Quadrature amplitude modulation

Quadrature amplitude modulation (abbreviation: QAM, English Quadrature Amplitude Modulation) is a modulation method in the electronic communication equipment that combines amplitude modulation and phase modulation. It is counted in the literature mainly to the digital modulation method, although of the analog quadrature amplitude modulation exist among the label quadrature modulation.

General

In QAM, the carrier angular frequency ω with used twice with 90 ° phase shift. On two separate baseband signals to be modulated by means of a multiplicative mixture. Subsequently, the two modulated signals are added in order to obtain the transmission signal. The two baseband signals are in the English literature also referred to as in-phase component I is and Q quadrature component, of which the name is derived IQ modulation. It does not matter whether the two baseband signals are time and continuous values ​​, such as for example, represent the analog color difference signals in analog television, or in the context of the digital QAM a continuous sequence of symbols.

The two baseband signals I and Q can be chosen independent or dependent of each other. If the two basic signals depend on each other according to certain rules, one no longer speaks of a QAM, although an unchanged modulator structure is present. By the type of dependence can be with the modulator structure, all linear and non-linear modulation schemes, such as amplitude modulation ( AM), angle modulation, such as frequency modulation (FM) or Einseitenbandmodulationen as SSB or VSB modulation to realize. Software defined radios make use of the fact that fact.

Carry the two baseband signals I and Q are mutually independent information, strictly speaking, only in this case one speaks of the QAM, must be available for demodulation in the receiver, the carrier not only at the same frequency as in the modulator, but also in an identical phase. This is also referred to as a coherent demodulation. When not properly phase the two independent parts of the base band signals are superimposed, and would prevent proper reconstruction of the transmitted signal at the receiver due to their independence.

The correct phase position is ensured by additional procedures that go beyond the modulation method; the method to depend on the particular application. For example, in the analog QAM method burst signals as the color difference signals, or by using additional pilot tones. In the field of digital signal transmission periodically special synchronization sequences are transmitted in the data stream, which are known to the receiver. The receiver adjusts during synchronization as long as the phase position in the demodulator to the known synchronization sequences are received appropriately.

Because of the greater complexity of circuitry on the receiver side is the QAM modulation and analog quadrature modulation only in specific areas of application. Examples are the AM stereo process and the transfer of the two color difference signals in the analog color television of the NTSC or the PAL method.

The majority of applications of the QAM in the range of the digital signal transmission, in which digital data streams are allocated first to the I- and Q - branch. The individual bits are assigned to specific symbols and that symbol sequences converted by pulse shaping filter in a continuous waveform of the baseband I and Q signals.

Further developments of the QAM in the context of digital signal processing result of the coded modulation, such as trellis coded modulation (TCM ), where the channel coding, for example a convolutional code with the modulation method, such as, inter alia, the QAM functional " melts ".

Applications of the QAM in the context of digital signal processing, for example, lie in modems for data transfer and in the area of multi-carrier techniques such as DSL technology or digital terrestrial television according to the standard DVB -T.

Mathematical Background

The transmitted signal s (t), as shown in the illustration, by the following relationship

Of the two base band signals I (t ) and Q ( t) is formed in the modulator. Represents the angular frequency of the carrier frequency f

The demodulation requires an identical set to the transmitter phase. If there is an interruption-free transmission channel, the received signal r ( t) is equal to the transmitted signal s (t ), otherwise get the error components to the received signal to:

The error signal is described, inter alia, by the channel model. In the error- free case applies to the recovery of the baseband signal:

This results in the signal in addition to the desired baseband signal in addition mixing products with double frequency. These upper, unwanted frequency components are filtered by a low pass filter of ( TP) so that the original signal is formed at the output of the demodulator.

The formation of proceeds analogously:

And a subsequent low-pass filtering to form. And wherein the occurring constant factor of 1 /2 can be compensated for by a gain.

Quantized QAM

The quantized QAM extends the QAM generally shown above for method for transferring value-and time-discrete signal sequences, also known as a digital signal.

Constellation diagram

A distinction is made between digital QAM orthogonal grids and non-orthogonal grids. The two baseband I and Q signals in bandpass location are always orthogonal to each other, which allows the display of the symbols in the complex plane in the form of a constellation diagram. Is the phase aligned correctly on the phase position of the transmitter at the receiver, there is a rotation of the constellation diagram in the complex plane, with the result of corresponding reception error.

The number of available symbols, they represent points or regions in the complex plane is is expressed in the form of a number. For example, in the specification for a 64 QAM having a circumference of 64 QAM symbols.

The number of symbols in the complex I / Q plane is a power of two, in order to assign the individual symbols of a certain number of bits for binary communications. A high spectral efficiency, and thus this noise ratio (SNR ) is defined by a sufficiently large signal to noise ratio possible, a large number of symbols is used. Examples of QAM constellations with an even number of bits in binary symbol assignment are:

• Bit 2: the 4- QAM - this is identical to the QPSK or 4 -PSK is used, and 4 points in a 2 x 2 grid or on a circle, the application with DVB -S.
• Bit 4: For 16- QAM are 16 symbols used, such as use in ITU -R standard V.29 and DVB- T
• 6-bit: On 64- QAM 64 symbols are used, application for DVB-C and DVB- T
• 8 bits: At 256 -QAM 256 symbols are used, application for DVB-C
• 10 bit: 1024 -QAM
• 12-bit 4096 - QAM. This is the largest part of the specification of DVB- C2 currently angedachte QAM constellation, which under the best conditions - is barely detectable - at a signal to noise ratio of 36 dB. Also, the ITU- T G.hn standard used in addition to an extensive channel coding, a 4096 -QAM

In the case of an odd number of bits per symbol, the required power of two is allocated to grid can be achieved by a reduction of that constellation space which is spanned by the next square. It is connected thereto, a deterioration of the error rate, in many cases, which is why these QAM constellations are used less frequently. An example of use is in combination with the low- density parity - check codes ( LDPC ) before an 8- QAM, where in combination with LDPC coding a better overall efficiency than with other QAM constellations results.

