TELE3113 Analogue and DigitalCommunications

Pulse Modulation

Wei Zhang

w.zhang@unsw.edu.au

School of Electrical Engineering and Telecommunications

The University of New South Wales

What did we study

In previous lectures, we studied continuous-wave (CW)modulation:

Some parameter of a sinusoidal carrier wave is varied

continuously in accordance with the message signal.

Amplitude Modulation (AM, DSB-SC, SSB, VSB)

Angle Modulation (PM, FM)

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What will we study

Next, we will study Pulse Modulation:

Some parameter of a pulse train is varied in accordance

with the message signal.

Analogue pulse modulation: some feature of the pulse

(e.g. amplitude, duration, or position) is varied continuouslyin accordance with the sample value of the message signal.

Digital pulse modulation: the message signal is discretein both time and amplitude, thereby transmitting a sequence

of coded pulses.

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Sampling Process 1

Let ga(t) be a continuous-time (CT) signal that is sampled

uniformly at t = nT , generating the sequence g[n],

g[n] = ga(nT ), −∞ < n < ∞ (1)

where T is the sampling period and n is an integer.

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Sampling Process 2

ga(t)

ga(t) g[n]

gp(t)

p(t)

Sampling

p(t) is a periodic impulse train: p(t) =∑

∞

n=−∞δ(t − nT ).

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Sampling Process 3

p(t) can be expressed as a Fourier series as (see page 18 for

details)

p(t) =1

T

∞∑

k=−∞

exp(j(2π

T)kt). (2)

The sampling operation is a multiplication of the continuous-time

signal ga(t) by a period impulse train p(t):

gp(t) = ga(t) · p(t) = ga(t) ·

(

1

T

∞∑

k=−∞

exp(j(2π

T)kt)

)

. (3)

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FT of Sampled Signal

Assume Ga(jω) ⇔ ga(t), i.e., Ga(jω) = F [ga(t)]. From the

frequency-shifting property of the FT, we have

F [ga(t) · exp(j(2π

T)kt)] = Ga(j(ω − k

2π

T)). (4)

Next, taking FT on both sides of (3) and using (4), we get

Gp(jω) = F [gp(t)] =1

T

∞∑

k=−∞

Ga(j(ω − kωT )), −∞ < k < ∞ (5)

where ωT = 2πT denotes the angular sampling frequency.

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Sampling Theorem

Sampling theorem: Let ga(t) be a bandlimited signal with

Ga(jω) = 0 for |ω| > ωm. Then ga(t) is uniquely determined by

its samples ga(nT ), −∞ < n < ∞, if

ωT ≥ 2ωm, (6)

where

ωT =2π

T. (7)

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Signal Recovery 1

Question: Suppose that g[n] is obtained by uniformly sampling

a bandlimited analog signal ga(t) with a highest frequency ωm at

a sampling rate ωT = 2πT satisfying (6), can the original analog

signal ga(t) be fully recovered from the given sequence g[n]?

Answer: YES, ga(t) can be fully recovered by generating an

impulse train gp(t) and then passing gp(t) through an ideal low

pass filter (LPF) H(jω) with a gain T and a cutoff frequency ωc

satisfying ωm < ωc < ωT − ωm.

∧g a(t) g[n] gp(t) LPF

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Representation in Spectrum

Recovery

mω− mω

)( ωjGp

mω Tω− Tω2 Tω

cω

ω

••• •••

ω

)( ωjGa

Sampling

LPF

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Signal Recovery 2

Taking the inverse FT of the frequency response of the ideal LPF

H(jω):

H(jω) =

T, |ω| ≤ ωc

0, |ω| > ωc

(8)

Then, the impulse response h(t) of the LPF is given by

h(t) =1

2π

∫

∞

−∞

H(jω)ejωtdω =T

2π

∫ ωc

−ωc

ejωtdω

=sin(ωct)

πt/T. (9)

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Signal Recovery 3

Consider the impulse train gp(t) be expressed as

gp(t) = ga(t) · p(t) = ga(t) ·

(

∞∑

n=−∞

δ(t − nT )

)

=∞∑

n=−∞

ga(nT )δ(t − nT ) =∞∑

n=−∞

g[n]δ(t − nT ). (10)

Therefore, the output of the LPF is given by the convolution of

gp(t) with the impulse response h(t):

ga(t) =∞∑

n=−∞

g[n]h(t − nT ). (11)

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Signal Recovery 4

Substituting h(t) from (9) in (11) and assuming for simplicity

ωc = ωT /2 = π/T , we arrive at

ga(t) =

∞∑

n=−∞

g[n]sin[π(t − nT )/T ]

π(t − nT )/T

=

∞∑

n=−∞

g[n] · sinc(t − nT

T), (12)

where sinc(x) is defined as sinc(x) = sin(πx)/(πx).

The reconstructed analog signal ga(t) is obtained by shifting in

time the impulse response of the LPF h(t) by an amount nT and

scaling it an amplitude by the factor g[n] for −∞ < n < ∞ and

then summing up all shifted versions.TELE3113 - Pulse Modulation. Sept. 1, 2009. – p.12/18

PAM

Pulse-amplitude modulation (PAM): The amplitudes of

regularly spaced pulses are varied in proportion to the

corresponding sample values of a continuous message signal.

Generation of PAM:

Natural Sampling: easy to generate, only an analog switch

required.

Flat-Top Sampling: generated by using a sample-and-hold

(S/H) type of electronic circuit.

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Natural Sampling

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Flat-top Sampling

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PDM and PPM

Pulse-duration modulation (PDM): The duration of the

pulses are varied according to the sample values of the

message signal. Also referred to as pulse-width modulation

or pulse-length modulation.

Pulse-position modulation (PPM): The leading or trailing

edge of each pulse is varied in accordance with the

message signal.

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PDM and PPM

PDM

PPM

Pulse train

Message Signal

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Derivation of Eq. (2)

Using Fourier series, p(t) can be expressed as

p(t) =

∞∑

k=−∞

ck exp(j2πk

Tt),

whereck =

1

T

∫ T/2

−T/2

p(t) exp(−j2πk

Tt)dt

=1

T

∫ T/2

−T/2

(

∞∑

n=−∞

δ(t − nT )

)

exp(−j2πk

Tt)dt

=1

T

∫ T/2

−T/2

δ(t) exp(−j2πk

Tt)dt

=1

T. (13)

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