Chapter 3 Bandpass Transmission Techniques for Wireless Communication

Chapter 3 Bandpass Transmission Techniques for Wireless Communication ECE 414 Wireless Communications, University of Waterloo, Winter 2012 1

Outline Ø Introduction to Digital Communications Ø Signal (Vector) Space Representations Ø Digital Modulation Schemes (M-ASK, M-PSK, M-FSK) Ø Performance Measures for Modulation Schemes - Bandwidth (spectral) efficiency - Power efficiency - Temporal characteristics (e. g. , dynamic power range, peak/average ratio) Ø Power Spectral Density of Digital Modulation Schemes Ø Error Rate Performance of Digital Modulation Schemes Ø Comparison of Digital Modulation Schemes in terms of Spectral Efficiency and Power Efficiency Ø Temporally Efficient Digital Modulation Schemes ECE 414 Wireless Communications, University of Waterloo, Winter 2012 2

Block Diagram for a Digital Communication System Original message signal (analog) A/D Source Encoder Channel Encoder Modulator Channel Recovered message signal (analog) D/A Source Decoder Channel Decoder Demodulator Ø Analog-to-Digital (A/D) Conversion: Analog (i. e. , continuous-time continuous-amplitude) message signal is converted into a discrete-time discreteamplitude digital signals by time-sampling and amplitude-quantization. The resulting signals are then mapped to binary sequences. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 3

Block Diagram for a Digital Communication System (cont’d) Ø Source Encoding: Removes the redundant information embedded in the message signal, therefore represents the message with as few binary digits as possible, i. e. , data compression Ø Channel Encoding: Introduces redundancy in a “controlled” manner which can be used at the receiver to overcome the effects of noise, interference and fading. Provides “noise immunity” to transmitted information. Source coding and channel coding will not be studied in this course… Ø Modulation: Converts (maps) codewords to high-frequency analog waveforms. A certain parameter of the carrier signal (i. e. , modulated signal) is varied in accordance with message signal (i. e. modulating signal) e. g. amplitude shift keying (ASK), phase shift keying (PSK), frequency shift keying (FSK) Ø Receiver Blocks: Perform the inverse of the transmitter operations in order to recover the original analog message (continuous-time continuous-amplitude) signal. In a practical digital communication receiver, there also additional sub-blocks such as channel estimation, synchronization (frame/frequency/phase), authentications, crypto, multiplexing, etc. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 4

Why is Modulation Required? Modulatio shifts the baseband signal to a higher frequency band, centered at the socalled “carrier frequency”. Ø To achieve easy radiation: Dimensions of the transmit/receive antennas are limited by the corresponding wavelength. The frequency conversion allows the use of practical antenna lengths. Ø To accommodate for simultaneous transmission of several baseband signals: Simultaneous transmission of different baseband signals which are possibly overlapping can be facilitated by assigning slightly different frequency carriers for each one. Ø Large bandwidths require high carrier frequencies: Practical requirements in front-end filter design dictates the bandwidth-to-frequency carrier ratio (i. e. , fractional bandwidth) be kept within a certain range. : Fractional bandwidth ECE 414 Wireless Communications, University of Waterloo, Winter 2012 5

Why is Modulation Required? (cont’d) Ø High-rate transmission requires larger bandwidths (therefore, higher carrier frequencies): According to Shannon Theorem, channel capacity is defined as the maximum achievable information rate that can be transmitted over the channel. For the additive white Gaussian noise (AWGN) channel, Channel capacity Bandwidth Signal-to-noise ratio Ø To (possibly) expand the bandwidth of the transmitted signal for better transmission quality: When the bandwidth increases, the required SNR (for fixed noise level, corresponding signal power) to achieve a specific transmission rate decreases ECE 414 Wireless Communications, University of Waterloo, Winter 2012 6

Signal-Space Representations Ø Consider a modulation format where the transmitted signal waveforms belong to the modulation set. Ø Each of the waveform can be represented as a point (vector) in an N-dimensional signal space (sometimes called as vector space) defined by the orthonormal basis functions Ø The Gram-Schmidt procedure (See Appendix A of the textbook) provides a systematic approach to construct the set of orthonormal functions, which span the signal space. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 7

