Communication Systems 1

Batch 07, Spring 2010

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 Signals (Review)
• Continuous-time analog signals
▫ Exists for every instant of time (Continuous-time)
▫ The amplitude may take any real value (analog)

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 Signals
• Continuous-time digital signals
▫ Exists for every time instant
▫ Amplitude takes only some finite values

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 Signals
• Discrete-time analog signals
▫ Exists for specific time intervals
▫ Amplitude may take any value

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 Signals
• Discrete-time digital signals
▫ Exists for specific time intervals
▫ Amplitude also may take specific values

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 Digital Communication
• We have to process the digital signals
• Advantages
▫ Ease of processing, ease of algorithm implementation,
better regeneration
• AM, FM,PM
▫ Communication and modulation schemes used for
continuous-time analog signal
• For digital communication, we have to convert analog
signals to digital signals
▫ The first step of the conversion is sampling
▫ The second step is the quantization
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 Sampling
• The process of converting continuous-time
analog signals to discrete-time analog signals is
called sampling
▫ This process is followed by Quantization which
converts discrete-time analog signal to discrete-
time digital signal
• Analog signal is converted to digital for
modulation/transmission
• At the receiver, the digital signal is converted
back to analog
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 Sampling theory
• The samples of the signals are taken at appropriate
intervals
• The signal can be reconstructed from the sampled
waveform using low pass filter
• In order to ensure a good reconstruction, the
original signal must be sampled at an appropriate
rate as defined in the Nyquist sampling theorem
• A real-valued band-limited signal having no spectral
components above a frequency of B Hz is determined
uniquely by its values at uniform intervals spaced no
 1 
greater
  than
 2B 
seconds apart.
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 Sampling block diagram
• Consider a band-limited signal f(t) having no
spectral component above B Hz.
• Let each rectangular sampling pulse have unit
amplitudes, occurring at interval of T seconds.

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 Impulse Sampling
• The simplest method to sample a signal is to
multiply it by a delta function (impulse)

Sampled waveform
Signal waveform

0
1 201
0
1 201

Impulse sampler

0
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1 Communication Systems Engineering
201 Department
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 Increased sampling time T
Sampled waveform Sampled waveform

0 0
1 201 1 201

Sampled waveform Sampled waveform

0 0
1 201 1 201

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 Math
• g(t) is continuous-time signal with a bandwidth
of B Hz
▫ If we multiply it with a delta function
▫ The output is given by

▫ Zero everywhere and a delta function at time to,

with amplitude equal to g(t) at to
• This represents only getting one sample of g(t)
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 Cont..
• If we want to sample g(t) at every Ts seconds,
then we have to repeat this process periodically
• We need a train of delta functions that occurs
every Ts seconds

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 Cont..
• Sampled signal is given by

• So, the sampled signal is equal to the sum of the

delta functions having magnitude equal to g(t) at
the time when delta function occours
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Institute of Space Technology Communication Systems Engineering Department
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 Spectrum of the sampled signal
• Spectrum of g(t)

• Spectrum of the sampled signal can be found out by

taking the Fourier series of the delta pulse train
(periodic signal)
• Then takes the Fourier transform independently
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Institute of Space Technology Communication Systems Engineering Department
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 Recovery – Digital to analog
• Using low pass filter

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 Aliasing
• If the signal was sampled at a sampling rate of
lower than 2 times the bandwidth of the signal
( called Nyquist sampling rate), the different
frequency components can interfere with each
other
▫ It is impossible to recover the original signal
▫ Also called under sampling

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 Aliasing

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 Aliasing in time domain
• Frequency of signals = 500 Hz, Sampling frequency =
2000Hz

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 • Frequency of signals = 1100 Hz, Sampling frequency =
2000Hz

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 • Frequency of signals = 1500 Hz, Sampling frequency =
2000Hz

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 Example: Aliasing of Sinusoidal Signals EE 541/451 Fall 2006

Frequency of signals = 1800 Hz, Sampling frequency =

2000Hz

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 • Frequency of signals = 2200 Hz, Sampling frequency =
2000Hz

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 Avoid aliasing
• If a signal is time limited, it is not band-limited
• If a signal is band-limited, it is not time-limited
• In real world, all the signals are time limited, so they
are not band limited
▫ So they cannot fulfill the nyquist criteria
▫ Aliasing always occurs
• Soloution
▫ Low-pass filter the signal before samplin
▫ Filter has half the bandwidth of the sampling frequency
▫ Such a low pass filter is called anti-aliasing filter

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 Avoid aliasing
• Band-limiting signals (by filtering) before
sampling.
• Sampling at a rate that is greater than the
Nyquist rate
Anti-aliasing A/D fs(t)
f(t)
filter conversion

Sampling

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 Real Sampling
• Sampling with rectangular waves instead of
pulses
Signal waveform Sampled waveform

0
0
1 201 401 601 801 1001 1201 1401 1601 1801 2001 1 201 401 601 801 1001 1201 1401 1601 1801 2001

Natural sampler

0
1 201 401 601 801 1001 1201 1401 1601 1801 2001

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 PAM (Pulse Amplitude Modulation)

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 Pulse Width Modulation (PWM)

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 Pulse Position Modulation (PPM)

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 Pulse Code Modulation (PCM)

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