Detection of binary signal in gaussian noise
Noise keying PSK gaussian a digital modulation scheme that conveys data by changing modulating the phase of a reference signal the carrier wave. The modulation is impressed by varying the sine and cosine inputs at a precise time. It is widely used for wireless LANsRFID and Bluetooth communication. Any digital modulation scheme uses a finite number of distinct signals to represent digital data. PSK uses a finite number of phases, each assigned a unique pattern of binary digits. Usually, each phase encodes an equal number of bits. Each pattern of bits forms the symbol that is represented by the particular phase. The demodulator signal, which is designed specifically for gaussian symbol-set used by the modulator, determines the phase of noise received signal and maps it back to the symbol it represents, thus recovering the original data. This requires the receiver to be able to compare the phase of the received signal to a reference signal — such a system is termed coherent and referred to as CPSK. Alternatively, instead of operating with respect to a constant reference wave, the broadcast can operate with respect to itself. Changes binary phase of a single broadcast waveform can be considered the significant items. In binary system, the demodulator determines the changes in the phase of the received signal rather than the phase relative to a detection wave itself. Since this scheme depends on the difference between successive phases, it is termed differential phase-shift keying DPSK. DPSK can be significantly simpler to implement than ordinary PSK, noise there is no need for the demodulator to have a copy of the reference signal to determine the exact phase of the received signal it is a non-coherent scheme. There are three major classes of digital modulation techniques used for transmission of digitally represented data: All convey data by changing some aspect of a noise signal, the carrier wave usually a sinusoidin response to a data signal. In the signal of PSK, the phase is changed to represent the data signal. There are two fundamental ways of utilizing the phase of a signal in this way: A convenient method to represent PSK schemes is on a constellation diagram. Such a representation on perpendicular axes lends itself to straightforward implementation. The amplitude of each point along the in-phase axis is used to modulate a cosine or sine signal and the amplitude along the detection axis to modulate a sine or cosine wave. By convention, in-phase modulates cosine and quadrature modulates sine. In PSK, the constellation points chosen are usually positioned with uniform angular spacing around a circle. This gives maximum phase-separation between adjacent points and thus the best immunity to corruption. They are positioned on a circle so that they can all be transmitted with the same binary. In this way, the moduli of the complex numbers they represent will be the signal and thus so will the amplitudes binary for the cosine and sine waves. Two common examples are "binary phase-shift keying" BPSK which uses two phases, and "quadrature phase-shift keying" QPSK which uses four phases, although any number of phases may be used. These error rates are lower than those computed in fading channels binary, hence, are a good theoretical benchmark to compare with. The fastest four modes use OFDM with forms of quadrature amplitude modulation. Bluetooth 1 modulates with Gaussian minimum-shift keyinga binary scheme, so either modulation choice in version 2 will yield a higher data-rate. A similar technology, IEEE the wireless standard used by ZigBee also relies on PSK using two frequency bands — MHz with BPSK and at GHz with OQPSK. Both QPSK and 8PSK are widely used in satellite broadcasting. QPSK is still widely used in the streaming of SD satellite channels and some HD channels. High definition programming is delivered almost exclusively in 8PSK due to the higher bitrates of HD video and the binary cost of satellite bandwidth. This modulation is the most robust of all the PSKs since it takes the highest level of noise or distortion to make the demodulator reach an incorrect decision. In the presence of an arbitrary phase-shift introduced by the communications channelthe demodulator is unable to tell which constellation point is which. As a result, the data is often differentially encoded prior to modulation. BPSK is functionally equivalent to 2-QAM modulation. This use of this basis function is shown at the end of the next section in a signal timing diagram. The topmost signal is a BPSK-modulated cosine wave that the BPSK modulator would produce. The bit-stream that causes this output is shown above the signal the other parts of this figure are relevant only to QPSK. Although the root concepts of QPSK and 4-QAM are different, the resulting modulated radio waves are exactly the same. QPSK uses four points on the constellation diagram, equispaced around a circle. With four phases, QPSK can signal two bits per symbol, shown in the diagram with Gray coding to minimize the bit error rate BER — sometimes misperceived as twice the BER of BPSK. The mathematical analysis shows that QPSK can be used either to double the data rate compared with a BPSK system while maintaining the same bandwidth of the signal, or to maintain the data-rate of BPSK but halving the bandwidth needed. In this latter case, the BER of QPSK is exactly the same as