ECE 342 Communication Theory

Fall 2005, Class Notes

Prof. Tiffany J Li

www: http://www.eecs.lehigh.edu/∼jingli/teach

email: jingli@ece.lehigh.edu

Analog Communications

Modulation and Communication Systems

• Modulation is a process that causes a shift in the range of frequencies

in a signal. In effect, modulation converts the message signal from

lowpass to bandpass.

• Two types of communication systems.

– Baseband communication: does not use modulation.

∗ The term baseband is used to designate the band of frequen-

cies of the signal delivered by the source or the input transducer.

– In telephony, the baseband is the audio band (band of voice

signals) of 0 to 3.5kHz.

– In television, the baseband is the video band occupying 0 to

4.3MHz.

– For digital data or pulse-code modulation (PCM) using bipo-

lar signaling at a rate Rb pulses per second, the baseband is 0

to Rb Hz.

∗ Baseband signals cannot be transmitted over a radio link but

are suitable for transmission over a pair of wires, coaxial cables,

or optical fibers. Examples include:

– local telephony communication

– short-haul PCM between two exchanges

– long-distance PCM over optical fibers

– Carrier communication: uses modulation.

∗ Eg: amplitude modulation, frequency modulation, phase mod-

ulation.

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∗ A comment about pulse-modulated signals (including pulse

amplitude modulation (PAM), pulse width modulation (PWM),

pulse position modulation (PPM), pulse code modulation (PCM)

and delta modulation (DM)): despite the term modulation,

these signals are baseband signals. Pulse-modulation schemes

are really baseband coding schemes, and they yield baseband

signals. These signal must still modulate a carrier in order to

shift their spectra.

Objectives of Modulation

• To translate the frequency of the lowpass signal to the passband of the

channel so that the spectrum of the transmitted bandpass signal will

match the passband characteristics of the channel. (Eg. in transmission

of speech over microwave links, the transmission frequencies must be

increased to the gigahertz range.)

• To reduce the size of the antennas. (To obtain efficient radiation of

electromagnetic energy, the antenna must be longer than 1/10 of the

wavelength.)

• To accommodate for the simultaneous transmission of signals from sev-

eral message sources (e.g. frequency-division multiplexing).

• To expand the bandwidth of the transmitted signal in order to increase

its noise and interference immunity in transmission over a noisy channel

(i.e. use modulation to exchange transmission bandwidth for the signal-

to-noise ratio).

Energy Spectral Density and Power Spectral Density

• Three ways of computing the energy of a signal g(t):

Eg =∫ ∞

−∞|g(t)|2dt = Rg(0) =

∫ ∞

−∞|G(f)|2df

where Rg(τ) = g(τ) ∗ g∗(−τ) is the (time) auto correlation function of

g(t).

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• Gg(f)∆= |G(f)|2 is called the energy spectral density of a signal g(t).

– |G(f)|2 equals the Fourier Transform of the time autocorrelation

function Rg(τ) = g(τ) ∗ g∗(−τ) =∫∞−∞ g(t)g∗(t − τ)dt.

– It represents the amount of energy per unit bandwidth present in

the signal at various frequencies.

– Energy of the signal can be computed by integrating the energy

spectral density over all frequencies:

Eg =∫ ∞

−∞Gg(f)df

– If g(t) is passed through a filter/channel with the impulse response

h(t), and the output signal is y(t), then the energy of the output

signal is

Ey(t) =∫ ∞

−∞|G(f)|2|Y (f)|2df

and the energy spectral density of the output signal y(t) is

Gy(f) = Gg(f)Gh(f) = |G(f)|2|H(f)|2

• Sg(f) ∆= limT→∞1T|G(f)|2 is called the power spectral density (PSD)

of a signal g(t)

– Sg(f) equals the Fourier Transform of the time-average autocorre-

lation function Rg(τ) = limT→∞1T

∫ T/2−T/2 g(t)g∗(t − τ)dt.

– It represents the amount of power per unit bandwidth present in

the signal at various frequencies.

– The power of the signal can be computed by integrating the PSD

over all frequencies:

Pg =∫ ∞

−∞Sg(f)df

– If g(t) is passed through a filter/channel with the impulse response

h(t), and the output signal is y(t), then the power of the output

signal is

Py(t) = limT→∞

1

T

∫ T/2

−T/2|G(f)|2|Y (f)|2df

and the energy spectral density of the output signal y(t) is

Sy(f) = Sg(f)Sh(f)

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Modulation

• Consider a lowpass, power-type signal, m(t), of bandwidth, W (that is,

M(f) = 0, for |f | > W ). Let Pm = limT→∞1T

∫ T/2−T/2 |m(t)|2dt denote the

power of the signal. The message signal m(t) is transmitted through

the communication channel by impressing it on a sinusoidal carrier

signal of the form

c(t) = Accos(2πfct + φc)

where Ac is the carrier amplitude,

fc is the carrier frequency,

φc is the carrier phase.

