Pulse modulation schemes aim to transfer an analog signal over an analog channel as a two-level signal by modulating a pulse wave. Some schemes also allow digital transfer of the analog signal with a fixed bit rate. Pulse modulation includes analog-over-analog methods like PAM, PWM, and PPM as well as analog-over-digital methods like PCM, DPCM, ADPCM, DM, and delta-sigma modulation. Sampling is the reduction of a continuous signal to a discrete signal by taking values at points in time. The Nyquist-Shannon sampling theorem states that a bandlimited signal can be perfectly reconstructed from samples if the sampling rate is at least twice the highest frequency in the signal.

1. Pulse Modulation

2. • Pulse modulation schemes aim at transferring
a narrowband analog signal over an analog
baseband channel as a two-level signal by
modulating a pulse wave.

3. • Some pulse modulation schemes also allow the
narrowband analog signal to be transferred as a
digital signal (i.e. as a quantized
discrete-time signal) with a fixed bit rate, which
can be transferred over an underlying digital
transmission system, for example some line code.
These are not modulation schemes in the
conventional sense since they are not
channel coding schemes, but should be
considered as source coding schemes, and in
some cases analog-to-digital conversion
techniques.

4. Types
• Analog-over-analog methods:
• Pulse-amplitude modulation (PAM)
• Pulse-width modulation (PWM) AND Pulse-depth modulation
(PDM)
• Pulse-position modulation (PPM)
• Analog-over-digital methods:
• Pulse-code modulation (PCM)
– Differential PCM (DPCM)
– Adaptive DPCM (ADPCM)
• Delta modulation (DM or Δ-modulation)
• Delta-sigma modulation (∑Δ)
• Continuously variable slope delta modulation (CVSDM), also called
Adaptive-delta modulation (ADM)
• Pulse-density modulation (PDM)

5. Sampling Theory

6. Sampling process:
• In signal processing, sampling is the reduction
of a continuous signal to a discrete signal.
• A sample refers to a value or set of values at a
point in time and/or space.
• A theoretical ideal sampler produces samples
equivalent to the instantaneous value of the
continuous signal at the desired points.

7. 7
Periodic (Uniform) Sampling
• Sampling is a continuous to discrete-time conversion
• Most common sampling is periodic
• T is the sampling period in second
• fs = 1/T is the sampling frequency in Hz
• Sampling frequency in radian-per-second Ωs=2πfsrad/sec
• Use [.] for discrete-time and (.) for continuous time signals
• This is the ideal case not the practical but close enough
– In practice it is implement with an analog-to-digital converters
– We get digital signals that are quantized in amplitude and time
[ ] ( ) ∞<<∞−= nnTxnx c
-3 -2 2 3 4-1 10

8. 8
Periodic Sampling
• Sampling is, in general, not reversible
• Given a sampled signal one could fit infinite continuous signals through the samples
0
-1
20 40 60 80 100
-0.5
0
0.5
1
• Fundamental issue in digital signal processing
– If we loose information during sampling we cannot recover it
• Under certain conditions an analog signal can be sampled without
loss so that it can be reconstructed perfectly

9. Sampling rate
• The sampling rate, sample rate, or sampling
frequency (fs) defines the number of samples
per unit of time (usually seconds) taken from a
continuous signal to make a discrete signal.

10. sampling period
• The sampling period is the time difference
between two consecutive samples.
• It is the inverse of the sampling frequency.
• For example: if the sampling frequency is
44100 Hz, the sampling period is 1/44100 =
2.2675736961451248e-05 seconds: the
samples are spaced approximately 23
microseconds apart.

