Wireless Communication Systems

Wireless Communication Systems

@CS.NCTU

Lecture 2: Modulation and Demodulation Reference: Chap. 5 in Goldsmith’s book

Instructor: Kate Ching-Ju Lin (林靖茹)

1

Modulation

2

From Wikipedia:

The process of varying one or more properties of a

periodic waveform with a modulating signal that typically contains information to be transmitted.

0 5 10 15 20 25 30 35

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0 5 10 15 20 25 30 35

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0 5 10 15 20 25 30 35

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0 5 10 15 20 25 30 35

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

modulate

Example 1

3

= bit-stream?

(a) 10110011 (b) 00101010 (c) 10010101

Example 2

4

= bit-stream?

(a) 01001011 (b) 00101011 (c) 11110100

Example 3

5

= bit-stream?

(a) 11010100 (b) 00101011

(c) 01010011 (d) 11010100 or

00101011

Types of Modulation

Amplitude ASK

Frequency FSK

Phase

PSK

Modulation

• Map bits to signals

wireless channel TX

transmitted Signal s(t)

1 0 1 1 0

bit stream

modulation

Demodulation

• Map signals to bits

RX

1 0 1 1 0

demodulation

received signal x(t) wireless

channel TX

transmitted Signal s(t)

1 0 1 1 0

bit stream

modulation

Analog and Digital Modulation

• Analog modulation

⎻ Modulation is applied continuously

⎻ Amplitude modulation (AM)

⎻ Frequency modulation (FM)

• Digital modulation

⎻ An analog carrier signal is modulated by a discrete signal

⎻ Amplitude-Shift Keying (ASK)

⎻ Frequency-Shift Keying (FSK)

⎻ Phase-Shift Keying (PSK)

⎻ Quadrature Amplitude Modulation (QAM)

9

Advantages of Digital Modulation

• Higher data rate (given a fixed bandwidth)

• More robust to channel impairment

⎻ Advanced coding/decoding can be applied to make signals less susceptible to noise and fading

⎻ Spread spectrum techniques can be applied to deal with multipath and resist interference

• Suitable to multiple access

⎻ Become possible to detect multiple users simultaneously

• Better security and privacy

⎻ Easier to encrypt

10

Modulation and Demodulation

• Modulation

⎻ Encode a bit stream of finite length to one of several possible signals

• Delivery over the air

⎻ Signals experience fading and are combined with AWGN (additive white Gaussian noise)

• Demodulation

⎻ Decode the received signal by mapping it to the closest one in the set of possible transmitted signals

11

Transmitter Receiver

+ x(t) n(t) AWGN Channel

i 1 K s(t)

m ={b ,...,b } ^

1 K

m ={b ,...,b }^ ^

Figure 5.1: Communication System Model

channel is analog, the message must be embedded into an analog signal for channel transmission. Thus, each message m_i ∈ M is mapped to a unique analog signal sⁱ(t) ∈ S = {s¹(t), . . . , s_M(t)} where sⁱ(t) is defined on the time interval [0, T ) and has energy

E_si = ^{! T}

0 s²_i(t)dt, i = 1, . . . , M. (5.1)

Since each message represents a bit sequence, each signal si(t) ∈ S also represents a bit sequence, and detection of the transmitted signal s_i(t) at the receiver is equivalent to detection of the transmitted bit sequence. When messages are sent sequentially, the transmitted signal becomes a sequence of the corresponding analog signals over each time interval [kT, (k + 1)T ): s(t) = ^"_ksi(t − kT), where sⁱ(t) is the analog signal corresponding to the message m_i designated for the transmission interval [kT, (k +1)T ).

This is illustrated in Figure 5.2, where we show the transmitted signal s(t) = s₁(t) + s₂(t − T) + s¹(t − 2T ) + s₁(t − 3T) corresponding to the string of messages m¹, m₂, m₁, m₁ with message m_i mapped to signal si(t).

