The evolution of wireless cellular technology from 1G to 4G has a similar aim that is capable to deliver high data rate signal so that it can transmit high bit rate multimedia content in cellular mobile communication. Thus, it has driven many researches into the application of higher order modulations.

One of the focuses of this project is to study and compare the different types of Digital Modulation technique that widely being used in the LTE systems. Hence, before being able to design and evaluate this in computer simulation. A study is carried out on digital modulation and drilled down further on QPSK modulation schemes, and followed by the QAM modulation schemes.

### What is modulation?

There are several definitions on modulation taken from several references as follows:

Modulation is defined as the process by which a carrier wave is able to carry the message or digital signal (series of ones and zeroes).

Modulation is the process of facilitating the transfer of information over a medium. Voice cannot be sent very far by screaming. To extend the range of sound, we need to transmit it through a medium other than air, such as a phone line or radio. The process of converting information (voice in this case) so that it can be successfully sent through a medium (wire or radio waves) is called modulation.

Modulation is the process of varying a carrier signal, typically a sinusoidal signal, in order to use that signal to convey information. One of the three key characteristics of a signal is usually modulated: its phase, frequency or amplitude.

There are 2 types of modulations: Analog modulation and digital modulation. In analog modulation, an information-bearing analog waveform is impressed on the carrier signal for transmission whereas in digital modulation, an information-bearing discrete-time symbol sequence (digital signal) is converted or impressed onto a continuous-time carrier waveform for transmission. 2G wireless systems are realized using digital modulation schemes.

### Why Digital Modulation?

The move to digital modulation provides more information capacity, compatibility with digital data services, higher data security, better quality communications, and quicker system availability. Developers of communications systems face these constraints:

- available bandwidth
- permissible power
- inherent noise level of the system

The RF spectrum must be shared, yet every day there are more users for that spectrum as demand for communications services increases. Digital modulation schemes have greater capacity to convey large amounts of information than analog modulation schemes.

### Different types of Digital Modulation

As mentioned in the previous chapter, there are three major classes of digital modulation techniques used for transmission of digitally represented data:

- Amplitude Shift Keying (ASK)
- Frequency Shift Keying (FSK)
- Phase Shift Keying (PSK)

All convey data by changing some aspect of a base signal, the carrier wave (usually a sinusoid) in response to a data signal. For ASK, FSK, and PSK the amplitude, frequency and phase are changed respectively.

### Bit rate and symbol rate

To understand and compare different PSK and QAM modulation format efficiencies, it is important to first understand the difference between bit rate and symbol rate. The signal bandwidth for the communications channel needed depends on the symbol rate, not on the bit rate.

Bit rate is the frequency of a system bit stream. Take, for example, a radio with an 8 bit sampler, sampling at 10 kHz for voice. The bit rate, the basic bit stream rate in the radio, would be eight bits multiplied by 10K samples per second or 80 Kbits per second. (For the moment we will ignore the extra bits required for synchronization, error correction, etc.).

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A Quadrature Phase Shift Keying (QPSK) signal. The states can be mapped to zeros and ones. This is a common mapping, but it is not the only one. Any mapping can be used. The symbol rate is the bit rate divided by the number of bits that can be transmitted with each symbol. If one bit is transmitted per symbol, as with BPSK, then the symbol rate would be the same as the bit rate of 80 Kbits per second. If two bits are transmitted per symbol, as in QPSK, then the symbol rate would be half of the bit rate or 40 Kbits per second. Symbol rate is sometimes called baud rate. Note that baud rate is not the same as bit rate. These terms are often confused. If more bits can be sent with each symbol, then the same amount of data can be sent in a narrower spectrum. This is why modulation formats that are more complex and use a higher number of states can send the same information over a narrower piece of the RF spectrum.

### Phase Shift Keying (PSK)

PSK is a modulation scheme that conveys data by changing, or modulating, the phase of a reference signal (i.e. the phase of the carrier wave is changed to represent the data signal). A finite number of phases are used to represent digital data. Each of these phases is assigned a unique pattern of binary bits; usually each phase encodes an equal number of bits. Each pattern of bits forms the symbol that is represented by the particular phase.

There are two fundamental ways of utilizing the phase of a signal in this way:

- By viewing the phase itself as conveying the information, in which case the demodulator must have a reference signal to compare the received signal’s phase against; (PSK) or
- By viewing the change in the phase as conveying information – differential schemes, some of which do not need a reference carrier (to a certain extent) (DPSK).

