Modulation Methods/Implementation of OFDM Systems - Revision history

Noah at 16:05, 11 March 2023

2023-03-11T16:05:26Z

Guenter at 16:33, 23 January 2023

2023-01-23T16:33:27Z

Guenter at 15:27, 19 January 2023

2023-01-19T15:27:37Z

Hwang at 11:33, 19 January 2023

2023-01-19T11:33:42Z

Hwang at 11:33, 19 January 2023

2023-01-19T11:33:05Z

Hwang at 11:31, 19 January 2023

2023-01-19T11:31:57Z

Hwang at 11:31, 19 January 2023

2023-01-19T11:31:19Z

Hwang at 11:30, 19 January 2023

2023-01-19T11:30:25Z

Hwang at 11:29, 19 January 2023

2023-01-19T11:29:08Z

Hwang at 11:27, 19 January 2023

2023-01-19T11:27:22Z

@@ Line 86: / Line 86: @@
 #Further one recognizes&nbsp; "intersymbol interference"&nbsp; $\rm (ISI)$&nbsp; in the green framed time interval &nbsp;$0 ≤ t < τ_1$: &nbsp; <br> &nbsp; &nbsp; &nbsp; The red predecessor symbol &nbsp;$k-\hspace{-0.08cm}1$ &nbsp; $($frequency&nbsp; $3 · f_0)$&nbsp; interferes with the black symbol &nbsp;$k$ &nbsp; $($frequency $1 · f_0)$. }}
-==Guard gap to reduce intersymbol interference==
+==Guard interval to reduce intersymbol interference==
 <br>
-A first possible solution for the second problem&nbsp; $\rm (ISI)$&nbsp; is the introduction of a guard gap of length &nbsp;$T_{\rm G}$:
+A first possible solution for the second problem&nbsp; $\rm (ISI)$&nbsp; is the introduction of a guard interval of length &nbsp;$T_{\rm G}$:
-[[File: P_ID1643__Mod_T_5_6_S4b_1_neu.png|right|frame|Principle of the&nbsp; "guard gap"]]
+[[File: P_ID1643__Mod_T_5_6_S4b_1_neu.png|right|frame|Principle of the&nbsp; "guard interval"]]
 *Here,&nbsp; the signal between two symbols is set to zero for the duration of the protection time &nbsp;$T_{\rm G}$.&nbsp;
-*As a result,&nbsp; possible pulse trailers of symbol&nbsp; $k-\hspace{-0.08cm}1$&nbsp; no longer extend into the following symbol &nbsp;$(k)$,&nbsp; provided that the guard gap is selected&nbsp; "wider"&nbsp; than the maximum channel delay.
+*As a result,&nbsp; possible pulse trailers of symbol&nbsp; $k-\hspace{-0.08cm}1$&nbsp; no longer extend into the following symbol &nbsp;$(k)$,&nbsp; provided that the guard interval is selected&nbsp; "wider"&nbsp; than the maximum channel delay.
 *The new frame duration &nbsp;$T_{\rm R}$ &ndash; i.e. the distance between successive transmitted symbols &ndash; is thus given by
@@ Line 99: / Line 99: @@
 {{GraueBox|TEXT=
 $\text{Example 2:}$&nbsp;
-This diagram again shows the real part of the received OFDM signal,&nbsp; but now with&nbsp; "guard gap".&nbsp; The assumptions of &nbsp;[[Modulation_Methods/Implementation_of_OFDM_Systems#Intercarrier_interference_and_intersymbol_interference|$\text{Example 1}$]]&nbsp; have been kept.
+This diagram again shows the real part of the received OFDM signal,&nbsp; but now with&nbsp; "guard interval".&nbsp; The assumptions of &nbsp;[[Modulation_Methods/Implementation_of_OFDM_Systems#Intercarrier_interference_and_intersymbol_interference|$\text{Example 1}$]]&nbsp; have been kept.
-[[File:EN_Mod_T_5_6_S4b_v2.png |right|frame| OFDM reception signal over multipath channel with guard gap]]
+[[File:EN_Mod_T_5_6_S4b_v2.png |right|frame| OFDM reception signal over multipath channel with guard interval]]
 The diagram shows:
 # In addition,&nbsp; $T_{\rm G} = T/4$&nbsp; is set,&nbsp; which corresponds to the limiting case &nbsp;$T_{\rm G} = τ_{\rm max}$&nbsp; for the present channel.
-#&nbsp;By using a guard gap of corresponding width, &nbsp;  intersymbol interference&nbsp; $\rm (ISI)$&nbsp; can be avoided &nbsp; &rArr; &nbsp; in interval &nbsp;$k$&nbsp; only one frequency occurs.
+#&nbsp;By using a guard interval of corresponding width, &nbsp;  intersymbol interference&nbsp; $\rm (ISI)$&nbsp; can be avoided &nbsp; &rArr; &nbsp; in interval &nbsp;$k$&nbsp; only one frequency occurs.
 #&nbsp;'''But''':&nbsp;  Intercarrier interference&nbsp; $\rm (ICI)$&nbsp; cannot be prevented by this,&nbsp; because the symbols still have a transient phase and thus jumps.
@@ Line 111: / Line 111: @@
-The&nbsp; "guard gap"&nbsp; approach will not be considered further.&nbsp;  Rather,&nbsp; a better alternative is presented in the next section.}}
+The&nbsp; "guard interval"&nbsp; approach will not be considered further.&nbsp;  Rather,&nbsp; a better alternative is presented in the next section.}}
 ==Cyclic Prefix==
@@ Line 119: / Line 119: @@
 *For this,&nbsp; the end of a symbol in the time interval&nbsp; $T \ – \ T_{\rm G} ≤ t < T$&nbsp; is prefixed again to the actual symbol.
 *This procedure thus generates a&nbsp; &raquo;'''cyclic prefix'''&laquo;.
-*As with the&nbsp; "guard gap",&nbsp; the interval duration increases from symbol duration &nbsp;$T$&nbsp; to the new frame duration &nbsp;$T_{\rm R} = T + T_{\rm G}$.&nbsp;
+*As with the&nbsp; "guard interval",&nbsp; the interval duration increases from symbol duration &nbsp;$T$&nbsp; to the new frame duration &nbsp;$T_{\rm R} = T + T_{\rm G}$.&nbsp;
 *The new number of samples of the extended discrete-time signal in the &nbsp;$k$–th interval is then:
 :$$N_{\rm{total}} = N + N_{\rm{G}} = N \cdot (1 + T_{\rm{G}} /T) .$$
@@ Line 127: / Line 127: @@
 {{GraueBox|TEXT=
 $\text{Example 3:}$&nbsp; The diagram shows the operation of the guard interval in the continuous-time case.&nbsp;
-The parameters from the consideration of the guard gap in&nbsp; [[Modulation_Methods/Implementation_of_OFDM_Systems#Intercarrier_interference_and_intersymbol_interference|$\text{Example 1}$]]&nbsp; still apply,&nbsp; although only one symbol&nbsp; $($with frequency &nbsp;$f_0)$ is now considered.&nbsp;
+The parameters from the consideration of the guard interval in&nbsp; [[Modulation_Methods/Implementation_of_OFDM_Systems#Intercarrier_interference_and_intersymbol_interference|$\text{Example 1}$]]&nbsp; still apply,&nbsp; although only one symbol&nbsp; $($with frequency &nbsp;$f_0)$ is now considered.&nbsp;
 [[File:EN_Mod_T_5_6_S5b_neu.png  |right|frame| Received OFDM signal over multipath channel with cyclic prefix]]