T

0 2T 3T 4T

s (t)

1 1 1

s (t−T)2

s (t−2T) s (t−3T)

s(t)

...

m1 m

1 m

1

m2

Figure 5.2: Transmitted Signal for a Sequence of Messages

In the model of Figure 5.1, the transmitted signal is sent through an AWGN channel, where a white Gaussian noise process n(t) of power spectral density N₀/2 is added to form the received signal x(t) = s(t) + n(t). Given x(t) the receiver must determine the best estimate of which si(t) ∈ S was transmitted during each transmission interval [kT, (k +1)T ). This best estimate for s_i(t) is mapped to a best estimate of the message m_i(t) ∈ M and the receiver then outputs this best estimate ˆm = {ˆb¹, . . . , ˆbK} ∈ M of the transmitted bit sequence.

The goal of the receiver design in estimating the transmitted message is to minimize the probability of message error:

P_e =

#M

i=1

p( ˆm ̸= mⁱ|mⁱ sent)p(m_i sent) (5.2) over each time interval [kT, (k + 1)T ). By representing the signals {sⁱ(t), i = 1, . . . , M} geometrically, we can solve for the optimal receiver design in AWGN based on a minimum distance criterion. Note that, as we saw in previous chapters, wireless channels typically have a time-varying impulse response in addition to AWGN. We will consider the effect of an arbitrary channel impulse response on digital modulation performance in Chapter 6, and methods to combat this performance degradation in Chapters 11-13.

127

modulate demodulate

Band-pass Signal Representation

• General form

• Amplitude is always non-negative

⎻ Or we can switch the phase by 180 degrees

• Called the canonical representation of a band-pass signal

12

𝑎 𝑡

2𝜋𝑓_&𝑡 + 𝜙 𝑡 𝑠 𝑡

amplitude frequency phase

s(t) = a(t)cos(2⇡f

t + (t))

In-phase and Quadrature Components

• : In-phase component of s(t)

• : Quadrature component of s(t)

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Amplitude: a(t) = q

s

(t) + s

(t) Phase: (t) = tan

( s

(t)

s

(t) ) s

(t) = a(t) cos( (t))

s

(t) = a(t) sin( (t))

s(t) = a(t) cos(2⇡f

t + (t))

= a(t)[cos(2⇡f

t) cos( (t)) sin(2⇡f

t) sin( (t))]

= s

(t) cos(2⇡f

t) s

(t) sin(2⇡f

t)

• We can also represent s(t) as

•

• s’(t) is called the complex

envelope of the band-pass signal

• This is to remove the annoying in the analysis

Band-Pass Signal Representation

𝑎 𝑡 𝜙 𝑡

𝑠′ 𝑡

s

(t) = s

(t) + js

(t) s(t) = <[s

(t)e

]

e

exp(iθ) = cos(θ)+jsin(θ)

s(t) = s

(t) cos(2 f (t)t) s

(t) sin(2 f (t)t)

I

Q

Types of Modulation

• Amplitude

⎻ M-ASK: Amplitude Shift Keying

• Frequency

⎻ M-FSK: Frequency Shift Keying

• Phase

⎻ M-PSK: Phase Shift Keying

• Amplitude + Phase

⎻ M-QAM: Quadrature Amplitude Modulation

s(t) = Acos(2πf

t+ 𝜙 )

Amplitude Shift Keying (ASK)

• A bit stream is encoded in the amplitude of the transmitted signal

• Simplest form: On-Off Keying (OOK)

⎻ ‘1’àA=1, ‘0’àA=0

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TX RX

signal s(t)

1 0 1 1 0 1 0 1 1 0

bit stream b(t)

modulation demodulation

M-ASK

• M-ary amplitude-shift keying (M-ASK)

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s(t) = A

cos(2 f

t) , if 0 t T

0 , otherwise,

where i = 1, 2, · · · , M

A

is the amplitude corresponding to bit pattern i

Example: 4-ASK

• Map ‘00’, ‘01’, ‘10’, ’11’ to four different amplitudes

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Amplitude-Shift Keying (ASK) Modulation on Mac

(a) 0

... ...

