Search results
From LNTwww
Create the page "Signal Representation: Exercises" on this wiki! See also the search results found.
- ...<h3> <a href=":Category:Signal_Representation:_Exercises">Signal Representation</a> </h3> <p>58 exercises ⇒ 250 subtasks</p>3 KB (266 words) - 16:05, 9 October 2022
- ...<h3> <a href=":Category:Signal_Representation:_Exercises">Signal Representation</a> </h3> <p>58 exercises ⇒ 250 subtasks</p>3 KB (266 words) - 10:12, 8 October 2022
- ...oscillation« and the limiting case »direct current signal«. ...and their description by the analytical signal and the equivalent low-pass signal.3 KB (421 words) - 19:31, 24 March 2023
- ==Time signal representation== ...bsp; $t$ from $-\infty$ to $+\infty$. Such a signal is the borderline case of a [[ Signal_Representation/Harmonic_Oscilla8 KB (1,233 words) - 16:55, 8 June 2023
- ...iated demodulators: Phase modulation and frequency modulation. Signal-to-noise ratio with angle modulation. *The description builds on the books »[[Signal Representation]]« and »[[Linear_and_Time_Invariant_Systems]]«4 KB (435 words) - 13:25, 3 April 2023
- |Nächste Seite=Direct Current Signal - Limit Case of a Periodic Signal ...of a »DC signal« as a limiting case of a periodic signal,6 KB (885 words) - 16:13, 8 June 2023
- ...etail with stochastic signals and their modelling. Knowledge of stochastic signal theory is an important prerequisite for understanding the following books, ...www}$-books, which describe the [[Signal Representation|»representation of deterministic signals«]] as well as the [[Linear_and_T4 KB (460 words) - 13:24, 3 April 2023
- {{BlaueBox|TEXT=Based on the book [[Signaldarstellung|»Signal Representation«]], here it is described how to mathematically capture the infl The filter influence on a random signal is only dealt with in the last chapter of the book [[Theory_of_Stocha4 KB (486 words) - 18:38, 1 November 2023
- ...presentation]]" and [[Theory_of_Stochastic_Signals|"Stochastic Signal Theory"]] $($Books 1 and 3$)$ is assumed for understanding this {{Collapse1| header=Digital Signal Transmission under Idealized Conditions4 KB (527 words) - 11:31, 31 March 2023
- |Nächste Seite=Equivalent Low-Pass Signal and Its Spectral Function ...nbsp; $f_{\rm T}$ is much larger than the bandwidth of the band-pass signal $x(t)$.13 KB (2,150 words) - 17:48, 19 June 2023
- |Vorherige Seite=Direct Current Signal - Limit Case of a Periodic Signal ...as well as in many natural sciences. The diagram shows an exemplary signal waveform.15 KB (2,344 words) - 17:39, 9 June 2023
- |Untermenü=Time and Frequency-Discrete Signal Representation |Vorherige Seite=Equivalent Low-Pass Signal and Its Spectral Function18 KB (2,937 words) - 14:51, 22 June 2023
- ...system, both deterministic signals and stochastic signal occur. A »'''deterministic signal'''« exists, if its time function $x(t)$ can b16 KB (2,524 words) - 14:46, 22 June 2023
- ...e stray components $(2\sigma^2)$ the better suited for digital signal transmission is the mobile communication channel. == Example of signal behaviour with Rice fading==7 KB (1,061 words) - 16:55, 3 February 2023
- |Untermenü=Time and Frequency-Discrete Signal Representation |Vorherige Seite=Discrete-Time Signal Representation20 KB (3,186 words) - 17:57, 28 June 2023
- *Many of the fundamentals covered in the book »[[Signal Representation]]« are still valid. Differences arise from the radio chan ===Exercises and multimedia===4 KB (530 words) - 13:27, 3 April 2023
- The two input signals and the output signal have the following characteristics: ...low-frequency message signal and has the spectrum $Q(f)$. This signal is continuous in value and time and limited to the frequency range $|17 KB (2,756 words) - 19:09, 12 January 2023
- ...al Representation» it was explained that for any deterministic signal $x(t)$ a »spectral function« $X(f)$&nbs ...mation about the spectral function is already contained in the time domain representation, even if not always immediately recognizable. The same facts apply to14 KB (2,235 words) - 18:21, 2 November 2023