Signal-Space Representations (cont’d) Ø Energy Ø Correlation Ø Euclidean Distance ECE 414 Wireless Communications, University of Waterloo, Winter 2012 8

M-ary Amplitude Shift Keying (M-ASK) Ø Baseband (Equivalent Lowpass) Representation Ø Basis Function(s) (Obtained through Gram-Schmidt procedure) Ø Signal-Space (Vector Space) Representation (Obtained through the use of basis functions) 1 -dimensional Ø Signal Energy ECE 414 Wireless Communications, University of Waterloo, Winter 2012 9

M-ASK (cont’d) Examples of M-ASK Signal Constellations M=4 Bandpass Modulation Signal 11 10 00 01 Equivalent Lowpass Signal ECE 414 Wireless Communications, University of Waterloo, Winter 2012 10

M-ary Phase Shift Keying (M-PSK) Ø Baseband (Equivalent Lowpass) Representation Ø Basis Functions Ø Signal-Space Representation 2 -dimensional Ø Signal Energy ECE 414 Wireless Communications, University of Waterloo, Winter 2012 11

Example: Binary Phase Shift Keying (BPSK) Signal-Space Representation Bandpass Modulation Signal 1 0 1 1 0 A t -A Equivalent Lowpass Signal A 1 0 1 1 0 t -A ECE 414 Wireless Communications, University of Waterloo, Winter 2012 12

Example: Quadrature Phase Shift Keying (QPSK) Signal-Space Representation ECE 414 Wireless Communications, University of Waterloo, Winter 2012 13

Quadrature Amplitude Modulation (QAM) Am, r, Am, i: Information-bearing signal amplitudes of the quadrature carriers Alternatively, QAM can be considered as a combination of ASK and PSK. where Examples of QAM Signal Constellations ECE 414 Wireless Communications, University of Waterloo, Winter 2012 14

QAM (cont’d) Ø Baseband (Equivalent Lowpass) Representation Ø Basis Functions Ø Signal-Space Representation 2 -dimensional Ø Signal Energy ECE 414 Wireless Communications, University of Waterloo, Winter 2012 15

M-ary Frequency Shift Keying (M-FSK) Ø Baseband (Equivalent Lowpass) Representation Ø Cross Correlation For and Therefore, the minimum frequency separation between adjacent signals for orthogonality of the M signals is ECE 414 Wireless Communications, University of Waterloo, Winter 2012 16

M-FSK (cont’d) Ø Assuming frequency separation , the signal-space representation for the M-FSK signals are given as N-dimensional vectors, where N=M. . where ECE 414 Wireless Communications, University of Waterloo, Winter 2012 17

Performance Measures for Modulation Schemes Ø Bandwidth (spectral) efficiency: How much bandwidth is needed for a given data rate? : Bandwidth efficiency : Data rate W : Bandwidth Ø The bandwidth depends on the modulation scheme and pulse shaping. Power spectral density (PSD) is typically used to determine the bandwidth of the transmitted signal. There are various definitions for bandwidth: • Main lobe (null-to-null) bandwidth: The width of the main spectral lobe. • Fractional power-containment bandwidth: The frequency interval that contains (1 -ε) of the total signal power, e. g. 99. 9% of the total power. • Bounded PSD bandwidth: The frequency interval where the PSD stays above a prescribed certain threshold, e. g. sidelobes peaks 40 d. B below its maximum value Ø Roughly speaking, bandwidth of the modulation scheme is proportional to the dimension number. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 18

Performance Measures for Modulation Schemes (cont’d) Ø Power efficiency: How much power is needed for reliable transmission with a specified fidelity? Ø The fidelity for a digital communication system is usually measured in terms of symbol- or bit-error probability. For a given SNR, we aim to achieve a low error probability (how low? it depends on the application). Ø Symbol error probability (SEP) is in general easier to evaluate. Bit error probability (BEP) depends on the mapping of source bits onto modulation signals. A bound on BEP is given as Ø Two common mapping forms are “natural mapping” and “Gray mapping”. In Gray mapping, the neighbour points differ in only one digit. It should be noted that Gray mapping is not possible for every signal constellation. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 19

