CSE414 Digital Communications Chapter 4 Bandpass Modulation and Demodulation/Detection Bandpass Modulation Page 1 1
Bandpass Modulation n Baseband transmission is conducted at low frequencies n Passband transmission is to send the signal at high frequencies n Signal is converted to a sinusoidal waveform, e.g. s A(t)cos[ ω 0 t +φ(t) ] where ω 0 is called carrier frequency is much higher than the highest frequency of the modulating signals, i.e. messages n Bits are encoded as a variation of the amplitude, phase, frequency, or some combination of these parameters. 3 ypes of Bandpass Modulation 4 Page
Bandpass Examples 5 CSE414 Digital Communications Phasor Representation of a Sinusoid Page 3 3
Phasor Representation of Sinusoidal Signals Using Euler identity e jω 0 t = t cos ω 0t + j sin ω 0 Inphase (I) Component Quadrature (Q) Component he unmodulated carrier wave c cos(ω 0 t) is represented as a unit vector rotating in a counter-clockwise direction at a constant rate of ω 0 radians/s. Amplitude Modulation (AM) A double side band, amplitude modulated (DSB-AM) signal is represented by s( t) = cos ω0t (1 + cos ωmt) where c cos(ω 0 t) is the carrier signal and x cos(ω m t) is the information bearing signal. An equivalent representation of DSB-AM signal is given by s( t) = cosω t [1 + = Re ( e 0 jω0t 1 jωmt jωmt { e [1 + ( e + e )]} he phasor representation of the DSB-AM signal is shown as 1 jωmt + e jωmt )] he composite signal rotates in a counter-clockwise direction at a constant rate of ω 0 radians/s. However, the vector expands and shrinks depending upon the term ω m t. Page 4 4
Frequency Modulation (FM) A frequency modulated (FM) signal is represented by Assuming that the information bearing signal x cos(ω m t), the above expression reduces to s( t) = cos For narrow band FM = cos [ ω t + k x( t ] s t) = cos f ) dt ( 0 k f [ ω0t + sin( ω t) ] ω m m k f k f ( ω0t) cos( sin( ωmt) ) sin( ω0t) sin( sin( ωmt) ) ω m ω m s cos( ω 0 t) β sin( ω 0 t)sin(ω m t), β = k f ω m <<1 { } "1+ β e jω mt β e jω mt { # $ %} = Re e jω0t β e jω 0t " 1 e jω mt 1 e jω mt # $ % = Re e jω 0t Frequency Modulation () he phasor representation of a narrowband FM signal is given by s t) jω t β jωmt β jωmt { e [ 1+ e e ]} 0 ( = Re he phasor diagram of the narrowband FM signal is shown as he composite signal speeds up or slows down according to the term ω m t. Page 5 5
Phase Shift Keying he general expression for M-ary PSK is [ ω t + φ ( t) ] 0 t, i = 1, M E si ( t) = cos 0 i, where the phase term φ i πi/m. he symbol energy is given by E and is the duration of the symbol. he waveform and phasor representation of the -ary PSK (binary PSK) is shown below. Frequency Shift Keying he general expression for MFSK is E [ ω t + φ] 0 t, i = 1, M si ( t) = cos i, where the frequency term ω i has M discrete values and phase φ is a constant. he symbol energy is given by E and is the duration of the symbol. he frequency difference (ω i+1 ω i ) is typically assumed to be an integral multiple of π/. he waveform and phasor representation of the 3-ary FSK is shown below. Page 6 6
Amplitude Shift Keying he general expression for M-ary ASK is s i E i (t) cos[ ω 0 t +ϕ] 0 t,i =1,, M where the amplitude term E i ( t) has M discrete values and frequency ω 0 and phase φ is a constant. he waveform and phasor representation of the -ary ASK (binary ASK) is shown below. Amplitude Phase Keying he general expression for M-ary APK is s i E i (t) cos[ ω 0 t +ϕ i (t)] 0 t,i =1,, M where both the signal amplitude and phase vary with the symbol. he waveform and phasor representation of the 8-ary APK is shown below. Page 7 7
