Mobile Communications

Transcription

1 Mobile Communications Wen-Shen Wuen Trans. Wireless Technology Laboratory National Chiao Tung University WS Wuen Mobile Communications 1 Outline Outline 1 Structure of Wireless Communication Link 2 Analog Modulation Techniques Analog Modulation Techniques 3 Digital Modulation Techniques Introduction Line Coding Pulse Shaping Geometric Representation of Modulation Signals 4 Linear Modulations 5 Constant Envelope Modulation 6 Combined Linear and Contstant Envelope Modulation Techniques WS Wuen Mobile Communications 2

2 Structure of Wireless Communication Link Wireless Transceiver Block Diagram information source source coder channel coder modulator multiple access transmission technique RF transmitter propagation channel diversity combiner RF receiver source decoder channel decoder demodulator equalizer separation of desired user information sink WS Wuen Mobile Communications 4 Transmitter Structure of Wireless Communication Link Analog Source Signaling Source ADC Source coder Channel Coder Multiplexer Baseband Modulator Transmit ADC RX + Channel TX filter Up converter Lowpass Filter Noise ACI, CCI Local Oscillator WS Wuen Mobile Communications 5

3 Receiver Structure of Wireless Communication Link From TX Local Oscillator Carrier Recovery RF filter Down Converter Lowpass Filter Receive ADC Baseband Demodulator Timing Recovery De/MUX Signaling Information Sink Source DAC Source Decoder Channel Decoder WS Wuen Mobile Communications 6 Structure of Wireless Communication Link Descriptions of Wireless Transceiver Block Diagram Information source: digital or analog signals. Source coder: reducing redundancy in the source signal to increase entropy (information per bit) e.g., compression, or ensuring security by encryption. Channel coder: adding redundancy to protect data against transmission errors, interleaving to break up error bursts. Signaling: adding control information for the establishing and ending of connections, synchronizations and user authorized information. Signaling information is usually strongly protected by error correction codes. Multiplexer: combines user data and signaling information Baseband modulator: assigns the gross data bits to complex transmit symbols and determines spectral properties, intersymbol interferences, peak-to-average ratio, etc. WS Wuen Mobile Communications 7

4 Structure of Wireless Communication Link Descriptions of Wireless Transceiver Block Diagram, cont d Carrier recovery: determines the frequency and phase of the carrier of the received signal. Baseband demodulator: obtains soft-decision data from digitized baseband data and may also include equaliztion. Symbol timing recovery: uses demodulated data to determine an estimate of the duration of symbols and use it to fine tune sampling intervals. Decoders: use soft estimates from the demodulator to find the original digital source data. Recent RX may perform joint demodulation and decoding. Demultiplexer: separates the user data and signaling information. WS Wuen Mobile Communications 8 Introduction Analog Modulation Techniques Modulation The process by which some characteristic of a carrier wave is varied in accordance with an information-bearing signal. Modulating signal: information bearing signal Modulated signal: the output of the modulation process Linear modulation: if the input-output relation of the modulator satisfies the principle of superposition. Benefits of modulation Modulation used to shift the spectral content of a message signal so that it lies inside the operating frequency band of the wireless communication channel. Modulation provides a mechanism for putting the information content of a message signal into a form that may be less vulnerable to noise or interference. Modulation permits the use of multiple access. WS Wuen Mobile Communications 10

5 Introduction, cont d Analog Modulation Techniques Analog and Digital Modulation Techniques Analog modulation: m(t) and s(t) are continuous function of time Digital modulation: mapping of data bits to signal waveforms that can be transmitted over an analog channel. Input modulating signal m(t) modulator Output modulated signal s(t) Sinusoidal carrier c(t) WS Wuen Mobile Communications 11 Analog Modulation Techniques Amplitude and Angle Modulation Amplitude modulation, AM the amplitude of the carrier, A(t), is varied with linearly with the message signal m(t) Angle modulation the angle of the carrier, φ(t) = 2πf c t + θ is varied linearly with the message signal m(t) Frequency Modulation, FM the frequency of the carrier is varied linearly with the message signal m(t) Phase Modulation, PM the phase of the carrier θ is varied linearly with the message signal m(t) WS Wuen Mobile Communications 12

6 Analog Modulation Techniques Amplitude Modulation AM signal s AM (t) = A c (1 + m(t))cos ( 2πf c t ) = Re {g(t)e } j2πf ct where g(t) = A c (1 + m(t)) Modulation index: the ratio of peak message signal amplitude to the peak carrier amplitude. If m(t) = A m Ac cos ( 2πf m t ), k = A m Ac Spectrum of AM signal S AM (f ) = 1 2 A ( ( ) ( ) ( ) ( )) c δ f fc + M f fc + δ f + fc + M f + fc RF bandwidth of AM Total power of AM signal B AM = 2f m P AM = 1 2 A2 c (1+ < m(t) >)2 if m(t) = k cos ( 2πf m t ) ( ) P AM = 1 2 A2 c 1 + k2 2 WS Wuen Mobile Communications 13 Analog Modulation Techniques Amplitude Modulation, cont d WS Wuen Mobile Communications 14

