Modulation Methods/Pulse Code Modulation - Revision history

Hwang at 13:29, 23 January 2023

2023-01-23T13:29:10Z

Hwang at 12:44, 18 January 2023

2023-01-18T12:44:34Z

Guenter at 13:44, 13 January 2023

2023-01-13T13:44:21Z

Guenter at 13:41, 13 January 2023

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Guenter at 13:39, 13 January 2023

2023-01-13T13:39:25Z

Hwang at 15:51, 9 January 2023

2023-01-09T15:51:52Z

Hwang at 15:51, 9 January 2023

2023-01-09T15:51:35Z

Hwang at 15:43, 9 January 2023

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Hwang at 15:37, 9 January 2023

2023-01-09T15:37:57Z

Hwang at 15:35, 9 January 2023

2023-01-09T15:35:28Z

@@ Line 246: / Line 246: @@
 *Here,&nbsp; the&nbsp; "dual code"&nbsp; is used &nbsp; &rArr; &nbsp; the quantization intervals&nbsp; $\mu$&nbsp; are numbered consecutively from&nbsp; $0$&nbsp; to&nbsp; $M-1$&nbsp; and then written in simple binary.&nbsp; With&nbsp; $M = 8$&nbsp; for example&nbsp; $\mu = 6$ &nbsp; ⇔ &nbsp; '''110'''.
-*The three symbols of the binary coded signal&nbsp; $q_{\rm C}(t)$&nbsp; are obtained by replacing&nbsp; '''0'''&nbsp; by&nbsp; '''L'''&nbsp; ("Low") and&nbsp; '''1'''&nbsp; by&nbsp; '''H'''&nbsp; ("High").&nbsp; This gives in the example the sequence&nbsp; "'''HHL HHL LLH LHL HLH LHH'''".
+*The three symbols of the binary encoded signal&nbsp; $q_{\rm C}(t)$&nbsp; are obtained by replacing&nbsp; '''0'''&nbsp; by&nbsp; '''L'''&nbsp; ("Low") and&nbsp; '''1'''&nbsp; by&nbsp; '''H'''&nbsp; ("High").&nbsp; This gives in the example the sequence&nbsp; "'''HHL HHL LLH LHL HLH LHH'''".
 *The bit duration&nbsp; $T_{\rm B}$&nbsp; is here shorter than the sampling distance&nbsp; $T_{\rm A} = 1/f_{\rm A}$&nbsp; by a factor&nbsp; $N = {\rm log_2}(M) = 3$.&nbsp;  So,&nbsp; the bit rate is&nbsp; $R_{\rm B} = {\rm log_2}(M) \cdot f_{\rm A}$.
 *If one uses the same mapping in decoding&nbsp; $(v_{\rm C} &nbsp; ⇒ &nbsp; v_{\rm Q})$&nbsp; as in encoding &nbsp; $(q_{\rm Q} &nbsp; ⇒ &nbsp; q_{\rm C})$,&nbsp; then,&nbsp; if there are no transmission errors: &nbsp; &nbsp; $v_{\rm Q}(ν \cdot T_{\rm A}) = q_{\rm Q}(ν \cdot T_{\rm A}). $

@@ Line 152: / Line 152: @@
 One can see from this plot:
 #The spectrum&nbsp; $P_{\rm R}(f)$&nbsp; is in contrast to&nbsp; $P_δ(f)$&nbsp; not a Dirac comb&nbsp; $($all weights equal $1)$,&nbsp; but the weights here are evaluated to the function&nbsp; $G_{\rm R}(f)/T_{\rm A} = T_{\rm R}/T_{\rm A} \cdot {\rm sinc}(f\cdot T_{\rm R})$.
-#Because of the zero of the&nbsp; $\rm sinc$-function,&nbsp; the Dirac lines vanish here at&nbsp; $±4f_{\rm A}$.
+#Because of the zero of the&nbsp; $\rm sinc$-function,&nbsp; the Dirac delta lines vanish here at&nbsp; $±4f_{\rm A}$.
 #The spectrum&nbsp; $Q_{\rm A}(f)$&nbsp; results from the convolution with&nbsp; $Q(f)$.&nbsp; The rectangle around&nbsp; $f = 0$&nbsp; has height&nbsp; $T_{\rm R}/T_{\rm A} \cdot Q_0$,&nbsp; the proportions around&nbsp; $\mu \cdot f_{\rm A} \ (\mu ≠ 0)$&nbsp; are lower.
 #If one uses  for signal reconstruction an ideal,&nbsp; rectangular low-pass