Digital Modulation Summary PSK s i E cos! ω t + πi $ 0 " # M % & 0 t,i =1,, M FSK s i E cos ω [ t +ϕ i ] 0 t,i =1,, M ASK s i E i (t) cos[ ω 0 t +ϕ] 0 t,i =1,, M QAM s i E i (t) cos[ ω 0 t +ϕ i (t)] 0 t,i =1,, M Detection of Signals in Gaussian Noise Decision Regions: Assume that the received signal r(t) is given by r( t) = s ( t) + n( t) r( t) = s ( t) + n( t) symbol 1 symbol he task of the detector is to decide which symbol was transmitted from r(t). 1 For equi-probable binary signals corrupted with AWGN, the minimum error decision rule is equivalent to choosing the symbol such that the distance d(r,s i ) = r s i is minimized. Procedure: 1. Pick an orthonormal basis functions for the signal space.. Represent s 1 (t) and s (t) as vectors in the signal space. 3. Connect tips of vectors representing s 1 (t) and s (t). 4. Construct a perpendicular bisector of the connecting lines. 5. he perpendicular bisector divides D plane in regions. 6. If r(t) is located in R1, choose s 1 (t) as transmitted signal 7. If r(t) is located in R, choose s (t) as transmitted signal 8. he figure is referred to as the signal constellation Page 8 8
Detection of Signals in Gaussian Noise () Correlator Receiver for M-ary ransmission (1) Approach 1: Use correlator implementation of matched filter. Decision Rule: Use signal s i (t) that results in the highest value of z i (t). Page 9 9
Correlator Receiver for M-ary ransmission () Approach : Use Basis functions {ψ i (t)}, 1 <= i <= N, N <= M, to represent signal space Each signal s i (t) is represented as a linear combination of the basis functions si ( t) = ai1ψ1 ( t) + aiψ( t) + + ainψ N ( t), 1 i M Decision Rule: Pick signal s i (t) whose coefficient a ij best match z j (). CSE414 Digital Communications Coherent & Non-Coherent Detection Page 10 1 0
Definitions n Coherent detection the receiver exploits knowledge of the carrier s phase to detect the signal n Require expensive and complex carrier recovery circuit n Better bit error rate of detection n Non-coherent detection the receiver does not utilize phase reference information n Do not require expensive and complex carrier recovery circuit n Poorer bit error rate of detection n Differential systems have important advantages and are widely used in practice 1 Coherent Receiver n Carrier recovery for demodulation n Received signal r Acos(ω c t +ϕ)+ n(t) n Local carrier cos(ω c t + ˆϕ) n Carrier recovery phase lock loop circuit Δϕ = ϕ ˆϕ 0 n Demodulation leads to recovered baseband signal Y s(t +τ )+ n(t) n iming recovery for sampling n Align receiver clock with transmitter clock, so that sampling à no ISI Y k = s k + n k Page 11 11
Non-Coherent Receiver n No carrier recovery for demodulation n Received signal r Acos(ω c t +ϕ)+ n(t) n Local carrier cos(ω c t + ˆϕ) n No carrier recovery Δϕ = φ = ϕ ˆϕ 0 n Demodulation leads to recovered baseband signal Y s(t +τ )e jφ + n(t) n iming recovery for sampling n Align receiver clock with transmitter clock, sampling results in Y k = s k e jφ + n k could not recover transmitted symbols properly from Y k CSE414 Digital Communications Coherent Detection Page 1 1
Binary PSK (1) In coherent detection, exact frequency and phase of the carrier signal is known. Binary PSK: 1. he transmitted signals are given by s 1 s E cos ω [ 0t +ϕ], 0 t E cos[ ω t +ϕ + π 0 ], 0 t = E cos[ ω t +ϕ 0 ], 0 t. Pick the basis function ψ ( t) = 1 cos[ ω t ] t 0 + φ, 0 3. Represent the transmitted signals in terms of the basis function s ( t) = 1 s ( t) = Eψ ( t), 1 Eψ ( t), 1 Binary PSK () 4. Draw the signal constellation for binary PSK s (t) R s 1 (t) R 1 ψ 1 (t) 5. Divide the signal space into two regions by the perpendicular to the connecting line between tips of vectors s1 and s. 6. he location of the received signal determines the transmitted signal. Page 13 1 3
M-ary PSK (1) M-ary PSK: 1. he transmitted signals are given by s i E cos [ ω t + 0 πi M ], 0 t,i =1,, M. Pick the basis function ψ ( t) = 1 ψ ( t) = cos sin [ ω t], 0 0 t [ ω t], 0 t 0 3. Represent the transmitted signals in terms of the basis function s ( t) = a i = i1 ψ ( t) + a 1 E cos i ( πi ) ( ) sin( πi ψ t + E ) ψ ( t), M ψ ( t), 1 M i = 1,, M M-ary PSK () 4. Draw the signal constellation for MPSK. he following illustrates the signal constellation for M = 4. 5. Divide the signal space into two regions by the perpendicular to the connecting line between tips of signals vectors. 6. he location of the received signal determines the transmitted signal. 7. Note that the decision region can also be specified in terms of the angle that the received vector makes with the horizontal axis. Page 14 1 4
M-ary PSK (3) M-ary PSK (4) Page 15 1 5