11. Instantaneous Sampling

12. Instantaneous sampling
• Consider an arbitrary signal g (t) of finite
energy, which is specified for all time.
Suppose that we sampled the signal g (t)
instantaneously and at a uniform rate, once
every Ts seconds. Consequently, we obtain an
infinite sequence of samples spaced Ts
seconds a part. W e refer to Ts as t he
sampling period, and to its reciprocal fs = 1 /Ts
as the sampling rate. This idle form of
sampling called instantaneous sampling

13. Sampling Theorem
Sampling of
Band-Limited Signals

14. Band-Limited Signals
Ω
Yc(jΩ)
Band-Limited
Band-Unlimited
Ω
Xc(jΩ)
ΩN−ΩN
1

15. Sampling of Band-Limited Signals
Band-Limited
T
kjjX
T
jX s
k
scs
π
=ΩΩ−Ω=Ω ∑
∞
−∞=
2
),(
1
)(
Ω
Xc(jΩ)
ΩN−ΩN
1
Ω
Ωs−Ωs 2Ωs 3Ωs
−2Ωs−3Ωs
S(jΩ)
2π/T
Ω
4Ωs−4Ωs
2Ωs 6Ωs
−2Ωs−6Ωs
S(jΩ)2π/T
Sampling with
Higher Frequency
Sampling with
Lower Frequency

16. Nyquist–Shannon sampling theorem
• Sampling is the process of converting a signal
(for example, a function of continuous time or
space) into a numeric sequence (a function of
discrete time or space).
• Shannon's version of the theorem states:
If a function x(t) contains no frequencies higher
than B hertz, it is completely determined by
giving its ordinates at a series of points
spaced 1/(2B) seconds apart.

17. • In other words, a bandlimited function can be perfectly
reconstructed from a countable sequence of samples if
the bandlimit, B, is no greater than ½ the sampling rate
(samples per second).
• The theorem also leads to a formula for reconstruction of
the original function from its samples.
• When the bandlimit is too high (or there is no bandlimit),
the reconstruction exhibits imperfections known as
aliasing.
• The Poisson summation formula provides a graphic
understanding of aliasing and an alternative derivation of
the theorem, using the perspective of the function's
Fourier transform.

18. Nyquist Theory
x(t) must contain no sinusoidal component at
exactly frequency B, or that B must be strictly
less than ½ the sample rate.

19. Recoverability
Band-Limited
T
kjjX
T
jX s
k
scs
π
=ΩΩ−Ω=Ω ∑
∞
−∞=
2
),(
1
)(
Ω
Xc(jΩ)
ΩN−ΩN
1
Ω
Ωs−Ωs 2Ωs 3Ωs
−2Ωs−3Ωs
S(jΩ)
2π/T
Ω
4Ωs−4Ωs
2Ωs 6Ωs
−2Ωs−6Ωs
S(jΩ)2π/T
Sampling with
Higher Frequency
Sampling with
Lower Frequency
Ωs > 2ΩN
Ωs > 2ΩN
Ωs < 2ΩN
Ωs < 2ΩN

20. Case 1: Ωs > 2ΩN T
kjjX
T
jX s
k
scs
π
=ΩΩ−Ω=Ω ∑
∞
−∞=
2
),(
1
)(
Ω
Xc(jΩ)
ΩN−ΩN
1
Ω
Ωs−Ωs 2Ωs 3Ωs
−2Ωs−3Ωs
S(jΩ)
2π/T
1/T
Ω
Ωs−Ωs 2Ωs 3Ωs
−2Ωs−3Ωs
Xs(jΩ)

21. Case 1: Ωs > 2ΩN T
kjjX
T
jX s
k
scs
π
=ΩΩ−Ω=Ω ∑
∞
−∞=
2
),(
1
)(
Ω
Xc(jΩ)
ΩN−ΩN
1
Ω
Ωs−Ωs 2Ωs 3Ωs
−2Ωs−3Ωs
S(jΩ)
2π/T
1/T
Ω
Ωs−Ωs 2Ωs 3Ωs
−2Ωs−3Ωs
Xs(jΩ)
Passing Xs(jΩ) through a low-
pass filter with cutoff
frequency ΩN < Ωc< Ωs− ΩN ,
the original signal can be
recovered.
Passing Xs(jΩ) through a low-
pass filter with cutoff
frequency ΩN < Ωc< Ωs− ΩN ,
the original signal can be
recovered.
Xs(jΩ) is a periodic
function with
period Ωs.
Xs(jΩ) is a periodic
function with
period Ωs.

