Implementation of Digital Modulation Techniques on MatlabCHAPTER 1INTRODUCTIONIn communication systems, major requirement is to use such modulation/demodulation technique, which consumes less bandwidth of channel and made the receiver side simple to estimate symbol timing so that optimal symbol decisions may be made. Today with the development of high speed and computationally powerful digital signal processing (DSP) chips there is increasing interest in moving digital communication functions to the digital domain. Implementing modems with an all-digital design may reduce front-end analog circuitry and decrease the burden on the analog-to-digital (A/D) converter while increasing the computational burden of the DSP. In this project Gaussian Minimum Shift Keying technique, is used for modulation and demodulation, which is currently being in use in GSM and DECT systems. Receiver algorithms have been implemented which operates on sampled data to recover symbol timing

1.1

Problem Description

For optimal performance, the incoming communications signal must be sampled at the instant of least inter-symbol interference (ISI). This is critical for GMSK where ISI is purposely introducing in order to compress the bandwidth of the signal. It is common to use either analog or digital processing to adjust the phase of the A/D so that the received signal is strobe at the appropriate instant. An alternative approach relies on asynchronous, digitized data from a free-running A/D converter, to recover estimates of the synchronous samples. Since the transmitter and receiver are unlikely to be perfectly synchronized it is necessary for the receiver to continuously track and correct for any timing discrepancies between the signaling rate and the A/D clock. When many samples are taken per symbol,

the strobe sample may be estimated by computing a timing error and selecting the sample closest to the strobe instant. Over-sampling, however, may reduce the data rate of the system. On the other hand, if sampling is reduced to a few times per symbol period, performance may be significantly affected. A potential solution is to determine interpolated samples from the real, asynchronous samples. Interpolation, however, introduces additional computational complexity.

1.2

Background on Symbol Synchronizationcodeword synchronization, frame synchronization; clock

In digital communications there are distinct types of synchronization, including carrier synchronization, synchronization, etc. Several papers have studied the issue of clock or symbol synchronization and various algorithms have been proposed for performing timing recovery. In the past these algorithms have focused on updating the sampling frequency or phase of the A/D converter with a feed-forward or feed-back loop to reduce the timing error. Floyd Gardner proposed a simple algorithm for detection of a timing error for synchronous, BPSK/QPSK data. Gardner's algorithm also requires just two samples per symbol, but the algorithm uses a total of only three samples for computing the timing error. One of the two samples is assumed to be the synchronized strobe sample.

1.3

Project Outline

The challenge of this project is to design and implement a system that will generate and detect a GMSK signal in real-time so that the effects of asynchronous sampling on modem performance may be examined. The setup must also provide a method for simulating channel effects as well as a way to test the performance of the system. The purpose of this project is to evaluate the performance of a one-bit deferential detection of a GMSK signal using Gardner's algorithm for phase correction with over sampling.

Performance will be compared for the two cases of over-sampling and Nyquist sampling with a free running A/D clock. The trade-off between sampling rate and computational complexity will be examined for the two approaches. In order to realize this experiment, a system has been designed and constructed which includes a GMSK modulator, demodulator, A/D clock. Chapter II discusses the basic concepts of digital communications, which help us in this project. Chapter III discusses the specifics of GMSK modulation and demodulation and Chapter IV describes the Matlab algorithm for generating the random data. In addition Chapter IV also goes into detail about the synchronization and interpolation algorithms.

CHAPTER 2DIGITAL COMMUNICATION SYSTEMDigital communication systems are becoming increasingly attractive because of the ever growing demand for data communications and because digital transmission offers data processing options and flexibilities not available with analog transmission. Digital system is often treated in the context of a mobile radio system, in which case signal transmission typically suffers from a phenomenon called fading, the principal feature of the digital communication system (DCS) is that during a finite interval of time, it sends a waveform from a finite set of possible waveforms, in contrast to an analog communications systems, which sends a waveform from an infinite variety of waveform shapes with theoretically infinite resolution. In DCS, the objective is at eh receiver end is not to reproduce a transmitted waveform with precision: instead, the objective is to determine from a noise-perturbed signal which waveform from the finite set of waveforms was sent by the transmitter. An important measure of system performance in a DCS is the probability error (PE).

2.1. Why DigitalWhy are communications systems going digital? There are many reasons. The primary advantage is the ease with which digital signals, compared with analog signals, are regenerated. Digital circuits are less subject to distortion and interference than are analog circuits. Because digital circuits operate in one of two states- fully on or fully off- to be meaningful, a disturbance must be large enough to change the circuit operating point from one state to the other. Such two-state operation facilitates signal regeneration and thus prevents noise and other disturbances from accumulating in transmission, analog signals, however, are not two-state signals: they can take an infinite variety of shapes. With analog circuits, even a small disturbance can render the reproduced waveform unacceptably distorted. Once the analog is distorted, the distortion cannot be removed by

amplification. Because accumulated noise is irrevocably bound to analog signals, they cannot be perfectly regenerated. With digital techniques, extremely low error rates producing high signal fidelity are possible through error detection and correction but similar procedures are not available with analog. There are other important advantages to digital communications. Digital circuits are more reliable and can be produced at a lower cost than analog circuits. Also, digital hardware lends itself to more flexible implementation than analog hardware. The combining of digital signals using timedivision multiplexing (TDM) is simpler than the combining of analog signals using frequency-division multiplexing (FDM). Different types of digital signals (data, telegraph, telephone, television) can be treated as identical signals in transmission and switching- a bit is a bit. Digital techniques lend themselves naturally to signal processing functions that protect against interference and jamming, or that provide encryption and privacy. One disadvantage of a digital communication system is non-graceful degradation. When the signal-to-noise ratio drops below a certain threshold, the quality of service can change suddenly from very good to very poor. In contrast, most analog communication systems degrade more gracefully.

2.2. Typical Block Diagram and TransformationsThe functional block diagram shown in figure 2.1 illustrates the signal flow and the signal processing steps through a typical digital communication system (DCS).Information Source Message Symbols

Format

Source encode

Encrypt

Channel Encode

Multiplex

Pulse modulate

Bandpass Modulate

Frequency spread

Multiple Access

X M T

Bit Stream

Synchro nization

Digital Baseband

Digital Bandpass

C H A N N E L

Format

Source decode

Decrypt

Channel decode

Demultipl ex

Detect

Demodulate & sample

Frequency despread

Multiple Access

R C V

Message Symbols

Optional Essential

Fig.2.1.

Digital Communication System

In the Figure 2.1, the upper blocks- format, source encode, encrypt, channel encode, multiplexing, pulse modulate, band pass modulate, frequency spread, and multiple access- denote signal transformation from the source to the transmitter (XMT). The lower blocks denote signal transformation from the receiver (RCV) to the sink, essentially reversing the signal processing steps performed by the upper blocks. The modulation and demodulation/detect blocks together are called a modem. The term modem can be thought of as the brain of the system and the transmitter and the receiver can be thought as the muscles of the system.

Figure 2.1 illustrates a kind of reciprocity between the blocks in the upper transmitter part of the figure and those in the lower receiver part. The signal processing steps that take place in the transmitter are, for the most part, reversed in the receiver. In the figure 2.1 the input information source is converted to binary digital (bits): the bits are then groups to form digital messages or message symbols. Each such symbol (mi = 1 M) can be regarded as a member of a finite alphabet set containing M members. Thus, for M = 2, the message symbol mi is binary (meaning it contains just one bit). Even though binary symbols fall within the general definition of M-ary, nevertheless the name M-ary is usually applied to those cases where M>2: hence, such symbols are each made up of a sequence of two or more bits. For systems that use channel coding (error correction coding), a sequence of message symbols becomes transforms to a sequence of a channel symbols, where each channel symbol is denoted ui. Because a message symbol or a channel symbol can consist of a single bit or grouping of bits, a sequence of such symbols is also described as a bit stream, as shown in Figure 2.1. Consider the key signal processing blocks shown in Figure 2.1: only formatting, modulation, demodulation/detection, and synchronization are essential for a DCS. Formatting transforms the source information into bits, thus assuring compatibility between the information and the signal processing within the DCS. From this point of in the figure up to the pulse modulation block, the information remains in the form of a bit stream. Modulation is the process by which message symbols or channel symbols are converted to waveforms that are compatible with the requirements imposed by the transmission channel. Pulse modulation is a essential step because each symbol to be transmitted must first be transformed from a binary representation (voltage levels) to a base-band waveform. The term base-band refers to a signal whose spectrum extends from dc up to some finite value, usually less than a few megahertz. The pulse modulation block usually includes filtering for minimizing the transmission bandwidth. When pulse modulation is applied to binary symbols the resulting binary waveform is called a pulsecode-modulation (PCM) waveform. When pulse modulation is applied to non-binary

symbols the resulting waveform is called an M-ary pulse. After pulse modulation, each message symbol or channel symbol takes the form of a base-band waveform gi(t), where i =1,.., M. For an application involving RF transmission, the next important step is band-pass modulation; it is required whenever the transmission medium will not support the propagation of pulse-like waveforms. For such cases, the medium requires a band-pass waveform si (t), where i =1,M. the term band-pass is used to indicate that the base-band waveform gi (t) is frequency translated by a carrier wave to a frequency that is much larger than the spectral content of gi (t). As si (t) propagates over the channel, it is impacted by the channel characteristics, which can be described in terms of the channels impulse response hc (t) at various points along the signal route, additive random noise distorts to received signal r (t), so that its reception must be termed a corrupted version of the signal si(t) that was launched at the transmitter. The received signal r(t) can be expressed as r(t) = si(t) * hc(t) + n(t) I = 1,.,M (2.1)