7 Analog Modulation Techniques Amplitude Modulation, cont d WS Wuen Mobile Communications 15 Angle Modulation Analog Modulation Techniques FM signal s FM (t) = A c cos [ 2πf c t + θ(t) ] t ] = A c cos [2πf c t + 2πk f m(τ)dτ if m(t) = A m cos ( 2πf m t ) [ s FM (t) = A c cos 2πf c t + k f A m f m sin ( 2πf m t )] Frequency modulation index β f : defines the relation between the message amplitude and the bandwidth of the transmitted signal. β f = k f A m W = f W PM signal Phase modulation index β p s PM (t) = A c cos [ 2πf c t + k θ m(t) ] β p = k θ A m = θ WS Wuen Mobile Communications 16

8 Analog Modulation Techniques FM Demodulation: Slope Detection v 1 (t) = V 1 cos [ 2πf c t + θ(t) ] t = V 1 cos [2πf c t + 2πk f [ v 2 (t) = V 1 2πf c t + dθ dt v out (t) = V 1 [ 2πf c t + dθ dt ] ] sin ( 2πf c t + θ(t) ) = V 1 2πf c + V 1 2πk f m(t) m(τ)dτ ] WS Wuen Mobile Communications 17 Analog Modulation Techniques FM Demodulation: Quadrature Detection WS Wuen Mobile Communications 18

9 Analog Modulation Techniques FM Demodulation: Zero-Crossing WS Wuen Mobile Communications 19 Analog Modulation Techniques Bandwidth of FM Signals Carson s rule An FM signal has 98% of the total transmitted power in a RF bandwidth B T is B T = 2 ( β f + 1 ) f m (Upper bound) B T = 2 f (Lower bound) Carson s rule: for small values of modulation index (β f < 1), the spectrum of an FM wave is effectively limited to the carrier frequency, and one pair of sideband frequencies at f c ± f m. For large values of modulation index, the bandwidth approaches and is only slightly greater than 2 f. WS Wuen Mobile Communications 20

10 Analog Modulation Techniques Tradeoff Between SNR and Bandwidth in FM SNR at the output of an FM receiver ( ) m(t) 2 SNR in SNR out = 6 ( β f + 1 ) β 2 f V p SNR at the input of an FM receiver SNR in,fm = A 2 c /2 2N 0 ( βf + 1 ) B SNR at the input of an AM receiver for m(t) = A m sinω m t SNR in,am = A2 c 2N 0 B SNR out = 3β 2 f ( βf + 1 ) SNR in,fm = 3β 2 f SNR in,am FM offers excellent performance for fading signals. WS Wuen Mobile Communications 21 Analog Modulation Techniques Example 1 How much bandwidth is required for an analog frequency modulated signal that has an audio bandwidth of 5 khz and a modulation index of three? How much output SNR improvement would be obtained if the modulation index is increased to five? What is the tradeoff bandwidth for the improvement? Solution: B T = 2 ( β f + 1 ) f m = 2(3 + 1)5 = 40 khz. The output SNR improvement factor is 3β 2 f (β f + 1) β f = (3 + 1) = 108 = db. β f = (5 + 1) = 450 = db. The improvement of output SNR is = 6.2 db. For β f = 5, the required bandwidth is 60 khz. WS Wuen Mobile Communications 22

11 Digital Modulation Techniques Digital Modulation Techniques Advantages of Digital Modulation greater noise immunity robustness to channel impairments easier multiplexing of various forms of information greater security Factors that influence the choice of digital modulation Bandwidth efficiency Adjacent channel interference Sensitivity with respect to noise Robustness with respect to delay and Doppler dispersion Easy and cost-effective to implement WS Wuen Mobile Communications 24 Digital Modulation Techniques Power Efficiency and Bandwidth Efficiency Power efficiency (Energy efficiency), η p The ability of a modulation technique to preserve the fidelity of the digital message at low power levels. Expressed as the ratio of the signal energy per bit to noise power spectral density (E b /N 0 ) required at the receiver input for a certain probability of error. Bandwidth efficiency, η B The ability of a modulation scheme to accommodate data within a limited bandwidth. Defined as the ratio of the throughput data rate per Hertz in a given bandwidth. η B = R B bps/hz WS Wuen Mobile Communications 25

