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Digital Modulation
• Continuous-wave(CW) modulation (recap):
– A parameter of a sinusoidal carrier wave is varied
continuously in accordance with the message signal.
∗ Amplitude
∗ Frequency
∗ Phase
• Digital Modulation:
– Pulse Modulation: Analog pulse modulation: A periodic
pulse train isused as a carrier. The following parameters of
the pulse are modified in accordance with the message
signal. Signal is transmitted at discrete intervals of time.
∗ Pulse amplitude
∗ Pulse width

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∗ Pulse duration
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– Pulse Modulation: Digital pulse modulation: Message signal

represented in a form that is discrete in both amplitude and
time.
∗ The signal is transmitted as a sequence of coded pulses
∗ No continuous wave in this form of transmission

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Analog Pulse Modulation

• Pulse Amplitude Modulation(PAM)
– Amplitudes of regularly spaced pulses varied in proportion
to the corresponding sampled values of a continuous
message signal.
– Pulses can be of a rectangular form or some other
appropriate shape.
– Pulse-amplitude modulation is similar to natural sampling,
where the message signal is multiplied by a periodic train of
rectangular pulses.
– In natural sampling the top of each modulated rectangular
pulse varies with the message signal, whereas in PAM it is
maintained flat. The PAM signal is shown in Figure 1.

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s(t)

m(t)

T t

Ts

Figure 1: PAM signal

• Mathematical Analysis of PAM signals

– Let s(t) denote the sequence of the flat-top pulses generated
in the manner described Figure 1. We may express the
PAM signal as
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+∞
X
s(t) = m(nTs )h(t − nTs )
n=−∞

– h(Ts ) is a standard rectangular pulse of unit amplitude and

duration T , defined as follows


= 1, 0≤t≤T


h(t) = = 12 , t = 0, t = T


= 0, otherwise


– The instantaneously sampled version of m(t) is given by

+∞
X
mδ (t) = m(nTs )δ(t − nTs )

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n=−∞
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– Therefore, we get

Z +∞
mδ (t) ∗ h(t) = mδ (τ )h(t − τ ) dτ
−∞
Z +∞ +∞
X
= m(nTs )δ(τ − nTs )h(t − τ ) dτ
−∞ n=−∞

+∞
X Z +∞
= m(nTs ) δ(τ − nTs )h(t − τ ) dτ
n=−∞ −∞

using the shifting property of the delta function, we obtain

+∞
X
s(t) = mδ (t) ∗ h(t) = m(nTs )h(t − nTs )
n=−∞

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S(ω) = Mδ (ω) ∗ H(ω)

∞
ωs X
= M (ω − kωs )H(ω)
2π
k=−∞

−jω T2
ω T2


– Since, we use flat top samples, H(ω) = T sinc e .
This results in distortion and a delay of T2 . To correct this
1
the magnitude of the equalizer is chosen as T

.
T sinc ω 2
– The message signal m(t) can be recovered from the PAM
signal s(t) as shown in Figure 2.

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s(t) Reconstruction Equalizer

m(t)
filter

Figure 2: recovering message signal from PAM signal

• Other forms of Pulse Modulation

1. Pulse-duration modulation(PDM), also referred to as
Pulse-width modulation, where samples of the message
signal are used to vary the duration of the individual pulses
in the carrier.
2. Pulse-position modulation(PPM), where the position of a
pulse relative to its unmodulated time of occurence is varied
in accordance with the message signal. It is similar to FM.

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The other two types of modulation schemes are shown in
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Figure 3.
m(t)

t
0

(a)

t
(b)

(c) t

(d) t

(a) modulating wave

(b) Pulse carrier
(c) PDM wave
(d) PPM wave

Figure 3: other pulse modulation schemes

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Pulse Digital Modulation

Pulse Code Modulation
• Discretisation of both time and amplitude.
• Discretisation of amplitude is called quantisation.
– Quantisation involves conversion of an analog signal
amplitude to discrete amplitude.
– Quantisation process is memoryless and instantaneous.
– A quantiser can be of uniform or non uniform.
∗ Uniform quantiser: the representation levels are uniformly
spaced
∗ Nonuniform quantiser: the representation levels are
non-uniformly spaced.

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Uniform Quantisers
• Quantiser type: The quantiser characteristic can be of midtread
or midrise quantizer. These two types are shown in Figure 4.
output output
level level

4 4

2 2

−4 −2 0 2 4 −4 −2 0 2 4 input
input level
−2 −2
level

−4 −4

(a) Midtread (b) Midrise

Figure 4: Different types of uniform quantisers

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• Quantization Noise: Quantisation introduces an error defined
as the difference between the input signal m and the output
signal v. The error is called Quantisation Noise.
• We may therefore write

Q=M −V
• Analysis of error in uniform quantisation:
– The input M has zero mean, and the quantiser assumed to
be symmetric =⇒ Quantiser output V and therefore the
quantization error Q also will have a zero mean.

