ece 544: digital communications (session 37)

ECE 544: Digital Communications " "(Session 37)

Complex Baseband Models for Passband Communication Systems!

J. V. Krogmeier Purdue University, West Lafayette

November 21, 2014

Contents"

 Review of Linear Analog Modulation and Demodulation"

 Passband Communications" Complex Envelope for Deterministic

Energy Signals" Complex Envelope and LTI Filtering" Complex Envelope for Wide-Sense

Stationary Random Signals"

AM DSB-SC"• Modulator:

• Spectrum: (assuming energy signal for message)

Xdsb(f) =Ac

2M(f − fc) +

Ac

2M(f + fc).

Transmission BW = 2W .

• Power: (assuming power signal for message) power[xdsb(t)] = 0.5A2cPm.

m(t)

Ac cos(2πfct)

xdsb(t) = Acm(t) cos(2πfct)

f

M(f)

−W W

M0

fc−fc fc −W fc +W

Xdsb(f)

f

AcM0/2

AM-LC"• Modulator: (ka sometimes called the amplitude sensitivity of the modula-

tor.)

• Spectrum: (assuming energy signal for message)

Xam−lc(f) = 0.5Ac [δ(f − fc) + δ(f + fc)]+0.5kaAc [M(f − fc) +M(f + fc)]

Transmission BW = 2W .

• Power: (assuming power signal for message) power[xam−lc(t)] = 0.5A2c(1+

k2aPm).

• E.D. Requirement: If 1+kam(t) > 0 for all t then the envelope of xam−lc(t)may be used to recover m(t).

f

M(f)

−W W

M0

fc−fc fc −W fc +W

f

Xam−lc(f)(0.5Ac)

0.5kaAcM0

(0.5Ac)

m(t)

Ac cos(2πfct)

ka

1

xam−lc(t) = [1 + kam(t)]Ac cos(2πfct)

SSB"

m(t) sidebandfilter

Ac cos(2πfct)

xdsb(t)xssb(t)

• Modulator: Several architectures producing modulated signal:

xssb(t) = Acm(t) cos(2πfct)±Acm̂(t) sin(2πfct)

* m̂(t) is the Hilbert Transform of m(t).

* the plus sign in the “±” is chosen for lower sideband SSB and the

minus sign is chosen for upper sideband SSB.

Sideband filtering modulator:

SSB"

m(t)

H(f) = −jsgn(f)

Quadrature

Oscillator

Ac sin(2πfct)

Ac cos(2πfct)

Acm(t) cos(2πfct)

Acm̂(t) sin(2πfct)m̂(t)

+

−

+

+

xssb−lsb(t)

xssb−usb(t)

• Modulator: Several architectures producing modulated signal:

xssb(t) = Acm(t) cos(2πfct)±Acm̂(t) sin(2πfct)

* m̂(t) is the Hilbert Transform of m(t).

* the plus sign in the “±” is chosen for lower sideband SSB and the

minus sign is chosen for upper sideband SSB.

Phase-shift modulator:

SSB"• Spectrum: (assuming energy signal for message)

Xssb(f) =Ac

2M(f − fc) +

Ac

2M(f + fc)±

Ac

j2M̂(f − fc)∓

Ac

j2M̂(f + fc)

=Ac

2[1∓ sgn(f − fc)]M(f − fc) +

Ac

2[1± sgn(f + fc)]M(f + fc)

where the top sign choice corresponds to lower sideband SSB and thebottom sign choice corresponds to upper sideband SSB. Transmission BW= W .

• Power: (assuming power signal for message) power[xdsb(t)] = A2cPm.

f

M(f)

−W W

M0

B A

fc−fc fc −W fc +W

fAB

fc−fc fc −W fc +W

fBA

Upper Sideband (USB)

Lower Sideband (LSB)

Demodulators"• DSB-SC, AM-LC, and SSB may all be demodulated coherently using the

following block diagram:

• AM-LC can also be demodulated non-coherently (i.e., without a phasecoherent local carrier reference) using a square law device or an envelopedetector.