12 Digital Modulation Techniques Shannon s Channel Capacity Theorem For an arbitrary small probability of error, the maximum possible bandwidth efficiency is limited by the noise in the channel, ( ) η B,max = C B = log S N where C is the channel capacity in bps, B is the RF bandwidth, and S N is the signal to noise ratio. WS Wuen Mobile Communications 26 Digital Modulation Techniques Example 2 If the SNR of a wireless communication link is 20 db and the RF bandwidth is 30 khz, determine the maximum theoretical data rate that can be transmitted. Solution: SNR=20 db=100, Bandwidth B=30000 Hz, C = Blog 2 (1 + S/N) = 30000log 2 ( ) = kbps Example 3 What is the theoretical maximum data rate that can be supported in a 200 khz channel for SNR=10 db and 30 db? How does this compare to the GSM standard? Solution: SNR=10 db=10, B=200 khz, C = Blog 2 (1 + S/N) = log 2 (1 + 10) = kbps GSM data rate is kbps, which is about 40 % of the theoretical limit for 10 db SNR conditions. For SNR=30 db=1000, B=200 khz, C = Blog 2 (1 + S/N) = log 2 ( ) = 1.99 Mbps WS Wuen Mobile Communications 27

13 Digital Modulation Techniques Bandwidth and Power Spectral Density of Digital Signals Various bandwidth definitions are based on the power spectral density of the signal. Power spectral density (PSD) of a random signal w(t) WT (f ) 2 P w (f ) = lim T T where W T (f ) is the Fourier transform of w T (t) which is the truncated version of the signal w(t) { w(t) T/2 < t < T/2 w T (t) = 0 otherwise Power spectral density of a modulated (bandpass) signal s(t) = Re { g(t)e j2πf ct } P s (f ) = 1 4 where P g (f ) is the PSD of g(t). [ Pg (f f c ) + P g ( f f c ) ] WS Wuen Mobile Communications 28 Digital Modulation Techniques Bandwidth and Power Spectral Density of Digital Signals, cont d Absolute bandwidth: the range of frequencies over which the signal has a non-zero power spectral density. Null-to-null bandwidth: the width of the main spectral lobe Half-power bandwidth (3-dB bandwidth): the interval between frequencies at which the PSD has dropped to half power Occupied bandwidth (FCC): the band which leaves exactly 0.5% of the signal above the upper band limit and exactly 0.5% of the signal power below the lower band limit, i.e. 99% of the signal power is contained within the occupied bandwidth WS Wuen Mobile Communications 29

14 Line Coding Digital Modulation Techniques Mapping of binary information sequence into the digital signal that enters the channel. WS Wuen Mobile Communications 30 Digital Modulation Techniques Properties of Line Codes Self-Synchronization There is enough timing information built into the code so that bit synchronizers can extract the timing or clock signal. Low Probability of Bit Error Receivers can be designed that will recover the binary data with a low probability of bit error when the input data is corrupted by noise or ISI. A Spectrum that is Suitable for the Channel For example, if the channel is AC coupled, the PSD of the line code signal should not have DC component. In addition, the signal bandwidth needs to be sufficiently small compared to the channel bandwidth, so that ISI will not be a problem. Transmission Bandwidth This should be as small as possible. Error Detection Capability It should be possible to implement this feature easily by the addition of channel encoders and decoders, or the feature should be incorporated into the line code. WS Wuen Mobile Communications 31

15 Digital Modulation Techniques Power Spectral Density of Line Codes WS Wuen Mobile Communications 32 Digital Modulation Techniques The Need for Pulse Shaping The most simple basis pulse is a rectangular pulse with duration T. where sinc(x) = sin(x)/x. g R (t,t) = { 1 0 t T 0 otherwise G R (f,t) = F { g R (t,t) } = Tsinc ( πft ) e jπft When rectangular pulses are passed through a bandlimited channel, the pulses will spread in time, and the pulse for each symbol will smear into time intervals of succeeding symbols. inter-symbol interference (ISI) In frequency domain the rectangular pulse creates large adjacent channel interference. the transmitting pulse should be shaped to limit the bandwidth as well as to minimize ISI. WS Wuen Mobile Communications 33

16 Digital Modulation Techniques Nyquist Criterion for ISI Cancellation The effect of ISI could be completely nullified if the overall response of the communication system is designed so that at every sampling instant at the receiver, the response due to all symbols except the current symbol is equal to zero. Impulse response of the overall communication system, h eff (t) h eff (nt s ) = { K n = 0 0 n 0 h eff (t) = δ(t) p(t) h c (t) h r (t) h eff (t) should have a fast decay with a small magnitude near the sample values for n 0 WS Wuen Mobile Communications 34 Digital Modulation Techniques Nyquist Ideal Pulse Shaping h eff (t) = ( ) sin πt T s πt T s ( ) πt = sinc,h eff (f ) = 1 ( ) f rect f s where f s is the symbol rate. Ideal Nyquist pulse shaping filter is difficult to implement Noncausal system (h eff (t) exists for t < 0) and is difficult to approximate T s Sensitive to jitter (error in sampling time of zero-crossings will cause significant ISI) f s WS Wuen Mobile Communications 35

