows that binary phase modulation  $\rm (BPSK)$  can be counted as a linear modulation method.  In general, (analog) phase modulation  $\rm (PM)$  is,  with a few exceptions,  non-linear with respect to the source signal.
  • For the diagrams,  different carrier phases were chosen for display reasons in the section  "ASK"   ⇒   "sine"  and here for  "BPSK"  ⇒   "minus cosine".  However,  this arbitrary determination is not a restriction.  Both methods work in the same way for other carrier phases.

Demodulation and detection of BPSK signals


Due to the constant envelope of the BPSK signal,  the demodulation must always be coherent here.  The  $\text{same block diagram}$  can be assumed as for coherent ASK demodulation.

Signals with BPSK modulation and coherent demodulation

$\text{Example 2:}$  The diagram shows from top to bottom

  1. the bipolar source signal  $q(t)$,
  2. the received signal  $r(t) = s(t)$  with ideal channel,
  3. $b(t)$  after multiplication by the receiver-side carrier  $z_{\rm E}(t) = 2 \cdot z(t)$,
  4. the detection signal  $d(t)$  after  "integration"  by the matched filter,
  5. the sink signal  $v(t)$.


A comparison with the corresponding  $\text{signals}$  in the coherent demodulation of the ASK shows:

  • The rectangular signals  $q(t)$  and  $v(t)$  are now bipolar.


  • For the detection signal at BPSK,  compared to ASK,  the following applies:
$$d_{\rm BPSK}(t) =2 \cdot d_{\rm ASK}(t)-s_0.$$
  • In the considered attenuation-free, distortion-free and noise-free case,  all detection samples are  $d(ν · T) = ±s_0$.  Therefore,  the decision threshold  $E = 0$  should be used here.


  • One can see the double distance of the BPSK detection samples  (circle marks)  from the threshold,  which decisively improves  $\text{error probability}$


DPSK – Differential Phase Shift Keying


DPSK transmitter

The diagram on the right shows the block diagram of the modulator for  "Differential Phase Shift Keying"  $\rm (DPSK)$.

  • The bipolar source signal  $q(t)$  with the amplitude coefficients  $q_ν ∈ \{-1, +1\}$  is transformed according to this mapping into the signal  $m(t)$  with the amplitude coefficients
$$m_{\nu} = m_{\nu -1} \cdot q_{\nu} \in \{ -1, +1\}$$
before it is fed to the BPSK modulator.


  • If  $q_ν = m_{ν-1}$,  the mapped amplitude coefficient is  $m_ν = +1$.  In contrast,  $m_ν = -1$  indicates that the amplitude coefficients  $q_ν$  and  $m_{ν-1}$  are different.


  • An essential advantage of differential binary phase modulation is that the resulting signal  $s(t)$  can be demodulated even without knowledge of the carrier phase  $ϕ_{\rm T}$,  see next section.  Although the exact carrier frequency  $f_{\rm T}$  must be known to the receiver,  it is still referred to as an  "incoherent PSK demodulator",  and sometimes as a  "differential coherent PSK demodulator".


Signals at the DPSK transmitter

$\text{Example 3:}$  The diagram on the right shows

  1. the bipolar source signal  $q(t)$,
  2. the mapping signal  $m(t)$,  and
  3. the DPSK transmitted signal  $s(t)$.


In the following,  $v_ν$  denotes the coefficients after the decision,  which should coincide with the transmission-side amplitude coefficients  $q_ν$. 

The following principle can be seen:

  • Whenever the receiver detects a phase jump,  it decides  $v_ν = -1$:
$$v_3 = v_5 =v_6 =-1.$$
  • If no phase jump is recognizable,  $v_ν = +1$  is set:
$$v_1 = v_2 =v_4 =+1.$$

Below the phase values  $ϕ_{\rm S}$  are given for the first six symbols:

  • Additional phase rotation on the channel,  e.g. by  $70.3π$,  does change the absolute phase values to  $69.8π$,  $69.8π$,  $70.8π$,  $70.8π$,  $69.8π$  and  $70.8π$.
  • However,  the phase difference of adjacent symbols is preserved, so the differential coherent demodulation still works. 
  • A corresponding demodulator is presented in the following section.

Differential coherent demodulation of the DPSK signal


The graphic shows the block diagram of a transmission system with DPSK modulation  ("Differential Phase Shift Keying")  and differential coherent demodulation.  In bullet point form,  the given system can be described as follows:

DPSK modulation and differential coherent demodulation
  • Ignoring the modulation with the carrier signals  $z(t)$  and  $2 · z(t)$  resp.,  the symbol  $m_ν = m_{ν-1} · q_ν$  is present in the interval  $ν$  at input  (1)  of the multiplier highlighted in yellow,  and the symbol  $m_{ν-1}$  is present at input   (2).
  • The multiplication of  (1)  and  (2)  gives the desired result:
$$v_ν = m_{ν-1} · q_ν · m_{ν-1} = q_ν.$$
It is taken into account here that  $m_{ν-1} ∈ \{+1, –1\}$  holds.
  • The matched filter with frequency response  $H_{\rm E}(f)$  eliminates the unwanted components around the doubled carrier frequency  $(\pm f_{\rm T})$  which result from the twofold multiplication by transmitter-side carrier  $z(t)$  and receiver-side carrier  $2 · z(t)$.  With rectangular pulse  $g_q(t)$   ⇒   $H_{\rm E}(f)$  can be realized by an integrator.


