Lecture Slides On Cpfsk.pdf

Shift Keying Modulation Techniques B. Sainath [email protected]

Department of Electrical and Electronics Engineering Birla Institute of Technology and Science, Pilani

March 4, 2019

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

1 / 37

Outline 1

What is Bandpass Transmission ?

2

Shift Keying Modulation

3

Phase Shift Keying

4

M−ary Phase Shift Keying

5

M−ary Quadrature Amplitude Modulation

6

Symbol Error Probability: Craig’s Formula

7

CPM & CPFSK

8

Minimum Shift Keying

9

References & Further Reading B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

2 / 37

What is Bandpass Transmission? Baseband transmission bit stream in the form of a discrete PAM signal is transmitted directly

Bandpass transmission incoming bit stream (eg. PCM encoded speech) is modulated onto RF carrier with fixed frequency limits imposed by bandpass channel e.g. satellite communication, mobile communication

Figure: Bandpass data transmission: an illustrative block diagram. B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

3 / 37

Shift Keying Modulation Schemes Modulation performed by switching (or keying) some characteristic of RF sinusoidal carrier w.r.t incoming symbols

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

4 / 37

Binary Signaling

L = 1 ⇒ two symbols

Pulse shape p(t) = 1, 0 < t < T Let a(k) = {−1, +1}

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

5 / 37

Binary Signaling

L = 1 ⇒ two symbols

Pulse shape p(t) = 1, 0 < t < T Let a(k) = {−1, +1} x(t) = a(k ), (k − 1)T < t < kT RF carrier’s characteristic is modulated in accordance with the baseband signal

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

5 / 37

Amplitude Shift Keying (ASK)

Also called ON-OFF signaling; Analogous to AM RF carrier of fixed amplitude and fixed frequency for bit duration → bit ‘1’ Switched off carrier for bit duration → bit ‘0’

Transmitted bandpass signal y(t) = (1 + x(t)) cos(2πfc t), for all t Application used in optical communication

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

6 / 37

Binary ASK Probability of Error in AWGN Channel Q

(0, 0)

(

√

Eb , 0)

I

Figure: Constellation diagram of ASK.

Let Eb denote bit energy = q  Eb SEP = Q 2N0

A2c Tb 2

joules

Solution by geometric approach (discussed in class) B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

7 / 37

Binary Frequency Shift Keying (BFSK)

This is analogous to FM Transmitted bandpass signal y (t) = cos(2πfc t + x(t)2πf0 t), for all t RF carrier with frequency fc + f0 ⇒ bit ‘1’ RF carrier with frequency fc − f0 ⇒ bit ‘0’

Application used in Amateur radio, emergency broadcasts

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

8 / 37

Binary FSK Probability of Error in AWGN Channel Q √ (0, Eb)

0

√ ( Eb, 0)

I

Assume that signals representing the bits are orthogonal Assume coherent q  reception SEP = Q

Eb N0

Solution by geometric approach B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

9 / 37

Binary Phase Shift Keying (BPSK)

This is analogous to PM PSK: Most popular and widely used modulation technique Transmitted bandpass signal y (t) = cos(2πfc t + βx(t)), for all t RF carrier with phase − π2 ⇒ bit ‘1’ RF carrier with phase + π2 ⇒ bit ‘0’

Application satellite communication, wireless LANs, RFID, and Bluetooth

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

10 / 37

Binary PSK Probability of Error in AWGN Channel Q

s0 √ (− Eb , 0) 0

s1 √ ( Eb , 0)

I

BPSK: also called antipodal signaling (correlation coefficient ρ = -1) Assume coherent reception q  2Eb SEP = Q N0 Solution by geometric approach

General formula: SEP = Q B. Sainath (BITS, PILANI)

q

Eb (1−ρ) N0



RT

s (t)s (t) dt , where ρ = √0 E 0 √1 E

Bandpass Data Transmission & Reception

s0 (t)

s1 (t)

March 4, 2019

11 / 37

Bandwidth of BPSK and BFSK

PSK modulated signal bandwidth 2×

1 Tb

FSK modulated signal bandwidth (2 ×

1 Tb

) + (f1 − f2 )

f1 − f2 >

2 Tb

where f1 = fc + f0 and f2 = fc − f0

fc in terms of f1 and f2 ?