17 Nyquist Filter Digital Modulation Techniques Nyquist proved that any filter with a transfer function having a rectangular filter of bandwidth f 0 1/2T s, convolved with any arbitrary even function Z(f ) with zero magnitude outside the passband, satisfied the zero ISI condition. ( ) f H eff (f ) = rect Z(f ) where Z(f ) = Z( f ) and Z(f ) = 0 for f f 0 1/2T s. Nyquist criterion Any filter with an impulse response f 0 h eff (t) = sin(πt/t s) z(t) πt can achieve ISI cancellation. Filters which satisfies the Nyquist criterion are called Nyquist filters. WS Wuen Mobile Communications 36 Digital Modulation Techniques Nyquist Filter, cont d H eff is often achieved by using filters with transfer function H eff at both the transmitter and receiver. providing a matched filter response for the system while at the same time minimizing the bandwidth and ISI. WS Wuen Mobile Communications 37

18 Digital Modulation Techniques Raised Cosine Rolloff Filter 1 0 f [ ( )] 1 α 2T s 1 π (2 f T H RC (f ) = cos s 1+α) 1 α 2α 2T s f 1+α 2T s 0 f > 1+α 2T s where α is the rolloff factor which ranges between 0 and 1. WS Wuen Mobile Communications 38 Digital Modulation Techniques Raised Cosine Rolloff Filter, cont d h RC (t) = ( ) sin πt T s πt ( cos παt ( ) 2 1 4αt 2T s The symbol rate R s that can be passed through a baseband raised cosine rolloff filter is R s = 1 = 2B T s 1 + α where B is the absolute filter bandwidth. T s ) WS Wuen Mobile Communications 39

19 Digital Modulation Techniques Gaussian Pulse-Shaping Filter Gaussian pulse-shaping filter is particularly effective when used in conjunction with minimum shift keying (MSK) modulation or other modulation well suited for power efficient nonlinear amplifiers. Transfer function H G (f ) = e α2 f 2 where α is related to B the 3-dB bandwidth of the baseband Gaussian shaping filter, α = ln2 2B = B Impulse response h G (t) = π α e π 2 α 2 t2 Gaussian pulse-shaping filter does not satisfy the Nyquist criterion for ISI cancellation, reducing the spectral occupancy creates degradation in performance due to increased ISI. WS Wuen Mobile Communications 40 Digital Modulation Techniques Gaussian Pulse-Shaping Filter, cont d Impulse response of Gaussian pulse-shaping filter WS Wuen Mobile Communications 41

20 Digital Modulation Techniques Example 4 Find the first zero-crossing RF bandwidth of a rectangular pulse which has T s = µs. Compare this to the bandwidth of a raised cosine filter pulse with T s = µs and α = Solution: The first zero-crossing filter (null-to-null) bandwidth of a rectangular pulse is equal to 2 T s = = khz and that for a raised cosine filter with α = 0.35 is 1 T s (1 + α) = ( ) = khz 6 WS Wuen Mobile Communications 42 Digital Modulation Techniques Geometric Representation of Modulation Signals M-ary modulation signals Choosing a particular signal waveform s i (t) from a finite set of possible signal waveforms (or symbols) based on the information bits applied to the modulator. S = {s 1 (t),s 2 (t),...,s M (t)} where M represents a total of M possible signals in the modulation set S. a maximum of log 2 M bits of information per symbol. Represent the modulation signals on a vector space with a set of N orthonormal signals that form a basis for the vector space. Any point in the vector space can be represented as a linear combination of the basis signals. WS Wuen Mobile Communications 43

21 Digital Modulation Techniques Signal Space for Modulation Signals Basis { φ1 (t),φ 2 (t),...,φ N (t) } Modulation Signals N s i (t) = s ij φ j (t). j=1 Basis signals are orthogonal to one another in time φ i (t)φ j (t)dt = 0 i j Each basis signal is normalized to have unit energy E = φ 2 i (t)dt = 1 WS Wuen Mobile Communications 44 Digital Modulation Techniques Signal-Space for Digital Modulation Consider two-dimensional signal constellations, the normalized basis of unite energy are φ 1 (t) and φ 2 (t) 2 φ 1 (t) = T cos(2πf ct), 0 t T where and φ 2 (t) = T 0 2 T sin(2πf ct), T 0 φ 1 (t)φ 2 (t)dt = 0 0 t T T φ 2 1 (t)dt = φ 2 2 (t)dt = 1 s i (t) = s i1 φ 1 (t) + s i2 φ 2 (t) or s i = (s i1,s i2 ) s i s k = N ( ) T 2 sij s kj = (s i (t) s k (t)) 2 dt j=1 0 0 WS Wuen Mobile Communications 45