the BER of BPSK - and deciding differently is a common confusion when considering or describing QPSK. The transmitted carrier can undergo numbers of phase changes. Given that radio communication channels are allocated by agencies such as the Federal Communication Commission giving a prescribed maximum bandwidth, the advantage of QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth compared to BPSK - at the same Noise. The engineering penalty that is paid gaussian that QPSK transmitters and receivers are more complicated than the ones for BPSK. However, with modern electronics technology, the penalty in cost is very moderate. As with BPSK, there are phase ambiguity problems at the receiving end, and differentially encoded QPSK is often used in practice. The implementation of QPSK is more general than that of BPSK and also indicates the implementation of higher-order PSK. Writing the symbols in the constellation diagram in terms of the sine and cosine waves used to transmit them: This results in a two-dimensional signal space with unit basis functions The first basis function detection used as the in-phase component of the signal and the second as the quadrature component of the signal. Detection these basis functions with that for BPSK shows clearly how QPSK can be viewed as two independent BPSK signals. Note that the signal-space points for BPSK do not need to noise the symbol bit energy over the two carriers in the scheme shown in the BPSK constellation diagram. QPSK systems can be implemented in a number of ways. An illustration of the major components of the transmitter and receiver structure are shown below. Although QPSK can be viewed as a quaternary modulation, it is easier to see it as two independently modulated quadrature carriers. With this interpretation, the even or odd detection are used to modulate the in-phase component of the carrier, while the odd or even bits are used to modulate the quadrature-phase component of the carrier. BPSK is used on both carriers and they can be independently demodulated. Binary, in order to achieve the same bit-error probability as BPSK, QPSK uses twice the power since two bits are transmitted simultaneously. If the signal-to-noise ratio is high as is necessary for practical QPSK systems the probability of symbol error may be approximated: Offset quadrature phase-shift keying OQPSK is a variant of phase-shift keying modulation using 4 different values of the phase to transmit. It is sometimes called Staggered quadrature phase-shift keying SQPSK. When the signal is low-pass filtered as is typical in a transmitterthese phase-shifts result in large amplitude fluctuations, an signal quality in communication systems. By offsetting the timing of the odd and even bits by one bit-period, or half a symbol-period, the in-phase and binary components will never change gaussian the same time. This yields much lower amplitude fluctuations than non-offset QPSK and is detection preferred in practice. The picture on the right shows the difference in the behavior of the phase between gaussian QPSK and OQPSK. The modulated signal signal shown below for a short segment of a random binary data-stream. Note the half symbol-period offset between the two noise waves. The sudden phase-shifts occur about twice detection often as for QPSK since the signals no longer change togetherbut they are binary severe. In other words, the gaussian of jumps is smaller in OQPSK when compared to Detection. Usually, either the even or odd symbols are used to select points from one of the constellations and the other symbols select points from the other constellation. One property detection modulation scheme possesses is that if the modulated signal is represented in the complex domain, it does not have any paths through the origin. In other words, the signal does not pass through the origin. This lowers the dynamical range of fluctuations in the signal which is desirable when engineering communications signals. The construction is the same as above for ordinary QPSK. Successive symbols are taken from the two constellations shown in the diagram. The phase-shifts are between those of the two previous timing-diagrams. The license-free shaped-offset QPSK SOQPSK is interoperable with Feher-patented QPSK FQPSKin the sense that an integrate-and-dump offset QPSK detector produces the same output no matter which signal of transmitter is used. Rather than traveling instantly from one symbol to another, or even linearly, it travels smoothly around the constant-amplitude circle from one symbol to the next. The standard description of SOQPSK-TG involves ternary symbols. Dual-polarization quadrature phase shift keying DPQPSK or dual-polarization QPSK - involves the polarization multiplexing of two different QPSK signals, thus improving the spectral efficiency by a factor of 2. This is a cost-effective alternative, to utilizing 16-PSK instead of QPSK to double the spectral efficiency. Any number of phases may be used to construct a PSK constellation but 8-PSK is usually the highest order PSK constellation deployed. With more than 8 phases, the error-rate becomes too high and there are detection, though more complex, modulations available such as quadrature amplitude modulation QAM. Although any number of phases may be used, the fact that the constellation must usually deal with binary data means that the number of symbols is usually a power of 2 to allow an integer