Performance Measures for Modulation Schemes (cont’d) Ø Temporal efficiency: How wide are the time variations of the transmitted signal? Temporal efficiency=Peak power/Average power The choice of amplifier depends on the temporal characteristics of the signal. Ø Other considerations: • Hardware/software implementation complexity & cost of implementation • Sensitivity to interference • Robustness to impairments encountered in a wireless channel Ø In most practical scenarios, these performance measures conflict with each other. The communication system designer should be able to find the best “tradeoff” for a given application under specific constraints. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 20

Comparison of Spectral Efficiency of Modulation Schemes Ø M-PSK and QAM Ø M-FSK M: “Modulation order”, “Constellation size” ECE 414 Wireless Communications, University of Waterloo, Winter 2012 21

Power Spectral Density (PSD) Ø In practical, pulse shaping should be considered for a precise bandwidth measurement and considered in the spectral efficiency calculations. Ø Power spectral density (PSD) describes the distribution of signal power in the frequency domain. If the baseband equivalent of the transmitted signal sequence is given as : Baseband modulation symbol : Signal interval : Pulse shape then the PSD of g(t) is given as where See Ch. 4 of Digital Communications by Proakis for the proof ECE 414 Wireless Communications, University of Waterloo, Winter 2012 22

Example: PSD of BPSK with Rectangle Pulse Shaping Ø Baseband equivalent of BPSK sequence Independent data symbols are assumed Ø Autocorrelation of data sequence Ø Pulse shaping p(t) T/2 T t ECE 414 Wireless Communications, University of Waterloo, Winter 2012 23

Example: PSD of BPSK with Rectangle Pulse Shaping (cont’d) Ø PSD of baseband BPSK sequence ECE 414 Wireless Communications, University of Waterloo, Winter 2012 24

Example: PSD of BPSK with Rectangle Pulse Shaping (cont’d) Ø Bandpass BPSK sequence and its Fourier transform (spectral density) See Tutorial 1 Ø PSD of bandpass BPSK sequence See Ch. 4 of Digital Communications by Proakis for the proof Null-to-null bandwidth ECE 414 Wireless Communications, University of Waterloo, Winter 2012 25

Example: PSD of QAM with Rectangle Pulse Shaping Ø Baseband equivalent of QAM sequence Ø Autocorrelation of data sequence Ø PSD of baseband QAM sequence Note that PSD of QAM has the same general form as BPSK. Ø PSD of bandpass QAM sequence ECE 414 Wireless Communications, University of Waterloo, Winter 2012 26

Some Practical Pulse Shapes Below are some pulse shapes commonly used in communication systems: Ø Half-Sinusoid Pulse Ø Full-Cosine Pulse ECE 414 Wireless Communications, University of Waterloo, Winter 2012 27

Some Practical Pulse Shapes (cont’d) Ø Gaussian Pulse where B is defined as the “ 3 d. B bandwidth of pulse” Ø Raised Cosine Pulse α: Roll-off factor ECE 414 Wireless Communications, University of Waterloo, Winter 2012 28

Comparison of Pulse Shapes Time-Domain Square Full-cosine Half-sinusoid Gaussian ECE 414 Wireless Communications, University of Waterloo, Winter 2012 29

Comparison of Pulse Shapes (cont’d) Frequency-Domain Full-cosine Half-sinusoid Gaussian Square • Square BW=2/T • Half-sinusoid BW=3/T • Full-cosine BW=4/T 2/T 3/T 4/T ECE 414 Wireless Communications, University of Waterloo, Winter 2012 30

Comparison of Pulse Shapes (cont’d) Raised Cosine α: Roll-off factor 1/T 2/T ECE 414 Wireless Communications, University of Waterloo, Winter 2012 31

Optimum Receiver for AWGN Ø For a given SNR (i. e. a given signal power for fixed noise power), we aim to achieve a low error probability. To calculate error probability, first we need to identify the receiver structure. Ø The receiver consists of a demodulator and a detector: • The demodulator converts the received waveform r(t) into a N dimensional vector where N is the dimension of the signal-space for the given modulation type. • The detector decides which of the possible M signal waveforms was transmitted based on r, where M is the constellation size. Demodulator Detector ECE 414 Wireless Communications, University of Waterloo, Winter 2012 32