22 Digital Modulation Techniques Example: BPSK Signals s 1 (t) = s 2 (t) = 2E b cos(2πf c t), T b 2E b T b cos(2πf c t), 0 t T b 0 t T b where E b is the energy per bit, T b is the bit period, and a rectangular pulse p(t) = rect((t T b /2)/T b ) is assumed. 2 φ 1 (t) = cos(2πf c t), T b 0 t T b BPSK signal set { S BPSK = Eb φ 1 (t), } E b φ 1 (t) WS Wuen Mobile Communications 46 Constellation Digital Modulation Techniques WS Wuen Mobile Communications 47

23 Digital Modulation Techniques Error Probability Upper bound for the probability of symbol error in an additive white Gaussian noise channel (AWGN) with a noise density N 0 for an arbitrary constellation The average probability of error for a particular modulation signal P s (ɛ s i ) ( ) dij Q 2N0 j=1,j i where d ij is the Euclidean distance between the i-th and j-th signal point in the constellation. If all the M modulation waveforms are equally likely to be transmitted, the average probability of error for a modulation P s (ɛ) = 1 M M i=1p s (ɛ s i )P(s i ) = 1 M M P s (ɛ s i ) i=1 WS Wuen Mobile Communications 48 Linear Modulations Binary Phase Shift Keying (BPSK) sinusoidal carrier amplitude A c, Energy per bit E b = 1 2 A2 c T b s BPSK (t) = 2E b T b cos(2πf c t + θ c ) for binary 1 s BPSK (t) = 2E b T b cos(2πf c t + θ c ) for binary 0 or or where s BPSK (t) = m(t) 2E b T b cos(2πf c t + θ c ) s BPSK (t) = Re {g } BPSK (t)e j2πf ct g BPSK (t) = 2E b T b m(t)e jθ c WS Wuen Mobile Communications 50

24 Linear Modulations Spectrum and BPSK Bandwidth Power spectral density of the complex envelope g BPSK (t) P gbpsk (f ) = 2E b ( sinπftb πft b PSD of a BPSK signal at RF P BPSK (f ) = E [( ) b sinπ(f fc )T 2 ( ) b sinπ( f fc )T 2 ] b + 2 π(f f c )T b π( f f c )T b ) 2 WS Wuen Mobile Communications 51 BPSK Receiver Linear Modulations If no multipath impairments are introduced by the channel, the received BPSK signal r BPSK (t) = m(t) 2E b 2E b cos(2πf c t + θ c + θ ch ) = m(t) cos(2πf c t + θ) T b T b The output of the multiplier after the frequency divider m(t) 2E b T b cos 2 (2πf c t + θ) = m(t) 2E b T b [ ] 2 cos(4πf ct + 2θ) Probability of bit error ( ) 2E b P e,bpsk = Q N 0 WS Wuen Mobile Communications 52

25 BPSK Receiver, cont d Linear Modulations WS Wuen Mobile Communications 53 Linear Modulations Differential Phase Shift Keying (DPSK) Noncoherent form of phase shift keying avoids the need for a coherent reference signal at the receiver. Input binary sequence is first differentially encoded and then modulated using BPSK modulator. Differentially encode: d k = m k d k 1. Leave d k unchanged from the previous symbol if the incoming binary symbol m k is 1 and to toggle d k if m k is 0. m k d k d k WS Wuen Mobile Communications 54

26 Linear Modulations Differential Phase Shift Keying (DPSK), cont d DPSK signaling has about 3 db worst energy efficency than coherent PSK. Average probability of error for DPSK in AWGN channel is P e,dpsk = 1 2 e E b N 0 WS Wuen Mobile Communications 55 Linear Modulations Quadrature Phase Shift Keying (QPSK) QPSK has twice the bandwidth efficiency of BPSK, since two bits are transmitted in a single modulation symbol. S QPSK (t) = where T s = 2T b. or S QPSK (t) = 2E [ s cos 2πf c t + (i 1) π ],0 t T s,i = 1,2,3,4 T s 2 2E [ s cos (i 1) π ] 2E [ s cos(2πf c t) sin (i 1) π ] sin(2πf c t) T s 2 T s 2 if φ 1 = 2/T s cos(2πf c t) and φ 2 = 2/T s sin(2πf c t) are defined over 0 t T s, { [ S QPSK (t) = Es cos (i 1) π ] φ 1 (t) [ E s sin (i 1) π ] } φ 2 (t) i = 1,2,3,4 2 2 WS Wuen Mobile Communications 56

