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Digital Modulation
Kate Ching-Ju Lin (林靖茹)
Academia Sinica
Modulation
§  Map bits to signals
TX bit stream x(t) 1 0 1 1 0 modula7on signal s(t) wireless channel Demodulation
§  Map signals to bits
TX bit stream x(t) 1 0 1 RX 1 0 1 1 1 0 demodula7on modula7on signal s(t) 0 wireless channel Considerations
§  Data rate
–  Bits per second
§  Bandwidth requirement
–  MHz
§  Power efficiency
–  ∑t|s(t)|2
§  Bit error rate
–  Related to SNR (Eb/N0)
§  Hardware cost
Sinusoid with Phase Shift
§  Sinusoidal carrier with center frequency fc
–  s(t) = cos(2πfct)
§  Sinusoid with phase shift
–  s(t) = cos(2πfct+𝜙)
Sinusoid with Phase Shift
§  Sinusoidal carrier with center frequency fc
–  s(t) = cos(2πfct)
§  Sinusoid with phase shift
–  s(t) = cos(2πfct+𝜙)
= cos(𝜙)cos(2πfct)-sin(𝜙)sin(2πfct)
Sinusoid with Phase Shift
§  Sinusoidal carrier with center frequency fc
–  s(t) = cos(2πfct)
§  Sinusoid with phase shift
–  s(t) = cos(2πfct+𝜙)
= cos(𝜙)cos(2πfct)-sin(𝜙)sin(2πfct)
= sI*cos(2πfct) – sQ*sin(2πfct)
Sinusoid with Phase Shift
§  Sinusoidal carrier with center frequency fc
–  s(t) = cos(2πfct)
§  Sinusoid with phase shift
–  s(t) = cos(2πfct+𝜙)
= cos(𝜙)cos(2πfct)-sin(𝜙)sin(2πfct)
= sI*cos(2πfct) – sQ*sin(2πfct)
Sinusoid with Phase Shift
§  Sinusoidal carrier with center frequency fc
–  s(t) = cos(2πfct)
§  Sinusoid with phase shift
–  s(t) = cos(2πfct+𝜙)
= cos(𝜙)cos(2πfct)-sin(𝜙)sin(2πfct)
= sI*cos(2πfct) – sQ*sin(2πfct)
= sI*cos(2πfct) – sQ*cos(2πfct+π/2)
§  sI and sQ are in-phase and quadrature
components of the signal s(t), respectively
Modulator
Demodulator
Constellations
§  cos(2πfct+𝜙)
= cos(𝜙)cos(2πfct)-sin(𝜙)sin(2πfct)
= sI*cos(2πfct) – sQ*sin(2πfct)
§  Constellation point on I-Q plane
–  (sI,sQ) = (cos(𝜙), sin(𝜙))
𝜙=0 𝜙=π/4 Q Q I 𝜙=π/2 Q I 𝜙=π I Q I Delay in )me domain = Phase shi1 in frequency domain = Rota)on in I-­‐Q plane Types of Modulation
§  s(t) = Acos(2πfct+𝜙))
§  Amplitude
–  ASK: Amplitude Shift Keying
§  Frequency
–  FSK: Frequency Shift Keying
§  Phase
–  M-PSK: Phase Shift Keying
§  Amplitude + Phase
–  M-QAM: Quadrature Amplitude Modulation
Amplitude Shift Keying (PSK)
§  Represent samples using different amplitudes
–  ‘1’àA=1, ‘0’àA=0
TX bit stream s(t) 1 0 1 RX 1 0 modula7on signal s(t) 1 0 1 1 0 demodula7on PSK
§  Pros
–  Easy to implement
–  Energy efficient
–  Low bandwidth requirement
§  Cons
–  Low data rate
•  bit-rate = baud rate
1 baud 1 second –  High error probability
•  Hard to pick a right threshold
Types of Modulation
§  s(t) = Acos(2πfct+𝜙)
§  Amplitude
–  ASK: Amplitude Shift Keying
§  Frequency
–  FSK: Frequency Shift Keying
§  Phase
–  M-PSK: Phase Shift Keying
§  Amplitude + Phase
–  M-QAM: Quadrature Amplitude Modulation
Frequency Shift Keying (FSK)
§  Represent samples using different frequencies
–  ‘1’àf=f1, ‘0’àf=f2
RX TX bit stream s(t) 1 0 1 1 0 modula7on signal s(t) 1 0 1 1 0 demodula7on FSK
§  Pros
–  Easy to implement
–  Better noise immunity than ASK
§  Cons
–  Low data rate
•  Bit-rate = baud rate
–  Require higher bandwidth
•  BW(min) = Nb + Nb
Types of Modulation
§  s(t) = Acos(2πfct+𝜙)