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

12 / 37

Bandwidth of BPSK and BFSK

PSK modulated signal bandwidth 2×

1 Tb

FSK modulated signal bandwidth (2 ×

1 Tb

) + (f1 − f2 )

f1 − f2 >

2 Tb

where f1 = fc + f0 and f2 = fc − f0

fc in terms of f1 and f2 ? fc =

f1 +f2 2

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

12 / 37

BPSK: Coherent RX

Figure: Coherent receiver of BPSK.

Alternatively, correlation receiver

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

13 / 37

M−ary Phase Shift Keying Transmitted signal representation & examples (in class) Quadrature PSK: M = 4 ; Octa-PSK or 8-PSK: M = 8

Figure: Constellation diagram of QPSK.

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

14 / 37

QPSK: Transmission & Reception

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

15 / 37

Signal-Space Diagram of 8-PSK

Figure: Constellation diagram of 8PSK.

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

16 / 37

Multilevel Signaling:16-QAM Q. Compute average energy per symbol

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

17 / 37

Symbol Error Probability: Craig’s Formula for MPSK

m , sin2

π M

MPSK modulation (AWGN) 1 SEP = π

Z 0

(M−1)π M


exp −mγ csc2 θ dθ

γ is received SNR

Check for BPSK (discussed in class)

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

18 / 37

Continuous Phase Modulation (CPM) Phase discontinuity in coherent digital phase modulation techniques ⇒ poor spectral efficiency (Why?) In CPM, carrier phase is modulated continuously Advantages

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

19 / 37

Continuous Phase Modulation (CPM) Phase discontinuity in coherent digital phase modulation techniques ⇒ poor spectral efficiency (Why?) In CPM, carrier phase is modulated continuously Advantages Phase continuity yields high spectral (bandwidth) efficiency Constant envelope yields

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

19 / 37

Continuous Phase Modulation (CPM) Phase discontinuity in coherent digital phase modulation techniques ⇒ poor spectral efficiency (Why?) In CPM, carrier phase is modulated continuously Advantages Phase continuity yields high spectral (bandwidth) efficiency Constant envelope yields excellent power efficiency

Drawback:

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

19 / 37

Continuous Phase Modulation (CPM) Phase discontinuity in coherent digital phase modulation techniques ⇒ poor spectral efficiency (Why?) In CPM, carrier phase is modulated continuously Advantages Phase continuity yields high spectral (bandwidth) efficiency Constant envelope yields excellent power efficiency

Drawback: high implementation complexity of optimal receiver

Figure: Illustrating Modulation types.

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

19 / 37

Conventional FSK

Memoryless & does not have continuous phase (why?) Binary FSK: Modulated signal switches instantaneously between two sinusoids with different frequencies M−ary FSK: 2k oscillators, k denotes number of bits in a symbol Let Rb denote transmission rate Drawback:

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

20 / 37

Conventional FSK

Memoryless & does not have continuous phase (why?) Binary FSK: Modulated signal switches instantaneously between two sinusoids with different frequencies M−ary FSK: 2k oscillators, k denotes number of bits in a symbol Let Rb denote transmission rate Drawback: Oscillators tuned to desired frequencies & selecting one of M frequencies according to k −bit symbol transmitted in T = Rk duration b

Abrupt switching from one oscillator to another in successive intervals results in large spectral side lobes outside main spectral band of signal ⇒ requires large frequency band ⇒ spectrally inefficient

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

20 / 37

Continuous Phase FSK

CPFSK: Form of CPM & variant of FSK Need for CPFSK

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

21 / 37

Continuous Phase FSK

CPFSK: Form of CPM & variant of FSK Need for CPFSK Maintaining phase continuity improves spectral efficiency

CPFSK signal s(t) = Ac cos(2πfc t + φ(t)), where Ac = φ(t) continuous, fc =

q

2Eb Tb

f1 +f2 2

two frequencies f1 and f2 transmitted to represent symbols 1 & 0

In 0 ≤ t ≤ Tb , φ(t) = φ(0) ±

πh t Tb

‘+’ corresponds to symbol 1, ‘-’ corresponds to symbol 0 Parameter h = Tb (f1 − f2 ) ← deviation ratio (modulation index)

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

21 / 37

Continuous Phase FSK

At t = Tb , φ(Tb ) − φ(0) =?