HLP (f)

cos(2πfct)

xdsb(t)or

xam−lc(t)or

xssb(t)

∝ m(t)

QAM, Demodulation, Spectral Efficiency"

  AM-DSB and AM-LC are each able to transmit a single baseband message. In doing so they require a transmission bandwidth equal to twice the bandwidth of the message. "

  Using a considerably more complex scheme AM-SSB is able to accomplish the same message transmission using a transmission bandwidth equal to the bandwidth of the message."

  The bandwidth efficiency is doubled by SSB relative to DSB or LC."

QAM, Demodulation, Spectral Efficiency"

  QAM uses the quadrature oscillator / mixers from the phase-shift modulator implementation of SSB to transmit a pair of independent messages. In this fashion it achieves the same bandwidth efficiency as SSB but with simplicity similar to that of AM-DSB."

QAM, Demodulation, Spectral Efficiency"• Modulator:

xqam(t) = AcmI(t) cos(2πfct) +AcmQ(t) sin(2πfct).

Quadrature

Oscillator

Ac sin(2πfct)

Ac cos(2πfct)

+

+

mI(t)

mQ(t)

AcmI(t) cos(2πfct)

AcmQ(t) cos(2πfct)

xqam(t)

QAM (contʼd.)"• Spectrum:

Xqam(f) =Ac

2MI(f − fc) +

Ac

2MI(f + fc) +

Ac

j2MQ(f − fc)−

Ac

j2MQ(f + fc)

=Ac

2[MI(f − fc)− jMQ(f − fc)] +

Ac

2[MI(f + fc) + jMQ(f + fc)]

Transmission BW = 2W .

• Power: power[xqam(t)] = 0.5A2c(PmI + PmQ).

f−W W

MI(f)Mi0

fc−fc fc −W fc +W

f

AcMi0/2

Real{Xqam(f)}

fc

−fc

fc −W fc +W

f

Imag{Xqam(f)}

AcMq0/2

f−W W

MQ(f)Mq0

−AcMq0/2

QAM (contʼd.)"

• Demodulator: QAM can be coherently demodulated with a pair of quadra-ture coherent AM demodulators.

Quadrature

Oscillatorxqam(t)

cos(2πfct)

sin(2πfct)

HLP (f)

HLP (f)

∝ mI(t)

∝ −mQ(t)

Passband Communications"

• The vast majority of communication systems are passband systems. Thismeans that the the transmitted information bearing signal s(t) has itsenergy restricted to a band of frequencies located around some nominalcarrier frequency and above and relatively far away from dc (baseband).

• A simple channel model is as shown below:

h(t)channels(t)

hBP (t)

Nw(t)

r(t)y(t)

Passband Communications"

h(t)channels(t)

hBP (t)

Nw(t)

r(t)y(t)

• Here:

– The information bearing signal is s(t) and the channel output is r(t),

the signal that the receiver must use to estimate the desired mes-

sage information. Typically, s(t) has energy restricted to a known

frequency band.

– The receiver channel filter is represented by hBP (t) ↔ HBP (f) which

is assumed to have a bandpass characteristic just sufficient to pass

s(t) without distortion.

– The communication channel itself is represented by some filtering

h(t) ↔ H(f) and the addition of a white Gaussian noise denoted by

Nw(t).

– The channel filter is here indicated as LTI but it could be generalized.

Note that if it is LTI then it is no loss of generality to consider its

passband to be fully contained in the of hBP (t) ↔ HBP (f).

Passband Communications"• If we assume that the information bearing signal is created using the usual

QAM modulator and that a QAM demodulator is used to recover it, thenthe standard passband modulator-channel-demodulator is as shown in thefigure below. All signals and impulse responses are real-valued.

• Note that the model allows for the possibility of frequency and phase offsetbetween transmitter and receiver.

