Multirate Digital Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 210

5.2.1 The Down-Sampler and the Up-Sampler

A reduction in the sampling rate by a factor of M (where M is a positive integer) is
achieved by retaining every Mth sample and discarding the other samples. The
device that performs this operation is called a down-Sampler, and the output of this
downsampler is a sequence whose sampling rate is 1/M times that of the input
sequence. Figure 5.1 shows a block diagram of the down-sampler. Its operation in
the time-domain can be described mathematically as

y ð n Þ¼ x ð nM Þ: ð 5 : 1 Þ

According to the above equation, all input samples with indices equal to integer
multiple of M are kept, while all others are discarded.
MATLAB: down-sampling can be achieved using either the ''downsample''
command, or equivalently, the vector command: y ¼ x ð 1 : M : length ð x ÞÞ:
Figure 5.2 shows a sinusoid with frequency = 0.0356 Hz downsampled by a factor
of M = 3. It is obvious that the sampling interval T s of the downsampled signal
(Fig. 5.2 b) is M = 3 times larger than the the original signal period, T s .
An increase in the sampling rate by a factor of L (where L is a positive integer)
is achieved by inserting L - 1 zero samples between each of the existing input
signal samples. Mathematically, up-sampling can be expressed as

y ð n Þ¼ x ð n = L Þ;

n ¼ 0 ; L ; 2L ; ...

ð 5 : 2 Þ

0 ; otherwise

From ( 5.2 ) it is apparent that the number of samples in the upsampled signal is
L times the number of input signal samples. Figure 5.3 shows the block diagram of
an up-sampler.
MATLAB: Up-sampling (expanding) can be implemented in MATLAB using
the command ''up-sample'', or alternatively, by using the following vector com-
mand scripts:

y ¼ zeros ð 1 ; L length ð x ÞÞ ;
y ð 1 : L : length ð y ÞÞ ¼ x;

Fig. 5.1 Block diagram of a
down-sampler

M

y ( n )

x ( nM )

x ( n )

=

f 1

1

^

s

f =

f

==

s

M MT

T

s