Where * represents a convolution operation and n(t) represents a noise process. In the reverse direction, the receiver front end and/or the demodulator provides frequency down-conversion for each band pass waveform r (t). The demodulator restores r(t) to an optimally shaped base-band pulse z(t) in preparation for detection. Typically, there can be several filters associated with the receiver and demodulator-filtering to remove unwanted high frequency terms, and filtering for pulse shaping. Equalization can be described as a filtering option that is used in pr after the demodulator to reverse ant degrading effects on the signal that were caused by the channel. Equalization becomes essential whenever the impulse response of the channel hc (t) is so poor that the received signal is badly distorted. An equalizer is implemented to compensate for any signal distortion caused by a non-ideal hc (t). Finally, the sampling step transforms the shaped pulse z (t) to a sample z(t), and the detection step transforms z(t) to an estimate of the channel symbol u^I . The terms demodulation and detection are interchangeably.

Demodulation is defined as recovery of a waveform (base-band pulse), and the detection is defined as decision-making regarding the digital meaning of that waveform. The other processing steps within the modem are design options for specific system needs. Source coding produces analog-to-digital (A/D) conversion and removes redundant information. A typical DCS would either use the source coding option, or it would use the simpler formatting transformation. A system would not use both source coding and formatting, because the former already includes the essential step of digitizing the information. Encryption, which is used to provide communication privacy, prevents unauthorized users from understanding messages and from injecting false messages into the system. Channel coding, for a given data rate; can reduce the probability of error. PE, or reduce the required signal-to-noise ratio to achieve a desired PE at the expense of transmission bandwidth or decoder complexity. Multiplexing and multiple access procedures combine signals that might have different characteristics or might originate from different sources, so that they can share a portion of the communications resource. Frequency spreading can produce a signal that is relatively invulnerable to interference and can be used to enhance the privacy of the communicators. It is also a valuable technique used for multiple accesses. The signal processing blocks shown in Figure 2.1 represents a typical arrangement: however, these blocks are sometimes implemented in a different order. For example, multiplexing can take place prior to channel encoding, or prior to modulation, or- with a two-step modulation process- it can be performed between the two modulation steps. Similarly frequency spread can take place at various locations along the upper portion of the Figure 2.1: its precise location depends on the particular technique used. Synchronization and its key element, a clock signal, are involved in the control of all signals processing within the DCS. For simplicity, the synchronization block in Figure 2.1 is drawn any connecting lines, when infact it actually plays a role in regulating the operation of almost every block shown in the figure.

Following are the basic transformations of the digital communication system (DCS). 1. Formatting and source coding. 2. Base-band signaling. 3. Band-pass signaling. 4. Equalization. 5. Channel coding. 6. Multiplexing and multiple access. 7. Spreading. 8. Encryption. 9. Synchronization.

2.3. Spectral DensityThe spectral density of a signal characterized the distribution of the signals energy or power in the frequency domain. This concept is particularly important when considering filtering in communication systems. To evaluate the signal and noise at the filter output energy spectral density (ESD) or Power spectral density (PSD) is used.

2.3.1 Energy Spectral DensityThe total energy of a real-valued energy signal x (t), defined over the interval (-, ), is described by the Ex = x2 (t) dt (2.2)

Using Parsevals theorem, we can relate to energy of such a signal expressed in the time domain to the energy expressed in the frequency domain, as Ex = x2 (t) dt = |X (f)|2 dt (2.3)

Where X (f) is the Fourier transform f the non-periodic signal x (t). Let x (f) denote the squared magnitude spectrum, defined as x (f) = |X (f)|2 (2.4)

The quantify x (f) is the waveform energy spectral density (ESD) of the signal x(t). Therefore, from Equation 2.3, we can express the total energy of x(t) by integrating the spectral density with respect to frequency: Ex = x (f) df (2.5)

This equation states that the energy of a signal is equal to the area under the x (f) versus frequency curve. Energy spectral density describes the signal energy per unit bandwidth measured in joules/hertz. There are equal energy contributions from both positive and negative energy frequency components, since for a real signal, x (t), |X (f)| is an even function of frequency. Therefore, the ESD is symmetrical in frequency about the origin, and thus the total energy of the signal x (t) can be expressed as Ex = 2 x(f) df 0 (2.6)

2.3.2 Power Spectral DensityThe average power Px of a real-valued power signal x(t) is defined as T/2 Px = lim 1/To x2 (t) dt T -T/2

(2.7)

If x(t) is a periodic signal with period To, it is classified as a power signal. The expression for the average power of a periodic signal takes the form of, where the time average is taken over the signal period To, as follows

To/2 Px = 1/To x2 (t) dt -To/2 Parsevals theorem for a real-valued periodic signal takes the form To/2 Px = 1/To x2 (t) dt = |cn|2 -To/2

(2.8)

(2.9)

Where the |cn| terms are the complex Fourier series coefficients of the periodic signal. The power spectral density (PSD) function Gx (f) of the periodic signal x (t) is a real, even, and nonnegative function of frequency that give the distribution of the power of x (t) in the frequency domain, defined as Gx (f) = |Cn|2 (f nfo) (2.10)

The above equation defines the power spectral density of a periodic signal x(t) as a succession of the weighted delta functions. Therefore, the PSD of a periodic signal is a discrete function of frequency. Using the PSD defined in Equation (2.10), we can now write the average normalized power of a real-valued signal is Px = Gx (f) df = 2 Gx (f) df - -

(2.11)

The Equation (2.10) describes the PSD of periodic signals only. If x(t) is a non periodic signal it cannot be expressed by Fourier series, and if it is a non periodic power signal (having infinite energy) it may not have a Fourier transform. However, we can still express the power spectral density of such signals in the limiting sense. If we form truncated version xT(t) of the non periodic power signal x(t) by observing I only in the interval (-T/2,T/2), then x T(t) has finite energy ad

has a proper Fourier transform Xt(f). the PSD of the non periodic x(t0 can then be defined in the limit as Gx(f) = lim 1/T |XT(f)|2 T (2.12)

2.4. Noise in Communication SystemsThe term noise refers to unwanted electrical signals that are always present in electrical systems. The presence of noise superimposed on a signal tends to obscure or mask the signal; it limits the receivers ability to make correct symbol decisions; and thereby limits the rate of information transmission. Noise arises from a variety of sources, both men made and natural. The man made noise includes such sources as spark-plug ignition noise, switching transients, and other radiating electromagnetic signals. Natural noise includes such elements as the atmosphere, the sun, and other galactic sources. Good engineering design can eliminate much of the noise or its undesirable effect through filtering, shielding, the choice of modulation, and the selection of an optimum receiver site. For example, sensitive radio astronomy measurements are typically located at remote desert locations, far from the man made noise sources. However, there is one natural source of noise, called thermal or Johnson noise, which cannot be eliminated. Johnson noise is caused by the thermal motion of electrons in all dissipative componentsresistors, wires, and so on. The same electrons that are responsible for electrical conduction are also responsible for thermal noise. We can describe thermal noise as a zero-mean Gaussian random process. A Gaussian process n(t) is a random function whose value n at any arbitrary time t is statically characterized by the Gaussian probability density function p(n) = 1/2 exp[-1/2 (n/ )2] (2.13)

Where 2 is the variance of n. the normalized or standardized Gaussian density function of a zero-mean process is obtained by assuming that = 1. This normalized pdf is shown in Figure 2.2

We will often represent a random signal as the sum of a Gaussian noise random variable and a dc signal. That is, z=a+n (2.14)

Where z is the random signal, a is the dc component, and n is the Gaussian noise random variable. The pdf p (z) is then expressed as p(z) = 1/2 exp[-1/2 (z-a/ )2] (2.15)

where, as before, 2 is the variance o n. the Gaussian distribution is often used as the system noise model because of a theorem, called the central limit theorem, which stats that under very general conditions the probability distribution of the sum of j statistically dependent random variables approaches the Gaussian distribution as j . Therefore, even though individual noise mechanisms might have other than Gaussian distributions, the aggregate of many mechanisms will tend toward the Gaussian distribution.