- ...D41__Sto_T_3_1_S1_neu.png |right|frame| Signal and PDF of a Gaussian noise signal]] The graphic shows a section of a stochastic noise signal $x(t)$ whose instantaneous value can be taken as a continuous r11 KB (1,672 words) - 11:32, 12 April 2023
- The »'''Fourier series'''« of a periodic signal $x(t)$ is defined as follows: ...e »'''basic circular frequency'''« of the periodic signal $(T_0$ is the period duration$)$.}}20 KB (3,164 words) - 18:21, 15 November 2023
- *the structure and representation of »digital filters« (non-recursive and recursive), ...operties of the »Wiener-Kolmogorow filter« for the signal reconstruction.17 KB (2,779 words) - 11:17, 22 December 2022
- |Vorherige Seite=Signal Representation |Nächste Seite=Signal Classification17 KB (2,614 words) - 19:56, 6 June 2023
- [[File:EN_Mod_T_2_5_S1.png|right|frame|Spectrum (of the analytical signal) for vestigial sideband amplitude modulation]] In this schematic representation, one recognizes:8 KB (1,275 words) - 11:52, 13 January 2023
- ...tem}$ with the signal $s(t)$ at the input and the output signal $r(t)$. For the sake of simplicity, let $s(t)$ and ...verse Fourier transform}$]] of $H(f)$. The output signal results from the convolution:10 KB (1,542 words) - 15:41, 29 January 2023
- $\text{Example 1:}$ The graph shows two signal waveforms with uniform amplitude distribution. ...given. Rather, this sawtooth signal represents a deterministic signal.}}11 KB (1,638 words) - 19:01, 20 February 2024
- ...itter-side carrier signal $z(t)$ and the receiver-side carrier signal $z_{\rm E}(t) =2\cdot z(t)$ are omitted. ...;$z(t)$ and AWGN noise, characterized by the additive Gaussian noise signal $n(t)$. The second interference component (interference f13 KB (2,071 words) - 17:13, 19 January 2023
- ...lated signal $s(t)$ requires twice the bandwidth of the source signal $q(t)$. In this representation, one can see:14 KB (2,244 words) - 14:43, 18 January 2023
- *Instead of a broadband, strongly modulated signal, a large number of narrowband, mutually orthogonal subcarriers ...t data rates for the subcarriers. This means that a high-rate source signal must be split into several signals of significantly lower symbol rate for t18 KB (2,900 words) - 18:31, 23 January 2023
- ...r_Transform_(DFT)|"Discrete Fourier Transform"]] of the book "Signal Representation" and is identical to the one there [[Aufgaben:Exercise_5.2:_Inverse_D ...s thus describes the discrete form of the Fourier correspondence of the DC signal:7 KB (1,085 words) - 20:22, 7 March 2023
- Untermenü=Signal Distortion and Equalization ...' is understood to be undesirable deterministic changes in a message signal caused by a transmission system. }}14 KB (2,172 words) - 17:30, 9 November 2023
- [[File:EN_Sig_T_3_4_S1.png|right|frame|On calculation of signal and spectrum of the LTI output]] ...X(f)$ is applied to the input, then the spectrum of the output signal is:16 KB (2,526 words) - 17:50, 15 June 2023
- * $m_2$ corresponds to the »signal power« $($referred to the unit resistance $1 \ Ω)$. #For discrete time, the random signal $x(t)$ is replaced by the random sequence $〈x_ν〉$.15 KB (2,185 words) - 19:37, 19 February 2024
- |Untermenü=Time and Frequency-Discrete Signal Representation *the »windowing» that may falsely limit the signal $x(t)$.12 KB (2,000 words) - 17:16, 24 June 2023
- |Nächste Seite=Analytical Signal and Its Spectral Function ...f band-pass signals« from the »equivalent low-pass signal«,17 KB (2,652 words) - 16:37, 19 June 2023
- |Untermenü=Time and Frequency-Discrete Signal Representation ...the distortion of the spectrum of a periodic and thus temporally unlimited signal due to the implicit time limit of the »Discrete Fourier Transfo16 KB (2,591 words) - 17:02, 27 June 2023
- ==Signal-to-noise power ratio in PM== [[File:EN_Mod_T_3_3_S1.png|right|frame|Signal-to-noise power ratio in phase modulation]]10 KB (1,563 words) - 15:29, 13 January 2023
- *In the book »Signal Representation« you were familiarized with the mathematical description of det ...ear Time-Invariant Systems« now describes what changes a signal or its spectrum undergoes through a transmission system and how these17 KB (2,685 words) - 18:50, 1 November 2023