FSK n A typical set of FSK is described by: E is the energy content of s i (t) over each symbol duration, and (ω i+1 -ω i ) is typically assumed to be an integral multiple of π/. he phase term is an arbitrary constant and can be set equal to zero. n s i E cos ω [ t +ϕ i ] 0 t,i =1,, M Assume that basis functions form an orthonormal set, i.e. ψ i cosω jt j =1,, N a ij E cos(ω t) " $ t cos(ω j t)dt = # 0 %$ E 0 for i = j otherwise 31 CSE414 Digital Communications Non-coherent Detection Page 16 1 6
Binary FSK Quadrature Receiver n Implemented with correlators, but based on energy detector without exploiting phase information 33 FSK Envelope Detector n Implemented with bandpass filters followed by envelope detectors. n Envelope detector consists of a rectifier and a lowpass filter 34 Page 17 1 7
Minimum one Spacing for Orthogonal FSK n FSK is usually implemented as orthogonal signaling. n Not all FSK signaling is orthogonal, how can we tell if the tone in a signaling set form an orthogonal set? n o form an orthogonal set, they must be uncorrelated over a symbol time n Minimum tone spacing for orthogonal FSK: n Any pair of tones in the set must have a frequency separation that is a multiple of 1/ hertz 35 Activity 1 Consider two waveforms cos(π f 1 t +φ) andcos(π f t) to be used for non-coherent FSK-signaling, where f 1 >f. he symbol rate is equal to 1/ symbols/s, where is the symbol duration and ϕ is a constant arbitrary angle from 0 to π. Prove that the minimum tone spacing for non-coherent detected orthogonal FSK signaling is 1/. 36 Page 18 1 8
CSE414 Digital Communications Error Performance for Binary Systems Probability of Bit Error for Coherently Detected BPSK n n n n For BPSK, the symbol error probability is the bit error probability. Assume n For transmitting s i (t) (i=1,), the received signal is r(t)=s i (t)+n(t) where n(t) is an AWGN process. n Any degradation effects due to channel-induced ISI or circuit-induced ISI have been neglected. he antipodal signals are: s 1 Eψ 1 (t) s Eψ 1 (t) " $ # 0 t % $ he decision rule are error probability are: s 1 (t) if z( ) > γ 0 = 0 " P s (t) otherwise B = Q (1 ρ)e % b $ # N ' 0 & = Q " E % b $ # N ' 0 & 38 Page 19 1 9
Activity Find the bit error probability for a BPSK system with a bit rate of 1Mbit/s. he received waveforms s 1 Acosω 0 t and s Acosω 0 t are coherently detected with a matched filter. he value of A is 10mV. Assume that the single-sided noise power spectral density is N 0 =10-11 W/ Hz and that signal power and energy per bit are normalized relative to a 1 ohm load. 39 Probability of Bit Error for Coherently Detected BFSK n n n n For BFSK, the symbol error probability is the bit error probability. Assume n For transmitting s i (t) (i=1,), the received signal is r(t)=s i (t)+n(t) where n(t) is an AWGN process. n Any degradation effects due to channel-induced ISI or circuit-induced ISI have been neglected. For orthogonal signals are: s 1 Acosω 0 t s Acosω 1 t he error probability is:! # " 0 t $# " P B = Q $ # (1 ρ)e b N 0 % ' & = Q " $ # E b N 0 % ' & 40 Page 0 0
Bit Error Probability for Several Binary Systems 41 CSE414 Digital Communications Error Performance for M-ary Systems Page 1 1
M-ary Signaling n n Modulator produces one of M= k waveforms n Binary signaling is the special case where k=1 Vectorial view of MPSK signaling 43 Symbol Error Performance for M-ary Systems (M>) n For large energy-to-noise ratios, the symbol error performance P E (M), for equally likely, coherently detected M-ary PSK signaling: " P E (M ) Q$ # E s N 0 sin π % M ' & where E s = E b (log M ) is the energy per symbol, and M = k 44 Page
Symbol Error Performance for MFSK n he symbol error performance P E (M), for equally likely, coherently detected M-ary orthogonal signaling can be upper bounded as: # P E (M ) (M 1)Q % $ E s N 0 & ( ' where E s = E b (log M ) is the energy per symbol, and M is the size of the symbol set. 45 Bit Error Probability versus Symbol Error Probability n For multiple phase signaling and utilizing Gray code assignment, P B P E log M n For orthogonal signaling, Binary assignment Gray code assignment P B P E = k 1 k 1 = M M 1 P lim B = 1 k P E 46 Page 3 3