2.4.1 White NoiseThe primary spectral characteristic of thermal noise is that its power spectral density is the same for all frequencies of interest in most communication systems; in other words, a thermal noise source emanates an equal amount of noise power per unit bandwidth at all frequencies- from d to about 1012 Hz. Therefore, a simple model for thermal noise assumes that its power spectral density Gn(f) is flat for all frequencies, denoted as Gn(f) = No/2 watts/hertz (2.16)

Where the factor 2 is included to indicate that Gn(f) is a two-sided power spectral density. When the noise power has such a uniform spectral density we refer to its 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 radiation. The autocorrelation function of white noise is given by the inverse Fourier transform of the noise power spectral density, denoted as follows; Rn() = img-1{Gn(f)} = No/2(()) (2.17)

Thus the autocorrelation of white nose is a delta function weighted by the factor No/2 and occurring at =0, note that Rn() is zero for 0 that is, any two different samples of white noise, no matter how close together in time they are taken, are uncorrelated. The average power Pn of white noise is infinite because its bandwidth is infinite. This can be seen by combining Equation (2.11) and (2.17) to yield Pn = No/2 df = -

(2.18)

Although white noise is a useful abstraction, no noise process can truly be white; however, the noise encountered in many real systems can be assumed to be approximately white. We can only observe such noise after it has passed through a real system which will have a finite bandwidth. Thus, as long as the bandwidth of the noise is appreciably larger then that of the system, the noise can be considered to have an infinite bandwidth. The delta function in Equation (2.17) means that the noise signal n(t) is totally decor related from its time-shifted version, for any > 0. Equation (2.18) indicates that any two different samples of a white noise process are uncorrelated. Since thermal noise is a Gaussian process and the samples are uncorrelated, the noise samples are also independent. Therefore, the effect on the detection process of a channel with additive white Gaussian noise (AWGN) is the noise affects each transmitted symbol independently. Such a channel is called memory less channel. The term additive means that the noise is simply superimposed or added to the signal- that there are no multiple mechanisms at work.

2.5. Signals and Noise2.5.1. Error-Performance Degradation in Communication SystemsThe task of the detector is to retrieve the bit stream from the received waveform, as error free as possible, notwithstanding the impairments to which the signal may have been subjected. There are two primary causes for error-performance degradation. The first is the effect of filtering at the transmitter, channel, and receiver. A non-ideal system transfer function causes symbol smearing or intersymbol interference (ISI). Another cause for error-performance degradation is electrical noise and interference produced by a variety of sources, such as galaxy and atmospheric

noise, switching transients, inter modulation noise, as well as interfering signals from other sources.

2.5.2.

The basic SNR Parameter for Digital Communication SystemsIn digital communications, we use Eb/No. a normalized version of SNR, as a figure of merit. Eb is bit energy and can be described as signal power S times the bit time Tb. No is noise power spectral density, and can be described as noise power N divided by bandwidth W. since the bit time and bit rate Rb are reciprocal, we can replace Tb with 1/Rb and write Eb/No = STb/(N/W) = (S/Rb)/(N/W) (2.19)

Data rate, in units of bits per second, is one of the most recurring parameters in digital communications. To just emphasize that Eb/No is just a version of S/N normalized by bandwidth and bit rate, as follows: Eb/No = (S/N)(W/R) (2.20)

The dimensionless ratio Eb/No is a standard quality measure for digital communications system performance. Therefore, required Eb/No can be considered a metric that characterizes the performance of one system versus another; the smaller the required Eb/No, the more efficient is the detection process for a given probability of error.

2.6.

Inter-symbol Interference

Besides noise, another problem in digital communication system is the intersymbol interference. As we have mentioned the base band waveform can not be transmitted through the channel directly. One of the major reasons is that the bandwidth is a limited resource. Many users transmit information through the channel simultaneously so that we have to limit the bandwidth of the waveform before it can be transmitted. This can be done in the signal transform. Lets consider again. For the matter of convenience, we simplify the structure here.

Fig 2.10Typical Base band system When the system is implemented, we may lump all the filters in the system to one overall equivalent system transfer function

where G(f) represents the transmitter transform filter, C(f) is the system function of the channel and H(f)r is combination of signal recover and observer. In the binary NRZ system, we have shown that the detector only need compare the sampled value with preset threshold and recover the symbol waveform. However, due to the requirement of limit bandwidth, the situation becomes complicated. As we know from signal processing, limit bandwidth will cause infinite expansion in the time domain. Our signal will be perturbed by the other signals in addition to noise in the channel. As an example, we can design the symbol waveform such that the spectrum only takes up 2/3T in the frequency domain. This shape is illustrated in the following graph

fig 2.11The band limited waveforms before detection (a)Rectangle function(b)Pulse shape

Using the conclusion of former section, the optimum detection will examine the output value at the end of symbol period T. Now, if we sample the waveforms at T, the value we get does not only include the amplitude of current symbol but also the tails of the adjacent symbols. This phenomenon is called Intersymbol Interference (ISI). Even we transmit the signal through a noise absent channel; the effect of ISI will still cause the detection error at the receiver. In fact, ISI is the major obstacle to achieve high-speed communication. Thus, minimizing ISI is another important consideration in digital communication system.

Fig 2.12Sampled value at the end of symbol period The combined waveforms are seriously distorted. Such distortions even worse than the noise in the channel. Remember, increasing the power will help us overcome the noise in the system. However, increasing signal energy will not help to avoid the ISI because all the symbols increase their amplitude at the same time. It can be seen an easiest method to reduce the ISI is that we can design the waveform so that at sampling time, the summation of tails of adjacent is zero, the sampled value will only contain the useful information. In this way, we can eliminate the disturbance of ISI. This method is known as pulse shaping.

There are various filters throughout the system- in the transmitter, in the receiver, and in the channel. At the transmitter, the information +symbols, characterized as impulses or voltage levels modulate pulses that are then filtered to comply with some bandwidth constraint. For base-band systems, the channel has distributed reactance that distorts the pulses. Some band-pass systems, such as wireless systems, are characterized by fading channels, which behave like undesirable filters manifesting signal distortion. When the receiving filter is configured to compensate for the distortion caused by both the transmitter and the channel, it is often referred to as an equalizing filter or a receiving/equalizing filter. Figure illustrates a convenient model for the system, lumping all the filtering effects into one overall equivalent system transfer function H(f) = Ht(f) Hc(f) Hr(f) (2.21)

Where Ht(f) characterizes the transmitting filter, Hc(f) the filtering within the channel, and the Hr(f) the receiving/equalizing filter. The characteristics H(f), then, represents the composite system transfer function due to all the filtering at various locations throughout the transmitter/channel/receiver chain. In a binary system with a common PCM waveform, such as NRZ-L, the detector makes a symbol decision by comparing a sample of the received pulse to a threshold; for example, the detector in figure given above decides that a binary one was sent if the received pulse is positive, and that a binary zero was sent, if the received pulse is negative. Due to the effects of system filtering, the received pulses can overlap one another. The tail of the pulse can smear into adjacent symbol intervals, thereby interfering with that detection process and degrading the error performance; such interference is termed inter-symbol interference (ISI). Even in the absence of noise, the effects of filtering and channel-induced distortion lead to ISI. Sometimes Hc(f) is specified, and the problem remains to determine H t(f) and Hr(f), such that the ISI is minimized at the output of Hr(f). Nyquist investigated the problem of specifying a received pulse shape so that no ISI occurs at the detector. He showed that the theoretical minimum system bandwidth needs in order to detect Rs symbols/s, without ISI, is Rs/2 hertz. This occurs when the system transfer function H(f) is made rectangular. For base-band systems, when H(f) is

