Digital Communication Systems

ELG4179: Wireless Communication Fundamentals © S.Loyka

Digital Communication Systems Review of ELG3175: • • • • •

Fourier Transform/Series Spectra Modulation/demodulation, baseband/bandpass signals Digital signaling Intersymbol interference (ISI)

Use your notes/textbook and the Appendix. Generic modulation formats: • AM, PM and FM • properties

Why digital? • better spectral efficiency • better noise immunity (quality) • allows DSP (not expensive)

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Block diagram of a digital communication system (Tx)

(Rx) P.M. Shankar, Introduction to Wireless Systems, Wiley, 2002.

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Digital Communications Digital source: select a message m from a set of possible messages {m1, m2 .........mN } (alphabet). Digital modulator: each message mi is represented via a corresponding (time-limited) symbol si (t ) , 0 ≤ t ≤ Ts .

mi → si (t ) , 0 ≤ t ≤ T

(7.1)

Transmitted sequence:

x(t ) = ∑ i si (t − iT ) ,

(7.2)

Pulse-amplitude modulation (PAM): si (t ) = ai p(t ) ,

x(t ) = ∑ i ai p (t − iT )

(7.3)

i.e. each message is encoded by amplitude ai : mi → ai . Baseband system: each si (t ) or p (t ) is a baseband pulse. Bandpass (RF) system: pulses are bandpass (or RF).

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Baseband Digital Signaling Also known as “line coding”: each si (t ) or p (t ) is a baseband pulse. Examples: • Binary data representation: 0 or 1. • Unipolar modulation: high level (e.g., 5V) / zero, or 1/0 • Bipolar modulation: +high level / -high level, or +1/-1

1 R= = bit (data) rate [bit/s] Tb x(t ) = ∑ i ai p(t − iT )

L.W. Couch II, Digital and Analog Communication Systems, Prentice Hall, 2001.

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Spectral Characteristics Spectrum is a very limited and expensive resource in wireless communications. Wireless systems are band-limited. How to select p (t ) ? • • • •

Rectangular pulse, Sinc pulse, Raised cosine pulse, Others.

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Example: Rectangular Pulse signal

1

1, t < T / 2 x ( t ) = Π (t ) =  0, t > T / 2 Sx ( f ) = ∫

+∞

−∞

Sx ( f ) = T

x (t ) e

− j 2 πft

sin πfT = Tsinc( fT ) πfT

−

dt

T 2

T 2

t

Sx ( f )

spectrum

T

∆f = ? −

Lecture 6

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1 T

1 T

f

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A2T A2 2 P( f ) = sinc ( fT ) + δ( f ) 2 2

R=

1 T

P ( f ) = A2T sinc 2 ( fT )

∆f = ?

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Digital Signaling over Bandlimited Channels: Pulse Shaping and ISI Practical channels are band-limited -> pulses spread in time and are smeared into adjacent slots. This is intersymbol interference (ISI).

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Transmission over a Band-Limited Channel baseband transmission system

Input PAM signal:

x (t ) = ∑ n an sT (t − nT )

(7.4)

y (t ) = ∑ n an sR (t − nT ),

(7.5)

Output signal:

where sR (t ) = sT (t ) * hT (t ) * hC (t ) * hR (t ) = sT (t ) * h(t ) is the Rx symbol. Sampled (at t=mT ) output ym = y (mT ) :

ym = ∑ n an sm − n = am s0 + ∑ n ≠ m an sm − n ,

(7.6)

where T = symbol interval, sm = sR (mT ) .

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Pulse Shaping to Eliminate ISI • Nyquist (1928) discovered 3 methods to eliminate ISI: – zero ISI pulse shaping, – controlled ISI (eliminated later on by, say, coding), – zero average ISI (negative and positive areas under the pulse in an adjacent interval are equal). • Zero ISI pulse shaping:

 s0 , n = 0 ⇒ s ( nT ) =   0, n ≠ 0 ym = am s0 + ∑ n ≠ m an sm − n ⇒ ym = am s0

(7.7)

• Pulse shapes: - rectangular pulse - sinc pulse - raised cosine pulse Example: sinc pulse,

s(t ) = sinc( Rt ) ⇒ s(nT ) = sinc(n) = 0, n ≠ 0 where R=1/T = transmission (symbol) rate [symbols/s].