A convenient way to represent PSK schemes is on a constellation diagram. This shows the points in the Argand plane where, in this context, the real and imaginary axes are termed the in-phase and quadrature axes respectively due to their 90° separation. 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) wave and the amplitude along the quadrature axis to modulate a sine (or cosine) wave.

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 energy. In this way, the moduli of the complex numbers they represent will be the same and thus so will the amplitudes needed 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. Since the data to be conveyed are usually binary, the PSK scheme is usually designed with the number of constellation points being a power of 2.

#### Applications of PSK and QAM

Owing to PSK’s simplicity, particularly when compared with its competitor quadrature amplitude modulation (QAM), it is widely used in existing technologies.

The most popular wireless LAN standard, IEEE 802.11b, uses a variety of different PSKs depending on the data-rate required. At the basic-rate of 1 Mbit/s, it uses DBPSK. To provide the extended-rate of 2 Mbit/s, DQPSK is used. In reaching 5.5 Mbit/s and the full-rate of 11 Mbit/s, QPSK is employed, but has to be coupled with complementary code keying. The higher-speed wireless LAN standard, IEEE 802.11g has eight data rates: 6, 9, 12, 18, 24, 36, 48 and 54 Mbit/s. The 6 and 9 Mbit/s modes use BPSK. The 12 and 18 Mbit/s modes use QPSK. The fastest four modes use forms of quadrature amplitude modulation.

The recently-standardised Bluetooth will use p / 4-DQPSK at its lower rate (2 Mbit/s) and 8-DPSK at its higher rate (3 Mbit/s) when the link between the two devices is sufficiently robust. Bluetooth 1 modulates with Gaussian minimum shift keying, a binary scheme, so either modulation choice in version 2 will yield a higher data-rate. A similar technology, ZigBee (also known as IEEE 802.15.4) also relies on PSK. ZigBee operates in two frequency bands: 868-915MHz where it employs BPSK and at 2.4GHz where it uses OQPSK.

Notably absent from these various schemes is 8-PSK. This is because its error-rate performance is close to that of 16-QAM – it is only about 0.5dB better – but its data rate is only three-quarters that of 16-QAM. Thus 8-PSK is often omitted from standards and, as seen above, schemes tend to ‘jump’ from QPSK to 16-QAM (8-QAM is possible but difficult to implement).

#### QPSK

QPSK is a multilevel modulation techniques, it uses 2 bits per symbol to represent each phase. Compared to BPSK, it is more spectrally efficient but requires more complex receiver.

#### Constellation Diagram for QPSK

The constellation diagram for QPSK with Gray coding. Each adjacent symbol only differs by one bit. Sometimes known as quaternary or quadriphase PSK or 4-PSK, QPSK uses four points on the constellation diagram, equispaced around a circle. With four phases, QPSK can encode two bits per symbol, shown in the diagram with Gray coding to minimize the BER – twice the rate of BPSK. Figure 2.5 depicts the 4 symbols used to represent the four phases in QPSK. Analysis shows that this may be used either to double the data rate compared to a BPSK system while maintaining the bandwidth of the signal or to maintain the data-rate of BPSK but halve the bandwidth needed.

#### Four symbols that represents the four phases in QPSK

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) bits 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.

As a result, the probability of bit-error for QPSK is the same as for BPSK:

However, with two bits per symbol, the symbol error rate is increased:

If the signal-to-noise ratio is high (as is necessary for practical QPSK systems) the probability of symbol error may be approximated:

As with BPSK, there are phase ambiguity problems at the receiver and differentially encoded QPSK is more normally used in practice.

As written above, QPSK, are often used in preference to BPSK when improved spectral efficiency is required. QPSK utilizes four constellation points, each representing two bits of data. Again as with BPSK the use of trajectory shaping (raised cosine, root raised cosine etc) will yield an improved spectral efficiency, although one of the principle disadvantages of QPSK, as with BPSK, is the potential to cross the origin, that will generate 100% AM.