such a filter with a single-sided bandwidth1/2T(the ideal Nyquist filter), its impulse response, the inverse Fourier transform of H(f) is of the form h(t) = sinc(t/T) This sinc(t/T)-shaped pulse is called the ideal Nyquist pulse its multiple lobes comprise a main lobe and sidelobes called pre- and post main lobe tails that are infinitely long. Nyquist established that if each pulse of a received sequence is of the form sinc(t/T), the pulses can be detected without ISI. There are two successive pulses, h(t) and h(t-T). even though h(t) has long tails, the figure sows a tail passing through zero amplitude at the instant (t = T) when h(t-T) is to be sampled, and likewise all tails pass through zero amplitude when another pulse of the sequence h(t kT),k =+1,+2,. and -2,-1,.. are to be sampled. Therefore, assuming that the sample timing is perfect, there will be no ISI degradation introduced. For base-band systems, the bandwidth required to detect 1/T pulses (symbols) per second is equal to 1/2T; in other words, a system with bandwidth W = 1/2T = Rs/2 hertz can support a maximum transmission rate of 2W = 1/T = Rs symbols/s without ISI. Thus, for ideal Nyquist filtering, the maximum possible symbol transmission rate per hertz, called the symbol rate packing, is 2 symbols/s/Hz. It should be clear from the rectangular-shaped transfer function of the ideal Nyquist filter and the infinite length of its corresponding pulse, that such ideal filters are not realizable; they can only be approximated. The names Nyquist filter and Nyquist pulse are often used to describe the general class of filtering and pulse-shaping that satisfies zero ISI at the sampling points. A Nyquist filter is one whose frequency transfer function can be represented by a rectangular function convolved with any real even-symmetric, frequency function. A Nyquist filter is one whose shape can be represented by a sinc(t/T) function multiplied by another time function. Hence, there are a countless number of Nyquist filters and corresponding pulse shapes. Amongst the class of Nyquist filters, the most popular ones are the raised cosine and the root-raised cosine.

2.6.1

PULSE SHAPING TO REDUCE ISI

To proceed our analysis, we consider the bipolar-NRZ system again. Now, since the pulse shaping has been used, we may represent the shape of each symbol by

KI represents the amplitude of the symbol waveforms. The signal waveforms will pass through the channel. The white noise is added in the channel. The matched filter will process the distorted signal as we discussed in the former section. Then, the receiver will sample the output of matched filter at the end of symbol period T. This value is fed into the detector to compare with the preset threshold. The decision is made on the basis of the value of the sampled value and the threshold. Thus, we can express the output of the matched filter to be

Where p (t) is the overall system response involves symbol waveform, channel and matched filter. ) (t n is the noise output. The sampled value of jth symbol can be shown as

The first component of contains the information of the symbol to be detected. The second term is known as ISI and the third term is the sampled noise output. We have shown that by using matched filter, the effect of noise

Can be reduced to minimal. Now, the task is to cancel the effect of ISI. Thus the pulse shape should satisfy the condition that

Generally, the distribution of k I varies in different situations and often unknown to the system. Thus a stronger condition can be given as

If we assume the p(t)is normalized so that p(0)=1. Then the condition given by can be replaced by

This condition is known as the Nyquist pulse-shaping criterion. The equivalent frequency domain condition can be shown by

Now, we assume p(f) is band limited to W that p(f)=0 when W f > . We distinguish three cases. Firstly, when T 1 /2 T. In such case, there exist many choices of shape that can be used. The expressed as frequently used shape is the raised cosine spectrum. It can be

parameter in equation is called the roll-off factor which ranges from 0 to 1.The actual bandwidth taken by the signal can be shown to be

where W0

is the ideal Nyquist bandwidth. Bandwidth occupied out of the

Nyquist bandwidth is called the excess bandwidth. Obviously, specifies how much excess bandwidth taken by the signal pulse. As a special case, when =0, we get the ideal Nyquist shape which is a rectangle function in the frequency domain.

2.6.2

Raised Cosine filter

The equivalent express in the time domain for (3-12) can be shown as

Fig 2.15 Raised cosine function For a given data rate, specifies the extension in the frequency domain and steepness in the time domain. When =0, this corresponds to the ideal Nyquist pulse. When =1, the bandwidth doubles the Nyquist bandwidth but the tail is quite small. However, if other users transmit information in adjacent bandwidth, the symbol may smear in the frequency domain which causes interference and degrades the system. Therefore, should be selected carefully in different system and different situations. The trade off between bandwidth and waveform is as follows: The larger is the value of , the more bandwidth of symbol waveform and the small tail of symbol in the time domain.

2.6.3 Square root Raised cosine FilterIf our channel is non-distortion channel, i.e., C(F)=1 , then the receiver pulse can be simplified as

As we have discussed, the matched filter is used in our receiver to achieve the best performance and matched filter is the complex conjugate form of the symbol waveform in the frequency domain. Thus

In order to get the raised cosine function at the receiver, the signal waveform must be designed to satisfy. We can realize such function by

p(f) is the raised cosine filter function and 0 t is the delay required to ensure the causality of the system. Such shape is called square raised cosine function. We may neglect the delay of the waveforms. The expression for the square-root raised-cosine function can be proved to be

Fig 2.16 Square root raised cosine filterSquare-root function is not zero crossing at the sampling time T. However, with matched filter at the receiver, the matched filter output of signal to be detected is a raised cosine function. The signal to noise ratio is maximized at the sampling time T. Next, if we analyze the output of other symbol waveforms, the output is also raised cosine function. However, due to different delay, the sampled values are the shifted raised cosine function. The time shift is the nT. As the property of raised cosine function, p (nT)=0 when n o, the effect of ISI is cancelled in our detection. Thus, by using square-root cosine filter, we achieve the maximal signal to noise ratio at the output and free of ISI.We also found, in a band limited system, ISI can degrade the performance of the system. Nyquist criterion can help us to find the waveforms to reduce ISI. The result of the project is the basic knowledge in digital communication. It can extend to advanced and more complicated system.

2.7.

Eye Pattern

An eye pattern is the display that results from measuring a systems response to baseband signals in a prescribed way. On the vertical plates of oscilloscope we connect the receivers response to a random pulse sequence. On the horizontal plates we connect a saw tooth wave at the signaling frequency. In other words, the horizontal time base of the oscilloscope is set equal to the symbol (pulse) duration. This setup superimposes the waveform in each signaling interval into a family of traces in a single interval (0, T). Figure 2.3 illustrates the eye pattern that results for binary antipodal (bipolar pulse) signaling. Because the symbols stem from a random source; they are sometimes positive and sometimes negative, and the persistence of the cathode ray tube display allows us to see the resulting pattern shaped as an eye. The width of the opening indicates the time over which sampling for detection might be performed. Of course, the optimum sampling time corresponds to the maximum eye opening, yielding the greatest protection against noise. If there were no filtering in the system- that is, the bandwidth corresponding the transmission of these data pulses were infinite-then the system response would yield ideal rectangular pulse shapes. In that case, the pattern would look like a box rather than an eye.

Fig.2.3. Eye Diagram In the Figure 2.3, the range of the amplitude differences labeled DA is a measure of distortion caused by ISI, and the range of time differences of the zero crossings labeled

JT is a measure of the timing jitter. Measures of noise margin MN and sensitivity-totiming error ST are also shown in the figure. In general, the most frequent use of the eye pattern is for qualitatively assessing the extent of the ISI. As the eye closes, ISI is increasing; as the eye opens, ISI is decreasing.

2.8. WHY MODULATE?Digital modulation is the process by which digital symbols are transformed into waveforms that are compatible with the characteristics of the channel. In the case of baseband modulation, these waveforms usually take the form of shaped pulses. But in the case of band-pass modulation the shaped pulses modulate a sinusoid called a carrier wave, or simply a carrier; for a radio transmission the carrier is converted to an electromagnetic (EM) field for propagation to the desired destination, one might ask why it is necessary to use a carrier for the radio transmission of base-band signals. The answer is as follows. The transmission of EM fields through space is accomplished with the use of antennas. The size of the antenna depends on the wavelength and the application. For cellular telephones, antennas are typically /4 in size, where wavelength is equal to c/f, and c, the speed of light, is 3 x 10 8 m/s. consider sending a base-band signal( say, f=3000Hz) by coupling it on an antenna directly without a carrier wave. How large would the antenna have to be? Let us size it by using the telephone industry benchmark of /4 as the antenna dimension. For the 3,000 Hz base-band signal, /4 = 2.5 x 104 m 15 miles. To transmit a 3,000 Hz signal through space, without carrier-wave modulation, an antenna that spans 15 miles would be required. However, if the base-band information is first modulated on a higher frequency carrier, for example a 900 MHz carrier, the equivalent antenna diameter would be about 8 cm. for this reason, carrier-wave or band-pass modulation is an essential step for all systems involving radio transmission. Band-pass modulation can provide other important benefits in signal transmission. If more than one signal utilizes a signal cannel, modulation may be used to separate the different signals. Such a technique is known as frequency-division multiplexing. Modulation can be used to minimize the effects of interference. A class of such modulation schemes, known as spread-spectrum modulation, requires a system

bandwidth much larger than the minimum bandwidth that would be required by the message. Modulation can also be used to place a signal in a frequency band where design requirements, such as filtering and amplification, can easily meet. This is the case when radio-frequency (RF) signals are converted to an intermediate frequency (if) in a receiver.