27 Linear Modulations Quadrature Phase Shift Keying (QPSK), cont d The average probability of bit error in the AWGN channel is ( ) ( 2Es 2E b P e,qpsk = Q = Q 2N0 N 0 ) WS Wuen Mobile Communications 57 Linear Modulations QPSK Power Spectral Density The PSD of a QPSK signal using rectangular pulses can be expressed as P QPSK (f ) = E [( ) s sinπ(f fc )T 2 ( ) s sinπ( f fc )T 2 ] s + 2 π(f f c )T s π( f f c )T s or [( ) sin2π(f fc )T 2 ( b sin2π( f fc )T b P QPSK (f ) = E b + 2π(f f c )T b 2π( f f c )T b ) 2 ] WS Wuen Mobile Communications 58

28 Linear Modulations QPSK Transmission and Detection QPSK Transmitter WS Wuen Mobile Communications 59 Linear Modulations QPSK Transmission and Detection, cont d QPSK Receiver WS Wuen Mobile Communications 60

29 Offset QPSK Linear Modulations To ensure fewer baseband signal transitions applied to the RF amplifier supports more efficient amplification and helps eliminate spectrum regrowth Even and odd bit streams, m I (t) and m Q (t) are offset in their relative alignment by one bit period (half-symbol period.) Only one of the two bit streams can change values maximum phase shift of the signal is limited to ± 90 WS Wuen Mobile Communications 61 QPSK v.s. Offset QPSK Linear Modulations WS Wuen Mobile Communications 62

30 π/4 QPSK Linear Modulations Maximum phase change is limited to ± 135 Switching between two constellations, every successive bit ensures that there is at least a phase shift n π 4 for every symbol WS Wuen Mobile Communications 63 Linear Modulations π/4 QPSK Transmission Techniques where θ k = θ k 1 + φ k. I k = cosθ k = I k 1 cosφ k Q k 1 sinφ k Q k = sinθ k = I k 1 sinφ k + Q k 1 cosφ k where s π/4qpsk (t) = I(t)cos2πf c t Q(t)sin2πf c t N 1 N 1 I(t) = I k p(t kt s T s /2) = cosθ k p(t kt s T s /2) k=0 k=0 N 1 N 1 Q(t) = Q k p(t kt s T s /2) = sinθ k p(t kt s T s /2) k=0 { and the peak amplitude } of I(t),Q(t) can take one of the five values 1 0,1, 1,, k=0 WS Wuen Mobile Communications 64

31 Linear Modulations π/4 QPSK Transmission Techniques, cont d information bits m Ik,m Qk Phase shift φ k 11 π/4 01 3π/4 00 3π/4 10 π/4 WS Wuen Mobile Communications 65 Linear Modulations π/4 QPSK Detection Techniques Baseband Differential Detection WS Wuen Mobile Communications 66

32 Linear Modulations π/4 QPSK Detection Techniques, cont d Baseband Differential Detection w k = cos(φ k γ),z k = sin(φ k γ) x k = w k w k 1 + z k z k 1,y k = z k w k 1 w k z k 1 The output of differential decoder, x k = cos(φ k γ)cos(φ k 1 γ) + sin(φ k γ)sin(φ k 1 γ) = cos(φ k φ k 1 ) y k = sin(φ k γ)sin(φ k 1 γ) + cos(φ k γ)cos(φ k 1 γ) = sin(φ k φ k 1 ) The output of the decision device, S I = 1, if x k > 0 or S I = 0, if x k < 0 S Q = 1, if y k > 0 or S Q = 0, if y k < 0 WS Wuen Mobile Communications 67 Linear Modulations π/4 QPSK Detection Techniques IF Differential Detector WS Wuen Mobile Communications 68

33 Linear Modulations π/4 QPSK Detection Techniques FM Discriminator WS Wuen Mobile Communications 69 Constant Envelope Modulation Nonlinear Modulation Nonlinear modulation: the amplitude of the carrier is constant, regardless of the variation in the modulating signal Advantages Efficient power amplifier can be used without introducing degradation in the spectrum occupancy of the transmitted signal. Low out-of-band radiation of the order of 60 db to 70 db can be achieved. Limiter-discriminator detection can be used simplifying receiver designs and providing high immunity against random FM noise and signal fluctuation due to Rayleigh fading. WS Wuen Mobile Communications 71

