TE331: Principles of Analogue

Telecommunications

Lecture #8
Pulse Modulation

January 19 TE331: PRINCIPLES OF ANALOGUE TELECOMMUNICATIONS 1

 Contents

 Sampling Theory

 Pulse Amplitude Modulation (PAM)

 Pulse Width Modulation (PWM)

 Pulse Position Modulation (PPM)

 pulse Code Modulation (PCM)

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 Sampling Theory
 The process of transforming an analog info
into a form compatible with DCS starts with
sampling
 Sampling is the process of converting
continuous-time analog signal into a discrete-
time signal by taking the samples at discrete
time intervals
 Sampling analog signals makes them discrete
in time but still continuous valued

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 Sampling Theory
Original
Analog signal

Sampled
signal

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 Sampling Theory
 Let the spectrum of a signal 𝑥(𝑡) be strictly band-
limited to 𝑓𝑚 𝐻𝑧 as shown

A signal of finite energy band-limited can be

uniquely determined from its values sampled at
uniform intervals
 sin 2f m (t  nTs )
x(t )   x(nTs )
n 2f m (t  nTs )
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 Sampling Theory
The sinc function

 sin 2πf m (t  nTs ) 

x (t )   x ( nTs )   x ( nTs )sinc( 2 f m (t  nTs )
n   2πf m (t  nTs ) n 

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 Sampling Theorem
 Sampling theorem states that: a band limited signal
which has no frequency components higher than 𝑓𝑚
can be recovered completely from a set of samples
taken at a rate greater or equal to twice of 𝑓𝑚 samples
per second
 Nyquist criterion: if sampling is done at proper rate, no
information is lost about the original signal. minimum
sampling rate is called Nyquist rate (2𝑓𝑚 )
 Sampling rate (sampling frequency 𝑓𝑠 ): rate at which
the signal is sampled, expressed as the number of
samples per second.

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 Sampling Theorem
 Sampling interval (𝑇𝑠 ): the time that separates
sampling points. If the signal is slowly varying,
then fewer samples per second will be required
than if the signal is rapidly varying
 Oversampling: too many samples per second
 Undersampling: sampling at too low rate (below
Nyquist rate)
 Aliasing: phenomenon (distortion of the original
signal) occurring as a result of undersampling

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 Instantaneous (Ideal) Sampling
 Also known as impulse sampling
 Consider a signal 𝑥𝑠 𝑡 obtained by
instantaneous sampling a signal 𝑥 𝑡 at a
periodic interval 𝑇𝑠

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 Instantaneous Sampling
 Theinstantaneous sampled signal 𝑥𝑠 𝑡 can be
obtained by taking the product of the signal 𝑥 𝑡
with a periodic train of impulse function 𝛿𝑇𝑠 (𝑡)
(Comb/switching function)

Train of impulse function selects values at specific

intervals
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 Instantaneous Sampling
 The instantaneous sampled signal can be
expressed in time domain by
 
xs (t )  x(t ) T (t )  x(t )  (t  nTs )   x(nTs ) (t  nTs )
n n
s

 In frequency domain, taking Fourier transform

(frequency convolution) we have
1  1 
X s ( f )  X ( f )  ( f  nf s )   X ( f  nf s )
Ts n Ts n

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 Instantaneous Sampling
 Alternatively,𝑋𝑠 (𝑓) can be derived using the
time-shift property of Fourier transform i.e.
1 
X s ( f )   X ( nTs )exp(  j 2fnTs )
Ts n

 Note that; 𝑋𝑠 (𝑓)

– is a continuous spectrum
– repeats at a rate equals to sampling rate 𝑓𝑠

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 Instantaneous Sampling

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 Natural Sampling
 This
is a more practical way of accomplishing the
sampling of bandlimited analog signal
 In Practice it is not possible to create a train of
impulses, thus non ideal approach to sampling
must be used where we can approximate a train
of impulses using a train of very thin rectangular
pulses
 Ifwe multiply 𝑥(𝑡) by a series of rectangular
pulses 𝑝(𝑡) we obtain a gated waveform that
approximates the ideal sampled waveform,
known as natural sampling
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 Natural Sampling

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 Natural Sampling

 Theswitching signal p(t )  h(t  nTs ) is called a
n
periodic rectangular pulse train.