• We say that the message signal m(t) modulates the carrier signal c(t)

in either amplitude, frequency, or phase if after modulation, the ampli-

tude, frequency, or phase of the signal become functions of the message

signal. This results in amplitude modulation (AM), frequency

modulation (FM), or phase modulation (PM). The latter two

types of modulation are similar, and belong to the class of modulation

known as angle modulation.

• Four types of amplitude modulation:

– Double-Sideband Suppressed-Carrier AM (DSB-SC)

– Conventional Double-Sideband AM (DSB, a.k.a. conventional AM)

– Single-Sideband AM (SSB)

– Vestigial-Sideband AM (VSB)

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Double-Sideband Suppressed-Carrier AM

• lowpass message signal m(t), bandwidth W ; also assume the average

of m(t) is zero (which is a valid assumption for many signals including

audio signals).

• carrier signal c(t) = Accos(2πfct) (without loss of generality, we as-

sumed the carrier phase is 0)

• DSB-SC AM modulated signal

u(t) = m(t)c(t) = Acm(t)cos(2πfct)

– u(t) has bandwidth 2W , i.e. double the bandwidth of m(t).

– u(t) has double sidebands: upper sideband |f | > fc; lower side-

band |f | < fc.

– u(t) does not contain a carrier component (hence carrier suppressed).

All the transmitted power is contained in the modulating (message)

signal m(t).

• Spectrum of DSB-SC signals: Assume m(t) ⇐⇒ M(f),

U(f) = F [Acm(t)cos(2πfct)] =Ac

2[M(f − fc) + M(f + fc)]

• Power content of DSB-SC signals

Pu = limT→∞

1

T

∫ T/2

−T/2u2(t)dt

= limT→∞

1

T

∫ T/2

−T/2A2

cmt(t)cos2(2πfct)dt

=A2

c

2lim

T→∞

1

T

∫ T/2

−T/2m2(t)dt +

A2c

2lim

T→∞

1

T

∫ T/2

−T/2m2(t)cos(4πfct)dt

=A2

c

2Pm (1)

(The overall integral of m2(t)cos(4πfct) is almost zero. Since the result

of the integral is divided by T , and T becomes very large, the second

term in the equation becomes zero.)

• Demodulation of DSB-SC signals:

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– Requires a phase-coherent or synchronous demodulator. That

is, the phase φ of the locally generated sinusoid should ideally be

equal to the phase of the received-carrier signal.

– Steps (synchronous demodulator):

∗ STEP 1. first multiply received signal r(t) by a locally gener-

ated sinusoid cos(2πfct + φ)

∗ STEP 2. pass the product signal through an ideal lowpass filter

with the bandwidth W

∗ Mathematically:

r(t) = u(t) = Acm(t)cos(2πfct) assuming noiseless channel

r(t)cos(2πfct + φ) = Acm(t)cos(2πfct)cos(2πfct + φ)

=1

2Acm(t)cos(φ) +

1

2Acm(t)cos(4πfct + φ)

︸ ︷︷ ︸

filtered out

(2)

The power in the demodulated signal is decreased by a factor

of cos2φ =⇒ in need for a synchronous demodulator

• Two ways to generate phase-locked sinusoidal signals at the receiver:

Phase-locked loop (PLL) and using a pilot tone.

• The method of pilot tone:

– Add a pilot tone, i.e. a carrier component, into the transmitted

signal.

– The power of the pilot tone, A2p/2 is selected to be significantly

smaller than that of the modulated signal u(t).

– The transmitted signal is still double sideband, but no longer car-

rier suppressed.

– At the receiver, a narrow band filter tuned to frequency fc filters

out the pilot signal component; its output is used to multiply the

received signal.

– Note that the presence of the pilot signal results in a DC component

in the demodulated signal; this needs to be subtracted out in order

to recover m(t).

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– Disadvantages of pilot tone: a certain portion of the transmitted

signal power is allocated to the transmission of the pilot.

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Conventional DSB AM

• A conventional AM modulated signal consists of a large carrier compo-

nent, in addition to the double-sideband AM modulated signal:

u(t) = Ac[1 + m(t)]cos(2πfct)

where the message signal is constrained to satisfy |m(t)| ≤ 1. (If

m(t) < −1, the AM signal is over-modulated, causing a more com-

plex demodulation process.)

• For convenience, let us express m(t) as

m(t) = amn(t)

where mn(t) = m(t)/ max |m(t)|, and a is called the modulation in-

dex and 0 < a < 1. Hence

u(t) = Ac[1 + amn(t)]cos(2πfct)

• Spectrum of the conventional AM signal:

U(f) = F [Ac[1 + amn(t)]cos(2πfct)]

=Aca

2[Mn(f − fc) + Mn(f + fc)] +

Ac

2[δ(f − fc) + δ(f + fc)]

• Power of the conventional AM signal:

Pu =A2

c

2(1 + Pm) =

A2c

2(1 + a2Pmn

)

• Demodulation:

– No need for a synchronous demodulator.