22. Case 2: Ωs < 2ΩN
T
kjjX
T
jX s
k
scs
π
=ΩΩ−Ω=Ω ∑
∞
−∞=
2
),(
1
)(
Ω
Xc(jΩ)
ΩN−ΩN
1
1/T
Ω
2Ωs−2Ωs 4Ωs 6Ωs
−4Ωs−6Ωs
S(jΩ)2π/T
Ω
2Ωs−2Ωs 4Ωs 6Ωs
−4Ωs−6Ωs
Xs(jΩ)

23. Case 2: Ωs < 2ΩN
T
kjjX
T
jX s
k
scs
π
=ΩΩ−Ω=Ω ∑
∞
−∞=
2
),(
1
)(
Ω
Xc(jΩ)
ΩN−ΩN
1
1/T
Ω
2Ωs−2Ωs 4Ωs 6Ωs
−4Ωs−6Ωs
S(jΩ)2π/T
Ω
2Ωs−2Ωs 4Ωs 6Ωs
−4Ωs−6Ωs
Xs(jΩ)
AliasingAliasing
No way to recover
the original signal.
No way to recover
the original signal.
Xs(jΩ) is a periodic
function with
period Ωs.
Xs(jΩ) is a periodic
function with
period Ωs.

24. Nequist Rate
Ω
Xc(jΩ)
ΩN−ΩN
1Band-Limited
Nequist frequency (ΩN)
The highest frequency of a band-limited signal
Nequist rate = 2ΩN

25. Nequist Sampling Theorem
Ω
Xc(jΩ)
ΩN−ΩN
1Band-Limited
Ωs > 2ΩN
Ωs < 2ΩN
Recoverable
Aliasing

26. Aliasing
• The Poisson summation formula shows that the
samples, x(nT), of function x(t) are sufficient to create
a periodic summation of function X(f).
• If the Nyquist criterion is not satisfied, adjacent copies
overlap, and it is not possible in general to discern an
unambiguous X(f). Any frequency component above
fs/2 is indistinguishable from a lower-frequency
component, called an alias, associated with one of the
copies. In such cases, the reconstruction technique
described below produces the alias, rather than the
original component.

27. The samples of several different sine waves can be identical, when at least one
of them is at a frequency above half the sample rate.

28. Mathematical Problems will be
provided later.

29. ©2000, John Wiley & Sons, Inc.
Haykin/Communication Systems, 4th Ed
Figure 3.1
The sampling process. (a) Analog signal. (b)
Instantaneously sampled version of the
analog signal.

30. ©2000, John Wiley & Sons, Inc.
Haykin/Communication Systems, 4th Ed
Figure 3.2
(a) Spectrum of a strictly band-limited signal
g(t). (b) Spectrum of the sampled version of g(t)
for a sampling period Ts = 1/2 W.

31. ©2000, John Wiley & Sons, Inc.
Haykin/Communication Systems, 4th Ed
Figure 3.5
Flat-top samples, representing an analog
signal.

32. ©2000, John Wiley & Sons, Inc.
Haykin/Communication Systems, 4th Ed
Figure 3.8
Illustrating two
different forms of
pulse-time modulation
for the case of a
sinusoidal modulating
wave.
(a) Modulating wave.
(b) Pulse carrier.
(c) PDM wave.
(d) PPM wave.

33. ©2000, John Wiley & Sons, Inc.
Haykin/Communication Systems, 4th Ed
Figure 3.10
Two types of quantization: (a) midtread and (b) midrise.

34. ©2000, John Wiley & Sons, Inc.
Haykin/Communication Systems, 4th Ed
Figure 3.11
Illustration of the quantization process. (Adapted from
Bennett, 1948, with permission of AT&T.)