2.9. Digital Band-pass Modulation TechniquesBand-pass modulation is the process by which an information signal is converted to a sinusoidal waveform; for digital modulation, such a sinusoidal of duration T is referred to as a digital symbol. The sinusoid has just three features that can be used to distinguish it from other sinusoids; amplitude, frequency, and phase. Thus band-pass modulation can be defined as the process whereby the amplitude, frequency, or phase of an RF carrier, or a combination of them, is varied in accordance with the information to be transmitted. The general form of the carrier wave is s(t) = A(t)cos (t) (2.22)

Where A(t) is the time-varying amplitude and (t) is the time-varying angle. It is convenient to write (t) = ot + (t) so that s(t) = A(t) cos [ot + (t)] (2.24) where o is the radian frequency of the carrier and (t) is the phase. The terms f and will be used to denote frequency. When f is used, frequency in hertz is intended; when is used, frequency in radians per second is intended. The two frequency parameters are related by = 2f. When the receiver exploits knowledge of the carriers phase to detect the signals, the process is called coherent detection; when the receiver does not utilize such phase reference information, the process is called non-coherent detection. In digital communications, the terms demodulation and detection are often used interchangeably, (2.23)

although demodulation emphasizes waveform recovery, and detection entails the process of symbol decision. In ideal coherent detection, there is available at the receiver a prototype of each possible arriving signal. These prototypes waveforms attempt to duplicate the transmitted signal set in every respect, even RF phase. The receiver is then said to be phase locked to the incoming signal. During demodulation, the receiver multiplies and integrates the incoming signal with each of its prototype replicas. Non-coherent demodulation refers to system employing demodulators that are designed to operate without knowledge of the absolute value of the incoming signals phase; therefore, phase estimation is not required. Thus the advantage of non-coherent over coherent systems is reduced complexity, and the price paid is increase probability of error (PE).

2.9.1 Phase Shift KeyingPhase shift keying (PSK) was developed during the early days of the deep-space program; PSK is now widely used in both military and commercial communications systems. The general analytic expression for PSK is Si(t) = (2E/T) cos [ot + (t)] 0 t T (2.25) i = 1,,M

where the phase term, (t) will have M discrete values, typically given by i(t) = 2i/M i= 1,,M

Fig. 2.4a PSK For the binary PSK (BPSK) example in Figure 2.4a, M is 2. the parameter E is symbol energy, T is symbol time duration, and 0 t T. in BPSK modulation, the modulating data signals shifts the phase of the waveform si(t) to one of two states, either zero or (180o). The waveform sketch in Figure 2.4a shows a typical BPSK waveform with its abrupt phase changes at the symbol transitions; if the modulating data stream were to consist of alternating ones and zeros, there would be such an abrupt change at each transition. The signal waveforms can be represented as vectors or phasors on a polar plot; the vector length corresponds to the signal amplitude, and the vector direction for the general M-ary case corresponds to the signal phase relative to the other M-1 signals in the set. For the BPSK example the vector picture illustrates the two 180o opposing vectors. Signal sets that can be depicted with such opposing vectors are called antipodal signal sets.

PSK CONSTELLATION:

2.9.2 Frequency Shift KeyingThe general analytic expression for FSK modulation is si(t) = (2Ei(t)/T)cos[it + ] 0 t T (2.26) i = 1,,M

Where the frequency term i has M discrete values, and the phase term is an arbitrary constant. The FSK waveform sketch in figure 2.4b illustrates the typical frequency changes at the symbol transitions. At the symbol transitions, the figure depicts a gentle shift from one frequency (tone) to another. This behavior is only true for a special class of FSK called continuous-phase FSK (CPFSK). In the general MFSK case, the change to a different tone can be quite abrupt, because

Fig. 2.4b FSK

There is no requirement for the phase to be continuous. In this example, M has been chosen equal to 3, corresponding to the same number of waveform types (3-ary). In practice, M is usually a nonzero power of 2 (2,4,8,16,..). The signal set is characterized by Cartesian coordinates, such that each of the mutually perpendicular axes represents a sinusoid with a different frequency. As described earlier, signal sets that can be characterized with such mutually perpendicular vectors are called orthogonal signals. Not all FSK signaling is orthogonal.

Fig: baud rate and bandwidth in FSK

2.9.3 Amplitude Shift KeyingFor the ASK example in figure 2.4c, the general analytic expression is Si(t) = (2Ei(t)/T)cos[ot + ] 0 t T i = 1,,M (2.27)

Fig. 2.4c ASK Where the amplitude term (2Ei(t)/T) will have M discrete values, and the phase term is an arbitrary constant. In figure 2.4c, M has been chosen equal to 2, corresponding to two waveform types. The ASK waveform sketch in the figure can describe a radar transmission example, where the two signal amplitude states would be (2E/T) and zero. The vector picture utilizes the same phase-amplitude polar coordinates as the PSK example. Here we see a vector corresponding to the maximum-amplitude state, and appoint the origin corresponding to the zero- amplitude state. Binary ASK signaling was one of the earliest forms of digital modulation used in radiotelegraphy. Simple ASK is no longer widely used in digital communications systems

Relationship between baud rate and bandwidth in ASK.

CHAPTER 3

GAUSSIAN MINIMUM SHIFT KEYING3.1. What is GMSK?From the viewpoint of mobile communication, the out-of-band radiation power in the adjacent channel should be generally suppressed 60~80 dB below that in the desired channel. To satisfy this severe requirement, manipulation on the output signal spectrum is needed. We know that the smoother the signal waveform is, the less high frequency components it contains. Thus making the signal waveform as smooth as possible may lead to a better spectrum. A continuous change in phase is necessary for a smooth waveform. To make the output power spectrum compact, the pre-modulation LPF should have the following properties: 1. Narrow bandwidth and sharp cutoff. 2. Lower overshoot impulse response. 3. Preservation of the filter output pulse area, which corresponds to a phase shift /2. Condition 1. is needed to suppress the high-frequency components, 2. is to protect against excessive instantaneous frequency deviation, and 3. is for coherent detection to be applicable as simple MSK. In the GSM standards, the pre-modulation Gaussian filtered minimum shift keying (GMSK) with coherent detection is suggested. Unlike the traditional MSK, signals in GMSK modulation first pass through a low pass Gaussian filter with a normalized 3 dB down bandwidth of 0.3, then the output signals are used to perform the phase modulation. Since the integral over the impulse response of the Gaussian filter is quite smooth, thus the phase of the modulated signal changes continuously over a symbol period. The

normalized pre-Gaussian bandwidth of 0.3 corresponds to a filter bandwidth of 81.25kHz for an aggregate data rate of 270.8 Kbps. With 200 kHz of carrier spacing and this data rate, the spectral efficiency of the system is 1.35 b/s/Hz (270.8/200). With the bit interval of 3.7 s , the GSM signal will encounter significant inter-symbol interference in the mobile radio path due to multipath. As a consequence, an adaptive equalizer is needed. GMSK has low out-of-band power and a constant envelope which give it desirable characteristics for wireless communications. The suppression of out-of-band power comes from the relatively sharp cut-off of the Gaussian pre-modulation filter. This chapter concerns itself with modulation and demodulation schemes used to generate and detect GMSK signals.

3.2. Background on GMSKGaussian Minimum Shift Keying (GMSK) is a digital modulation scheme commonly used in wireless, mobile communications. In GMSK, the phase of the carrier is continuously varied by an antipodal signal, which has been shaped by a Gaussian filter. Since it is a type of minimum shift keying (MSK), GMSK has a modulation index of 0.5 and may be demodulated using differential detection. The Gaussian filter concentrates the energy allowing for the desirable characteristic of low out-of-band power. Among the widely touted advantages of GMSK are its relatively narrow bandwidth, constant envelope modulation, and its suitability for both coherent and incoherent detection. The constant envelope allows GMSK to be less susceptible to a fading environment than amplitude modulation and requires only an inexpensive C-class amplifier. These advantages have allowed GMSK to gain acceptance as a part of the GSM standard for cellular land mobile radio channels. The advantages of GMSK as well as its wide-spread use in wireless and satellite communication systems make it a modulation scheme of considerable interest. There are a number of papers that describe GMSK modulation (and/or demodulation) in some detail. In GMSK modulation a bit stream is mapped to a nonreturn-to-zero (NRZ) sequence and then passed through a low-pass filter with a Gaussian

impulse response. The amount of ISI introduced depends on the bandwidth-time product, BTb, of the Gaussian transmit filter. The output of the filter is then used to modulate an IF (intermediate frequency) carrier. For GMSK signaling, demodulation may be accomplished through differential detection as in regular MSK.