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QPSK is also known as a method for transmitting digital information across an analog channel. Data bits are grouped into pairs, and each pair is represented by a particular waveform, called a symbol, to be sent across the channel after modulating the carrier. QPSK is also the most commonly used modulation scheme for wireless and cellular systems. It’s because it does not suffer from BER degradation while the bandwidth efficiency is increased. The QPSK signals are mathematically defined as:

#### Implementation of QPSK

QPSK signal can be implemented by using the equation stated below. The symbols in the constellation diagram in terms of the sine and cosine waves used to transmit them is being written below:

This yields the four phases p/4, 3p/4, 5p/4 and 7p/4 as needed. As a result, a two-dimensional signal space with unit basis functions

The first basis function is used as the in-phase component of the signal and the second as the quadrature component of the signal. Therefore, the signal constellation consists of the signal-space 4 points

The factors of 1/2 show that the total power is divide evenly among the two carriers. QPSK systems can be implemented in a few ways.

First, the dual data stream is divided into the in-phase and quadrature-phase components. These are then independently modulated onto two orthogonal basis functions. In this implementation, two sinusoids are used. Next, the two signals are superimposed, and the resulting signal is the QPSK signal. Polar non-return-to-zero encoding is also being used. These encoders can be located before for binary data source, but have been located after to illustrate the theoretical dissimilarity between digital and analog signals concerned with digital modulation. The matched filters can be substituted with correlators. Each detection device uses a reference threshold value to conclude whether a 1 or 0 is detected.

#### Quadrature Amplitude Modulation (QAM)

Quadrature amplitude modulation (QAM) is both an analog and a digital modulation scheme. It is a modulation scheme in which two sinusoidal carriers, one exactly 90degrees out of phase with respect to the other, which are used to transmit data over a given physical channel. Because the orthogonal carriers occupy the same frequency band and differ by a 90degree phase shift, each can be modulated independently, transmitted over the same frequency band, and separated by demodulation at the receiver. For a given available bandwidth, QAM enables data transmission at twice the rate of standard pulse amplitude modulation (PAM) without any degradation in the bit error rate (BER).

QAM and its derivatives are used in both mobile radio and satellite communication systems. The modulated waves are summed, and the resulting waveform is a combination of both phase-shift keying (PSK) and amplitude-shift keying, or in the analog case of phase modulation (PM) and amplitude modulation. In the digital QAM case, a finite number of at least two phases and at least two amplitudes are used. PSK modulators are often designed using the QAM principle, but are not considered as QAM since the amplitude of the modulated carrier signal is constant. In 16 QAM 4 different phases and 4 different amplitudes are used for a total of 16 different symbols. This means such a coding is able to transmit 4bit per second. 64-QAM yields 64 possible signal combinations, with each symbol representing six bits (2^6 = 64). The yield of this complex modulation scheme is that the transmission rate is six times the signaling rate.

This modulation format produces a more spectrally efficient transmission. It is more efficient than BPSK, QPSK or 8PSK while QPSK is the same as 4QAM. Another variation is 32QAM. In this case there are six I values and six Q values resulting in a total of 36 possible states (6×6=36). This is too many states for a power of two (the closest power of two is 32). So the four corner symbol states, which take the most power to transmit, are omitted. This reduces the amount of peak power the transmitter has to generate. Since 25 = 32, there are five bits per symbol and the symbol rate is one fifth of the bit rate. The current practical limits are approximately 256QAM, though work is underway to extend the limits to 512 or 1024 QAM. A 256QAM system uses 16 I-values and 16 Q-values giving 256 possible states. Since 2^8 = 256, each symbol can represent eight bits. A 256QAM signal that can send eight bits per symbol is very spectrally efficient. However, there is some drawbacks, the symbols are very close together and are thus more subject to errors due to noise and distortion. Such a signal may have to be transmitted with extra power (to effectively spread the symbols out more) and this reduces power efficiency as compared to simpler schemes.

BPSK uses 80 K symbols-per-second sending 1 bit per symbol. A system using 256QAM sends eight bits per symbol so the symbol rate would be 10 K symbols per second. A 256QAM system enables the same amount of information to be sent as BPSK using only one eighth of the bandwidth. It is eight times more bandwidth efficient. However, there is a drawback too. The radio becomes more complex and is more susceptible to errors caused by noise and distortion. Error rates of higher-order QAM systems such as this degrade more rapidly than QPSK as noise or interference is introduced. A measure of this degradation would be a higher Bit Error Rate (BER).