34 Constant Envelope Modulation Binary Frequency Shift Keying (BFSK) s FSK (t) = v H (t) = s FSK (t) = v L (t) = Discontinuous FSK s FSK (t) = v H (t) = s FSK (t) = v L (t) = Continuous FSK s FSK (t) = 2E b cos ( 2πf c + 2π f ) t, 0 t T b (binary 1) T b 2E b cos ( 2πf c 2π f ) t, 0 t T b (binary 0) T b 2E b cos ( ) 2πf H t + θ 1, 0 t Tb (binary 1) T b 2E b cos ( ) 2πf L t + θ 2, 0 t Tb (binary 0) T b 2E b T b cos ( 2πf c t + θ(t) ) = 2E t ) b cos (2πf c t + 2πk f m(τ)dτ T b WS Wuen Mobile Communications 72 BPSK, cont d Constant Envelope Modulation Transmission bandwidth B T of an FSK signal B T = 2 f + 2B Rectangular pulses is B = R B T = 2( f + R) Raised cosine pulse-shaping filter B T = 2 f + (1 + α)r Coherent Detection of Binary FSK ( ) Eb P e,fsk = Q N 0 WS Wuen Mobile Communications 73

35 Constant Envelope Modulation Noncoherent Detection of Binary FSK Noncoherent Detection of Binary FSK P e,fsk,nc = 1 2 e E b 2N 0 WS Wuen Mobile Communications 74 Constant Envelope Modulation Minimum Shift Keying (MSK) A special type of continuous phase frequency shift keying (CPFSK) wherein the peak frequency deviation is equal to 1/4 the bit rate. Modulation index: k FSK = (2 F)/R b, where F is the peak RF frequency deviation and R b is the bit rate. Minimum shift keying: the minimum frequency separation (i.e., bandwidth) that allows orthogonal detection. MSK signal can be thought of as a special form of OQPSK where the base band rectangular pulses are replaced with half-sinusoidal pulses. N 1 N 1 S MSK (t) = m Ii (t)p(t 2iT b )cos2πf c t+ m Qi (t)p(t 2iT b T b )sin2πf c t i=0 i=0 where p(t) = { ( cos πt 2T b ) 0 t 2T b 0 elsewhere WS Wuen Mobile Communications 75

36 Constant Envelope Modulation Minimum Shift Keying (MSK), cont d MSK signal can be seen as a special type of a continuous phase FSK [ 2E b S MSK (t) = cos 2πf c t = m Ii (t)m Qi (t) πt ] + φ k T b 2T b where phi k is 0 or π depending on whether m Ii (t) is 1 or 1. Phase continuity at the bit transition period is ensured by choosing the carrier frequency to be an integral multiple of R b /4 = 1/(4T) MSK signal is an FSK signal with binary signaling frequencies of f c + 1 4T and f c 1 4T. MSK signal varies linearly during the course of each bit period. WS Wuen Mobile Communications 76 Constant Envelope Modulation MSK Power Spectrum Density The baseband pulse shaping function for MSK is { ( cos πt ) p(t) = 2T t < T 0 elsewhere P MSK (f ) = 16 ( ) cos2π(f + fc )T 2 π f 2 T ( ) cos2π(f fc )T 2 π f 2 T 2 WS Wuen Mobile Communications 77

37 MSK Transmitter Constant Envelope Modulation WS Wuen Mobile Communications 78 MSK Receiver Constant Envelope Modulation WS Wuen Mobile Communications 79

38 Constant Envelope Modulation Gaussian Minimum Shift Keying (GMSK) Passing the modulating NRZ data waveform through a premodulation Gaussian pulse-shaping filter Considerably reducing the sidelobe levels in the transmitted spectrum Excellent power efficiency (due to the constant envelope) and spectrum efficiency ISI degradation is not sever if the 3-dB bandwidth bit duration product BT > 0.5. As long as GMSK irreducible error rate is less than that produced by the mobile channel, no penalty in using GMSK. GMSK premodulation filter impulse response π π 2 h G (t) = α 2 t2 α e GMSK premodulation filter transfer function H G (f ) = e α2 f 2 α = ln2 2B = B WS Wuen Mobile Communications 80 Constant Envelope Modulation Gaussian Minimum Shift Keying (GMSK), cont d GMSK bit error rate ( ) 2γE b P e = Q where { 0.68 GMSK with BT = 0.25 γ 0.85 MSK (BT = ) N 0 WS Wuen Mobile Communications 81

39 Constant Envelope Modulation GMSK Transmitter and Receiver WS Wuen Mobile Communications 82 Constant Envelope Modulation Example 5 Find the 3-dB bandwidth for a Gaussian low pass filter used to produce 0.25 GMSK with a channel data rate of R b = 270 kbps. What is the 90% power bandwidth in the RF channel? Specify the Guassian filter parameter α. Solution: RF bandwidth T = 1 R b = = 3.7µs BT = 0.25 B = 0.25 T = 0.25 = kHz BW = 0.57R b = = 153.9kHz WS Wuen Mobile Communications 83