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 Natural Sampling
A natural sampled signal can be expressed in time
domain by (from practical switching circuits)
 1 0  t 
xs (t )  x(t )  h(t  nTs ) h (t )  
n 0 otherwise
 Complex exponential Fourier series of the
switching signal is expresses as

p(t )  cn exp( j 2nf s t )
n
 Thus

xs (t )  x(t ) cn exp( j 2nf s t )
n

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 Natural Sampling
 From the frequency-shifting property, Fourier
transform of the sampled signal becomes

X s ( f )  cn X ( f  nf s )
n

 Note that; 𝑋𝑠 (𝑓)

– Consists of an infinite number of copies of 𝑋(𝑓)
shifted every 𝑓𝑠 𝐻𝑧
– The 𝑛𝑡ℎ copy is scaled by 𝑐𝑛 , and 𝑐0 = 1

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 Natural Sampling

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 Natural Sampling
 Reconstruction of the original signal from the
sampled signal is implemented by means of a
low-pass filter (LPF) called reconstruction filter
 The problem with a natural sampled waveform is
that
– The tops of the sample pulses are not flat
– It is not compatible with the digital system since the
amplitude of each sample has infinity number of
possible values
A technique used to alleviate this problem is flat
top sampling
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 Natural Sampling
a2
a1

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 Flat-Top Sampling
 Flat-top sampling of a signal is obtained by
instantaneous sampling at every sampling
period 𝑇𝑠 and holding the sample value for
duration of 𝑇 𝑠𝑒𝑐 (sample-and-hold circuit)

 InS/H, input signal is continuously sampled

and the value is held for as long as it takes for
A/D to acquire its value

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 Flat-Top Sampling
Natural
Sampling

Flat-top
Sampling

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 Flat-Top Sampling
 The flat-top sampled signal is defined by

 
x s (t )  g (t ) x (t )  (t  nTs ) ,
n
 1 0  t  T
g (t )  
0 otherwise

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 Flat-Top Sampling
 Taking the Fourier Transform of 𝑥𝑠 (𝑡)

X s ( f )  G ( f ) F x (t ) δ (t  nTs )

 n 
 1  
 G ( f )  X ( f ) δ ( f  nf s )
 Ts n 
1 
 G ( f )  X ( f  nf s ), where G ( f )  Tsinc( fT )exp(  jπfT )
Ts n
𝐺 𝑓 is a sinc function
 Flat top sampling becomes identical to ideal
sampling as the width of the pulses becomes
shorter
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 Aliasing Effects
 Aliasingeffects can easily be illustrated in the
frequency domain. Consider a baseband
spectrum in the figure

 Ifthe signal is under-sampled, the resulting

sampled signal will contain overlapping region

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 Aliasing Effects
Ways of eliminating aliasing effects
1. Sampling at higher sampling rate
2. Using antialiasing filters
– Pre-filtering: Signal is filtered prior to sampling to limit
𝑓𝑚 to 𝑓𝑠 2 or less
– Post-filtering: Appling LPF with cut-off frequency less than
𝑓𝑠 − 𝑓𝑚 on the sampled signal
 Note: Pre-filtering is considered good engineering
practice while post-filtering can result in loss of some
signal information

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 Practical Considerations
 Practicalsignals are time-limited which implies
that they are not band-limited.
 To avoid aliasing, an anti-aliasing (pre-filter)
low-pass filter processes a signal with cut-off
frequency equal to half the Nyquist rate.
 Realizablefilters require a nonzero transition
bandwidth which implies that sampling rate
must be much larger than baseband signal
bandwidth
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 Practical Considerations
 To minimize the sampling rate, which implies lower
transmission rates and less storage memory, small
transition bandwidth of filters is desired.
 A good engineering balance is to allow a transition
bandwidth 20% of the baseband signal bandwidth such that
f s  2.2 f m
 A sampling rate of 44,100 samples/s is used for a high
quality compact disc (CD) digital audio system for a music
signal with bandwidth of 20 kHz.
 A sampling rate of 8000 samples/s is used for digital
telephone systems for telephone quality speech with
bandwidth of 4 kHz.

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 Analog Pulse Modulation
Recall!
 Two main types of analog modulation schemes:
– Continuous wave (CW) modulation
– Pulse modulation
 In CW modulation, some parameter of a
sinusoidal carrier signal is varied continuously in
accordance with the message signal
 In pulse modulation, some parameter of a
periodic pulse train is varied in accordance with
the message signal.

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 Analog Pulse Modulation
Three basic types of Analog pulse modulation
are;
 Pulse amplitude modulation (PAM) – width
fixed, amplitude varies.
 Pulse position modulation (PPM) – width fixed,
position varies
 Pulse width modulation (PWM) – position
fixed, width varies

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 Analog Pulse Modulation

Message Signal

PAM

PWM

PPM
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 Pulse Amplitude Modulation
 In PAM,
– The carrier signal consists of a periodic train of
rectangular pulses
– The amplitudes of rectangular pulses vary with the
instantaneous sample values of analog message
signal
– Carrier frequency is the same as sampling
frequency
 The definition (expression) of PAM signal is
similar to flat-topped sampling

