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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.
3.1 What is modulation?
There are several definitions on modulation taken from several references as follows:
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. 
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 defined as the process by which a carrier wave is able to carry the message or digital signal (series of ones and zeroes). 
Modulations can be divided into two types:
Analog modulation and digital modulation. In analog modulation, an information-bearing analog waveform is impressed on the carrier signal for transmission while in digital modulation; an analog carrier signal is modulated by a digital bit stream. Digital modulation methods can be considered as digital-to-analog conversion, and the corresponding demodulation or detection as analog-to-digital conversion. The changes in the carrier signal are chosen from a finite number of M alternative symbols (the modulation alphabet). 4G wireless systems are realized using digital modulation schemes.
3.2 Advantages of Digital Modulation
The move to digital modulation provides higher data security, better quality communications, more information capacity, compatibility with digital data services, and quicker system availability. There are four primary criteria of choosing modulation schemes:
inherent noise level of the system
The RF spectrum must be shared, yet the demand for communications services is increasing as there are more users for that spectrum every day. Digital modulation schemes have greater capacity to convey large amounts of information than analog modulation schemes. This will help to meet the increasing demand of the communications services.
3.3 Different types of Digital Modulation
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.
3.4 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.).
Figure 3.1 Bit Rate & Symbol Rate
Figure 3.1 is an example of a state diagram of 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 the reason 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.
3.5 Phase Shift Keying (PSK)
PSK is a large class of digital modulation schemes and is widely used in the communication industry. PSK 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 (as shown in figure 3.2 below). 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.
Figure 3.2 Constellation Diagram
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.
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.
Figure 3.3 Constellation Diagram for QPSK
Figure 3.3 above shows 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 3.4 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.
Figure 3.4 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 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. 
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 .
18.104.22.168 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 Ï€/4, 3Ï€/4, 5Ï€/4 and 7Ï€/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. An illustration of the major components of the transmitter and receiver structure is shown below.
Figure 3.5 QPSK theoretical transmitter structure
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. 
Figure 3.6 QPSK receiver structure
3.6 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 90 degrees 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 90 degree 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.
Figure 3.7 QAM Constellations
Each constellation point in 64-level QAM (64-QAM) system is represented by a unique 6-bit symbol (26 = 64), which is Gray coded to minimize the decoded error probability. The yield of this complex modulation scheme is that the transmission rate is six times the signaling rate. The complex phasors of the constellation are decomposed into 8-level I and Q components. The amplitudes 7d, 5d, 3d, d, -d, -3d, -5d and -7d of the I and Q AM signals are assigned the 3-bit Gray codes 011, 010, 000, 001, 101, 100, 110 and 111, respectively. The three I and Q bits are denoted by i1, i2, i3 and q1, q2, q3 respectively. These bits are interleaved to give a 6-bit QAM symbol represented by i1, q1, i2, q2, i3, q3.  The 64-level QAM phasors are transmitted over the channel, where they become corrupted. They are demodulated using the decision boundaries shown in Figure 3.8.
The bits i1, q1, 12, q2, i3, q3, are recovered according to
if I, Q â‰¥ 0 then i1, q1 = 0
if I, Q < 0 then i1, q1 = 1,
For the most significant bits, and
if I, Q â‰¥ 4d then i2, q2 = 1
if -4d â‰¤ I, Q < 4d then i2, q2 = 0
if -4d > I, Q then i2, q2 = 1
For the next most significant bits, while finally for the least significant bits
if I, Q â‰¥ 6d then i3, q3 = 1
if 2d â‰¤ I, Q < 6d then i3, q3 = 0
if -2d â‰¤ I, Q < 2d then i3, q3 = 1
if -6d â‰¤ I, Q < -2d then i3, q3 = 0
if -6d > I, Q then i3, q3 = 1
Similarly to 16-QAM the position of the bits in the 6-bit QAM symbol has an effect on their error probabilities. In case of the i1, q1 bits for example, the phasor can be at a distance d, 3d, 5d or 7d from the decision boundary represented by the coordinate axis. Therefore their average protection distance is 16d/4 = 4d. 