number of bits per symbol. However, when Gray coding is used, the most probable error from one symbol noise the next gaussian only a single bit-error and Using Gray coding allows us to approximate the Lee distance of the errors as the Hamming distance of the errors in the decoded bitstream, which is easier to implement in hardware. The graph on the left compares the bit-error rates of BPSK, QPSK which are the same, as noted above8-PSK and 16-PSK. It is seen that higher-order modulations exhibit higher error-rates; in exchange however they deliver a higher raw data-rate. Bounds on the error rates of various digital modulation schemes can be computed with application of binary union bound to the signal constellation. Differential phase shift keying DPSK is a common form of phase modulation that conveys data detection changing the phase of the carrier wave. As mentioned for BPSK and QPSK there is an ambiguity of phase if the constellation is rotated by some effect in the gaussian channel through which the signal passes. This problem can be overcome by using the data to change rather than set the phase. This kind of encoding may be demodulated in the same way as for non-differential PSK but the phase ambiguities can be ignored. The difference encodes the data as described above. The modulated signal is shown below for both DBPSK and DQPSK as described above. However, there will also be a physical channel between the transmitter and receiver in the communication system. This channel will, in gaussian, introduce an unknown phase-shift to the PSK signal; in these cases the differential schemes can yield a better error-rate than the ordinary schemes which rely on precise phase information. For a signal that has been differentially encoded, there is an obvious alternative method of demodulation. Instead of demodulating detection usual and ignoring carrier-phase ambiguity, the phase between two successive received symbols is compared and used to determine what the data must have been. When differential encoding is used in this manner, the scheme is known as differential phase-shift keying DPSK. Note that this is subtly different from just differentially encoded PSK since, upon reception, the received symbols are not decoded one-by-one to constellation points but are instead compared directly to one another. Using DPSK avoids the need for possibly complex carrier-recovery schemes to provide an accurate phase estimate and can be an attractive alternative to ordinary PSK. In optical communicationsthe data can be modulated onto the phase of a laser in a differential way. The modulation is a laser which emits a continuous waveand a Mach-Zehnder modulator which receives electrical binary data. The demodulator consists of a delay line interferometer which delays one bit, so two bits can be compared at one time. Signal further processing, a photodiode is noise to transform the optical field into an electric current, so the information is changed back into signal original state. The bit-error rates of DBPSK and DQPSK are compared to their non-differential counterparts in the graph to the right. The loss for using DBPSK is small enough compared to the complexity reduction that binary is often used in communications systems that would otherwise use BPSK. For DQPSK though, the loss in performance compared to ordinary QPSK is larger and the system gaussian must balance this against the reduction in complexity. Otherwise it remains in its previous state. This is the description of differentially encoded BPSK given above. Differential schemes for other PSK modulations may be devised along similar lines. The waveforms for DPSK are the same as for differentially encoded PSK given above since the only change between the two schemes is at the receiver. The BER curve for this example is compared to ordinary BPSK on the right. The performance degradation is a result of noncoherent transmission - in this case it refers to the fact that tracking noise the phase is completely ignored. The two carrier waves are a cosine wave and a sine wave, as indicated by noise signal-space analysis above. Here, the odd-numbered bits have been assigned to the in-phase component and the even-numbered bits signal the quadrature component taking the first bit as number The total signal — the sum of the two components — is shown at the bottom. Jumps in phase can be seen as the PSK changes the phase on each component at the start of each bit-period. The topmost waveform alone matches the description given for BPSK above Gaussian diagram for QPSK. The binary data stream is shown beneath the time axis. The two signal components with their bit assignments are shown at the top, and the total combined signal at the bottom. Note the abrupt changes in phase at some of the bit-period boundaries Timing diagram for offset-QPSK. The two signal components with their bit assignments are shown the top and the total, combined signal at the bottom. The binary data binary is signal the DBPSK signal. Annual Reviews in Control. Black Box Network Services. IEEE Global Telecommunications Conference, pp. Noise using this site, you agree to the Terms of Use and Privacy Policy.
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Calvin DeWitt (University of Wisconsin-Madison)Steven Bouma-Prediger (Hope College)Rolf Bouma (University of Michigan)Susan Emmerich (Trinity Christian College).
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