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 Pulse Amplitude Modulation
 PAM signal is obtained by sampling the
message signal at every sampling period 𝑇𝑠
and holding the sample value for duration of
𝑇 𝑠𝑒𝑐
 The PAM signal is expressed as

s(t )  m( nTs ) g (t  nTs )
n 1 0  t  T
g (t )  
 
 g (t ) m(t )  (t  nTs )
n
 0 otherwise

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 Pulse Amplitude Modulation

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 Types of PAM
 The PAM signal are of two types;
– Double-polarity PAM – pulses are both
positive and negative
– Single - polarity PAM – all the pulses are
positive.
 In Single polarity, a DC bias is added to the
signal to make the pulses positive

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 PAM – Demodulation
 Demodulation of the PAM signal is
accomplished by the reconstruction LPF with
cut-off frequency equal to the bandwidth of
the message signal
A dc block is required to remove the spectral
impulse at 0

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 PAM – Demodulation
 The reconstructed signal (LPF output) is
processed by an equalizer to minimize the
aperture effect caused by 𝐺(𝑓)

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 PAM – Demodulation
 Frequency response of the equalizer is given
by
1 f
H eq ( f )  
Tsinc(Tf ) sin(fT )

 Adequate pulse resolution requires that;

1
BT   B
2T

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 PAM – Applications
PAM is used for efficient transmission of data in
terms of pulses rather than continuous
The applications of PAM includes;
 In Ethernet communication standard
– 100BASE-T2 using 5 level PAM modulations
– IEEE 802.3an standard for 10GBase-T
 Control of Light-Emitting Diodes (LEDs)
 In micro-controllers for generating control signals
 Digital television

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 PAM – Advantages
 Provides a means for converting an analog
signal to pulse code modulation (PCM) for
digital transmission.
 Provides a means for breaking an analog signal
into different timeslots for interleaving signals
from different sources over a common channel
(Time-Division Multiplexing)

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 PAM – Disadvantages
 Requiresvery wide bandwidth due to narrow
pulse width. (Transmission bandwidth is much
wider than the message signal)
 Noise performance is never better than that
achieved by base-band transmission of the
message signal
 Variations of amplitude of PAM signal causes
variations in the peak power required by the
transmitter with modulating signal.

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 Pulse Amplitude Modulation
Advantages of PAM
 Provides a means for converting an analog
signal to pulse code modulation (PCM) for
digital transmission.
 Provides a means for breaking an analog signal
into different timeslots for interleaving signals
from different sources over a common channel
(Time-Division Multiplexing)

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 Pulse Amplitude Modulation
Disadvantages of PAM
 Requires very wide bandwidth due to narrow
pulse width. (Transmission bandwidth is much
wider than the message signal)

 Noiseperformance is never better than that

achieved by base-band transmission of the
message signal

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 Pulse Width Modulation
 PWM performs sampling in time instead of
sampling in amplitude as in PAM.

 The information is coded into the pulse time

position within each switching interval.

 PWMis more often used for control than for

communication

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 Pulse Width Modulation
 PWM output is generated by a sawtooth signal
gating the input with varying pulse width

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 Pulse Width Modulation
Disadvantage
 The bandwidth requirements for PWM are
typically close to an order of magnitude higher
than PAM.
Advantage
 Simplifications in the switching power stage

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 Pulse Position Modulation
 PPM differs from PWM in that the value of
each instantaneous sample of a modulating
wave is caused to vary the position in time of a
pulse

 Eachpulse has identical shape independent of

the modulation depth.

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 Pulse Position Modulation
 The value of the signaldetermines the delay of
the pulse from the clock.

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 Pulse Position Modulation
Advantage
 The uniform pulse is simple to reproduce with a
simple switching power stage
Disadvantage
 The required power supply level of the switching
power stage is much higher than the required
load voltage.
 This affects performance on other parameters as
efficiency, complexity and audio performance.
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 Pulse Code Modulation
 Analog Pulse modulations (PAM, PWM, PPM)
represent analog signals by analog variations
in pulses
 In PCM, a signal value is represented by a
sequence of pulses (digits). PCM uses only two
pulse values, which represent 0 and 1.
 Width and spacing of pulses is constant. Value
of pulse is chosen from a small number of
values.
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 Pulse Code Modulation
 PCM is a method of converting an analog into
digital signals
 implies PAM - quantization by time and
quantization by amplitude
 After sampling process, the digital
representation of a signal requires: -
– Quantization of the amplitude of a sampled signal
– Encoding of each quantized sample value

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 Pulse Code Modulation
 The value that a signal has in certain time is
called a sample.
 Quantization by amplitude means that
according to the amplitude of sample one
quantization segment is chosen
 Encoding process transforms the recorded
quantization segments into code words
(binary numbers).

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 Pulse Code Modulation

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