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

22 / 37

Continuous Phase FSK

At t = Tb , φ(Tb ) − φ(0) =?  φ(Tb ) − φ(0) =

Transmission ‘symbol 1’

πh, for symbol 1, . −πh, for symbol 0

increases phase of CPFSK signal by πh radians

Transmission ‘symbol 0’ reduces phase of CPFSK signal by πh radians

Exercise: Plot φ(t) − φ(0) as a function of Tb (in steps of Tb )

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

22 / 37

CPFSK Spectral Density Spectral density of CPFSK (e.g., MSK) signal produced by random binary sequence falls off at least as f −4 at frequencies remote from the center of the signal band

Pulse shape g(t) =

q

2Eb Tb

cos



πt 2Tb



, −Tb ≤ t ≤ Tb

Prove that the PSD of g(t) Sg (f ) =

Ψg (f ) 32Eb = Tb π2

=⇒ |G(f )|2 ∝ f −4 for f >>



cos (2πTb f ) 16Tb2 f 2 − 1

2

1 Tb

QPSK signal spectral density fall off as f −2

CPFSK signal does not produce as much interference outside signal band compared to QPSK signal advantage over QPSK when operating with bandwidth-limited systems

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

23 / 37

Cyclostationary Signal: Representation & Analysis Signal representation (general expression) X x(t) = Ik g(t − kT ) ∀k

Instantaneous frequency f (t; I) ∝ x(t) X f (t; I) = h Ik g(t − kT ) ∀k

Instantaneous phase φ(t; I)

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

24 / 37

Cyclostationary Signal: Representation & Analysis Signal representation (general expression) X x(t) = Ik g(t − kT ) ∀k

Instantaneous frequency f (t; I) ∝ x(t) X f (t; I) = h Ik g(t − kT ) ∀k

Instantaneous phase φ(t; I) Z

t

φ(t; I) = 2π

f (τ ; I) dτ −∞ Z t

= 2πh

x(τ ) dτ −∞

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

24 / 37

Instantaneous Phase

Rt Let q(t) = −∞ g(τ ) dτ Q: For the pulse shown, determine φ(t; I) & sketch q(t)

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

25 / 37

Instantaneous Phase

Rt Let q(t) = −∞ g(τ ) dτ Q: For the pulse shown, determine φ(t; I) & sketch q(t) X φ(t; I) = 2πh Ik q(t − kT ) ∀k

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

25 / 37

Instantaneous Phase of CPM

In general

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

26 / 37

Instantaneous Phase of CPM

In general φ(t; I) = 2π

X ∀k

hk Ik q(t − kT ), nT ≤ t ≤ (n + 1)T ⇒ CPM

{Ik }: sequence of M−ary information symbols selected from ±1, ±2, . . . , ±(M − 1) hk : sequence of modulation indices (MIs) q(t): normalized waveform shape If hk = h, MI is fixed for all symbols Multi-h CPM: MI varies from symbol to symbol

Full-response CPM: g(t) = 0 for t > T Partial-response CPM: g(t) 6= 0 for t > T

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

26 / 37

CPFSK Modulated Signal Let fd denote peak frequency deviation & φ0 denote initial phase Let E denote signal energy Modulation index h , 2fd T Lowpass (baseband) signal representation r 2E xbb (t) = exp (jφ(t; I) + jφ0 ) T Q: Verify the following expressions ! r Z t 2E xbb (t) = exp j4πfd T x(τ ) dτ + jφ0 T −∞ Bandpass signal representation: r 2E xbp (t) = cos (jφ(t; I) + jφ0 ) T B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