- ==Block diagram and time domain representation== ...e sideband (DSB) or single sideband (SSB), with or without a carrier signal - the so-called »'''synchronous demodulator'''«&nbs27 KB (4,430 words) - 11:32, 13 January 2023
- *to provide a sink signal $v(t)$ at a spatially distant sink *that differs as little as possible from the source signal $q(t)$ .18 KB (2,734 words) - 19:01, 12 January 2023
- |Vorherige Seite=Analytical Signal and Its Spectral Function |Nächste Seite=Discrete-Time Signal Representation35 KB (5,623 words) - 16:11, 21 June 2023
- ...ier_integral|"Inverse Fourier transform"]] in the book "Signal Representation"): *The random signal $x(t)$ is applied to a (preferably) rectangular and19 KB (2,939 words) - 17:13, 22 December 2022
- ...ce transform of the output signal $y(t)$ considering the input signal $x(t)$ . $Y_{\rm L}(p)$ is characterized by $N ...}(p) = 1$ ⇒ $Y_{\rm L}(p) = H_{\rm L}(p)$, the output signal $y(t)$ then describes the [[Linear_and_Time_Invariant_Systems/S20 KB (3,260 words) - 17:03, 21 November 2023
- *A »'''two-way channel'''«: <br>The transmitted signal $s(t)$ reaches the receiver on a direct path with the path leng Thus, the following applies to the received signal:14 KB (2,197 words) - 15:44, 29 January 2023
- ...also called "correlation filter" – is used to prove the signal existence. *Let the useful component $g(t)$ of the received signal $r(t)=g(t)+n(t)$ be pulse-shaped and thus "energy-limited18 KB (2,987 words) - 11:22, 22 December 2022
- ...re decided simultaneously, statistical bindings between the received signal samples can be taken into account during detection, which results in ...bsp; and the signal $s(t)$. This measure is only for a simpler representation and is not a restriction.<br>23 KB (3,666 words) - 14:16, 11 July 2022
- ...en one comes from the periodic signal to the unique »aperiodic signal« – often also called »pulse«. ...ely large value $T_0$, one speaks of an »aperiodic signal«.22 KB (3,658 words) - 13:38, 14 June 2023
- We consider the following problem: a source signal $q(t)$, whose spectrum $Q(f)$ is bandlimited to the ...tion of frequency $f_{\rm T}$, which we will refer to as the carrier signal $z(t)$.25 KB (4,222 words) - 14:42, 18 January 2023
- ...bsp; matches the bit duration $T_{\rm B}$ of the binary source signal, and ...] with respect to the input signal $b(t)$ and the output signal $c(t)$: 15 KB (2,474 words) - 00:47, 23 March 2023
- ==General description and signal space allocation== ...e| Linear modulator with $\rm I$ and $\rm Q$–components; signal space for $\text{16-QAM}$]]27 KB (4,342 words) - 15:50, 13 January 2023
- *Let the source signal $q(t)$ be free of a DC component and limited in magnitude to &n *For simplicity of representation, the carrier phase is set to $\mathbf{ϕ_{\rm T} } = 0$ w31 KB (4,950 words) - 11:44, 13 January 2023
- |Vorherige Seite=Signal Classification == Representation by magnitude and phase==12 KB (1,973 words) - 17:45, 7 June 2023
- == Signal space representation of linear modulation == ...ation Methods", the structure of the optimal receiver and the signal representation by means of basis functions were treated by the example of baseband transmi33 KB (5,393 words) - 12:04, 17 November 2022
- |Untermenü=Signal Distortion and Equalization ...enuation curve must be constant for all frequencies contained in the input signal:18 KB (2,829 words) - 19:47, 14 November 2023
- ...l_of_digital_signal_transmission|"Information-theoretical model of digital signal transmission"]] the "mutual information" between the two ...ous mutual information $I(X;Y) = h(X) + h(Y) - h(XY)$. This representation is also found in the following diagram $($left graph$)$.21 KB (3,453 words) - 16:22, 28 February 2023
- ...ansmission to $\text{22.8 kbit/s}$ and to reproduce the speech signal at the receiver side in the best possible way. Different signal processing methods are used for speech encoding and decoding:30 KB (4,749 words) - 16:06, 20 February 2023