– Consider that input m has continuous amplitude in the

range (−mmax , mmax ). Assuming a uniform quantiser of
midrise type, we get the step size of the quantiser given by

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2mmax
∆= L
where, L is the total number of representation levels.
– The quantization error Q will have its sample values
bounded by −∆2 ≤q ≤ 2.
∆

– For small ∆, assume that the quantisation error Q is a

uniformly distributed random variable. Thus,
– The probability density function of quantisation error Q can
be given like this


1, −∆
≤q≤ ∆
∆ 2 2
fQ (q) =
0, otherwise

– Since we assume mean of Q is zero, the variance is same as

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the mean square value.
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2
σQ = E[Q2 ]
Z + ∆2
= q 2 fQ (q) dq
−∆
2

+∆
1
Z 2
= q 2 dq
∆ −∆
2

∆2
=
12
– Typically, the L-ary number k, denotes the kth
representation level of the quantiser,
– It is transmitted to the receiver in binary form.
– Let R denote the number of bits per sample used in the
construction of the binary code. We may then write

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L = 2R
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R = log2 (L)
Hence, we get

2mmax
∆= 2R
also,
2
σQ = 13 m2max 2−2R
– Let P denote the average power of the message signal m(t).
We may then express the output signal to noise ratio of a
uniform quantiser as

P
(SN R)O = 2
σQ
3P
= 2 .22R
mmax

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• Example: Sinusoidal Modulating Signal
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– Let the amplitude be Am .

A2m
– Power of the signal, P = .
2
– Range of the singal : −Am , Am .
2
– Hence, σQ = 13 A2m 2−2R .
A2m
– =⇒ (SN R)O = 1 2 2 −2R
3 Am 2
– Thus, (SN R)O = 32 22R or
– 10 log10 (SN R)O = 1.8 + 6RdB.

• The major issue in using a uniform quantiser is that no effort is

made to reduce quantisation power for a presumed set of levels.

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Non Uniform Quantisers
The main advantages of using non-uniform quantizer are:
colorblack
1. Protection of weak passages over loud passages.
2. Enable uniform precision over the entire range of the voice
signal.
3. Fewer steps are required in comparison with uniform quantizer.
A nonuniform quantiser is equivalent to passing the baseband
signal through a compressor and then applying the compressed
signal to a uniform quantizer. A particular form of compression law
that is used in practice is µ - law, which is defined as follows

log(1+µ|m|)
|v| = log(1+µ)

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where, m and v are the normalized input and output voltages and
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µ is a positive constant.
Another compression law that is used in practice is the so called A
- law defined by


A|m| 1

1+log A , 0 ≤ |m| ≤ A
|v| =
 1+log(A|m|) , 1
≤ |m| ≤ 1
1+log A A

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µ − law A − law
1 1

µ=100 A=100
A=2
µ=5

|v| µ=0 |v| A=1

0 1 0 1

|m| |m|

Figure 5: Compressions in non-uniform quantization

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Differential Pulse code Modulation(DPCM)

• In DPCM, we transmit not the present sample m[k], but d[k]
which is the difference between m[k] and its predicted value
m̂[k].
• At the receiver, we generate m̂[k] from the past sample values
to which the received d[k] is added to generate m[k]. Figure 6
shows the DPCM transmitter.

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m[k]
d[k] d [k] To channel
quantizer q
sum

− +

^
m [k] sum
q
+

Predictor
m [k]
q

Figure 6: DPCM Transmitter

• Analysis of DPCM: If we take the quantised version, the

predicted value as m̂q [k] of mq [k]. The difference
d[k] = m[k] − m̂q [k]
which is quantized to yield
dq [k] = d[k] + q[k]
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The predictor input mq [k] is

m̂q [k] = m̂q [k] + dq [k]

= m[k] − d[k] + dq [k]
= m[k] + q[k]

• The quantized signal dq [k] is now transmitted over the channel.

At the receiver, m̂[k] is predicted from the previous samples
and d[k] is added to it to get m[k]. A DPCM receiver is shown
in Figure 7

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d q[k] m q[k]
Σ

m q[k]

Predictor

Figure 7: DPCM Receiver

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Delta Modulation
• Delta Modulation uses a first order predictor.
• It is a one bit DPCM.
• DM quantizer uses only two levels(L = 2)
• The signal is oversampled(atleast 4 times the Nyquist rate) to
ensure better correlation among the adjacent samples.

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m(t)

staircase
T Approximation
s
m q(t)

t
0

Figure 8: Delta Modulation

• DM provides a staircase approximation to the oversampled

version of the message signal.
• The difference between the input and the approximation is
quantised into only two levels, namely, ±∆, corresponding to
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positive and negative differences.
• For this approximation to work, the signal should not change
rapidly.
• Analysis of Delta Modulation:
– Let m(t) denote the input signal, and mq (t) denote its
staircase approximation. For convenience of presentation,
we adopt the following notation that is commonly used in
the digital signal processing literature:

m[n] = m(nTs ), n = 0, ±1, ±2, · · ·

Ts is the sampling period and m(nTs ) is a sample of the

signal m(t) taken at time t = nTs , and likewise for the
samples of other continuous-time signals.

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– The basic principles of delta modulation is the following set
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of discrete-time relations:

e[n] = m[n] − mq [n − 1]
eq = ∆sgn(e[n])
mq [n] = mq [n − 1] + eq [n]

– e[n] is an error signal representing the difference between

the present sample m[n] of the input signal and the latest
approximation mq [n − 1] to it.
– eq [n] is the quantized version of e[n],
– sgn(.) is the signum function.
– The quantiser output mq [n] is coded to produce the DM
signal.

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– The transmitter and receiver of the DM is shown in Figure
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9.
+ε
m(t) + d (t) Sampler
d(t) q
Σ
−ε Comparator
− m q (t)

Integrator

Transmitter

d q(t)
m(t)
LPF

Integrator

Receiver

Figure 9: Transmitter and Receiver for Delta Modulation

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