E s i

i

= 2T cos 2 π f c t φ 1 t( )

0

s 0s 1 s 2s 3 (b)

φ 1 t( )

Figure 22.5 (a) M-ASK and (b) 4-ASK signal constellation diagrams.

0 0 0 1 1 0 1 1

0 Time

(a)

(c)

4-ary Time signal

3

-3 -1 1

0

(b)

m (t )

0 A -A -3A 3A

Time

T T

4-ASK signal

s (t ) Binary

sequence 1

Figure 22.6 4-ASK modulation: (a) binary sequence, (b) 4-ary signal, and (b) 4-ASK signal.

22.6

Pros and Cons of ASK

• Pros

⎻ Easy to implement

⎻ Energy efficient

⎻ Low bandwidth requirement

• Cons

⎻ Low data rate

§ bit-rate = baud rate

⎻ High error probability

§ Hard to pick a right threshold

1 baud

1 second

Bandwidth is the difference between the upper and lower frequencies in a

continuous set of frequencies.

Types of Modulation

• Amplitude

⎻ M-ASK: Amplitude Shift Keying

• Frequency

⎻ M-FSK: Frequency Shift Keying

• Phase

⎻ M-PSK: Phase Shift Keying

• Amplitude + Phase

⎻ M-QAM: Quadrature Amplitude Modulation

s(t) = Acos(2πf

t+ 𝜙 )

Frequency Shift Keying (FSK)

• A bit stream is encoded in the frequency of the transmitted signal

• Simplest form: Binary FSK (BFSK)

⎻ ‘1’àf=f₁, ‘0’àf=f₂

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TX

signal s(t)

1 0 1 1 0

bit stream

modulation

RX

1 0 1 1 0

demodulation

M-FSK

• M-ary frequency-shift keying (M-FSK)

• Example: Quaternary Frequency Shift Keying (QFSK)

⎻ Map ‘00’, ‘01’, ‘10’, ’11’ to four different frequencies

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s(t) = A cos(2 f

t) , if 0 t T

0 , otherwise,

where i = 1, 2, · · · , M

f

is the center frequency corresponding to bit pattern i

Pros and Cons of FSK

• Pros

⎻ Easy to implement

⎻ Better noise immunity than ASK

• Cons

⎻ Low data rate

§ Bit-rate = baud rate

⎻ Require higher bandwidth

§ BW(min) = N_b + N_b

Types of Modulation

• Amplitude

⎻ M-ASK: Amplitude Shift Keying

• Frequency

⎻ M-FSK: Frequency Shift Keying

• Phase

⎻ M-PSK: Phase Shift Keying

• Amplitude + Phase

⎻ M-QAM: Quadrature Amplitude Modulation

s(t) = Acos(2πf

t+ 𝜙 )

Phase Shift Keying (PSK)

• A bit stream is encoded in the phase of the transmitted signal

• Simplest form: Binary PSK (BPSK)

⎻ ‘1’à𝜙=0, ‘0’à𝜙=π

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TX RX

signal s(t)

1 0 1 1 0

bit stream s(t)

modulation

1 0 1 1 0

demodulation

Constellation Points for BPSK

• ‘1’à𝜙=0

• cos(2πf_ct+0)

= cos(0)cos(2πf_ct)- sin(0)sin(2πf_ct)

= s_Icos(2πf_ct) – s_Qsin(2πf_ct)

• ‘0’à𝜙=π

• cos(2πf_ct+π)

= cos(π)cos(2πf_ct)- sin(π)sin(2πf_ct)

= s_Icos(2πf_ct) – s_Qsin(2πf_ct)

I

Q

I Q

𝜙=π

(s_I,s_Q) = (1, 0)

‘1’à 1+0i (s_I,s_Q) = (-1, 0)

‘0’à -1+0i

‘0’ ‘1’

Demodulate BPSK

• Map to the closest constellation point

• Quantitative measure of the distance between the received signal s’ and any possible signal s