@@ Line 100: / Line 100: @@
 $\text{Example 2:}$&nbsp;
 This diagram again shows the real part of the received OFDM signal,&nbsp; but now with&nbsp; "guard gap".&nbsp; The assumptions of &nbsp;[[Modulation_Methods/Implementation_of_OFDM_Systems#Intercarrier_interference_and_intersymbol_interference|$\text{Example 1}$]]&nbsp; have been kept.
-[[File:EN_Mod_T_5_6_S5a.png |right|frame| OFDM reception signal over multipath channel with guard gap '''KORREKTUR''': duration]]
+[[File:EN_Mod_T_5_6_S4b_v2.png |right|frame| OFDM reception signal over multipath channel with guard gap]]
 The diagram shows:

@@ Line 156: / Line 156: @@
 *Under the simplifying assumption of a transmission spectrum  &nbsp;$S(f)$&nbsp; hard limited to&nbsp;$1/T$,&nbsp; the bandwidth efficiency – see [Kam04]<ref>Kammeyer, K.D.:&nbsp; Nachrichtenübertragung.&nbsp; Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>:
 :$$\beta  = \frac{ \text{symbol rate} }{ {\rm bandwidth} } = \frac{1/(T + T_{\rm G})}{1/T} = \frac{1}{{1 + T_{\rm{G}} /T}}.$$
-*However,&nbsp; in a system using the so-called&nbsp; "matched filter approach",&nbsp; increasing the frame duration from &nbsp;$T$&nbsp; to &nbsp;$T_{\rm G} + T$&nbsp; leads to a decrease in the signal-to-noise ratio if the impulse responses &nbsp;$g_{\rm S}(t)$&nbsp; and &nbsp;$g_{\rm E}(t)$&nbsp; of the transmission and reception '''KORREKTUR: receiver''' filters are matched to the symbol duration &nbsp;$T$.&nbsp;
+*However,&nbsp; in a system using the so-called&nbsp; "matched filter approach",&nbsp; increasing the frame duration from &nbsp;$T$&nbsp; to &nbsp;$T_{\rm G} + T$&nbsp; leads to a decrease in the signal-to-noise ratio if the impulse responses &nbsp;$g_{\rm S}(t)$&nbsp; and &nbsp;$g_{\rm E}(t)$&nbsp; of the transmission and  receiver filters are matched to the symbol duration &nbsp;$T$.&nbsp;
 *The resulting &nbsp;signal&ndash;to&ndash;noise ratio&nbsp; $\rm (SNR)$&nbsp;  of the overall system&nbsp; (in dB)&nbsp; can be calculated as follows,&nbsp; taking into account the guard interval:
 :$${\rm{SNR}}_{\hspace{0.08cm}{\rm{ {\rm{with} }\hspace{0.08cm} GI} } } = {\rm{SNR}}_{\hspace{0.08cm}{\rm{{\rm{without}}\hspace{0.08cm} GI}}} + 10 \cdot \lg (\beta ), \quad {\rm{where}}$$
@@ Line 188: / Line 188: @@
 *Using the convolution theorem for linear time-invariant systems,&nbsp; one can then also write the spectrum as a product of two discrete Fourier transforms:
 :$$R_\mu = {\rm{DFT} }\{ d_\nu \} \cdot {\rm{DFT} }\{ h_\nu \} = D_\mu \cdot H_\mu.$$
-*To compensate for the influence of the channel on the reception '''KORREKTUR: received?''' sequence,&nbsp; it is convenient to multiply the spectrum by the inverse transfer function &nbsp;$1/H_{\mu}$.&nbsp;
+*To compensate for the influence of the channel on the received sequence,&nbsp; it is convenient to multiply the spectrum by the inverse transfer function &nbsp;$1/H_{\mu}$.&nbsp;
 *This&nbsp; "zero forcing"&nbsp; approach leads to the ideal signal reconstruction in the noise-free case.&nbsp; The equalization can be done point by point:
 :$$\hat {D}_\mu = \frac{1}{ {H_\mu } } \cdot R_\mu.$$}}
@@ Line 203: / Line 203: @@
 *The vector of a channel with &nbsp;$L$&nbsp; echoes is &nbsp;$\mathbf h = (h_0$, ... , $h_L)$.&nbsp; The transmission matrix with &nbsp;$N$&nbsp; rows and &nbsp;$N + L$&nbsp; columns is:
 :$${\rm\bf{H}} = \left( {\begin{array}{*{20}c}   {h_0 } & {h_1 } &  \cdots  & {h_L } & {} & {} & {}  \\  {} & {h_0 } & {h_1 } &  \cdots  & {h_L } & {} & {}  \\  {} & {} &  \ddots  &  \ddots  & {} &  \ddots  & {}  \\   {} & {} & {} & {h_0 } & {h_1 } &  \cdots  & {h_L }  \\ \end{array}} \right).$$
-*Here, &nbsp;$N$&nbsp; indicates the number of carriers and hence the number of time samples of the IDFT.&nbsp; With the transmit '''KORREKTUR: transmitted''' vector &nbsp;${\bf d} = (d_0$, &nbsp;...&nbsp; , $d_{N–1})$&nbsp; the receive '''KORREKTUR: received''' vector is:
+*Here, &nbsp;$N$&nbsp; indicates the number of carriers and hence the number of time samples of the IDFT.&nbsp; With the transmitted vector &nbsp;${\bf d} = (d_0$, &nbsp;...&nbsp; , $d_{N–1})$&nbsp; the received  vector is:
 :$$\bf r = d · H.$$
-*Considering the cyclic prefix,&nbsp; the extended transmit '''KORREKTUR: transmitted''' vector is obtained:
+*Considering the cyclic prefix,&nbsp; the extended transmitted vector is obtained:
 :$${\rm\bf{d}}_{{\rm{ext}}} = (d_{N - N_G } , \ \ldots \ ,d_{N - 1} ,d_0 , \ \ldots \ ,d_{N - 1} ).$$
 *Now,&nbsp; one could extend the above transmission matrix &nbsp;$\bf H$&nbsp; likewise accordingly on &nbsp;  $(N + N_{\rm G})$&nbsp; rows &nbsp; and &nbsp; $(N + L + N_{\rm G})$&nbsp; columns &nbsp; as well as remove the prefix at the receiver again,&nbsp; which is not to be pursued here further.
@@ Line 230: / Line 230: @@
    \end{array}} \right) .$$
-*The Discrete Fourier Transform&nbsp; $\rm (DFT)$&nbsp; can be represented by &nbsp;$1/N · \bf F$&nbsp; and its inverse&nbsp; $\rm (IDFT)$&nbsp; by &nbsp;$\rm \bf F^{\star}$ such that for the transmit '''KORREKTUR: transmitted''' vector: &nbsp;$\rm {\bf d} = {\bf D} · {\bf F}^{\star}$.
+*The Discrete Fourier Transform&nbsp; $\rm (DFT)$&nbsp; can be represented by &nbsp;$1/N · \bf F$&nbsp; and its inverse&nbsp; $\rm (IDFT)$&nbsp; by &nbsp;$\rm \bf F^{\star}$ such that for the transmitted  vector: &nbsp;$\rm {\bf d} = {\bf D} · {\bf F}^{\star}$.
-*The &nbsp;$N$&nbsp; spectral coefficients are described by the vector &nbsp;${\bf D} = 1/N · {\bf d} · {\bf F}$&nbsp; and the reception '''KORREKTUR: received''' vector is &nbsp;${\bf r} = {\bf d} · {\bf H}_{\rm C} = {\bf D} · {\bf F}^{\star} · {\bf H}_{\rm C}$.
+*The &nbsp;$N$&nbsp; spectral coefficients are described by the vector &nbsp;${\bf D} = 1/N · {\bf d} · {\bf F}$&nbsp; and the  received vector is &nbsp;${\bf r} = {\bf d} · {\bf H}_{\rm C} = {\bf D} · {\bf F}^{\star} · {\bf H}_{\rm C}$.
-*The (discrete) Fourier transform &nbsp;$\rm \bf R$&nbsp; of the reception '''KORREKTUR: received''' vector &nbsp;$\rm \bf r$&nbsp; can then be written in the following way:
+*The (discrete) Fourier transform &nbsp;$\rm \bf R$&nbsp; of the received vector &nbsp;$\rm \bf r$&nbsp; can then be written in the following way:
 :$${\rm\bf{R}} = \frac{1}{N} \cdot {\rm\bf{r}} \cdot {\rm\bf{F}} = {\rm\bf{D}} \cdot \left( {\begin{array}{*{20}c}
     {H_0 } & {} & {} & {}  \\
@@ Line 246: / Line 246: @@
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp;
-*The reception '''KORREKTUR: received''' symbol on the&nbsp; $\mu$–th carrier is: &nbsp;
