Quadrature Amplitude Modulation free essay sample

Because of the  orthogonality  property of the carrier signals, it is possible to detect the modulating signals independently. In the ideal case  I(t)  is demodulated by multiplying the transmitted signal with a cosine signal: [pic] Using standard  trigonometric identities, we can write it as: [pic] Low-pass filtering  ri(t)  removes the high frequency terms (containing  4? f0t), leaving only the  I(t)  term. This filtered signal is unaffected by  Q(t), showing that the in-phase component can be received independently of the quadrature component. Similarly, we may multiply  s(t)  by a sine wave and then low-pass filter to extract  Q(t). The phase of the received signal is assumed to be known accurately at the receiver. If the demodulating phase is even a little off, it results in  crosstalk  between the modulated signals. This issue ofcarrier synchronization  at the receiver must be handled somehow in QAM systems. The coherent demodulator needs to be exactly in phase with the received signal, or otherwise the modulated signals cannot be independently received. We will write a custom essay sample on Quadrature Amplitude Modulation or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page For example  analog television  systems transmit a burst of the transmitting colour subcarrier after each horizontal synchronization pulse for reference. Analog QAM is used in  NTSC  and  PAL  television systems, where the I- and Q-signals carry the components of chroma (colour) information. Compatible QAM or  C-QUAM  is used in  AM stereo  radio to carry the  stereo difference  information. [edit]Fourier analysis of QAM In the  frequency domain, QAM has a similar spectral pattern to  DSB-SC  modulation. Using the  properties of the Fourier transform, we find that: [pic] here  S(f),  MI(f) and  MQ(f) are the Fourier transforms (frequency-domain representations) of  s(t),  I(t) and  Q(t), respectively. [edit]Quantized QAM [pic] [pic] Digital 16-QAM with example constellation points. Like many digital modulation schemes, the  constellation diagram  is a useful representation. In QAM, the constellation points are usually arranged in a squa re grid with equal vertical and horizontal spacing, although other configurations are possible (e. g. Cross-QAM). Since in digitaltelecommunications  the data are usually  binary, the number of points in the grid is usually a power of 2 (2, 4, 8 . Since QAM is usually square, some of these are rare—the most common forms are 16-QAM, 64-QAM and 256-QAM. By moving to a higher-order constellation, it is possible to transmit more  bits  per  symbol. However, if the mean energy of the constellation is to remain the same (by way of making a fair comparison), the points must be closer together and are thus more susceptible to  noise  and other corruption; this results in a higher  bit error rate  and so higher-order QAM can deliver more data less reliably than lower-order QAM, for constant mean constellation energy. If data-rates beyond those offered by 8-PSK  are required, it is more usual to move to QAM since it achieves a greater distance between adjacent points in the I-Q plane by distributing the points more evenly. The complicating factor is that the points are no longer all the same amplitude and so the  demodulator  must now correctly detect both  phase  and  amplitude, rather than just phase. 64-QAM and 256-QAM are often used in  digital cable  television and  cable modem  applications. In the United States, 64-QAM and 256-QAM are the mandated modulation schemes for  digital cable  (see  QAM tuner) as standardised by the  SCTE  in the standard  ANSI/SCTE 07 2000. Note that many marketing people will refer to these as QAM-64 and QAM-256. In the UK, 16-QAM and 64-QAM are currently used for  digital terrestrial television  (Freeview  and  Top Up TV) and 256-QAM is planned for Freeview-HD. Communication systems designed to achieve very high levels of  spectral efficiency  usually employ very dense QAM constellations. One example is the  ITU-T  G. n  standard for networking over existing home wiring (coaxial cable,  phone lines  and  power lines), which employs constellations up to 4096-QAM (12 bits/symbol). Another example is  VDSL2  technology for copper twisted pairs, whose constellation size goes up to 32768 points. [edit]Ideal structure [edit]Transmitter The following picture shows the ideal structure of a QAM transmitter, wit h a  carrier frequency  f0  and the frequency response of the transmitters filter  Ht: [pic] First the flow of bits to be transmitted is split into two equal parts: this process generates two independent signals to be transmitted. They are encoded separately just like they were in an  amplitude-shift keying  (ASK) modulator. Then one channel (the one in phase) is multiplied by a cosine, while the other channel (in quadrature) is multiplied by a sine. This way there is a phase of 90 ° between them. They are simply added one to the other and sent through the real channel. The sent signal can be expressed in the form: [pic] where  vc[n]  and  vs[n]  are the voltages applied in response to the  nth  symbol to the cosine and sine waves respectively. [edit]Receiver The receiver simply performs the inverse process of the transmitter. Its ideal structure is shown in the picture below with  Hr  the receive filters frequency response  : [pic] Multiplying by a cosine (or a sine) and by a low-pass filter it is possible to extract the component in phase (or in quadrature). Then there is only an  ASK  demodulator and the two flows of data are merged back. In practice, there is an unknown phase delay between the transmitter and receiver that must be compensated by  synchronization  of the receivers local oscillator, i. e. the sine and cosine functions in the above figure. In mobile applications, there will often be an offset in the relative  frequency  as well, due to the possible presence of a Doppler shift proportional to the relative velocity of the transmitter and receiver. Both the phase and frequency variations introduced by the channel must be compensated by properly tuning the sine and cosine components, which requires a  phase reference, and is typically accomplished using a  Phase-Locked Loop (PLL). In any application, the low-pass filter will be within  hr  (t): here it was shown just to be clearer. [edit]Quantized QAM performance The following definitions are needed in determining error rates: M  = Number of symbols in modulation constellation Eb  = Energy-per-bit Es  = Energy-per-symbol =  kEb  with  k  bits per symbol N0  =  Noise  power spectral density  (W/Hz) Pb  =  Probability  of bit-error Pbc  = Probability of bit-error per carrier Ps  = Probability of symbol-error Psc  = Probability of symbol-error per carrier [pic]. Q(x)  is related to the  complementary Gaussian error function  by:  [pic], which is the probability that  x  will be under the tail of the Gaussian  PDF  towards positive  infinity. The error rates quoted here are those in  additive  white  Gaussian noise  (AWGN). Where  coordinates  for constellation points are given in this article, note that they represent a  non-normalised  constellation. That is, if a particular mean average energy were required (e. g. unit average energy), the constellation would need to be linearly scaled. [edit]Rectangular QAM [pic] [pic] Constellation diagram  for rectangular 16-QAM. Rectangular QAM constellations are, in general, sub-optimal in the sense that they do not maximally space the constellation points for a given energy. However, they have the considerable advantage that they may be easily transmitted as two  pulse amplitude modulation  (PAM) signals on quadrature carriers, and can be easily demodulated. The non-square constellations, dealt with below, achieve marginally better bit-error rate (BER) but are harder to modulate and demodulate. The first rectangular QAM constellation usually encountered is 16-QAM, the constellation diagram for which is shown here. A  Gray coded  bit-assignment is also given.