1. Sampling Theorem
Dr. G.Aarthi,
Associate Professor, School of Electronics Engineering

2. INTRODUCTION
• Modulation is the process of frequency translation in which any
one parameter(Amplitude, frequency or phase) of high
frequency carrier signal is varied in accordance with
instantaneous value of low frequency modulating signal.
• Modulation is either analog or digital.
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3. Why Analog to Digital Transmission
•Analog transmission
• Transmission method of conveying voice, data, image, signal or
video information using a continuous signal.
• It could be the transfer of an analog source signal using an analog
modulation method such as FM or AM, or no modulation at all.
• Disadvantages
• High signal-to-noise ratio is required.
• In long distances, high output systems, analog is unattractive due
to attenuation problems .
• The effects of random noise can make signal loss and distortion
impossible to recover .

4. Digital Transmission
• Less Power needed to transmit over the same channel.
• Transmit longer distances.
• Compatibility with other digital systems
• A digital signal is superior to an analog signal because it is
more robust to noise .
• Easily be recovered, corrected and amplified.
• For this reason, the tendency today is to change an analog
signal to digital data

5. The Sampling Theorem
Impulse sampling of an analog voltage.

6. The Sampling Theorem
• A sampler is a mixer with a train of very narrow pulses as the
local oscillator input.
• If the analog input is sampled instantaneously at regular
intervals at a rate that is at least twice the highest analog
frequency
fs > 2fa(max)
• then the samples contain all of the information of the original
signal.

7. The Sampling Theorem
• The analog signal v(t) has a signal spectrum represented
by the Fourier transform V(f),
and the sampling signal
consists of instantaneous impulses every nTs sec, where n
= 0, +1, +2, …
• The Fourier transform of s(t) is
   






n
s
nT
t
t
s 
   
s
n
s
nf
f
T
f
S 
 




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8. The Sampling Theorem
• The time-domain product performed by the sampler
produces a sampled output spectrum given by
• where this spectrum consists of replicas of the analog
signal spectrum V(f), translated in frequency by each of
the sampling frequency harmonics.
   
s
n
s
s nf
f
V
T
f
V 
 



1

9. The Sampling Theorem
• The sampler is a wideband (harmonic) mixer producing
upper and lower sidebands at each harmonic of the sampling
frequency.
• Figure 1-a illustrates the correct way to sample: if sampling
is done at fs > 2fA(max) the upper and lower sidebands do not
overlap each other
• and the original information can be recovered by passing the
signal through a low-pass filter (see Figure 1c and d).

10. Figure 1. Sample spectra and their outputs. (a) fs > 2fA(max) Nyquist
criteria met. (b) fs < 2fA(max) Frequency foldover of “aliasing”
distortion occurs. (c) fs > 2fA(max) and recovery of original information
with low-pass filter. (d) The original analog signal spectrum following
recovery as in (c).

11. The Sampling Theorem
•However, if the sampling rate is less than the
Nyquist rate, fs < 2fA(max) the sidebands overlap, as
shown in Figure 1b.
•The result is frequency-folding or aliasing distortion,
which makes it impossible to recover the original
signal without distortion.

12. Sampling
 Analog signal is sampled every TS secs.
 Ts is referred to as the sampling interval.
 fs = 1/Ts is called the sampling rate or sampling frequency.
 There are 3 sampling methods:
 Ideal - an impulse at each sampling instant
 Natural - a pulse of short width with varying amplitude
 Flat top - sample and hold, like natural but with single
amplitude value
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13. Three different sampling methods for PCM
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