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Zero ISI: sinc Pulse sinc(t) pulse

0.9

0.6 0.3

0

0.3

3

2

1

0

1

2

3

t

s(t ) = sinc( Rt ) ⇒ s(nT ) = sinc(n) = 0, n ≠ 0 , R = 1/ T = transmission rate [symbols/s]. Sinc pulse allows to eliminate ISI at sampling instants.

However, it has some (2) serious drawbacks. Lecture 6

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Nyquist Criterion for Zero ISI • Provides a generic solution to the zero ISI problem. • A necessary and sufficient condition for zero ISI:

m  S f + ∑ s  T  = T m =−∞ ∞

(7.8)

W = Fmax Consider a channel badndlimited to [0,Fmax]. There are 3 possible cases: 1) 1/ T = R > 2Fmax no way to eliminate ISI. 2) R = 2Fmax only sinc( Rt ) eliminates ISI. The highest possible transmission rate f0 for transmission with zero ISI is 2Fmax , not Fmax (what a surprise!) 3) R < 2Fmax many signals may eliminate ISI in this case. Sinc pulse:

Lecture 6

∆f = Fmax =

R 1 = . Q: rectangular pulse? 2 2T

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The Raised Cosine Pulse When R < 2Fmax , raised cosine pulse is widely used. Its spectrum is

1− α  T f R , 0 ≤ ≤  2  πT  1 − α  1 − α 1+ α T  Src ( f ) =  1 + cos f R R f R − ≤ ≤ ,   α  2 2 2  2   1+ α f > R 0,   2 (7.9) where 0 ≤ α ≤ 1 is roll-off factor, The bandwidth above the Nyquist frequency f0/2 is called excess bandwidth. Time-domain waveform (shape) of the pulse is

s ( t ) = sinc ( t / T )

cos ( παt / T ) 2 2

1 − 4α t / T

2

(7.10)

Baseband bandwidth = ? RF (bandpass) bandwidth = ?

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Raised Cosine Spectrum

Time-Domain pulse 1

1

0.5

0.5

0

0

1

0.5

0

0.5

1

f /f0

alfa=1 alfa=0.5 alfa=0

3

2

1

0

alfa=1 alfa=0.5 alfa=0

1

2

3

t/T

Note that: (1) α = 0 , pulse reduces to sinc ( Rt ) and R = f 0 = 2 Fmax (2) α = 1, symbol rate R = Fmax (3) tails decay as t3 -> mistiming is not a big problem (4) smooth shape of the spectrum -> easy to design filter Lecture 6

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Bandpass (RF) Digital Signaling How to transform the baseband signal to the RF frequencies (for efficient wireless transmission)? Most popular approach: combine the baseband signaling with the DSB-SC modulation: (7.11)

x(t ) = s (t ) ⋅ Ac cos(ωct ) i.e. multiply the digitally-modulated message (baseband) s (t ) = ∑ i ai p(t − iT ) with the carrier Ac cos(ωct ) . Compare to the analog DSB-SC:

(7.12)

x(t ) = m(t ) ⋅ Ac cos(ωct )

where m(t ) is the message: (7.11) = (7.12) when m(t ) = s (t ) , i.e. s (t ) serves as an analog message to DSB-SC. Spectrum of the modulated bandpass signal: Sx ( f ) =

Ac  Sm ( f − f c ) + Sm ( f + f c )  2

Sm ( f )

(7.13)

Sx ( f ) Sm

Sm

f 0

Lecture 6

f

fc

− fc

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0

fc

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Binary Phase Shift Keying (BPSK) Assume rectangular pulse shape and random binary message:

s (t ) = ∑ i ai p(t − iT ) , ai = ±1, p(t ) = Π(t )

(7.14)

Then, the PSD of the modulated signal x(t ) = s (t ) ⋅ Ac cos(ωct ) is

Ac2T PBPSK ( f ) = sinc 2 (( f − f c )T ) 2

(7.15)

Bandwidth ∆f = ? RF vs. baseband badwidth?