3.3

GMSK Basics

Prior to discussing GMSK in detail we need to review MSK, from which GMSK is derived. MSK is a continuous phase modulation scheme where the modulated carrier contains no phase discontinuities and frequency changes occur at the carrier zero crossings. MSK is unique due to the relationship between the frequency of a logical zero and one: the difference between the frequency of a logical zero and a logical one is always equal to half the data rate. In other words, the modulation index is 0.5 for MSK, and is defined as

m= fxTFor example, a 1200 bit per second baseband MSK data signal could be composed of 1200 Hz and 1800 Hz frequencies for a logical one and zero respectively (see Figure 1).

Figure 1: 1200 baud MSK data signal; a) NRZ data, b) MSK signal.

Base band MSK, as shown in Figure 1, is a robust means of transmitting data in wireless systems where the data rate is relatively low compared to the channel BW. MX-COM devices such as the MX429 and MX469 are single chip solutions for base band MSK systems, incorporating modulation and demodulation circuitry on a single chip. An alternative method for generating MSK modulation can be realized by directly injecting NRZ data into a frequency modulator with its modulation index set for 0.5 (see Figure 2). This approach is essentially equivalent to base band MSK. However, in the direct approach the VCO is part of the RF/IF section, whereas in base band MSK the voltage to frequency conversion takes place at base band.

Figure 2: Direct MSK modulation

The fundamental problem with MSK is that the spectrum is not compact enough to realize data rates approaching the RF channel BW. A plot of the spectrum for MSK reveals sidelobes extending well above the data rate (see Figure 4). For wireless data transmission systems which require more efficient use of the RF channel BW, it is necessary to reduce the energy of the MSK upper sidelobes. Earlier we stated that a straightforward means of reducing this energy is lowpass filtering the data stream prior to presenting it to the modulator (pre-modulation filtering). The pre-modulation lowpass filter must have a narrow BW with a sharp cutoff frequency and very little overshoot in its impulse response. This is where the Gaussian filter characteristic comes in. It has an

impulse response characterized by a classical Gaussian distribution (bell shaped curve), as shown in Figure 3. Notice the absence of overshoot or ringing.

Figure 3: Gausssian filter impluse response for BT = 0.3 and BT = 0.5

Figure 3 depicts the impulse response of a Gaussian filter for BT = 0.3 and 0.5. BT is related to the filters - 3dB BW and data rate by

Hence, for a data rate of 9.6 kbps and a BT of 0.3, the filters -3dB cutoff frequency is 2880Hz. Still referring to Figure 3, notice that a bit is spread over approximately 3 bit periods for BT=0.3 and two bit periods for BT=0.5. This gives rise to a phenomena called inter-symbol interference (ISI). For BT=0.3 adjacent symbols or bits will interfere with each other more than for BT=0.5. GMSK with BT=0.5 is equivalent to MSK. In other words, MSK does not intentionally introduce ISI. Greater ISI allows the spectrum to be

more compact, making demodulation more difficult. Hence, spectral compactness is the primary trade-off in going from MSK to Gaussian pre-modulation filtered MSK. Figure 4 displays the normalized spectral densities for MSK and GMSK. Notice the reduced sidelobe energy for GMSK. Utlimately, this means channel spacing can be tighter for GMSK when compared to MSK for the same adjacent channel interference.

3.3.1 Performance MeasurementsThe performance of a GMSK modem is generally quantified by measurement of the signal-tonoise ratio (SNR) versus BER. SNR is related to Eb/N0 by

3.4 Comparison With Different Techniques3.4.1 Quadrature Phase Shift KeyingIf we define four signals, each with a phase shift differing by 90 0 then we have quadrature phase shift keying (QPSK). The input binary bit stream {dk}, dk = 0,1,2,..... arrives at the modulator input at a rate respectively. dI(t) = d0, d2, d4 ,... dQ(t) = d1, d3, d5 , ... (3.1) (3.2) 1/T bits/sec and is separated into two data streams dI(t) and dQ(t) containing odd and even bits

A convenient orthogonal realization of a QPSK waveform , s(t) amplitude modulating the in-phase and sine functions of a carrier wave as follows:

is achieved by

quadrature data streams onto the cosine and

s(t)=1/ 2 dI(t) cos (2pift + p/4) + 1/ 2 dQ(t) sin (2pift + p/4) Using trigonometric identities this can also be written as s(t)=A cos [2pift + p/4 + q(t)]. (3.4)

(3.3)

Problem In QPSKIn QPSK the carrier phase can change only once every 2T secs. If from one T interval to the next one, neither bit remains occurs. stream changes sign, the carrier phase phase unchanged. If one component aI(t) or aQ (t) changes sign, a

change of p/2 occurs. However if both components change sign then a phase shift of pi

Fig. 3.1 QPSK If a QPSK modulated signal undergoes filtering to reduce the lobes, the resulting waveform will no longer have occasional 180o shifts in spectral side

a constant envelop and in fact, the

phase will cause the envelope to go to zero momentarily.

3.4.2 Offset Quadrature phase shift KeyingIf the two bit streams I and Q are offset by a 1/2 bit interval, then the amplitude fluctuations are minimized since the phase never changes by 180o. This modulation scheme, Offset Quadrature Phase shift Keying (OQPSK) is obtained from QPSK by delaying the odd bit stream by half a bit interval with respect to the even bit stream. Thus the range of phase transitions is 0o and 90o (the possibility of a phase shift of 180o is eliminated) and occurs twice as often, but with half the intensity of the QPSK. While amplitude fluctuations still occur in the transmitter and receiver they have smaller magnitude. The bit error rate for QPSK and OQPSK are the same as for BPSK.

Fig.3.2 OQPSK When an OQPSK signal undergoes band limiting, the resulting but since the phase intersymbol

interference causes the envelop to droop slightly to the region of 90o phase transition, transitions of 180 have been avoided in OQPSK, the envelop will never go to zero as it does in QPSK.

3.4.3 Minimum Shift KeyingMinimum Shift Keying (MSK) is derived from OQPSK by replacing rectangular pulse in amplitude with a half-cycle defined as: S(t) = d(t) cos (pit/2T) cos 2pift + d(t) sin (pit/2T) sin 2pift. (3.5) the sinusoidal pulse. The MSK signal is

Fig.3.3 MSK The MSK modulation makes the phase change linear and limited to (p/2) over

a bit interval T. This enables MSK to provide a significant improvement over QPSK. Because of the effect of the linear phase change, the power spectral density has low side lobes that help to control adjacent-channel interference. However the main lobe becomes wider than the quadrature shift keying.

3.4.4 Power spectral densities:Figure displays the normalized spectral densities for QPSK MSK and GMSK. Notice the reduced side lobe energy for GMSK. Ultimately, this means channel spacing can be tighter for GMSK when compared to MSK for the same adjacent channel interference.

Fig.3.4 Power Spectral Density

3.3.