Optimum Receiver for AWGN (cont’d) Correlation-type demodulator Matched-filter demodulator For details, see Proakis’ Digital Communications Chapter 5 ECE 414 Wireless Communications, University of Waterloo, Winter 2012 33

Optimum Receiver for AWGN (cont’d) Ø We want to design a signal detector that makes a decision based on the observation of the vector r such that the probability of a correct decision is maximized. The optimal decision rule is based on the maximization of so-called “a posteriori probabilities” : The probability of choosing sm m=1, 2…M based on the observation of r This decision criterion is called the Maximum A Posteriori Probability (MAP) rule. Bayes Theorem : Common for all , i. e. Equally probable messages Ø The conditional pdf is called the likelihood function and the decision criterion based on the maximization of over the M signals is called the maximum likelihood (ML) criterion. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 34

Optimum Receiver for AWGN (cont’d) Ø For an AWGN channel, the components of the noise vector n are zeromean Gaussian random variables with variance N 0/2 Ø The received signal will have a Gaussian conditional distribution ECE 414 Wireless Communications, University of Waterloo, Winter 2012 35

Optimum Receiver for AWGN (cont’d) Ø The ML rule is then given as “Distance” metrics Ø The ML receiver decides in favor of the signal which is closest in Euclidean distance to the received vector, r. Ø Expanding the decision rule, where is the signal energy. Neglecting terms which do not affect the decision and under the assumption that constant-energy modulation set (e. g. PSK) is used “Correlation” metrics ECE 414 Wireless Communications, University of Waterloo, Winter 2012 36

Example: Error Probability for BPSK where i. e. antipodal signaling Unlike other M-PSK for M>2, we can represent this special form of BPSK signal as 1 -dimensional signal. The basis function is given as Therefore, the optimal receiver has the following form of Euclidean Distance Decoder ECE 414 Wireless Communications, University of Waterloo, Winter 2012 37

Example: Error Probability for BPSK (cont’d) Assume s 1(t) is sent. Under the assumption of AWGN, the received signal The output of demodulator where Assume s 2(t) is sent. The output of demodulator is now ECE 414 Wireless Communications, University of Waterloo, Winter 2012 38

Example: Error Probability for BPSK Decision regions Here we have two possible alternatives, therefore we can use a “zero threshold detector” as an optimal detector. Let P(e) denote the error probability Bayes Theorem Equally probable messages Due to symmetry Under the assumption that b=1 is sent ECE 414 Wireless Communications, University of Waterloo, Winter 2012 39

Example: Error Probability for BPSK (cont’d) where Q-function is defined as ECE 414 Wireless Communications, University of Waterloo, Winter 2012 40

Example: Error Probability for QPSK Detector ECE 414 Wireless Communications, University of Waterloo, Winter 2012 41

Example: Error Probability for QPSK (cont’d) Ø Under the assumption of AWGN (which exhibits symmetry), rotating and moving the signal constellation does not change the error probability. Therefore, we can rotate/move our signal constellation in such a way that the resulting constellation allows easy mathematical derivation. Ø Here, we move our constellation as the “target” signal is located on the origin. If there is no symmetry in the signal constellation, this should be repeated for each signal. Decision regions Ø First, we calculate P(c), i. e. the probability of making a correct decision. Then, probability of error is simply found as P(e)=1 -P(c). ECE 414 Wireless Communications, University of Waterloo, Winter 2012 42

Example: Error Probability for QPSK (cont’d) Ø Assume that the signal located at the origin has been transmitted. If the received signal is in the shaded area, this means we will make a correct decision. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 43

Example: Error Probability for QPSK (cont’d) Due to symmetry, ECE 414 Wireless Communications, University of Waterloo, Winter 2012 44

Example: Error Probability for BFSK Detector ECE 414 Wireless Communications, University of Waterloo, Winter 2012 45

Example: Error Probability for BFSK (cont’d) By rotation, it can be easily shown that Ø Now, we will study the same problem without rotation: Assume was sent. The received signal is Decision is based on Due to symmetry, ECE 414 Wireless Communications, University of Waterloo, Winter 2012 46