In any digital modulation system, if the input signal is distorted or severely attenuated the receiver will eventually lose symbol clock completely. If the receiver can no longer recover the symbol clock, it cannot demodulate the signal or recover any information. With less degradation, the symbol clock can be recovered, but it is noisy, and the symbol locations themselves are noisy. In some cases, a symbol will fall far enough away from its intended position that it will cross over to an adjacent position. The I and Q level detectors used in the demodulator would misinterpret such a symbol as being in the wrong location, causing bit errors. In the case of QPSK, it is not as efficient, but the states are much farther apart and the system can tolerate a lot more noise before suffering symbol errors. QPSK has no intermediate states between the four corner-symbol locations so there is less opportunity for the demodulator to misinterpret symbols. As a result, QPSK requires less transmitter power than QAM to achieve the same bit error rate.

#### Implementation of QAM

First, the incoming bits are encoded into complex valued symbols. Then, the sequence of symbols is mapped into a complex baseband waveform.

For implementation purposes, each complex multiplication above corresponds to 4 real multiplications. Besides, and will be the real and imaginary parts of = + iand assume that the symbols are generated as real and imaginary parts (as opposed to magnitude and phase, for example). After being derived, we will get and. From (1), x (t) becomes.

This can be understand as two parallel PAM systems, followed by “double-sideband” modulation by “quadrature carriers” and. This realization of QAM is called double-sideband quadrature-carrier (DSB-QC) modulation.

A QAM receiver must first demodulate the received waveform y(t). Assuming the scaling and receiver time reference discussed before, this received waveform is assumed to be simply y(t) = x(t) + n(t). Here, it is being understood that there is no noise, so that y(t) is simply the transmitted waveform x(t). The first task of the receiver is to demodulate x(t) back to baseband. This is done by multiplying the received waveform by both and. The two resulting waveforms are each filtered by a filter with impulse response q(t) and then sampled at T spaced intervals.

The multiplication by at the receiver moves the positive frequency part of x(t) both up and down in frequency by, and does the same with the negative frequency part. It is assumed throughout that both the transmit pulse p(t) and the receive pulse q(t) are in fact baseband waveforms relative to the carrier frequency (specifically, that and for). Thus the result of multiplying the modulated waveform x(t) by yields a response at baseband and also yields responses around and.

The receive filter q(t) then eliminates the double frequency terms. The effect of the multiplication can be seen by both at transmitter and receiver from the following trigonometric identity:

Thus the receive filter q(t) in the upper (cosine) part of the demodulator filters the real part of the original baseband waveform, resulting in the output Assuming that the cascade g(t) of the filters p(t) and q(t) is ideal Nyquist, the sampled output retrieves the real part of the original symbols without intersymbol interference. The filter q(t) also rejects the double frequency terms. The multiplication by similarly moves the received waveform to a baseband component plus double carrier frequency terms. The effect of multiplying by at both transmitter and receiver is given by Again, (assuming that p(t) * q(t) is ideal Nyquist) the filter q(t) in the lower (sine) part of the receiver retrieves the imaginary components of the original symbols without intersymbol interference.

Finally, from the identity, there is no crosstalk at baseband between the real and imaginary parts of the original symbols. It is important to go through the above argument to realize that the earlier approach of multiplying u(t) by for modulation and then by for demodulation is just a notationally more convenient way of doing the same thing. Working with sines and cosines is much more concrete, but is messier and makes it harder to see the whole picture.

#### Modulation and transmission of QAM

In general, the modulated signal can be represented by

Where the carrier cos(wct) is said to be amplitude modulated if its amplitude is adjusted in accordance with the modulating signal, and is said to be phase modulated if (t) is varied in accordance with the modulating signal. In QAM the amplitude of the baseband modulating signal is determined by a(t) and the phase by (t). The in phase component I is then given by

This signal is then corrupted by the channel. In this case is the AWGN channel. The received signal is then given by Where n(t) represents the AWGN, which has both the in phase and the quadrature component. It is this received signal which will be attempted to demodulate.

### Reference

- Fundamentals of Communication SystemsDescription: http://i.cmpnet.com/dspdesignline/2008/07/image046.gif, by John G. Proakis, Masoud Salehi Description: http://i.cmpnet.com/dspdesignline/2008/07/image046.gif
- Cross-layer resource allocation in wireless communications: techniques and Models from PHY and MAC Layer Interactionby Ana I. Pérez-Niera, Marc Realp Campalans
- Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmit
- OFDM for wireless multimedia communications by Richard Van Nee, Ramjee Prasad
- Modern Quadrature Amplitude Modulation by W.T Webb and L.Hanzo
- Digital Signal Processing in Communication Systems by Marvin E.Frerking

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