- An important concept in stochastic signal theory is the "random process". Below are some characteristics ...r physical and statistical properties, each of which yields a random signal $x_i(t)$.30 KB (4,674 words) - 11:07, 22 December 2022
- ...whose input the signal $x(t)$ is applied to. The output signal $y(t)$ is then obtained as the convolution product $x(t) #However, if the signal $x(t)$ has periodic components and thus the spectral function &24 KB (3,967 words) - 17:56, 19 November 2023
- Untermenü=Digital Signal Transmission under Idealized Conditions ...h some simplifying assumptions: a redundancy-free binary transmitted signal, no intersymbol interference. Although the description is mainl28 KB (4,573 words) - 14:51, 31 August 2022
- Untermenü=Digital Signal Transmission under Idealized Conditions ...h some simplifying assumptions: a redundancy-free binary transmitted signal, no intersymbol interference. Although the description is mainl28 KB (4,573 words) - 11:30, 12 April 2023
- #the »signal characteristics and spectral functions« of angle-modulated sign #the »signal-to-noise power ratio« of FM, which is more favorable than31 KB (4,943 words) - 14:43, 18 January 2023
- # the »signal-to-noise power ratio« as a very general quality criterion, ...lytical signal« and the »equivalent low–pass signal«.22 KB (3,220 words) - 15:09, 23 January 2023
- [[File:P_ID62__Sto_T_3_7_S1_neu.png |right|frame|Sample signal and PDF of a Rayleigh distributed random variable|class=fit]] One can see from this representation:12 KB (1,790 words) - 11:03, 22 December 2022
- *the basis functions and the signal space representation according to the chapter [[Digital_Signal_Transmission/Signals,_Basis *For the transmission, $M$ different signal forms $s_i(t)$ are available; for the indexing variable s39 KB (6,156 words) - 16:18, 23 January 2023
- ...(t)$ corresponds to the bit duration $T_q$ of the source signal, which is assumed to be binary and redundancy-free.<br> ...;$16\%$. Because of ${T_c}/{T_q} = 4/3$, the transmitted signal $s(t)$ is lower in frequency here than in uncoded transmission,26 KB (4,161 words) - 16:01, 23 January 2023
- ...irst 160 bytes of the file "LNTprofs.bmp". The file header (in hexadecimal representation from 0x00 to 0x0D) as well as the image data area (from 0x36) are highlight ...the hexadecimal representation and the trailing "d" indicates the decimal representation):15 KB (2,365 words) - 13:54, 17 April 2023
- ...espect to the signal component $d_{\rm S}(t)$ of the detection signal if it is fully compensated with $1/H_{\rm K}(f)$. ...ponent $d_{\rm S}(t)$ – originating from the transmitted signal $s(t)$ – has exactly the same shape as calculated in the20 KB (3,309 words) - 16:40, 28 June 2022
- ..._Viterbi $\text{Andrew J. Viterbi}$]. In a somewhat simplified representation – without convolutional encoder, interleaver and de-interleaver as *The spreading signal $c(t)$ causes "band spreading" by the spreading fac20 KB (3,049 words) - 18:30, 23 January 2023
- ...uo;'''impulse response'''« $h(t)$ is equal to the output signal $y(t)$ of the system if an infinitely short impulse with an inf ...ity condition »the output signal cannot start before the input signal« is:18 KB (2,887 words) - 19:21, 15 November 2023
- ...also takes the channel coding into account. In the case of a speech signal, for every $20\ \rm kbit/s$ speech frame $456$ bits ...demodulation of FSK) we refer to two exercises in the book "Digital Signal Transmission":17 KB (2,642 words) - 18:41, 20 February 2023
- '''Notes on the representation chosen here''' The time domain representation is obtained by applying the [[Signal_Representation/Fourier_Transform22 KB (3,399 words) - 19:37, 24 November 2023
- ...ure. In this process, an adaptation of the digital transmitted signal to the characteristics of the transmission channel is achieved through the ...edundancy-free coding« which leads to a non-binary transmitted signal,28 KB (4,566 words) - 16:53, 23 March 2023
- ...e$ of equidistant signal values after sampling has the best possible signal-to-noise power ratio $\rm (SNR)$.<br> #This gives for the samples of the continuous decorrelation filter output signal $d(t)$ at time $\nu \cdot T$: $d_{\nu} = a_{\nu}\c31 KB (4,689 words) - 13:20, 13 July 2022