@@ Line 8: / Line 8: @@
 == # OVERVIEW OF THE FOURTH MAIN CHAPTER # ==
 <br>
-The fourth chapter deals with the digital modulation methods&nbsp; &raquo;amplitude shift keying&laquo;&nbsp; $\rm (ASK)$,&nbsp; &raquo;phase shift keying&laquo;&nbsp; $\rm (PSK)$&nbsp; and&nbsp; &raquo;frequency shift keying&laquo;&nbsp; $\rm (FSK)$&nbsp; as well as some modifications derived from them.&nbsp; Most of the properties of the analog modulation methods mentioned in the last two chapters still apply.&nbsp; Differences result from the now required&nbsp; &raquo;decision component&laquo;&nbsp; of the receiver.
+The fourth chapter deals with the digital modulation methods&nbsp; &raquo;'''amplitude shift keying'''&laquo;&nbsp; $\rm (ASK)$,&nbsp; &raquo;'''phase shift keying'''&laquo;&nbsp; $\rm (PSK)$&nbsp; and&nbsp; &raquo;'''frequency shift keying'''&laquo;&nbsp; $\rm (FSK)$&nbsp; as well as some modifications derived from them.&nbsp; Most of the properties of the analog modulation methods mentioned in the last two chapters still apply.&nbsp; Differences result from the now required&nbsp; &raquo;decision component&laquo;&nbsp; of the receiver.
 We restrict ourselves here essentially to the&nbsp; &raquo;system-theoretical and transmission aspects&laquo;.&nbsp; The error probability is given only for ideal conditions.&nbsp; The derivations and the consideration of non-ideal boundary conditions can be found in the book&nbsp; "Digital Signal Transmission".
 In detail are treated:
-*the &nbsp;&raquo;pulse code modulation&laquo;&nbsp; $\rm (PCM)$&nbsp; and its components&nbsp; "Sampling"&nbsp; &ndash; &nbsp;"Quantization"&nbsp; &ndash; &nbsp; "Coding",
+#the &nbsp;&raquo;pulse code modulation&laquo;&nbsp; $\rm (PCM)$&nbsp; and its components&nbsp; "sampling"&nbsp; &ndash; &nbsp;"quantization"&nbsp; &ndash; &nbsp; "encoding",
-*the &nbsp;&raquo;linear modulation&laquo;&nbsp; $\rm ASK$,&nbsp; $\rm BPSK$,&nbsp; $\rm DPSK$&nbsp; and associated demodulators,
+#the &nbsp;&raquo;linear modulation&laquo;&nbsp; $\rm ASK$,&nbsp; $\rm BPSK$,&nbsp; $\rm DPSK$&nbsp; and associated demodulators,
-* the &nbsp;&raquo;quadrature amplitude modulation&laquo;&nbsp; $\rm (QAM)$&nbsp; and more complicated signal space mappings,
+# the &nbsp;&raquo;quadrature amplitude modulation&laquo;&nbsp; $\rm (QAM)$&nbsp; and more complicated signal space mappings,
-*the&nbsp;  &raquo;frequency shift keying&laquo;&nbsp; $\rm (FSK$)&nbsp; as an example of non-linear digital modulation,
+#the&nbsp;  &raquo;frequency shift keying&laquo;&nbsp; $\rm (FSK$)&nbsp; as an example of non-linear digital modulation,
-*the FSK with &nbsp;&raquo;continuous phase matching&laquo;&nbsp; $\rm (CPM)$,&nbsp; especially the&nbsp; $\rm (G)MSK$&nbsp; method.
+#the FSK with &nbsp;&raquo;continuous phase matching&laquo;&nbsp; $\rm (CPM)$,&nbsp; especially the&nbsp; $\rm (G)MSK$&nbsp; method.