27 / 37

Instantaneous Phase of the Carrier

Exercise: In nT ≤ t ≤ (n + 1)T , prove that φ(t; I) = θk + 2πh Ik q(t − kT ) θn = πh

n−1 X

Ik

k=−∞

θn ⇒ represents accumulation of all symbols up to (n − 1)T

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

28 / 37

Minimum Shift Keying (MSK): h =

1 2

All phase shifts are modulo 2π Interesting scenario h =

1 2

phase can take on only two values ± π2 at odd multiples of Tb phase can take on two values 0 and π at even multiples of Tb

When h = 12 , CPFSK is called MSK

Expression of MSK signal s(t) = Ac cos[φ(t)] cos(2πfc t) − Ac sin[φ(t)] sin(2πfc t) π t, 0 ≤ t ≤ Tb where φ(t) = φ(0) ± 2Tb Q. Expression for received signal at the Rx front end?

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

29 / 37

Minimum Shift Keying (MSK): h =

1 2

All phase shifts are modulo 2π Interesting scenario h =

1 2

phase can take on only two values ± π2 at odd multiples of Tb phase can take on two values 0 and π at even multiples of Tb

When h = 12 , CPFSK is called MSK

Expression of MSK signal s(t) = Ac cos[φ(t)] cos(2πfc t) − Ac sin[φ(t)] sin(2πfc t) π t, 0 ≤ t ≤ Tb where φ(t) = φ(0) ± 2Tb Q. Expression for received signal at the Rx front end?     π π x(t) = ±Ac cos t cos(2πfc t) ± Ac sin t sin(2πfc t) + w(t) 2Tb 2Tb

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

29 / 37

Phase Trellis Example

Figure: Phase trellis; boldfaced path represents the sequence 1101000.

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

30 / 37

Probability of Bit Error Probability of bit error: same as coherent QPSK However, MSK spectrum decays rapidly (f −4 ) MSK by Pasupathy S. (classic paper) (IEEE, 1979)

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

31 / 37

TX & Coherent RX of MSK

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

32 / 37

Gaussian MSK

Gaussian pre-modulation filter is used Pass line-encoded (eg. NRZ) bits through a Gaussian filter Use FM on the filtered pulses to generate GMSK signal q  2αEb Probability of bit error = Q N0 α depends on BT For example, for BT = 0.25, α ≈ 0.68.

GMSK with BT = 0.3 used in GSM GMSK offers better bandwidth efficiency but not power efficient Trade-off between power efficiency & bandwidth efficiency To confront ISI, GMSK requires equalization

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

33 / 37

PSD of MSK & GMSK

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

34 / 37

Exercise on GMSK

Notation: U[0,T ] shown in Figure. It denotes BOXCAR function which has constant amplitude of 1 in the defined interval.

U[0,T ] (t) 1 0

T

t

Figure: BOXCAR function

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

35 / 37

Exercise

Exercise. Pulse shape g(t) for GMSK can be obtained by smoothing 1 U[0,T ] , by passing through a rectangular pulse of MSK, pMSK (t) = 4T Gaussian filter. The filter has transfer function given by  2 ! cf H(f ) = exp − , W3 dB c=

q

ln 2 , 2

W3 dB is the 3 dB bandwidth

Derive the following: Impulse response h(t) of the Gaussian filter Convolution of pMSK (t) and h(t)

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

36 / 37

References & Further Reading

Digital communications by Proakis Communication Systems by Simon Haykin and Michael Moher, 5th edition Digital communication by Simon Haykin http://users.okan.edu.tr/didem.kivanc/courses/EEE306_ 2013_Spring/EEE306_Digital_Communication_2013_Spring_ Slides_10.pdf http://www-inst.eecs.berkeley.edu/˜ee120/fa16/ handouts/digital-comm-sp02.pdf

B. Sainath (BITS, PILANI)

Bandpass Data Transmission & Reception

March 4, 2019

37 / 37