We now assume that the channel causes a phase rotation by  $ϕ$,  which the receiver does not know  (red block): 

  1. For example,  assuming the transmitter-side carrier  $z(t) = \cos (2π · f_{\rm T} · t)$,  the received signal  $r(t)$  contains a multiplicative component with  $\cos (2π · f_{\rm T} · t + ϕ)$.  Thus,  the addition of the receiver-side carrier  $2 · z(t)$  is not phase synchronous.
  2. The signal  $r(t – T)$  delayed by a symbol duration  $T$  has the same phase  $ϕ$.  The correlation between  $2 · r(t) · z(t)$   and   $2 · r(t – T) · z(t – T )$  makes the decision result independent of the random phase  $ϕ$.  This type of demodulation is called  "differential-coherent".


Error probabilities - a brief overview


The error probabilities of the discussed digital modulation methods  (ASK,  BPSK,  DPSK)  are calculated in chapter  "Linear Digital Modulation - Coherent Demodulation"  of the book  "Digital Signal Transmission"  under different boundary conditions.  Here only some results are given without proof,  valid for

  1. a transmitted signal with the average energy  $E_{\rm B}$  per bit,
  2. AWGN noise with the  (one-sided)  noise power density  $N_0$,  and
  3. best possible receiver realization using the matched filter principle.


$\text{Here without proof:}$  Let us first consider the  »bit error probability«  of  $\text{Binary Phase Shift Keying}$   $\rm (BPSK)$  assuming a coherent receiver:

$$p_{\text{B, BPSK, coh} } = {\rm Q}\left ( \sqrt{ {2 \cdot E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right ) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{ {E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right ).$$
$${\rm Q} ({\it x}) = \frac{\rm 1}{\sqrt{\rm 2\pi} }\int_{\it x}^{+\infty}{\rm e}^{ {\it -u}^{\rm 2}/\rm 2}\,{\rm d} {\it u} \hspace{0.05cm},\hspace{0.3cm} {\rm erfc} ( {\it x} ) = \frac{\rm 2}{\sqrt{\rm \pi} }\int_{\it x}^{+\infty}{\rm e}^{ {\it -u}^{\rm 2} }\,{\rm d} {\it u} \hspace{0.05cm}.$$
$$p_{\text{B, ASK, coh} } = {\rm Q}\left ( \sqrt{{E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right ) ={1}/{2}\cdot {\rm erfc}\left ( \sqrt{ {E_{\rm B}}/{(2 \cdot N_0) } } \hspace{0.1cm}\right ).$$
If we plot the bit error probability over the quotient  $E_{\rm B}/N_0$  in double logarithmic scale,  the ASK curve is always  $3 \ \rm dB$  to the right of the BPSK curve.  This degradation is one of the reasons why ASK is rarely used in practice.
  • The decisive advantage of  $\text{Differential Phase Shift Keying}$  $\rm (DPSK)$  is that it  can be demodulated without knowledge of the carrier phase.  This simple realization is bought by an increased error probability compared to coherent BPSK:
$$p_{\text{B, DPSK, incoh} } = {1}/{2}\cdot {\rm e}^{ - {E_{\rm B} }/{N_0 } } .$$
$$p_{\text{B, ASK, incoh} } = {1}/{2}\cdot {\rm e}^{- {E_{\rm B} }/{(2N_0) } } .$$


The equations given here are to be evaluated in  "Exercise 4.8"

$\text{Example 4:}$ 

  • For  $\rm BPSK$  one needs  $10 · \lg \ E_{\rm B}/N_0 ≈ 8.4 \ \rm dB$,  to achieve the bit error probability  $p_{\rm B} = \rm 10^{–4}$.  However,  this always requires coherent demodulation.
  • With  $($differential coherent$)$  $\rm DPSK$,  about  $9.3 \ \rm dB$  are necessary for this, i.e. almost one decibel more.
  • With  $\rm ASK$  one needs even  $11.4 \ \rm dB$  $($with coherent demodulation$)$  or  $12.3 \ \rm dB$  $($with incoherent demodulation$)$.



Exercises for the chapter


Exercise 4.7: Spectra of ASK and BPSK

Exercise 4.7Z: Signal Shapes for ASK, BPSK and DPSK

Exercise 4.8: Different Error Probabilities

Exercise 4.8Z: BPSK Error Probability

Exercise 4.9: Costas Rule Loop