@@ Line 248: / Line 248: @@
 *The three symbols of the binary coded signal&nbsp; $q_{\rm C}(t)$&nbsp; are obtained by replacing&nbsp; '''0'''&nbsp; by&nbsp; '''L'''&nbsp; ("Low") and&nbsp; '''1'''&nbsp; by&nbsp; '''H'''&nbsp; ("High").&nbsp; This gives in the example the sequence&nbsp; "'''HHL HHL LLH LHL HLH LHH'''".
 *The bit duration&nbsp; $T_{\rm B}$&nbsp; is here shorter than the sampling distance&nbsp; $T_{\rm A} = 1/f_{\rm A}$&nbsp; by a factor&nbsp; $N = {\rm log_2}(M) = 3$.&nbsp;  So,&nbsp; the bit rate is&nbsp; $R_{\rm B} = {\rm log_2}(M) \cdot f_{\rm A}$.
-*If one uses the same mapping in decoding&nbsp; $(v_{\rm C} &nbsp; ⇒ &nbsp; v_{\rm Q})$&nbsp; as in coding '''KORREKTUR: encoding'''&nbsp; $(q_{\rm Q} &nbsp; ⇒ &nbsp; q_{\rm C})$,&nbsp; then,&nbsp; if there are no transmission errors: &nbsp; &nbsp; $v_{\rm Q}(ν \cdot T_{\rm A}) = q_{\rm Q}(ν \cdot T_{\rm A}). $
+*If one uses the same mapping in decoding&nbsp; $(v_{\rm C} &nbsp; ⇒ &nbsp; v_{\rm Q})$&nbsp; as in encoding &nbsp; $(q_{\rm Q} &nbsp; ⇒ &nbsp; q_{\rm C})$,&nbsp; then,&nbsp; if there are no transmission errors: &nbsp; &nbsp; $v_{\rm Q}(ν \cdot T_{\rm A}) = q_{\rm Q}(ν \cdot T_{\rm A}). $
 *An alternative to dual code is&nbsp; "Gray code",&nbsp; where adjacent binary values differ only in one bit.&nbsp; For&nbsp; $N = 3$:
 :&nbsp; $\mu = 0$:&nbsp; '''LLL''', &nbsp; &nbsp; $\mu = 1$:&nbsp; '''LLH''', &nbsp; &nbsp; $\mu = 2$:&nbsp; '''LHH''', &nbsp; &nbsp; $\mu = 3$: &nbsp; '''LHL''',
@@ Line 307: / Line 307: @@
 *The white dots mark the source signal&nbsp; $q(t)$.&nbsp; Without transmission error the sink signal&nbsp; $v(t)$&nbsp; has the same course when neglecting quantization.
-*Now,&nbsp; exactly one bit of the fifth sample&nbsp; $q(5 \cdot T_{\rm A}) = -0.715$&nbsp; is falsified,&nbsp; where this sample has been coded '''KORREKTUR: encoded''' as&nbsp; '''LLHL LHLL'''.
+*Now,&nbsp; exactly one bit of the fifth sample&nbsp; $q(5 \cdot T_{\rm A}) = -0.715$&nbsp; is falsified,&nbsp; where this sample has been encoded  as&nbsp; '''LLHL LHLL'''.
 <br><br><br><br><br>