√2 cos(2πf ′

ct + θ)

√2 sin(2πf ′

ct + θ)

h(t)channel

sI(t)

sQ(t)

s(t)hBP (t)

hLP (t)

hLP (t)

rI(t)

rQ(t)Nw(t)

√2 cos(2πfct)

−√

2 sin(2πfct)

r(t)y(t)

Complex Baseband Model for Passband Communications"

• The passband modulator-channel-demodulator can be drawn more com-pactly by using complex notation as shown in the figure below wheresL(t) = sI(t) + jsQ(t) and rL(t) = rI(t) + jrQ(t).

h(t)channels(t)

Re(·)

√2ej(2πf ′

ct+θ)

sL(t)hLP (t)hBP (t)

Nw(t) √2e−j2πfct

rL(t)r(t)y(t)

Switch to Handwritten (work out correspondence)"

Complex Baseband Model for Passband Communications"

• A complex baseband model for the passband system is a way to directlycompute the output rL(t) from the input sL(t) using baseband opera-tions, i.e., baseband LTI filtering and/or frequency conversions with small(relative to baseband) center frequencies. This has several benefits:

1. A baseband model is a simpler model.

2. A baseband model can be numerically simulated with much lowercomputation than can a passband model because the sampling rateis much lower.

3. A baseband model can form the basis for a discrete-time implemen-tation of a bandpass communications system.

sL(t)

NL(t)

rL(t)A BasebandLTI Filter

ej(2π(∆f t+∆θ)A Frequency and

Phase Offset(Complex) AWGN

Complex Baseband Model for Passband Communications"

• A complex baseband model for the passband system is a way to directlycompute the output rL(t) from the input sL(t) using baseband opera-tions, i.e., baseband LTI filtering and/or frequency conversions with small(relative to baseband) center frequencies. This has several benefits:

1. A baseband model is a simpler model.

2. A baseband model can be numerically simulated with much lowercomputation than can a passband model because the sampling rateis much lower.

3. A baseband model can form the basis for a discrete-time implemen-tation of a bandpass communications system.

sL(t)

NL(t)

rL(t)A BasebandLTI Filter

(Complex) AWGNej(2π(∆f t+∆θ)A Frequency and

Phase Offset

Complex Envelope for Deterministic Energy Signals"

• Frequency domain definition.

• Implementations: 1) filtering followed by down-conversion, 2) down-conversionfollowed by filtering.

• Time domain definition.

• Properties and other observations.

Frequency Domain Definition of Complex Envelope"

• Let x(t) be a real-valued deterministic signal and let x(t) ↔ X(f) denotea Fourier Transform pair. Since the time-domain signal is real-valued,the Fourier Transform has conjugate symmetry about f = 0: X(f) =X∗(−f). Define the complex envelope with respect to a carrier waveform√2 cos(2πfct) of some fixed center frequency fc > 0 by

√2

2XL(f − fc) = u(f)X(f)

�XL(f) =

√2u(f + fc)X(f + fc).

1. In the above definition u(·) denotes the usual unit step function.

2. The time domain complex envelope is defined as the inverse FourierTransform: xL(t) ↔ XL(f).

Switch to Handwritten (CE example for passband and for generic)"

Two Implementations"Block Diagram 1: (filtering followed by down-conversion)

The filter√2u(f), called a phase splitter, can be replaced by a complex bandpass

filter if x(t) is bandpass as illustrated in the figure below.

LTI Filter

2 u(f)

-j2 f tce

x(t) x (t)A L

x (t)

Two Implementations"Block Diagram 2: (down-conversion followed by filtering)

The filter√2u(f+fc), can be replaced by a real lowpass filter if x(t) is bandpass

as illustrated in the figure below.

LTI Filter

2 u(f+f )

-j2 f tce

x(t) Lx (t)

c

Time Domain Definition of Complex Envelope"The impulse response of the filter

√2u(f) is

√2

2

�δ(t) + j

1

πt

�.

This can be related to the notion of the Hilbert transform. The Hilbert transform

x̂(t) of a real-valued signal x(t) is the output of a LTI system driven by x(t).The impulse response and transfer function of the Hilbert transformer are:

1

πt↔ −jsgn(f).

Therefore, the time domain complex envelope can also be given as

xL(t) =

√2

2[x(t) + jx̂(t)] e−j2πfct

where the real-valued signal x̂(t) is the Hilbert transform of x(t) and xA(t) =√22 (x(t)+jx̂(t)) is called the analytic signal. In general, signals having frequency

content containing only positive frequencies are said to be analytic signals.

Properties and Other Observations"

(Continued next lecture session)