+*The received symbol on the&nbsp; $\mu$–th carrier is: &nbsp;
 :$$R_{\mu} = D_{\mu} · H_{\mu} \ \ (\mu = 0, \text{...}\ ,\ N–1).$$
 *This can thus be equalized using the &nbsp;"Zero Forcing"&nbsp; approach:
@@ Line 258: / Line 258: @@
 $\text{Example 5:}$&nbsp;
 We assume a system with &nbsp;$N = 4$&nbsp; carriers and a channel with &nbsp;$L = 2$&nbsp; echoes,
-*so that for the transmit '''KORREKTUR: transmitted''' vector &nbsp;${\bf d} = (d_0, d_1, d_2, d_3)$,&nbsp; and
+*so that for the transmitted vector &nbsp;${\bf d} = (d_0, d_1, d_2, d_3)$,&nbsp; and
 *for the channel impulse response &nbsp;${\bf h} = (h_0, h_1, h_2)$.&nbsp;
-'''(1)''' &nbsp; To represent the cyclic prefix,&nbsp; we use the cyclic transmission matrix  &nbsp;${\rm\bf{H} }_{\rm{C} }$,&nbsp; instead of the extended transmission '''KORREKTUR: transmitted?''' vector with the corresponding transmission matrix,&nbsp; resulting in the reception '''KORREKTUR: received''' vector &nbsp;${\rm \bf r}=  {\rm \bf d} \cdot {\rm \bf H}_{\rm{C} }$:&nbsp;
+'''(1)''' &nbsp; To represent the cyclic prefix,&nbsp; we use the cyclic transmission matrix  &nbsp;${\rm\bf{H} }_{\rm{C} }$,&nbsp; instead of the extended transmitted vector with the corresponding transmission matrix,&nbsp; resulting in the received vector &nbsp;${\rm \bf r}=  {\rm \bf d} \cdot {\rm \bf H}_{\rm{C} }$:&nbsp;
 :$${\rm\bf{H} }_{\rm{C} }  = \left( {\begin{array}{*{20}c}
     {h_0 } & {h_1 } & {h_2 } & { }  \\
@@ Line 282: / Line 282: @@
 r_3 = d_1 \cdot h_2 + d_2 \cdot h_1 + d_3 \cdot h_0.$$
-'''(2)''' &nbsp;  The&nbsp; (discrete)&nbsp; Fourier transform of the receive '''KORREKTUR: received''' vector is calculated to be
+'''(2)''' &nbsp;  The&nbsp; (discrete)&nbsp; Fourier transform of the received  vector is calculated to be
 :$${\rm\bf{R} } = \frac{1}{N} \cdot {\rm\bf{r} } \cdot {\rm\bf{F} } = {\rm\bf{D} } \cdot \left( {\begin{array}{*{20}c}
     {H_0 } & {} & {} & {}  \\
@@ Line 291: / Line 291: @@
    {\rm{e} }^{ - {\rm{j \hspace{0.05cm}\cdot \hspace{0.03cm} 2\pi } }\hspace{0.05cm}\cdot \hspace{0.03cm}l \hspace{0.05cm}\cdot \hspace{0.03cm} \mu /4} }  .$$
-'''(3)'''  &nbsp; For numerical calculations,&nbsp; we assume a known BPSK-encoded transmit '''KORREKTUR: transmitted?''' sequence &nbsp;$\rm \bf D$&nbsp; (in the frequency domain)&nbsp; and the following channel impulse response &nbsp;$\bf h$:&nbsp;
+'''(3)'''  &nbsp; For numerical calculations,&nbsp; we assume a known BPSK-encoded transmitted  sequence &nbsp;$\rm \bf D$&nbsp; (in the frequency domain)&nbsp; and the following channel impulse response &nbsp;$\bf h$:&nbsp;
 :$${\rm\bf{D} } = \frac{1}{N} \cdot {\rm\bf{d} } \cdot {\rm\bf{F} } =
 \left( D_0, D_1,D_2,D_3\right) = \left( +1,\ -1,\ +1,\ -1\right),$$
@@ Line 326: / Line 326: @@
     {e_3 }  \\
 \end{array}} \right) = \left( +1, -1, \;  +1, -1 \right).$$
-&rArr; &nbsp; This corresponds exactly to the transmit '''KORREKTUR: transmitted?''' symbol sequence &nbsp;$\bf D$.&nbsp; That is: <br>
+&rArr; &nbsp; This corresponds exactly to the transmitted  symbol sequence &nbsp;$\bf D$.&nbsp; That is: <br>
 :'''Knowing the channel,&nbsp; the signal can be completely equalized,&nbsp; requiring only a single multiplication per symbol&nbsp; (carrier)'''. }}