⎻ Find |s’-s| in the I-Q plane

I Q

s₁=1+0i n₁

n₀

n

=|s’-s

|=|s’-(1+0i)|

n

=|s’-s

|=| |s’-(-1+0i)|

since n

< n

map s’ to (1+0i)à‘1’

s’=a+bi s₀=-1+0i

I Q

s₁=1+0i

‘0’ ‘1’

s₀=-1+0i

Demodulate BPSK

• Decoding error

⎻ When the received signal is mapped to an incorrect symbol (constellation point) due to a large error

• Symbol error rate

⎻ P(mapping to a symbol s_j, j≠i | s_i is sent )

Given the transmitted symbol s

s’=a+bi

à incorrectly map s’ to

s

=(-1+0)à‘0’, when the

error is too large

SNR of BPSK

• SNR: Signal-to-Noise Ratio

• Example:

⎻ Say Tx sends (1+0i) and Rx receives (1.1 – 0.01i)

⎻ SNR?

I Q

s’ = a+bin

SN R = |s|

n

= |s|

|s s |

= |1 + 0i|

|(a + bi) (1 + 0i) |

SN R

= 10 log

(SN R)

SER/BER of BPSK

• BER (Bit Error Rate) = SER (Symbol Error Rate)

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SER = BER = P

= Q

✓ d

p 2N

◆

= Q

r 2E

N

!

= Q( p

2SN R)

From Wikipedia:

Q(x) is the probability that a normal (Gaussian) random variable will obtain a value larger than x standard deviations above the mean.

Minimum distance of any two cancellation points

Constellation point for BPSK

• Say we send the signal with phase delay π

31

cos(2j f

t + )

= cos(2j f

t) cos( ) sin(2j f

t) sin( )

= 1 cos(2j f

t) 0 sin(2j f

t)

=( 1 + 0i)e

Illustrate this by the constellation

point (-1 + 0i) in an I-Q plane I

Q

-1+0i Band-pass representation

Quadrature PSK (QPSK)

• Use four phase rotations 1/4π, 3/4π, 5/4π, 7/4π to represent ‘00’, ‘01’, ‘11’, 10’

32

A cos(2j f

t + /4)

=A cos(2j f

t) cos( /4) A sin(2j f

t) sin( /4)

=1 cos(2j f

t) 1 sin(2j f

t)

=(1 + 1i)e

A cos(2j f

t + 3 /4)

=A cos(2j f

t) cos(3 /4) A sin(2j f

t) sin(3 /4)

= 1 cos(2j f

t) 1 sin(2j f

t)

=( 1 + 1i)e

I Q

‘00’

‘10’

‘01’

‘11’

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

⎻ Demodulation by mapping the received signal to the closest constellation point

⎻ Double the bit-rate

• No free lunch:

⎻ Higher error probability (Why?)

I Q

‘00’

‘10’

‘01’

‘11’

Quadrature PSK (QPSK)

• Maximum power is bounded

⎻ Amplitude of each constellation point should still be 1

I Q

‘00’ = 1/√2(1+1i)

‘10’

‘01’

‘11’

1 2 1

2

− 1 2

− 1 2

Bits Symbols

‘00’ 1/√2+1/√2i

’01’ -1/√2+1/√2i

‘10’ 1/√2-1/√2i

‘11’ -1/√2-1/√2i

Higher Error Probability 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

I Q

1 n

✔ in BPSK

I Q

✗ In QPSK 1/√2 n

‘0’ ‘1’ ‘x1’ ‘x0’

Trade-off between Rate and SER

• Trade-off between the data rate and the symbol error rate

⎻ Denser constellation points

à More bits encoded in each symbol à Higher data rate

⎻ Denser constellation points

à Smaller distance between any two points à Higher decoding error probability

36

SEN and BER of QPSK

• SNR

: SNR per symbol; SNR

: SNR per bit

• SER: The probability that each branch has a bit error

• BER

37

BER = P

⇡ P

2

QPSK: M=4

SN R

⇡ SN R

log

M , P

⇡ P

log

M

E

is the bounded maximum power

SER = P

= 1 [1 Q( p

2SN R

)]