– We can use a simple envelope detector (i.e. a rectifier + lowpass

filter) followed by a transformer to demodulate it:

∗ STEP 1. Rectify the signal: i.e. eliminating the negative val-

ues. This would not affect the message signal since the envelop

(amplitude) 1 + m(t) > 0.

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∗ STEP 2. Pass the rectified signal through a low pass filter

whose bandwidth matches that of the message signal.

∗ STEP 3. Eliminate the DC component in the output of the

envelope detector through a transformer.

Note the output of the envelope detector (i.e. a rectifier and a

lowpass filter) is (ideally) d(t) = g1+g2m(t), where g1 is the DC

term (introduced by the carrier component at the transmitter)

and g2 is a gain factor due to the signal demodulator.

• Power efficiency:

η =useful power

total power=

Pm

1 + Pm=

a2Pmn

1 + a2Pmn

For practical signals, the efficiency is on the order of 25% or lower.

• Major advantage of the conventional AM is the simple demodulator

=⇒ widely used in AM radio broadcast.

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Single Sideband (SSB) AM

• The DSB spectrum has two sideband: upper side-band (USB) and low

side-band (LSB), both containing the complete information of the base-

band signal.

=⇒ sufficient to transmit only one sideband: Single-sideband trans-

mission. SSB transmission requires only one-half the bandwidth of the

DSB signal.

• The Single-sideband (suppressed-carrier) signals can be demodulated

using a synchronous demodulator in a way identical to demodulating

DSB-SC signals:

– Multiplying a USB (LSB) signal by cos(2πfct) shifts its spectrum

to the left and right by fc.

– Low-passing filtering of the signal yields the desired baseband sig-

nal.

• Generating SSB AM signals:

– In the frequency domain:

generate double-sideband signals and use an ideal bandpass filter

to filter one of the sidebands of a DSB signal.

– In the time domain:

Upper sideband: u(t) = Acm(t)cos(2πfct) − Acm̂(t)sin(2πfct)

Lower sideband: u(t) = Acm(t)cos(2πfct) + Acm̂(t)sin(2πfct)

where m̂(t) is the Hilbert transform of m(t).

• Hilbert transform of a signal m(t) results in a new signal ˆm(t) that

has exactly the same frequency components present in m(t) with the

same amplitude - except there is a 90o phase delay (i.e. a π/2 phase

shifter).

– In effect, after Hilbert transform, the spectrum of the signal at posi-

tive frequencies is multiplied by −j (i.e. ej2πf0t becomes ej2πf0t−π/2),

and the spectrum of the signal at negative frequencies is multiplied

by +j (i.e. e−j2πf0t becomes e−j(2πf0t−π/2))

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– This is equivalent to multiplying the spectrum (Fourier transform)

of the signal by −j sgn(f) (or filtering by a filter with response

−j sgn(f)):

if m(t) ⇐⇒ M(f)

then m̂(t) ⇐⇒ −j sgn(f)M(f)

– Since 1πt ⇐⇒ −jsgn(f), in the time domain, the operation of Hilbert

transform is equivalent to a convolutional operation:

m̂(t) =1

πt∗ m(t) =

1

π

∫ ∞

−∞

m(τ

t − τdτ

• The impact of a phase-offset in the demodulator:

– Consider for example USB signals:

r(t) = m(t) = Acm(t)cos(2πfct) − Acm̂(t)sin(2πfct)

r(t)cos(2πfct + φ) =1

2Acm(t)cos(φ) +

1

2m̂(t)sin(φ)

+double frequency terms

– The impact of phase-offset is two-fold:

- It reduces the amplitude of of the desired signal m(t) by cos(φ)

(same as DSB-SC)

- It also introduces an undesirable sideband signal (interference)

due to the presence of m̂(t)

– Hence, coherent demodulator is desired, and the use of pilot tone

is a very effective method in SSB Am systems (at the cost of some

power efficiency).

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Vestigial Sideband (VSB) AM

• Motivation:

– SSB AM is bandwidth efficient. The filter method to generate

SSB requires an exact sideband filter which has an extremely sharp

cutoff in the vicinity of the carrier (particularly when m(t) has a

large power concentrated in the vicinity of f = 0). Such filter

characteristics are very difficult to implement in practice.

– Relax the SSB system by allow vestige, which is a portion of the

unwanted sideband.

• Justification:

– VSB AM system can be implemented by generating a DSB-SC AM

signal and passing it through a sideband filter with the frequency

response H(f).

– To recover the original message spectrum, the VSB-filter charac-

teristics must satisfy:

H(f − fc) + H(f + fc) = constant for any |f | ≤ W

where W is the bandwidth of the message signal m(t).

– In practice, to minimize distortion of the message signal, the VSB-

filter should also have a linear phase over its passband.

• Practice:

– Suitable for signals that have a strong low-frequency component,

such as video signals. VSB used in standard TV broadcasting.

– A carrier component is generally transmitted so that the demodula-

tor can extract a phase-coherent reference for synchronous demod-

ulation. In TV broadcast, a large carrier component is transmitted.

and it is possible to recover the message using a simple envelope

detector (same as the conventional DSB AM in radio broadcast-

ing).

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