@@ Line 326: / Line 326: @@
     {e_3 }  \\
 \end{array}} \right) = \left( +1, -1, \;  +1, -1 \right).$$
-&rArr; &nbsp; This corresponds exactly to the transmit symbol sequence &nbsp;$\bf D$.&nbsp; That is: <br>
+&rArr; &nbsp; This corresponds exactly to the transmit '''KORREKTUR: transmitted?''' symbol sequence &nbsp;$\bf D$.&nbsp; That is: <br>
 :'''Knowing the channel,&nbsp; the signal can be completely equalized,&nbsp; requiring only a single multiplication per symbol&nbsp; (carrier)'''. }}

@@ Line 291: / Line 291: @@
    {\rm{e} }^{ - {\rm{j \hspace{0.05cm}\cdot \hspace{0.03cm} 2\pi } }\hspace{0.05cm}\cdot \hspace{0.03cm}l \hspace{0.05cm}\cdot \hspace{0.03cm} \mu /4} }  .$$
-'''(3)'''  &nbsp; For numerical calculations,&nbsp; we assume a known BPSK-encoded transmit sequence &nbsp;$\rm \bf D$&nbsp; (in the frequency domain)&nbsp; and the following channel impulse response &nbsp;$\bf h$:&nbsp;
+'''(3)'''  &nbsp; For numerical calculations,&nbsp; we assume a known BPSK-encoded transmit '''KORREKTUR: transmitted?''' sequence &nbsp;$\rm \bf D$&nbsp; (in the frequency domain)&nbsp; and the following channel impulse response &nbsp;$\bf h$:&nbsp;
 :$${\rm\bf{D} } = \frac{1}{N} \cdot {\rm\bf{d} } \cdot {\rm\bf{F} } =
 \left( D_0, D_1,D_2,D_3\right) = \left( +1,\ -1,\ +1,\ -1\right),$$