GMSK Modulation

A block diagram of a GMSK modulator is shown in Figure 4.1. For frequency shift keying, the output of the FM modulator, x (t), can be written as x(t) = Ao cos(wct + (t)) (3.6)

where Ao is the signal amplitude, wc is the carrier frequency in radians per second, and (t) is the transmit-filtered data phase. Generating (t) involves mapping binary symbols to an NRZ sequence whose elements may be denoted as ak E [1;-1]. The sequence is then spread into pulses and passed through a low-pass Gaussian filter as shown in Figure 5.1. In the case of two-bit differential detection, the symbols must be encoded before premeditation; however, when one-bit differential detection is employed, no encoding is necessary. The elements of ak are spread out over time so that there is a unit pulse symbol per bit interval. The time dependent NRZ sequence can be represented as a series of pulse functions d (t) = ak (t-kTb) k=- where Tb is the bit period and II(t) is the unit pulse function defined as 2Tb (t) = 1, |t| Tb (3.8) 0, otherwise The pulse train is convolved with a transmit Gaussian filter with an impulse response ht(t) = (k Bt e(pi2Btt2) ) (2) (3.9) (3.7)

Where k = pi (2/ln2) and Bt is the half-power (3 dB) bandwidth of the transmit Gaussian filter. Smaller bandwidth-time products correspond to narrower signal bandwidths, and consequently greater ISI. In this way performance may be traded for more channels. For the purposes of this study a BtTb of 0.5 is chosen and used consistently as the width of the

pre-modulation filter. The data phase, (t), is the time integral of the output of the Gaussian filter, i.e. t (t) =/2Tb [d(t) * h(t)]dt - = /2Tb akg(t-kTb)dt - k=- t

(3.10)

where * denotes convolution and g(t) = (t) * ht(t). The function g (t) can be represented as g (t) = erf(-kBt (t-(Tb/2)) + erf (kBt (t+(Tb/2))) , where erf(x) is the error-function defined as x erf(x) = 2/ exp(-u2)du 0 (3.13) t>0 (3.12)

where erf(x) = erf(-x) . The / 2 factor in equation (3.10) scales the phase such that the modulation scheme is minimum shift keying. In other words the modulation index is one half, indicating that the maximum frequency deviation about the carrier frequency is half the signaling rate. Generating GMSK samples with Matlab is a discrete-time realization of this continuoustime algorithm. A block diagram showing the Matlab algorithm for creating a base-band I and Q signal is shown in Figure 6, while Appendix B contains the actual Matlab code. From the figure it is noticed that there is a random bit generator creating a random sequence of 1's and 0's which a thousand bits long is. The 0's are mapped to -1's to produce the antipodal sequence, ak, then spread out over 10 samples per symbol. The resulting sequence is d(mT), where T is the time period per sample and m is the sample index. The sample index ranges from 0 to L, where L is the length of the data sequence in samples; in this case L = 10kmax. For the generation of the transmit GMSK sequence Tb = 10T and d(mT) = ak for k = m/10. The d(mT) pulse sequence is then convolved with a digital Gaussian filter which has a finite length is 2 bit periods; i.e. it is 21 samples long. The inter-symbol interference is caused almost entirely by the two adjacent bits. The output of the Gaussian filter is then summed such that

(mT) = /2Tb [ d(mT) * ht(mT)]

(3.14)

and the accumulated phase is modulated with a cosine and sine respectively to produce the I and Q components; i.e. I(mT) = cos( (mT)) and Q(mT) = sin( (mT)).

The two analog waveforms generated from the discrete-time data is denoted as I(t) and Q(t). Given that I(t) and Q(t) are defined as cos( (t)) and sin( (t)) respectively, equation (3.15) can be written as x(t) = I(t) cos(wct) - Q(t) sin(wct) (3.15)

3.4.

GMSK Demodulation

Determining the transmitted symbols involves finding the change in transmission phase over each bit period. From (3.10) this phase difference can be written as t b(t) = (t) (t-Tb) = /2Tb [d(t) * h(t)]dt t-Tb

(3.16)

From equations it is noted that the value of [d(t) * h(t)]dt does not exceed Tb so that the maximum possible change in phase over one bit period,( b(t))max, is /2 . The sign of the phase change corresponds with the transmitted symbol, ak. In order to recover (t), the carrier is first removed. The availability of the carrier frequency is used to convert the IF signal to base-band. The received signal is multiplied with the carrier from the vector signal modulator to separate the base-band components. Then a low-pass filter removes the carrier component, leaving I(t) and Q(t). Demodulation of the base-band signal is accomplished directly with a one-bit differential detection scheme. If

z (t) = I(t) + jQ(t) = Arej (t)

(3.17)

where Ar is the magnitude of the received vector, then the phase transition is D(t) = b(t) = img(z(t) z*(t-Tb)) (3.18)

where img(.) denotes the imaginary part and D(t) denotes the demodulated waveform. This demodulation algorithm is implemented in a straight-forward fashion with a one-bit delay, phase shift, and multiply. The one-bit differential detection algorithm is realized with the DSP using the received base-band I and Q samples after A/D conversion.

Gardner AlgorithmThe algorithm is based on delay differencing between the current sample and another sample delayed by half the symbol period, i.e.

xd=pr(t)-pr(t-T/2)..(1)By passing xd(t) through a square-law rectifier, the following is obtained

u(n) = x2d(t)=p2(t)+p2r(t-T/2)-2pr(t)pr(t-T/2)..(2)It is due to the above squaring that the operation of the Gardner algorithm becomes independent of the carrier phase. Substituting the early sampling time, te = nT + , and the late sampling time, tl=nT+ +T/2 gives

u(n)=pr2( +nT)-pr2( +[n-1]T)-2pr( +[n-1/2]T){pr( +[n-1]T)}(3)By simulation, it has been shown that the first two terms have a major contribution to the self-noise (see figure A) Therefore; the algorithm is approximated by dropping the Pr2 (.)Terms

u (n) = -pr ( +{n-1/2]T){pr( +nT)-pr( +[n-1]T)}.(4) At equilibrium, is zero. Therefore the above equation becomes U(n)=pr(n-1/2){pr(n)-pr(n-1)}..(5) The sign reversal in (5) is compensated by changing the sign of the loop gain factor

Figure A: Performance of the Gardner algorithm. Equation (4) is for BPSK. For (O) QPSK modulation scheme, the Gardner Algorithm becomes

u(n) = pr(n + 1 2)[pr(n) _ pr(n + 1)] + qi(n + 1 2)[qi(n) _ qi(n + 1)]

= fp(n + 1 2)[p_(n) _ p_(n + 1)]...(6)The block diagram of the Gardner algorithm is shown in Figure B. At the output of the matched filter, the sampling rate is reduced by a factor of N / 2, where

Figure B: Block diagram of the Gardner algorithm.N is the number of samples per transmitted symbol. The remaining 2 samples per symbol are used in (6) to generate an error sample. The demultiplexers separate the samples which occur at the symbol strobe times from the samples which occur half-way between the symbol strobe times. For QPSK modulation scheme, 2Ni + 3 real multiplications and 2Ni+2 real additions are performed per timing error estimate

The S-curve is shown in Figure C(a). It can be clearly seen that as the timing error increases, the gradient of the timing error detector decreases and, hence, the acquisition time increases. The longest acquisition time occurs when the normalized timing error is 0.025.. On the same figure, it has been shown that the operation of the algorithm is independent of the phase error . An interesting feature of the Gardner algorithm is that its gradient remains unaffected by additive white Gaussian noise. The gradient of the timing error detector remains at 1.48 (See Figure C(b)). The gradient of other synchronization algorithms is a function of the SNR; as SNR decreases, the gradient decreases. A low gradient means a slow acquisition time. The noise performance of the Gardner algorithm is shown in Figure 4.13(c). For increasing Es/N0the simulation results converge to Cramer-Rao bound

(7)

(a) S-curve

(b) Change in gradient of the error detector

as function of SNR

(c) Normalized PSD at fT = 0

Figure C: Performance of the Gardner algorithm.From Figure 4.13(c) it can be deduced that, even at medium to high SNR, there is selfnoise in this algorithm. When is less than 100%, the zero crossings of data transitions do not lie midway between the desired symbol strobe points. The average location is centred on the midway point, but any individual point can depart from the average, causing self-noise. The self-noise and the acquisition time are related to the loop gain factor . By simulations it has been found that the root mean square (RMS) of the selfnoise changes as =0.1 (8) Low values of result in smaller jitter at the expense of longer acquisition time. Even replacing the symbol strobe values in (6) with the hard decision on the symbol, as suggested in [Gar85b], does not eliminate the effect of noise in the timing loop. It is only with the roll off factor of 100% that the self-noise in the Gardner algorithm vanishes. Further simulations have shown that the gradient of the Gardner algorithm characteristics, K , is a function of the roll off factor by the following relationship)

(9) Therefore the algorithm is not suitable for very small roll off factors.

CHAPTER4MATLAB SIMULATION4.1. ModulationImplementing Gaussian Minimum Shift Keying in MatLab is just a discrete time realization of continuous-time algorithm. A block diagram of GMSK transmitter implementing in MatLab is shown in figure 4.1 while the appendix contains the complete code. From figure it is noticed that a random number generator generates the random bits 1s and 0s whose length is chosen according to user specified size.

C o s ()

*

90 Ra ndo m N u m b ers NRZ M a p p in g G a u s s ia n Filter I n teg ra te LO

-

S in ()

*

Fig. 4.1. Generation of GMSK signal Then the 1s are mapped to 1s and 0s to the -1s to produce the antipodal signal. These samples are spread out to 10 samples per symbol. The resulting signal becomes the d (mT), where T is the time sample per symbol and m is sample index. The sample index m ranges from 0 to L where L is the length of data sequence in samples in this case L = 10kmax. For generation of GMSK signal Tb = 10T and k = m/32.