- #The »distance-dependent attenuation of a radio signal« and various »path loss models«, In this representation, the radio signal reaches the mobile station via a direct path.25 KB (4,083 words) - 15:23, 29 January 2023
- ...ditions apply as for the last example. The graph shows the detection signal $d(t)$ after the Gaussian low-pass $($before the decision [[File:Dig_T_3_2_S1c_version2_ret.png|right|frame|Detection signal with Gaussian receiver filter]]27 KB (4,389 words) - 13:15, 10 October 2022
- ...f> link takes you to a list with a total of around $640$ exercises and $3100$ subtasks $($each with detailed sample solution$)$. ...courses with a total of thirty-six semester hours per week of lectures and exercises.19 KB (2,918 words) - 14:56, 22 January 2024
- |Untermenü=Digital Signal Transmission under Idealized Conditions ...el, so that $r(t) = s(t) + n(t)$ applies to the received signal. The two-sided noise power density $(N_0/2)$ is more suit32 KB (5,279 words) - 15:51, 23 January 2023
- == Representation in the state transition diagram == ...d_Trellis_Diagram#State_definition_for_a_memory_register|$\text{sequential representation}$]] in the previous section.27 KB (4,131 words) - 18:10, 7 December 2022
- ...n of non-ideal boundary conditions can be found in the book "Digital Signal Transmission". ...e amplitude modulation« $\rm (QAM)$ and more complicated signal space mappings,44 KB (7,142 words) - 15:29, 23 January 2023
- * in the first one an analog source signal $q(t)$ is present, and *in the second one a digital signal.28 KB (4,482 words) - 16:21, 17 January 2023
- ==Information-theoretical model of digital signal transmission == The entropies defined so far in general terms are now applied to digital signal transmission, whereby we assume a discrete memoryless channel $\rm (D42 KB (6,429 words) - 16:10, 28 February 2023
- In the binary representation this is: "$10011$". The graph shows the extraction of the CRC4 # As in the AMI code, each binary "'''0'''" is assigned the signal level $\rm 0\hspace{0.09cm} V$, while the binary "'''1'''" is a12 KB (1,926 words) - 17:05, 23 January 2023
- ...f digital signal transmission, which is based on basis functions and signal space constellations. This makes it possible to treat very different # the application of the signal space description to »carrier frequency systems«,27 KB (4,255 words) - 12:11, 5 April 2023
- ...for example, for the simulation of a binary or multi-level digital signal, but equally for the emulation of a channel with statistically indepe ...langle z_ν\rangle$ with the following properties established for our representation:14 KB (2,155 words) - 18:09, 6 February 2024
- The estimation of the phase angle from the incoming signal, which is required for coherent demodulation, is not possible o ...bility density functions of Rayleigh and Rice distributions. To this representation it is to be noted:28 KB (4,269 words) - 13:43, 10 October 2022
- ...al distinction of these functions with regard to time and frequency domain representation by lowercase and uppercase letters is thus excluded. ${\rm (2)}$ For the "frequency–time representation" you get the »'''time-variant transfer function'''«40 KB (6,350 words) - 18:38, 30 January 2023
- ...stributed signal to a linear filter for spectral shaping, the output signal is also Gaussian distributed. Only the distribution parameters such ...neu.png |right|frame|Gaussian distributed and uniformly distributed random signal]]19 KB (2,981 words) - 11:00, 22 December 2022
- With the nomenclature and representation form chosen for the fourth main chapter the following constellation results ...nbsp; $r = s + n$ is now a scalar and is composed of the transmitted signal $s \in \{s_0, \hspace{0.05cm}s_1\}$ and the noise term $n37 KB (5,706 words) - 12:04, 17 November 2022
- ...en|$\text{Gram–Schmidt process}$]] for the book "Digital Signal Transmission", which teaches the calculation of basis functions,  == Representation of SPC and RC as dual codes ==37 KB (5,068 words) - 11:32, 22 November 2022
