Waveform Encoding - Digital Communication: Principles and System Modelling - page 17

taking

F
F S
From Eq. (4.1) and (4.2), the correspondence of sampling is shown. The sequence
formed as S(n) and the input signal S(t) to the sampler, are correspondent.
Looking closely regarding the units of the different variations of frequencies,
generated as a bi-product of the process of sampling, we have the following derived
units.

f =

Unit of F cycles/s

Unit of F S =

=

samples/s

Therefore,

Unit of digital frequency, f cycles/sample

Unit of analog angular frequency,

=

, i.e., 2 π F = rad.cycles/s

Unit of digital angular frequency ω = rad/samples.

Here, if one decrease the spacing between the samples (T S ), i.e., increase the
rate of sampling F S , the reconstruction would be easier at the receiver side. But,
the bandwidth of transmission would be increased thereafter, it will affect the pro-
cessing time of the sampled signal. So, there is a trade-off between noise and
transmission bandwidth. However, it's really important to know the lower limit of
the choice of the sampling frequency for successful reconstruction.

2.2.1.1 Sampling Theorem
An analog signal can be reconstructed from its sampled values un-erroneously, if the
sampling frequency is at least twice the bandwidth of the analog signal.

Say an analog signal m(t) has three different frequency components f 1 ,f 2 , and f 3 ,or
combination of all of them, where f 1 <f 2 <f 3 then the bandwidth of that m(t) signal
must be f 3 . For ease of calculation, we are taking bandwidth of that signal B Hz, in
general.
Therefore we can say the spectrum M(

ω π ω

scale,
and band limited by B Hz in f scale. Now, sampled signal is nothing but the signal
obtained by multiplying m(t) by unit impulse train

) is band limited by 2 Bin

nTs (t). From the figure the

sampled signal is g(t) = m(t) × nTs (t). Time period of the impulse train is T s .

therefore frequency is f s = 1/ T s . Let's now expand the nTs (t) signal in Fourier

series so that we can study the spectrum of g(t) signal. As nT S ( t ) is even function

of time, By Fourier series,

∞

nT S ( t ) = a 0 +

1 (

a n cos n ω s t + b n sin n ω s t ) (2.3)

n =