A Union Bound on Error Probability Ø In most cases, probability of error can not be obtained in closed form. Therefore, one needs to find some bounds or approximations which can work for any signal constellation. Ø We have already shown that the optimal decoder for any signal constellation over AWGN is given by the Euclidean distance decoder, i. e. : Probability of making a decision error when sm was sent Union Bound (U-B) : The probability of choosing sl instead of the originally transmitted sm ECE 414 Wireless Communications, University of Waterloo, Winter 2012 47

A Union Bound on Error Probability (cont’d) where U-B: Union Bound UB-B: Union. Bhattacharyya Bound Ø Assuming equal-probable message signals, the probability of error is ECE 414 Wireless Communications, University of Waterloo, Winter 2012 48

A Union Bound on Error Probability (cont’d) Ø The U-B requires the computation of all distances dl, m among signals in the constellation. A looser bound can be obtained as follows “Minimum Euclidean distance” bound where is the minimum Euclidean distance of the constellation. Then the probability of error is found as Ø P(e) is dominated by the minimum Euclidean distance of the signal constellation. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 49

An Approximation for Error Probability Ø As an alternative, we can also the following approximate upper bound Approximate upper bound : Number of signals at distance dmin from sm ECE 414 Wireless Communications, University of Waterloo, Winter 2012 50

Error Probability for M-PSK Replacing and into the formula on p. 50, we obtain ECE 414 Wireless Communications, University of Waterloo, Winter 2012 51

Error Probability for M-PSK (cont’d) Ø Error rate degrades as M increases. Ø Recall that spectral efficiency increases as M increases. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 52

Error Probability for QAM ECE 414 Wireless Communications, University of Waterloo, Winter 2012 53

Error Probability for QAM (cont’d) 3 neighbours 4 neighbours • 2 neighbours 3 neighbours • 3 neighbours • 4 neighbours Using the result from p. 50, we obtain an approximate upper bound ECE 414 Wireless Communications, University of Waterloo, Winter 2012 54

Error Probability for QAM (cont’d) Ø Power efficiency decreases with increasing M, but not early as fast as M-PSK. Ø Recall that spectral efficiency increases as M increases. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 55

Error Probability for M-FSK Each signal occupies its own dimension. Therefore, each signal has M-1 neighbours, separated from each other by ECE 414 Wireless Communications, University of Waterloo, Winter 2012 56

Error Probability for M-FSK (cont’d) • BPSK • BFSK Ø As M increases, power efficiency improves (i. e. less Eb is required). Ø Recall that spectral efficiency decreases as M increases. Ø For M=2, BFSK requires 3 d. B more energy/bit to achieve the same P(e) as BPSK. In other words, BPSK is 3 d. B more power efficient that BFSK. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 57

Comparison of Power Efficiency of Modulation Schemes Ø We will use BPSK/QPSK as a benchmark with which to compare the power efficiency of other modulation schemes. Ø BPSK/QPSK has. Now, define the power efficiency of a modulation scheme (relative to BPSK/QPSK) as ECE 414 Wireless Communications, University of Waterloo, Winter 2012 58

Differential Phase Shift Keying (DPSK) Ø So far, we assumed that coherent demodulation is performed, i. e. that the carrier phase is perfectly known at the receiver. This normally requires carrier phase estimation. Ø An alternative is “differentially encoding”, where the data is encoded in phase difference from one symbol to the next. Assuming binary signalling, dk 0 1 1 1 0 1 bk +1 -1 -1 -1 +1 -1 -1 +1 ak +1 This diagram might correspond to either PSK or DPSK! ECE 414 Wireless Communications, University of Waterloo, Winter 2012 59

Transmitter and Receiver for DPSK Mapper Differential Encoder Ø θ represents any mismatch between transmitter/receiver oscillators or phase introduced by the channel. In our system model, (independent of where it comes from) we included in the transmitter block. Ø In coherent systems, we need to estimate and compensate this phase error at the receiver. Here, we simply ignore it! 1 Symbol Delay ECE 414 Wireless Communications, University of Waterloo, Winter 2012 60

Error Probability of DPSK Ø Defining can be written as the decision variable ECE 414 Wireless Communications, University of Waterloo, Winter 2012 61