§  Amplitude
–  ASK: Amplitude Shift Keying
§  Frequency
–  FSK: Frequency Shift Keying
§  Phase
–  M-PSK: Phase Shift Keying
§  Amplitude + Phase
–  M-QAM: Quadrature Amplitude Modulation
BPSK
§  Represent samples using different phases
–  ‘1’à𝜙=0, ‘0’à𝜙=π
RX TX bit stream s(t) 1 0 1 1 0 modula7on signal s(t) 1 0 1 1 0 demodula7on Constellation Points for BPSK
§  ‘1’à𝜙=0
§  ‘0’à𝜙=π
§  cos(2πfct+0)
= cos(0)cos(2πfct)-sin(0)sin(2πfct)
= sI*cos(2πfct) – sQ*sin(2πfct)
§  cos(2πfct+π)
= cos(π)cos(2πfct)-sin(π)sin(2πfct)
= sI*cos(2πfct) – sQ*sin(2πfct)
𝜙=0 Q 𝜙=π Q I (sI,sQ) = (1, 0) ‘1’à 1+0i I (sI,sQ) = (-­‐1, 0) ‘0’à -­‐1+0i Demodulate BPSK
§  Map to the closest constellation point
‘0’ Q n0 ‘1’ s’=a+bi n1 s=1+0i I n1=|s’-­‐(1+0i)|, n0=|s’-­‐(-­‐1+0i)| Since n1 < n0, map s’ to (1+0i) = ‘1’ Demodulate BPSK
§  Decoding error
‘0’ Q ‘1’ s’=a+bi s=1+0i I Incorrectly map s’ to (-­‐1+0) = ‘0’ SNR vs. BPSK BER
Q s’ = a+bi n SNR =
s'
2
n
2
=
s'
2
s'− s
2
=
a + bi
I 2
(a + bi) − (1+ 0i)
SNRdB = 10 log10 (SNR)
" E %
b
Bit error rate: Pb = Q $$
''
# N0 &
2
Quadrature PSK (QPSK)
§  Use 2 degrees of freedom in I-Q plane
§  Represent two bits as a constellation point
–  Rotate the constellations by π/2
–  Double the bit-rate
–  No free lunch: Higher error probability (Why?)
Q ‘01’ ‘00’ I ‘11’ ‘10’ Quadrature PSK (QPSK)
§  Maximum power is bounded
–  Amplitude of each point should still be 1
Q ‘01’ −
1
2
I 1
2
‘11’ ‘00’ = 1/√2(1+1i) 1
2
1
−
2
‘10’ Bits Symbols ‘00’ 1/√2+1/√2i ’01’ -­‐1/√2+1/√2i ‘10’ 1/√2-­‐1/√2i ‘11’ -­‐1/√2-­‐1/√2i Higher BER in QPSK
§  For a particular error n, the symbol could be
decoded correctly in BPSK, but not in QPSK
–  Why? Each sample only gets half power.
‘0’ Q ‘1’ n 1 ‘x1’ Q ‘x0’ I I n 1/√2 ✔ in BPSK ! 2E $( 1
2Eb +
b
Bit error rate: Pb = 2Q ##
&&*1− Q
N
2
N
"
0 %)
0 ,
✗ In QPSK Types of Modulation
§  s(t) = Acos(2πfct+𝜙)
§  Amplitude
–  ASK: Amplitude Shift Keying
§  Frequency
–  FSK: Frequency Shift Keying
§  Phase
–  M-PSK: Phase Shift Keying
§  Amplitude + Phase
–  M-QAM: Quadrature Amplitude Modulation
Quadrature Amplitude Modulation
§  Change both amplitude and phase
§  s(t)=Acos(2πfct+𝜙))
Q Bits ‘0000’ ‘0100’ ‘1100’ ‘1000’ ‘1000’ s1=3a+3ai ‘0001’ ‘0101’ ‘1101’ ‘1001’ a 3a ‘0011’ ‘0111’ ‘1111’ ‘1011’ ‘0010’ ‘0110’ ‘1110’ ‘1010’ 16-QAM
Symbols I ’1001’ s2=3a+ai ‘1100’ s3=a+3ai ‘1101’ s4=a+ai 2
expected power: E !" si #$ = 1
§  64-QAM: 64 constellation points, each with 8 bits
BER Comparison
~3dB §  Require extra 3dB to ensure Pb=0.001
Modulation in 802.11
§  802.11a
–  6 mb/s: BPSK + ½ code rate
–  9 mb/s: BPSK + ¾ code rate
–  12 mb/s: QPSK + ½ code rate
–  18 mb/s: QPSK + ¾ code rate
–  24 mb/s: 16-QAM + ½ code rate
–  36 mb/s: 16-QAM + ¾ code rate
–  48 mb/s: 64-QAM + ⅔ code rate
–  54 mb/s: 64-QAM + ¾ code rate
§  FEC (forward error correction)
–  k/n: k-bits useful information among n-bits of data
–  Decodable if any k bits among n transmitted bits are
correct
Bit-Rate Selection
54 48 36 24 18 12 6 throughputr = (1-PERr,SNR) * r = (1-BERr,SNR)N *r
r* = arg max throughputr
Bit-Rate Selection
best rate 54 48 36 24 18 12 6 Adapt bit-rate to dynamic RSSI