- For the necessary directional separation of the xDSL signal transmission according to ...zes mutual interference in the trunk cable. Thus, the stronger signal ISDN does not interfere with the parallel running xDSL and vice versa.35 KB (5,461 words) - 17:05, 24 April 2023
- |Untermenü=Time and Frequency-Discrete Signal Representation ==Exercises for the chapter==17 KB (2,644 words) - 16:38, 28 June 2023
- One can see from this representation: Here, for the output signal $y = x + n$, where $n$ describes a [[Theory_o27 KB (4,255 words) - 12:12, 22 November 2022
- *This sequence is the discrete-time equivalent of the continuous-time signal $X(t)$ after band-limiting and sampling. ...entation|"Dirac comb"]] $p_δ(t)$. The resulting discrete-time signal is thus:37 KB (5,856 words) - 16:28, 28 February 2023
- ...ert ≤ φ_{xx}(τ = 0)$, where $φ_{xx}(τ =0)$ indicates the signal power $P_x = {\rm E}\big[x^2(t)\big]$. We now apply the above calculation rule to the spreading signal24 KB (3,708 words) - 18:37, 23 January 2023
- ...on channel" $\rm (FCCH)$ corresponds to an unmodulated carrier signal with frequency $f_{\rm T} + Δf_{\rm A}$ $($carrier frequency + ...text{Frequency Modulator}$]] with a binary bipolar rectangular input signal.33 KB (5,164 words) - 18:48, 20 February 2023
- *In the following, all symbols are given in bipolar representation: ...block $\underline{x} = (0,\ 1,\ 1)$ ⇒ in bipolar representation: $(+1, \, -1, \, -1)$.<br>|class=fit]]36 KB (5,613 words) - 18:21, 12 January 2023
- ...al $q(t)$ can be analog as well as digital, just like the sink signal $v(t)$. All other signals in this block diagram, even those not ...|"Parameters of Digital Channel Models"]] in the book "Digital Signal Transmission" describes the modeling of the "digital channel".30 KB (4,551 words) - 17:18, 19 February 2023
- ...he Gilbert-Elliott model (red curve) in linear and logarithmic representation for the parameters ...Gilbert-Elliott model marked with red circles. One can see from this representation:51 KB (8,435 words) - 18:10, 24 October 2022
- *on the one hand for reasons of simplest possible representation, ...aquo;<b>D–transform</b>« belonging to the discrete time signal $\underline{x} = (x_0, x_1, x_2, \ \text{...}) $ reads:41 KB (5,999 words) - 19:50, 2 December 2022
- ...can be signals such as the real and imaginary parts of a phase modulated signal. <br>One recognizes from this representation:23 KB (3,664 words) - 15:38, 21 December 2022
- One can see from this representation: ...or the coaxial cable systems, $H_{β1}(f)$ does not lead to any signal distortion, but only to a time delay by the [[Linear_and_Time_I37 KB (5,664 words) - 19:52, 28 November 2023
- ...raquo;Line Coding« (additional redundancy to adapt the digital signal to the spectral characteristics of the transmission medium).aa ...p; of such a bit error very small, for example by using a very large signal energy. However, the bit error probability $p_{\rm B} = 0$ 37 KB (5,753 words) - 18:26, 18 November 2022
- A speech codec is used to reduce the data rate of a digitized speech or music signal. ...he process, redundancy and irrelevance are removed from the original signal.40 KB (6,238 words) - 18:15, 20 April 2023
- *The direct signal $\rm (DC)$ transmission factor is always assumed to be $H Special cases included in the general representation are:25 KB (3,925 words) - 19:19, 7 November 2023
- *the signal-to-noise ratio $\rm (SNR)$ $\rho= 1/\sigma^2 = 2 \cdot E_{\rm The error probability in double logarithmic representation is shown in the left graph.25 KB (3,813 words) - 16:42, 23 January 2023
- ...uo;'''probability density function'''« $\rm (PDF)$ the representation form $f_X(x)$ often used in the literature, where holds: *On the left is the result of a random process, on the right a deterministic signal with the same amplitude distribution.36 KB (5,684 words) - 15:29, 28 February 2023
- ...smission/Viterbi_Receiver| "Viterbi receiver"]] of the book "Digital Signal Transmission" with regard to its use for equalization. ...tatements in the last section. The tildes again indicate the bipolar representation.}}<br>43 KB (6,415 words) - 16:53, 23 January 2023