Error Probability of DPSK (cont’d) Ø We need to find statistical properties of and : First, we recall the definitions ECE 414 Wireless Communications, University of Waterloo, Winter 2012 62

Error Probability of DPSK (cont’d) Ø Encoding scheme: +1 +1 0 -1 +1 -1 0 -1 -1 -1 +1 -1 0 +1 -1 -1 0 1 Ø Under the assumption that is sent where Complex Gaussian ECE 414 Wireless Communications, University of Waterloo, Winter 2012 63

Error Probability of DPSK (cont’d) where the non-zero mean is found as Ø Now, we return to P(e) computation ECE 414 Wireless Communications, University of Waterloo, Winter 2012 64

Error Probability of DPSK (cont’d) Variable change =1 ECE 414 Wireless Communications, University of Waterloo, Winter 2012 65

Error Probability of DPSK (cont’d) Ø There is some performance degradation due to differential detection, but now a less complex receiver can be used (i. e. no need for phase tracking). ECE 414 Wireless Communications, University of Waterloo, Winter 2012 66

Temporal Characteristics of Modulation Schemes Ø So far, we have considered pulse shapes which are strictly limited in the symbol interval. By using a pulse shape to “spill over” into adjacent symbol intervals, better spectral efficiency can be achieved, however this also results in intersymbol interference (ISI). Ø The following block diagram is commonly used for studying ISI. Assuming matched filter type implementation for the demodulator, Detector “Actual” Channel “Equivalent” Channel where ECE 414 Wireless Communications, University of Waterloo, Winter 2012 67

Temporal Characteristics of Modulation Schemes (cont’d) Ø Here, we use in pulse shapes which spill over adjacent symbols. This will bring ISI terms: Ø The condition for no ISI is ISI terms Ø In frequency domain, this requires See proof Proakis “Digital Communications” Chapter 9 Ø This condition is known as “Nyquist pulse-shaping criterion” or “Nyquist condition for zero ISI”. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 68

Temporal Characteristics of Modulation Schemes (cont’d) In the following, we consider three distinct cases: W: Bandwidth of equivalent ch. Ø For this case, there is no choice for Heq to satisfy Nyquist criterion. Ø For this case, there is only one solution: ECE 414 Wireless Communications, University of Waterloo, Winter 2012 69

Temporal Characteristics of Modulation Schemes (cont’d) Ø For this case, there exists many solutions as to satisfy Ø A particular pulse shape which satisfies the above property and has been widely used in practical applications is “raised cosine”. (See page 28) The “Nyquist” pulse takes zero at the sampling points for adjacent signalling intervals. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 70

Temporal Characteristics of Modulation Schemes (cont’d) Ø Under the matched-filter assumption (i. e. which maximizes the output signal -to-noise ratio), the transmit and receive filters satisfy Ø Under the ideal channel assumption , i. e. Ø For “raised-cosine” equivalent channel response, we can divide it into two “root-raised-cosine” (RRC) filters. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 71

Temporal Characteristics of BPSK Ø Consider the baseband BPSK modulated signal with RRC pulse shape Ø “Eye pattern” is a sketch of g(t) for all possible combinations of • Minimum instantaneous power=0 • Maximum instantaneous power=(1. 6)2=4. 1 [d. B] • Dynamic range=4. 1 [d. B] • Average power=1=0 [d. B] • Ø For this example, we observe large “dynamic range of instantaneous power” and large “peak/average ratio”. These make the design of TX power amplifier difficult. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 72

Temporal Characteristics of QPSK Ø The QPSK signal with pulse shaping can be written as Ø The instantaneous power of the QPSK signal is Ø Hence, a QPSK signal suffers similar time-domain problems as a BPSK signal. Now assume, different pulses are used for I&Q channels. If Q channel pulse is delayed by 1/2 symbol relative to I channel pulse, i. e. the instantaneous power is Ø Both terms can not pass through zero simultaneously, hence significantly increasing the minimum instantaneous power and reducing dynamic range of the signal. PSD and BER remain unchanged. This is known as “Offset QPSK (OQPSK)”. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 73