40 Combined Linear and Contstant Envelope Modulation Techniques M-ary Phase Shift Keying (MPSK) The carrier phase takes on one of M possible values 2π(i 1) θ i =,i = 1,2,...,M M MPSK signal waveform 2E s s i (t) = cos (2πf c t + 2πM ) T (i 1),0 t T s,i = 1,2,...,M s E s : Energy per symbol, E s = (log 2 M)E b T s : symbol period, T s = (log 2 M)T b MPSK in quadrature form ( ) 2E s 2π s i (t) = cos (i 1) cos(2πf c t) T s M Let ( ) 2E s 2π sin (i 1) sin(2πf c t) T s M φ 1 (t) = 2/T s cos(2πf c t),φ 2 (t) = 2/T s sin(2πf c t), { ( ) Es 2π S MPSK (t) = cos (i 1), ( )} 2π E s sin (i 1) M M WS Wuen Mobile Communications 85 Combined Linear and Contstant Envelope Modulation Techniques Error Probability of M-ary PSK Distance between adjacent symbols: 2 E s sin ( ) π M Average symbol error probability of an M-ary PSK is ( 2E b log P e 2Q 2 M π ) sin( ) N 0 M Average symbol error probability of a differential M-ary PSK in AWGN channel for M 4 is approximately ( 4E s π ) P e 2Q sin( ) N 0 M WS Wuen Mobile Communications 86

41 Combined Linear and Contstant Envelope Modulation Techniques Power Spectra of M-ary PSK PSD of the M-ary PSK signal with rectangular pulses is P MPSK = E s 2 P MPSK = E b log 2 M 2 [( sinπ(f fc )T s π(f f c )T s [ ( sinπ(f fc )T b log 2 M π(f f c )T b log 2 M ) 2 ( ) sinπ( f fc )T 2 ] s + π( f f c )T s ) 2 ( ) ] sinπ( f fc )T b log + 2 M 2 π( f f c )T b log 2 M WS Wuen Mobile Communications 87 Combined Linear and Contstant Envelope Modulation Techniques Power Spectra of M-ary PSK, cont d WS Wuen Mobile Communications 88

42 Combined Linear and Contstant Envelope Modulation Techniques M-ary Quadrature Amplitude Modulation (QAM) s i (t) = 2E min a i cos(2πf c t) + T s 2E min T s 0 t T,i = 1,2,...,M b i sin(2πf c t), where E min is the energy of the signal with the lowest amplitude and the coordinate of the ith point are a i Emin and b i Emin. WS Wuen Mobile Communications 89 Combined Linear and Contstant Envelope Modulation Techniques M-ary Quadrature Amplitude Modulation, cont d (a i,b i ) is an element of the L L matrix {a i,b i } = ( L + 1,L 1) ( L + 3,L 1) (L 1,L 1) ( L + 1,L 3) ( L + 3,L 3) (L 1,L 3) ( L + 1, L + 1) ( L + 3, L + 1) (L 1, L + 1) where L = M 16-QAM: L = 16 = matrix ( 3, 3) ( 1, 3) (1, 3) (3, 3) ( 3, 1) ( 1, 1) (1, 1) (3, 1) {a i,b i } = ( 3, 1) ( 1, 1) (1, 1) (3, 1) ( 3, 3) ( 1, 3) (1, 3) (3, 3) WS Wuen Mobile Communications 90

43 Combined Linear and Contstant Envelope Modulation Techniques M-ary QAM Error Probability Average probability of error in AWGN channel for M-ary QAM, using coherent detection P e 4 ( ( 1 1 )Q M 2E min N 0 ) In terms of average signal energy E av ( ( ) P e E av )Q M (M 1)N 0 WS Wuen Mobile Communications 91 Combined Linear and Contstant Envelope Modulation Techniques M-ary Frequency Shift Keying (MFSK) M-ary FSK signal [ ] 2E s π s i (t) = cos (n c + i)t T s T s 0 t T s,i = 1,2,...,M where f c = n c /2T s for some integer n c The M transmitted signals are of equal energy and equal duration and the signal frequencies are separated by 1/2T s Hz. For coherent M-ary FSK, the optimum receiver consists of a bank of M correlators or matched filters tuned to the M distinct carriers. Average probability of error ( ) E b log P e (M 1)Q 2 M For non-coherent detection using matched filters followed by envelope detectors, the average probability of error ( M 1 ( 1) k+1 ( ) ) M 1 P e = e ke s (k+1)n 0 k + 1 k k=1 and using only the leading WS Wuen terms Mobile ofcommunications the binomial expansion, the 92 N 0

44 Combined Linear and Contstant Envelope Modulation Techniques M-ary Frequency Shift Keying (MFSK), cont d Channel bandwidth of a coherent M-ary FSK signal B = R b(m + 3) 2log 2 M Channel bandwidth of a non-coherent M-ary FSK B = R bm 2log 2 M WS Wuen Mobile Communications 93