Sampling and Quantization - Digital Speech: Coding for Low Bit Rate Communication Systems - page 23

The sampling theorem states that if a signal s a (t) has a band-limited Fourier
transform S a (jω) given by,

∞

s a (t)e − jωt dt S a (jω) = (3.2)

−∞

such that S a (jω) 2 πW then the analogue signal can be recon-

structed from its sampled version if T

= 0for | ω |≥

≤

1 / 2 W . W is called the Nyquist

frequency .
The effect of sampling is shown in Figure 3.1. As can be seen from
Figures 3.1b and 3.1c, the band-limited Fourier transform of the analogue
signal which is shown in Figure 3.1a is duplicated at every multiple of the
sampling frequency.
This is because the Fourier transform of the sampled signal is evaluated at
multiples of the sampling frequency which forms the relationship,

∞

1
T

S(e jωT )

= +

S a (jω j 2 πn/T) (3.3)

n =−∞

This can also be interpreted by looking into the time domain sampling process
where the input signal is regularly (at every sampling interval) multiplied

Magnitude

Analog Signal

(a)

− w w

Frequency

Magnitude

Over Sampled Signal

(b)

−w w

fs 2fs

Frequency

Magnitude

Under Sampled Signal

(c)

−w w

fs 2fs

Frequency

Figure 3.1 Effects of sampling: (a) original signal spectrum, (b) over sampled signal
spectrum and (c) under sampled signal spectrum