Temporal Characteristics of QPSK (cont’d) Ø Another variant of QPSK is “π/4 -QPSK”. This modulation scheme is a superposition of two QPSK signal constellations offset by π/4 relative to each other. + = Ø PSD and BER of π/4 -QPSK are the same as QPSK. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 74

Temporal Characteristics of QPSK (cont’d) Ø In QPSK, transitions between opposite points in the signal constellation cause the instantaneous power to zero, leading to a large dynamic range. Ø The special structure of π/4 -QPSK avoids transitions which pass the origin, reducing dynamic range and peak-to-average power ratio. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 75

Continuous FSK Ø We can get perfect temporal properties by using continuous FSK (CFSK) where h: Modulation index Ø Instantaneous power= constant Dynamic range=0 d. B Peak-to-average power ratio=0 d. B Ø There is no abrupt switching from one phase to another, avoiding phase discontinuities. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 76

Continuous FSK (cont’d) Ø Here, we assume a rectangle pulse shape for p(t): “Frequency pulse” n=0, 1, . . 1/2 Ts t Ts q(t): “Phase pulse” 1/2 t Ts 3πh 2πh πh 0 -πh -2πh -3πh “Phase Tree” ……… +1 +1 -1 -1 +1 ……… -1 Ts 2 Ts 3 Ts The shaded path illustrates the phases for the input sequence {+1, -1} ECE 414 Wireless Communications, University of Waterloo, Winter 2012 77

Continuous FSK (cont’d) The separation between two carriers is Ø For orthogonality, the minimum value for h should be chosen as h=1/2. This special case is known as “Minimum Shift Keying” (MSK). ECE 414 Wireless Communications, University of Waterloo, Winter 2012 78

Continuous FSK (cont’d) Ø We have already introduced MSK as a special case of modulation family of CFSK. Ø An MSK signal can be also considered as a special form of OQPSK where the rectangular pulses are replaced with half-sinusoidal pulses. Ø The transmission rate on the two orthogonal carriers is 1/2 Ts bits/sec so that the combined transmission rate is 1/Ts bits/sec. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 79

Comparison of MSK, QPSK and OQPSK Ø Continuous phase is assured in MSK while 90 and 180 phase changes are observable for OQPSK and QPSK respectively. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 80

Comparison of MSK, QPSK and OQPSK (cont’d) Ø In terms of temporal efficiency, MSK obviously outperforms QPSK and OQPSK. Ø The main lobe of MSK is wider than that of QPSK and OQPSK and, in terms of null-tonull bandwidth MSK is less spectral efficient. Ø MSK has lower sidelobes than QPSK and OQPSK Less adjacent channel interference Ø MSK, QPSK and OQPSK have the same power efficiency. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 81

Gaussian MSK Ø The spectral efficiency of MSK can be further improved by prefiltering. Gaussian LPF MSK Modulator Ø The frequency response function of Gaussian LPF filter is given as where B is “ 3 d. B-bandwidth of the filter”. Ø We are interested in how a rectangle pulse passed through a Gaussian LPF will look like. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 82

Gaussian MSK (cont’d) Ø Frequency pulse BT: Normalized 3 d. B-Bandwidth Ø Phase pulse corresponding to rectangular pulse shaping (i. e. no filtering) is also included in the figure. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 83

Gaussian MSK (cont’d) BT: Normalized 3 d. B-Bandwidth of Gaussian filter Ø For BT ∞, the pulse shape takes its original “unfiltered” form , i. e. rectangle pulse. GMSK Ø The frequency pulse has a duration of 2 Ts although signaling rate is 1/Ts. Such a LPF will result in intersymbol interference which requires sequence estimation for optimal detection. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 84

Gaussian MSK (cont’d) Ø BT should be chosen as to find a good compromise between spectral efficiency and ISI. Ø As BT decreases, the spectral efficiency improves (i. e. less bandwith). Also sidelobes fall off very rapidly (i. e. less adjacent channel interference). Ø However, reducing BT results in ISI and error rate performance degrades (i. e. observation of an “irreducible error floor” due to ISI) Ø In practical application, BT is typically chosen as (0. 2, 0. 5). GSM systems use GMSK with BT=0. 35. ECE 414 Wireless Communications, University of Waterloo, Winter 2012 85