@@ Line 55: / Line 55: @@
 *Further, it can be seen from the block diagram that there is no equivalent for&nbsp; "quantization"&nbsp; at the receiver-side.&nbsp; Therefore,&nbsp; even with error-free transmission,&nbsp; i.e.,&nbsp; for&nbsp; $v_{\rm C}(t) = q_{\rm C}(t)$,&nbsp; the analog sink signal&nbsp; $v(t)$&nbsp; will differ from the source signal&nbsp; $q(t)$.
-*As a measure of the quality of the digital transmission system,&nbsp;  we use the&nbsp; [[Modulation_Methods/Quality_Criteria#Signal.E2.80.93to.E2.80.93noise_.28power.29_ratio|$\text{Signal-to-Noise Power Ratio}$]] &nbsp; &rArr; &nbsp; in short: &nbsp; &raquo;'''Sink-SNR'''&laquo;&nbsp; as the quotient of the powers of source signal&nbsp; $q(t)$&nbsp; and fault '''KORREKTUR: error''' signal&nbsp; $ε(t) = v(t) - q(t)$:
+*As a measure of the quality of the digital transmission system,&nbsp;  we use the&nbsp; [[Modulation_Methods/Quality_Criteria#Signal.E2.80.93to.E2.80.93noise_.28power.29_ratio|$\text{Signal-to-Noise Power Ratio}$]] &nbsp; &rArr; &nbsp; in short: &nbsp; &raquo;'''Sink-SNR'''&laquo;&nbsp; as the quotient of the powers of source signal&nbsp; $q(t)$&nbsp; and error signal&nbsp; $ε(t) = v(t) - q(t)$:
 :$$\rho_{v} = \frac{P_q}{P_\varepsilon}\hspace{0.3cm} {\rm with}\hspace{0.3cm}P_q = \overline{[q(t)]^2},
 \hspace{0.2cm}P_\varepsilon = \overline{[v(t) - q(t)]^2}\hspace{0.05cm}.$$
@@ Line 125: / Line 125: @@
 *In&nbsp; &raquo;'''natural sampling'''&laquo;&nbsp; the sampled signal&nbsp; $q_{\rm A}(t)$&nbsp; is obtained by multiplying the analog source signal&nbsp; $q(t)$&nbsp; by&nbsp; $p_{\rm R}(t)$. &nbsp; Thus in the ranges&nbsp; $p_{\rm R}(t) = 1$,&nbsp; $q_{\rm A}(t)$&nbsp; has the same progression as&nbsp; $q(t)$.
-*In&nbsp; &raquo;'''discrete sampling'''&laquo;&nbsp; the signal&nbsp; $q(t)$&nbsp; is&nbsp; &ndash; at least mentally &ndash; first multiplied by the Dirac comb&nbsp; $p_δ(t)$.&nbsp; Then each Dirac delta impulse '''KORREKTUR: pulse'''&nbsp; $T_{\rm A} \cdot δ(t - ν \cdot T_{\rm A})$&nbsp; is replaced by a rectangular pulse&nbsp; $g_{\rm R}(t - ν \cdot T_{\rm A})$&nbsp; .
+*In&nbsp; &raquo;'''discrete sampling'''&laquo;&nbsp; the signal&nbsp; $q(t)$&nbsp; is&nbsp; &ndash; at least mentally &ndash; first multiplied by the Dirac comb&nbsp; $p_δ(t)$.&nbsp; Then each Dirac delta pulse &nbsp; $T_{\rm A} \cdot δ(t - ν \cdot T_{\rm A})$&nbsp; is replaced by a rectangular pulse&nbsp; $g_{\rm R}(t - ν \cdot T_{\rm A})$&nbsp; .
@@ Line 208: / Line 208: @@
 The second functional unit&nbsp; &raquo;'''Quantization'''&laquo;&nbsp; of the PCM transmitter is used for value discretization.
 *For this purpose the whole value range of the analog source signal&nbsp; $($e.g.,&nbsp; the range $± q_{\rm max})$&nbsp; is divided into&nbsp; $M$&nbsp; intervals.
-* Each sample&nbsp; $q_{\rm A}(ν ⋅ T_{\rm A})$&nbsp; is then assigned TO '''KORREKTUR: to??''' a representative&nbsp; $q_{\rm Q}(ν ⋅ T_{\rm A})$&nbsp; of the associated interval&nbsp; (e.g.,&nbsp; the interval center)&nbsp;.
+* Each sample&nbsp; $q_{\rm A}(ν ⋅ T_{\rm A})$&nbsp; is then assigned to  a representative&nbsp; $q_{\rm Q}(ν ⋅ T_{\rm A})$&nbsp; of the associated interval&nbsp; (e.g.,&nbsp; the interval center)&nbsp;.