Figure 3.8 Square 64-QAM phasor constellation.
3.6.3 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. The baseband waveform is then modulated up to pass band to get the transmitted waveform, given by
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 (3.6), x (t) becomes. (3.7)
This can be understand as two parallel PAM systems, followed by “double-sideband” modulation by “quadrature carriers” and. These are then summed (with the usual factor of 2), as shown in Figure 3.9. 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 resulting DSB-QC receiver is shown in Figure 3.10. 
Figure 3.9 DSB-QC modulation
Figure 3.10 DSB-QC demodulation
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. 
3.7 Noise and Interference
3.7.1 Additive White Noise Gaussian (AWGN)
The term noise refers to unwanted electrical signals that are always present in electrical systems . The term additive means the noise is superimposed or added to the signal that tends to obscure or mask the signal where it will limit the receiver ability to make correct symbol decisions and limit the rate of information transmission. Thus, AWGN is the effect of thermal noise generated by thermal motion of electron in all dissipative electrical components i.e. resistors, wires and so on . Mathematically, thermal noise is described by a zero-mean Gaussian random process where the random signal is a sum of Gaussian noise random variable and a dc signal that is
where pdf for Gaussian noise can be represented as follows where is the variance of n.
A simple model for thermal noise assumes that its power spectral density Gn (f) is a flat for all frequencies and is denoted as
where the factor of 2 is included to indicate that Gn (f) is a two-sided power spectral density. When noise power has such a uniform spectral density, it is referred as white noise. The adjective “white” is used in the same sense as it is with white light, which contains equal amounts of all frequencies within the visible band of electromagnetic (EM) radiation.
Since thermal noise is present in all communication systems and is a prominent noise source for most system, the thermal noise characteristics that are additive, white and Gaussian are most often used to model the noise in communication systems. 
3.8 Analysis of QPSK and 64-QAM
3.8.1 Error Performance (BER)
An important reference for the assessment of any modulation scheme is the bit error probability or bit error rate (BER) for the corresponding uncoded system. Unfortunately, for most non binary modulation techniques (e.g. M-QAM and M-PSK) an exact expression for BER is hard to find.
At high SNR and using Gray mapping, it is commonly assumed that an erroneous detected symbol differs from the correct one in only one bit. Consequently, the BER is approximated by the symbol error rate (SER) divided by the number of bits per symbol b.
Figure 3.11 compares different modulation method including QPSK (4-QAM) and 64-QAM on the basis of bit error probability versus. It shows that the bit error probability for these systems gets worse as M gets larger. This can be attributes to the signal points being crowded closer together in the two dimensional signal spaces with increasing M. In this figure, it can be observed that the BER for 64-QAM is considerably higher compared to QPSK (4-QAM). This is because 64-QAM is more susceptible to noise because the states are closer together so that a lower level of noise is needed to move the signal to a different decision point. 
Figure 3.11: Bit Error Rate vs SNR for various digital modulation schemes.
The overall BER can be improved in exchange for a slight loss of a system throughput or by adapting the transmission technique to the channel conditions on a timeslot-by-timeslot basis for serial modems in narrowband fading channels. This method has been shown to considerably improve the BER performance.
Listed below is the symbol that will be used to describe the performance equation.
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
Even though QPSK can be viewed as a quaternary modulation, however it is simpler to see it as two separately modulated quadrature carriers. With this understanding, 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. As a result, the probability of bit-error for QPSK is given as: 
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:
For 64-QAM bit error rate, the equation for M-QAM with square constellations is used; the BER approximation is given by,
In order to get the BER rate for 64-QAM, the value of M is being substituted with 64.