The d (mT) pulse signal is then converted to Gaussian shaped by passing this data through a Gaussian filter. Gaussian filter has a finite length of 2 bit periods with bandwidth to Time product i.e. BTb is 0.5. The intersymbol interference is almost among two bits. Coefficients of Gaussian filter are generated by using the impulse response equation of Gaussian filter with BTb = 0.5. k = pi*sqrt(2/log(2)); g[n] = (k*B/sqrt(pi))*(exp(-k^2)*(B^2)*(n.^2))); Where n = -Tb:Tb/M:Tb; Figure 4.2 shows the Gaussian pulse of BT = 0.5 (4.1) (4.2)

Fig. 4.2 Gaussian Pulse Figure 4.3 shows the data passing the signal through Gaussian filter

Fig. 4.3. Data after pulse shaping The output of the Gaussian filter is summed such that and shown in Figure 4.4 (mT) = pi/2*Tb [d(mT)*g(mT)] (4.3)

Fig.4.4 Accumulated Phase This accumulated phase of the signal is modulated with cosine and sine to form the I and Q components of the signal. I(mT) = cos[(mT)] Q(mT) = sin[(mT)] (4.5) (4.4)

Fig. 4.5. a) In phase, b) Quadrature components

These two waveforms are modulated with the carrier and transmitted to the channel after summation. x(mT) = I(mT)cos(wcmT) Q(mT)sin(wcmT) Figure 4.6 shows the waveforms of modulated carrier of 20 bits. (4.6)

Fig. 4.6 Modulated carrier

4.2. Demodulation

The block diagram shown in figure 4.7.describes the receiver for GMSK data. First the carrier is removed by multiplying the data by carrier and passing it through a low pass filter. r(t) = I(t)cos(wt) Q(t)sin(wt) (4.7)

*

LPF One Bit Differential Detection LPF Symbol Synchronization and sampling

90 LO

*

Fig. 4.7. GMSK Receiver r1(t) = [I(t)cos(wt) Q(t)sin(wt)] 2cos(wt) r1(t) = I(t) + I(t)cos(2wt) + Q(t)sin(2wt) (4.8) (4.9)

The high frequency components are removed by low pass filter and we have Inphase component that was originally transmitted. Similarly we have get quadrature component by multiplying the received signal by sin(wt). Signals with high frequency and noise are shown in Figure 4.8. r2(t) = Q(t) (4.10)

Fig.4.8. Received Noisy I,Q Components After getting I and Q components a one bit differential detection technique is employed to determine the actual data that was transmitted. Determining the transmitted symbols involves finding the change in transmission phase over each bit period. This phase difference can be written as

(t) = (t)- (t-Tb)

(4.11)

The equation (4.11) can simply be implemented as shown in figure 4.9.

Fig. 4.9. One bit differential Detection Waveform of the demodulated signal is shown in Figure 4.10.

Fig. 4.10. Demodulated Signal

4.3. Symbol SynchronizationOne possible approach to symbol synchronization, when using a free-running A/D clock, is to over-sample and select the sample closest to the instant of least ISI. If the symbol is not over-sampled then there could be more error in detection of data. When sampling at N = 10 the sample closest to the strobe instant is no less than 99% of the magnitude of the perfectly synchronized sample. When sampling at Nyquist rate the closest sample to strobe is only 75% of the peak value. This means this low sampled data is more susceptible to the noise. The method used here is to compute a timing error from the sampled data then adjust the sampling phase to correct the error. No attempt is made to adjust the phase or frequency of the A/D clock.

Gardners error function is a good candidate for computing the timing error since it only requires a total of three samples and a minimum sampling rate of two samples per bit. The timing error function used in this system is be written as T(nTs) = D((n-N/2)Ts)[sgn(D(nTs) sgn(D((n N)Ts))] (4.12) A block diagram of the algorithm used for over-sampling can be seen in Figure 4.11. Once a timing error has been determined for the current strobe, it is filtered with a step factor, u, and the output is used to control the selection of the next strobe instant. OutputN= 10

Data Buffer Timing error detection

Hard Decision 1s or -1s

Control

Fig. 4.11. Block Diagram of timing recovery algorithm

This block diagram is implemented in MatLab using this algorithm. %Symbol synchronization and original sampled signal while(time 0)*2-1;

error = (sign(demod(fix(time)))-sign(demod(fix(timeM))))*(demod(time-M/2)); shift = error*M*mu; steps = fix(shift); time = time + M-steps; num = num+1; end

CHAPTER5CONCLUSIONS AND FUTURE SCOPE OF THE PROJECTConclusion Of The projectThe challenge of this project is to design and implement a system that will generate and detect a GMSK signal in the effects of asynchronous sampling on modem performance. A system has been designed and constructed which includes a GMSK modulator, demodulator, A/D clock. Chapter II discusses the basic concepts of digital communications. Chapter III discusses the specifics of GMSK modulation and demodulation In addition Chapter IV also goes into detail about the synchronization and interpolation algorithms. In this project ,I have evaluated the performance of a one-bit deferential detection of a GMSK signal using Gardner's algorithm for phase correction with over sampling. The Performance of GMSK will be compared for two cases of oversampling and Nyquist sampling with a free running A/D clock. The trade-off between sampling rate and computational complexity will be examined for the two approaches.

Future Scope Of The ProjectThe proliferation of computers in today's society has increased the demand for transmission of data over wireless links. Binary data, composed of sharp "one to zero" and "zero to one" transitions, results in a spectrum rich in harmonic content that is not well suited to RF transmission. Hence, the field of digital modulation has been flourishing. Recent standards such as Cellular Digital Packet Data (CDPD) and Mobitex specify Gaussian filtered Minimum Shift Keying (GMSK) for their modulation method. GMSK provides a straightforward, spectrally efficient modulation method for wireless data transmission systems GMSK is a simple yet effective approach to digital modulation for wireless data transmission. To provide a good understanding of GMSK, we will review the basics of MSK and GMSK, as well as how GMSK is implemented in CDPD and Mobitex systems. GMSK modems reduce system complexity, and in turn lower system cost. There are, however, some important implementation details to be considered. This paper will cover some of these details, focusing on interfacing a single chip base band modem to the IF/RF section of a "typical" FM radio topology.

APPENDIX-A REFERENCES 1. Practical GMSK Data Transmission Application note MX.Com, INC. 2. Digital Communications Fundamentals and Applications (Second Edition) By Benard Sklar. 3. Modern Digital and Analog Communication System By B.P. Lathi. 4. A.Biran, MATLAB 5 for Engineers, Addison-Wesley, 1999. 5. Digital Communication fundamentals and Application 2nd edition book by Bernald Sklar. 6. Communication systems book by Simon Hykin. 7. www.mathworks.com 8. Bahai, A., and B. Saltzberg. Multicarrier Digital Communications: Theory and Applications of OFDM. New York: Kluwer Academic/Plenum Publishers, 1999 9. Van Nee, R., and R. Prasad. OFDM Wireless Multimedia Communications. Boston: Artech House, 2000. 10. Couch II, L. W. Digital and Analog Communication Systems. New Jersey: Prentice-Hall, 1997. 11. Keller, T., and L. Hanzo. Adaptive Multicarrier Modulation: A Convenient Framework for Time-Frequency Processing in Wireless Communications. Proceedings of the IEEE 88.5 (2000) 609 639. 12. Nikolai, K.D. Kammeyer, and A. Dekorsy .On the Bit Error Behaviour of Coded DS-CDMA with Various Modulation Techniques. IEEE 9th International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC-98) , Boston, USA, 8-11 September 1998. 13. J. Cheng and N. C. Beaulieu, ``Accurate DS-CDMA Bit Error Probability Calculation in Rayleigh Fading,'' IEEE Transactions on Wireless Communications, vol. 1, pp. 3-15, January 2002. 14. T.S. Rappaport, Wireless Communications: Principles and Practice. Piscataway., NJ: Prentice Hall, 1998.

15. B.Sklar, Digital Communications: Fundamentals and Applications, 2nd ed., NJ: Prentice Hall, 2001. 16. John G. Proakis, .Digital Communications., McGraw Hill Series in Electrical and Computer Engineering, Third Ed., 17. Clark and Gain, .Error-Correction Coding for Digital Communications., Plenum Press, 1981. 18. Simon, Omura, Scholtz, Levitt, .Spread Spectrum Communications: Volume I., Computer Science Press, Inc., 1985.

APPENDIX-B