@@ Line 319: / Line 319: @@
 ==Estimation of SNR degradation due to transmission errors==
 <br>
-Now we will try to&nbsp; (approximately)&nbsp; determine the SNR curve of the PCM system taking into account bit errors.&nbsp; We start from the following block diagram and further assume:
+Now we will try to&nbsp; (approximately)&nbsp; determine the SNR curve of the PCM system taking bit errors into account.&nbsp; We start from the following block diagram and further assume:
 [[File:EN_Mod_T_4_1_S7c.png |right|frame|For calculating the SNR curve  of the PCM system;&nbsp; bit errors are taken into account]]

@@ Line 307: / Line 307: @@
 *The white dots mark the source signal&nbsp; $q(t)$.&nbsp; Without transmission error the sink signal&nbsp; $v(t)$&nbsp; has the same course when neglecting quantization.
-*Now,&nbsp; exactly one bit of the fifth sample&nbsp; $q(5 \cdot T_{\rm A}) = -0.715$&nbsp; is corrupted,&nbsp; where this sample has been coded as&nbsp; '''LLHL LHLL'''.
+*Now,&nbsp; exactly one bit of the fifth sample&nbsp; $q(5 \cdot T_{\rm A}) = -0.715$&nbsp; is falsified,&nbsp; where this sample has been coded '''KORREKTUR: encoded''' as&nbsp; '''LLHL LHLL'''.
 <br><br><br><br><br>
 The results of the error analysis shown in the graph and the table below can be summarized as follows:
-*If only the last bit &nbsp; &rArr; &nbsp; "Least Significant Bit" &nbsp; &rArr; &nbsp; $\rm (LSB)$&nbsp; of the binary word is corrupted&nbsp; $($'''LLHL LHL<u>L</u> &nbsp; ⇒ &nbsp; LLHL LHL<u>H</u>''',&nbsp;  white curve$)$,&nbsp; then no difference from error-free transmission is visible to the naked eye. Nevertheless,&nbsp; the signal-to-noise ratio is reduced by &nbsp; $3.5 \ \rm dB$.
+*If only the last bit &nbsp; &rArr; &nbsp; "Least Significant Bit" &nbsp; &rArr; &nbsp; $\rm (LSB)$&nbsp; of the binary word is falsified&nbsp; $($'''LLHL LHL<u>L</u> &nbsp; ⇒ &nbsp; LLHL LHL<u>H</u>''',&nbsp;  white curve$)$,&nbsp; then no difference from error-free transmission is visible to the naked eye. Nevertheless,&nbsp; the signal-to-noise ratio is reduced by &nbsp; $3.5 \ \rm dB$.
 *An error of the fourth last bit leads to a clearly detectable distortion by eight quantization steps &nbsp; $($'''LLHL<u>L</u>HLL ⇒ LLHL<u>H</u>HLL''',&nbsp; green curve$)$: &nbsp; $v(5T_{\rm A}) \ - \ q(5T_{\rm A}) = 8/256 - 2 = 0.0625$&nbsp; and the signal-to-noise ratio drops to &nbsp; $10 · \lg \ ρ_υ = 28.2 \ \rm dB$.
-*Finally,&nbsp; the red curve shows the case where the&nbsp; $\rm MSB$&nbsp; ("Most Significant Bit")&nbsp; is corrupted: &nbsp; '''<u>L</u>LHLLHLL ⇒ <u>H</u>LHLLHLL''' &nbsp; &rArr;  &nbsp; distortion&nbsp; $v(5T_{\rm A}) \ - \ q(5T_{\rm A}) = 1$&nbsp; $($corresponding to half the modulation range$)$.&nbsp; The SNR is now only about &nbsp; $4 \ \rm dB$.
+*Finally,&nbsp; the red curve shows the case where the&nbsp; $\rm MSB$&nbsp; ("Most Significant Bit")&nbsp; is falsified: &nbsp; '''<u>L</u>LHLLHLL ⇒ <u>H</u>LHLLHLL''' &nbsp; &rArr;  &nbsp; distortion&nbsp; $v(5T_{\rm A}) \ - \ q(5T_{\rm A}) = 1$&nbsp; $($corresponding to half the modulation range$)$.&nbsp; The SNR is now only about &nbsp; $4 \ \rm dB$.
 *At all sampling times except&nbsp; $5T_{\rm A}$,&nbsp; $v(t)$&nbsp; matches exactly with&nbsp; $q(t)$&nbsp; except for the quantization error.&nbsp; Outside these points marked by yellow crosses,&nbsp; the single error at&nbsp; $5T_{\rm A}$&nbsp; leads to strong deviations in an extended range,&nbsp; due to the interpolation with the&nbsp; $\rm sinc$-shaped impulse response of the reconstruction low-pass&nbsp; $H(f)$.