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Digital DSB-SC Transmitter source

{m i } baseband

s (t )

×

modulator

BPF

x (t )

Ac cos(2 π f c t )

carrier

Digital baseband modulation: mi → ai , s (t ) = ∑ i ai p(t − iT ) Baseband (digitally-modulated) signal: s (t ) Bandpass (RF) modulated signal: x(t ) = s (t ) ⋅ Ac cos(ωct ) BPF: eliminates out-of-band emission * Demodulation (detection) ?

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Summary • Review of digital communications. • Baseband digital modulation (line coding). • Band-limited systems and intersymbol interference. • Bandpass (RF) modulation. • Spectra. • Modulators/demodulators (detectors), block diagrams. • Review of FT & modulation from ELG3175.

Reading:

Rappaport, Ch. 6 (6.1-6.10). L.W. Couch II, Digital and Analog Communication Systems, 7th Edition, Prentice Hall, 2007. (other editions are OK as well)

Other books (see the reference list). Note: Do not forget to do end-of-chapter problems. Remember the learning efficiency pyramid!

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Appendix 1: Baseband/Bandpass Signals and Modulation In what follows is a review of baseband/bandpass signals and modulation (DSB-SC), corresponding modulators and detectors as well as basic digital modulation techniques. This material was covered in ELG3175 and you are advised to consult your notes and the textbook of that course. A good textbook to follow is this: L.W. Couch II, Digital and Analog Communication Systems, 7th Edition, Prentice Hall, 2007. (other editions are OK as well)

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Baseband & Bandpass Signals

Lecture 5

• Baseband (lowpass) signal: spectrum is (possibly) nonzero around the origin (f=0) and zero (negligible) elsewhere:

S x ( f ) = 0,

f > f max

• Bandpass (narrowband) signal: spectrum is (possibly) nonzero around the carrier frequency fc and zero (negligible) elsewhere:

S x ( f ) = 0, Sx ( f )

f − fc > B Sx ( f )

∆f

f

f

fc

Lecture 6

∆f

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Lecture 6

DSB-SC: General Case DSB-SC signal: x ( t ) = Acm ( t ) cos ( 2πf ct ) Spectrum: S x ( f ) =

Ac  Sm ( f − f c ) + Sm ( f + f c )  2

What do you see on a spectrum analyzer? Bandwidth ? Power efficiency? PSD? Sm ( f )

Sx ( f ) Sm

Sm

f 0 Lecture 6

fc 12-Oct-17

f

− fc

0

fc 21(37)


Generation of DSB-SC

Lecture 6

Generation: Mixer. Not practical in many cases. Filtered conventional AM. Not practical.

Balanced modulator:

J.Proakis, M.Salehi, Communications Systems Engineering, Prentice Hall, 2002

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Demodulation of DSB-SC

Lecture 6

Demodulation - Costas loop: Details: Couch, Sec. 4.14, 5.4

θ = 0? = π / 2? = π?

Why 2 channels? L.W. Couch II, Digital and Analog Communication Systems, Prentice Hall, 2001.

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Lecture 6

Demodulation of DSB-SC Product detector + squaring carrier recovery loop:

+ BPF at fc L.W. Couch II, Digital and Analog Communication Systems, Prentice Hall, 2001.

+ BPF at 2fc

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Demodulation of DSB-SC Using a pilot tone:

Lecture 6

modulation

demodulation

J.Proakis, M.Salehi, Communications Systems Engineering, Prentice Hall, 2002

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Lecture 9

Amplitude Shift Keying (ASK) • Switch on-off the carrier:


• Signal representation: • Is it similar to something? • Signal spectrum?

x ( t ) = Ac m ( t ) cos ( 2πf ct ) binary ASK:

m ( t ) = 1 or 0

general case: a fixed number of levels Lecture 6

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Amplitude Shift Keying (ASK)

Ac Sx ( f ) =  Sm ( f − f c ) + Sm ( f + f c ) 2  +∞

• Square-wave message: Sm ( f ) =

∑

cn δ( f − nf 0 ),

n =−∞ π

• Random message PSD: • How to detect?