• Bit 3: In 8- QAM, the 8 points (9 points in a 3 x 3 grid, less the central position result in the required positions 8 - this modulation is similar to the 8 -PSK)
• 5 bits: For 32 -QAM the 32 points (36 points in a 6 × 6 grid less per one position per vertex yield the required 32 positions )

The non-orthogonal grid arrangements may offer advantages in QAM at the symbol synchronization at the receiver. At high error rate, therefore, certain symbols can be relatively easily removed from the constellation diagram dynamically. The bit rate associated reduction reduces the probability of transmission errors in the useful signal. The disadvantage of all non-orthogonal QAMs, however, compared to the orthogonal symbol arrangement in a poorer spectral efficiency. This is a result of the abandonment of the use of the highest possible packing density of the symbol arrangement in the complex plane.

For areas with high noise levels constellations are chosen with low symbol number. In almost all cases, an additional channel coding comes as a convolutional code for correction of transmission errors apply.

Constellation points of a 256- QAM

By not properly adjusted phase angle by 3.57 degree rotated constellation points of a 256- QAM. Even this slight rotation would lead to massive 256- QAM transmission errors

Non-orthogonal as QAM is used in fax machines, according to standard V.29

• 4 -QAM symbols received with additive Gaussian noise ( AWGN) of variance 0.01, 0.1 and 0.3

Noise variance 0.1

Noise variance 0.3

In each case, 5000 noisy reception values ​​per 4- QAM symbol were produced. At a variance of the noise of receiving values ​​0.3 are visible which are not in the appropriate quadrant. In a decision this leads to a symbol error. The number of incorrectly transmitted bits depends on the employed bit symbol mapping ( bit mapping).

Assignments

The assignment of bit strings to the individual transmit symbols in the constellation diagram can be temporally static or dynamic according to certain rules.

A static allocation is usually done so that adjacent symbols as possible differ only by one bit, as shown in the adjacent figure. For the systematic distribution of the bit sequences in the constellation diagram can be used, among other things, the Gray code. The essential characteristic of this code is that only one bit changes at each step. The use of the gray code assumes the use of a square constellation.

This assignment allows for efficient correction of transmission errors. The signal is in transmission superimposed by noise, which leads to dispersion of the signal points. For normal probability distributions of the noise (such as Gaussian distribution ), it is most likely that a signal point is shifted in the vicinity of an immediately adjacent signal point. By Gray coding ensures that in case such errors, only one bit is wrong. The absolute number of bit errors is minimized and an optionally existing forward error correction (ECC ) more accessible.

Dynamic mappings apply in the context of coded modulation. In these cases, the allocation of the bit pattern of the previous states, inter alia, or combinations of symbols is dependent. An example of a dynamic bit allocation in the constellation diagram, the already mentioned at Trellis Coded Modulation dar.

Modulator structure

In the quantized QAM modulator represented above general structure for the transmission of data sequences to the following function blocks is extended, as shown in the adjacent diagram:

• A splitter that divides the data stream generated by the data source S into two data streams I and Q each with half the bit rate.
• A pulse generator is ever provided for each branch. This allocates a specific number of bits, depending on the symbol extent a specific signal level at its output. For example, 2 bits are at 16 -QAM per branch summarized by the pulse generator and then form 4 stages with the bipolar level of -3, -1, 1 and 3 These numerical values ​​correspond directly to the corresponding deflections on the I- and Q -axis in the constellation diagram.
• In the sudden change between two different levels occurs at the output of the pulse generator on a jump that has unwanted spectral noise components. To minimize these pulses are "formed" in a pulse shaping filter H in order to obtain as uniform as possible over the I and Q component. For pulse-shaping filter, such as the raised cosine filters or Gaussian filters.

The pulse width, the information of 4 bits for 16- QAM carries, depends on the bandwidth of the transmission channel. If a large bandwidth available, the symbols can be transmitted in time scarcer than at low bandwidth. For a given bandwidth, the maximum symbol rate is determined by the inter-symbol interference, which may occur not just in the ideal case, because it prevents the differentiation of the temporally consecutive symbols.

Demodulator structure

The receiver quantized QAM is the counterpart of the transmitter represents the baseband signals obtained from the low pass filters the I and Q are each supplied to a matched filter H. The transfer function of this filter is designed to pulse shapes formed by the transmitter and thus allows an optimal interference suppression.

These two signals ( ADC) to a respective analog -to-digital converter is supplied. High levels of the different transmitting-end pulse generator are converted in the next step into the corresponding bit combination. Thereafter, the composition of the two streams for further data processing.