= 1 [1 Q(

r 2E

N

)]

= 1 [1 Q( p

SN R

)]

= 1 [1 Q(

r E

N

)]

M-PSK

38

I Q

‘10’

‘01’

‘11’

1 2 1

2

− 1 2

I Q

− 1 2

‘0’ ‘1’

I Q

‘111’

‘100’

‘010’

‘011’

‘001’

‘000’

‘100’ ‘101’

I Q

‘1111’

‘0000’

BPSK QPSK

8-PSK 16-PSK

M-PSK BER versus SNR

Denser constellation points à higher BER

Acceptable reliability

Types of Modulation

• Amplitude

⎻ M-ASK: Amplitude Shift Keying

• Frequency

⎻ M-FSK: Frequency Shift Keying

• Phase

⎻ M-PSK: Phase Shift Keying

• Amplitude + Phase

⎻ M-QAM: Quadrature Amplitude Modulation

s(t) = Acos(2πf

t+ 𝜙 )

Quadrature Amplitude Modulation

• Change both amplitude and phase

• s(t)=Acos(2πf

t+ 𝜙 )

• 64-QAM: 64 constellation points, each with 8 bits I

Q

‘1000’

‘1100’

‘0100’

‘0000’

‘1001’

‘1101’

‘0101’

‘0001’

‘1011’

‘1111’

‘0111’

‘0011’

‘1010’

‘1110’

‘0110’

‘0010’

Bits Symbols

‘1000’ s₁=3a+3ai

’1001’ s₂=3a+ai

‘1100’ s₃=a+3ai

‘1101’ s₄=a+ai

expected power: E s !

" #

$ = 1

a 3a

16-QAM

M-QAM BER versus SNR

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

Signal Encoder

90 degree

shift

×

× Message

Source

Band-pass Signal s(t) +

+

Map each bit into s_I(t) and s_Q(t)

Band-Pass Signal Transmitter

s(t) = s_I(t) cos(2⇡f_ct) s_Q(t) sin(2⇡f_ct)

s_I(t)

s_Q(t)

Σ

mixer

∼

cos(2πf_ct)

sin(2πf_ct)

Band-Pass Signal Receiver

Band- pass Filter

90 degree

shift

×

×

Message Sink Received

Signal plus noise

Filters out out- of-band signals and noises

x(t) = s(t) + n(t)

Low-pass Filter

Signal Detector

Low- pass Filter

0.5[A_cs_I(t) + n_I(t)]

0.5[A_cs_Q(t) + n_Q(t)]

∼

cos(2πf_ct)

sin(2πf_ct)

Detection

• Map the received signal to one of the possible transmitted signal with the minimum distance

• Find the corresponding bit streams

46

[s_I(t)+n(t)] + j[s_Q(t)+n(t)]

0000..00⁰

0000..01⁰ ...

0001..10⁰ ...

0111..11⁰

s¹_I(t) + js¹_Q(t) s²_I(t) + js²_Q(t)

...

s^k_I(t) + js^k_Q(t) ...

s^K_I (t) + js^K_Q(t)

received signal

possible transmitted

signals corresponding bit streams

……

closest

Announcement

• Install Matlab

• Teaming

⎻ Elevator pitch: 2 per group (Each group talks about 3-5 minutes. Each member needs to talk)

⎻ Lab and project: 3-4 members per group

⎻ Send your team members to the TA (張威竣)

• Sign up for the talk topic

⎻ Pick the paper (topic) according to your preference or schedule

⎻ Sign up from 18:00@Thu (will announce the url in the announcement tab of the course website)

⎻ Pick your top five choices (from Lectures 4-18)

⎻ FIFS

47

Quiz

• What are the four types of modulation introduced in the class?

• Say Tx sends (-1 + 0i) and Rx receives -(0.95+0.01i).

Calculate the SNR.

48