Lecture 6

12-Oct-17

1 1 R  n  j 2n cn = sinc   e , f 0 = = 2 2Tb 2 2


• Signal spectrum (FT):

Lecture 9

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Binary Phase Shift Keying

Lecture 9


• BPSK signal representation: x ( t ) = Ac cos ( ωct + ∆ϕ ⋅ m(t ) ) where m(t ) = ±1 is bipolar message. • Another form of the BPSK signal -> x ( t ) = Ac cos ∆ϕ cos ωc t − − Ac sin ∆ϕ ⋅ m(t )sin ωc t Lecture 6

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Binary Phase Shift Keying

Lecture 9

• Digital modulation index: βd = 2∆ϕ π

• Important special case βd = 1 (random message) • Compare with ASK! • How to detect?

PSD


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Frequency Shift Keying

Lecture 9

 Ac cos ( ω1t + θ1 ) , mark (1) • Discontinuous FSK: x ( t ) =  A cos ω t + θ , space (0) ( 2 2)  c


Not popular (spectral noise + PLL problems) Lecture 6

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Frequency Shift Keying

Lecture 9

• Continuous FSK:

t   x ( t ) = Ac cos  ωct + ∆Ω∫ m ( τ )d τ    0  


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Appendix 2: Autocorrelation and Power/Energy Spectral Density This material was covered in ELG3175 and ELG3126. You are advised to consult your notes and the textbook of that course. A good textbook to follow is this: L.W. Couch II, Digital and Analog Communication Systems, 7th Edition, Prentice Hall, 2007. (other editions are OK as well)

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Lecture 3

Power and Energy • Power Px & energy Ex of signal x(t) are:

1 Px = lim T →∞ 2T

T 2

∫

x (t ) dt

−T

∞

Ex =

∫

2

x (t ) dt

−∞

• • • •

Energy-type signals: E x < ∞ Power-type signals: 0 < Px < ∞ Signal cannot be both energy & power type! Signal energy: if x(t) is voltage or current, Ex is the energy dissipated in 1 Ohm resistor • Signal power: if x(t) is voltage or current, Px is the power dissipated in 1 Ohm resistor.

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Energy-Type Signals (summary)

Lecture 3

• Signal energy in time & frequency domains: ∞

Ex =

2

∫

∞

x (t ) dt =

−∞

∫

−∞

1 2 S x ( f ) df = 2π

∞

∫

2

S x ( ω) d ω

−∞

• Energy spectral density (energy per Hz of bandwidth):

Ex ( f ) = Sx ( f )

2

• ESD is FT of autocorrelation function: Rx ( τ ) = ∫

+∞

−∞

Lecture 6

x ( t ) x ∗ ( t − τ ) dt ↔ E x ( f )

12-Oct-17

Rx ( 0 ) = E x

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Lecture 3

Power-Type Signals: PSD • Definition of the power spectral density (PSD) (power per Hz of bandwidth): 2

∞ ST ( f ) Px ( f ) = lim ⇒ Px = ∫ Px ( f ) df < ∞ −∞ T →∞ T

• where xT (t ) is the truncated signal (to [-T/2,T/2]), t xT (t ) = x (t )Π  T

  x (t ), − T / 2 ≤ t ≤ T / 2  = 0, otherwise  

• and ST ( f ) is its spectrum (FT), ST ( f ) = FT { xT (t )} Lecture 6

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Lecture 3

Power-Type Signals • Time-average autocorrelation function: 1 Rx ( τ ) = lim T →∞ 2T

T

∫−T

x ( t ) x∗ ( t − τ ) dt

• Power of the signal: 1 Px = lim T →∞ 2T

T

∫−T

2

x ( t ) dt = Rx ( 0 )

• Wiener-Khintchine theorem : Px = ∫

∞

P −∞ x

Lecture 6

12-Oct-17

( f ) df

⇒ Px ( f ) = FT {Rx ( τ )}

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Lecture 3

Periodic Signals • Power of a periodic signal: ∞ +∞ 1 2 2 Px = ∫ x (t ) dt = ∑ cn = Rx (0) ← x ( t ) = ∑ cne jnω0t T T n =−∞ n =−∞

• Autocorrelation function: ∞ 1 2 ∗ Rx ( τ ) = ∫ x ( t ) x ( t − τ ) dt = ∑ cn e jnω0τ T T n =−∞

• Power spectral density (PSD): Px ( f ) = FT {Rx ( τ)} =

∞

∑ n =−∞

Lecture 6

12-Oct-17

n 2  cn δ  f −  T 

Prove these properties!

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