Sampling and Quantization - Digital Speech: Coding for Low Bit Rate Communication Systems - page 24

with a delta function. When converted to the frequency domain, the multi-
plication becomes convolution and the message spectrum is reproduced at
multiples of the sampling frequency.
We can clearly see that if the sampling frequency is less than twice the
Nyquist frequency, the spectra of two adjacent multiples of the sampling
frequencies will overlap. For example, if

1
T

f s < 2 W the analogue signal
image centred at 2 π/T overlaps into the base band image. The distortion
caused by high frequencies overlapping low frequencies is called aliasing .In
order to avoid aliasing distortion, either the input analogue signal has to be
band-limited to a maximum of half the sampling frequency or the sampling
frequency has to be increased to at least twice the highest frequency in the
analogue signal.
Given the condition 1 /T > 2 W , the Fourier transform of the sampled
sequence is proportional to the Fourier transform of the analogue signal in
the base band as follows:

=

1
T S a (jω)

< π
T

S(e jωT )

= | |

ω (3.4)

Using the above relationship, the original analogue signal can be obtained
from the sampled sequence using interpolation given by [1],

∞

s a (nT) sin [ π(t

− nT)/T ]

s a (t) = (3.5)

π(t − nT)/T

=−∞

n

which can be written as,

∞

s a (t) =

s a (nT)sinc(φ) (3.6)

n =−∞

where φ = π(t nT)/T .

Therefore, if the sampling frequency is at least twice the Nyquist frequency,
the analogue signal can be recovered completely from its sampled version
by adding together sinc functions centred on each sampling point and scaled
by the sampled value of the analogue signal. The sinc(φ) function in the
above equation represents an ideal low pass filter. In practice, the front
end band limitation before sampling is usually achieved by a low pass
filter which is less than ideal and may cause aliasing distortion due to its
roll-off characteristics. In order to avoid aliasing distortion, the sampling
frequency is usually chosen to be higher than twice the Nyquist frequency.
In telecommunication networks the analogue speech signal is band-limited
to 300 to 3400Hz and sampled at 8000Hz. This same band limitation and
sampling is used throughout this topic unless otherwise specified.

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