3.8.2 Power Consumption and Average Energy
The constellations of QPSK and 64-QAM are not normalized; in order to normalize them to an average power of one, each constellation has to be multiplied by the normalization factor listed in Table 3.1. This is to assume that all constellation points are equally likely to occur. Normalization factor for 64-QAM can be obtained from the derivation below:
In a general M-QAM constellation where and are even, the alphabets used are:
For example, considering a 64-QAM () constellation, and the alphabets are
To compute the average energy of the M-QAM constellation:
The sum of energy of the individual alphabets is being calculated.
Each alphabet is used times in the M-QAM constellation.
Thus, to find the average energy from constellation symbols, divide the product of (3.17) and above by M. The average energy is,
Plugging in the number for 64-QAM, will get
This is how we get the normalization factor of for 64-QAM.
Table 3.1 also gives the loss in the minimum squared Euclidean distance between two constellation points, divided by the gain in data rate of that particular QPSK/QAM type relative to BPSK. This value defines the maximum loss in Eb/No relative to BPSK that is needed to achieve a certain bit-error ratio in an uncoded QPSK/QAM link. The difference between QPSK and 64-QAM is about 8.45dB. While for larger constellation sizes, the Eb/No penalty of increasing the number of bits per symbol by 1 converge to 3dB.  This difference can be seen in the simulation result on the next chapter.
Maximum Eb/No loss relative to BPSK in dB
Table 3.1 QPSK, QAM normalization factors and normalized Euclidean distance differences. 
QPSK modems typically consume less power and require less processing than other technologies’ modems. In comparison, 64 QAM modulators and demodulators consume more power than QPSK modulators and demodulators.
Two schemes are used for reducing the peak-to-average power ratio of various modulation formats. First, data translation codes are used to avoid data sequences which cause large peaks in the transmitted signal. This approach was found to be most productive in quadrature amplitude modulation (QAM) formats. A reduction of up to 2.0 dB reduction in peak to- average power is obtained for 16 QAM modulations at the cost of a small decrease (10-15%) in throughput or, alternatively, a slight increase in occupied bandwidth.
Another scheme, an adaptive peak suppression algorithm is presented which further reduces the peak-to-average power ratios of the PSK and generalized PSK formats. It was found that using this scheme on the generalized PSK format produced a 4-bit/symbol modulation technique which was 2.8-4.0 dB superior to 16 QAM in terms of peak power over a Rayleigh fading channel.
3.8.3 Symbol Error Rate vs Eb/No
Figure 3.12: Symbol Error Rate vs SNR per bit (Eb/No) for digital modulation schemes
The ratio is a standard quality measurement for digital communications system performance. The ratio expresses the bit-energy-to-noise-density of the received signal. can be considered a metric that characterizes the performance of one communication system over another; the smaller required the more efficient is the system modulation and detection process for a given probability of error.
There is another related metric which is the ratio of symbol-energy-to-noise-density of the received signal. The relation between bit energy and symbol energy is reasonably straight forward. For M-PSK/M-QAM modulation, the number bits in each constellation symbol is,
Since each symbol carriesbits, the symbol to noise ratio is times the bit to noise ratio, in example, . Plugging in the above formula, the symbol Error Rate vs Bit Energy (SNR per bit, Eb/No) is given as,
3.8.4 Bandwidth Efficiency
When bandwidth efficiency is of primary importance, modulation schemes with small signal-space dimensionality per bit transmitted () are necessary. M-PSK or M-QAM is one such alternative. For large M, however, arrangement of the M constellation points on a circle becomes progressively less energy efficient. From Table 3.2, in the comparison between the two modulation methods, QPSK’s practical bandwidth efficiency is low which is around 2bps/Hz while for 64 QAM’s practical bandwidth efficiency is around 6 bps/Hz/s.
The minimum required bandwidth for transmitting symbols with symbol period T without causing intersymbol interference (ISI) is Hz. However, the spectral efficiency can be improved by either,
Filtering the unwanted half of the bandwidth from the passband PAM, resulting in a bandwidth requirement of Hz- called single
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