Fundamentals of Digital Image Processing - Control of Color Imaging Systems: Analysis and Design - page 71

y ( n )

H ( j ω) Q g ( n )

f ( n )

FIGURE 2.35 Downsampling by a factor of Q.

2.7.4 S AMPLING R ATE C ONVERSION BY A F ACTOR OF Q
To change the sampling rate by a factor of Q ,we first upsample the signal by a factor

of P and then downsample it by a factor of Q, as shown in Figure 2.36.
The two LPFs can be combined into one

¼ P
and a bandwidth given by the minimum of P and Q . The overall system is shown in
Figure 2.37. In this

filter with a passband gain of P 1

figure, the LPF bandwidth is given by

P

, Q

BW¼ min ( 2 : 116 )

2.7.5 E XAMPLES OF L OW -P ASS F ILTERS U SED FOR S AMPLING R ATE C ONVERSION
An ideal LPF is not realizable, so in practice it is approximated by a simple realizable
FIR

filter. For example, the following simple filters can be used for upsampling.

These filters are derived using a linear interpolation algorithm. For example, for

upsampling by a factor of P, the LPF would be an FIR filter of size 2P 1, given by

1
P

2
P

P 1 P 1

P

2
P

1
P

h ( n ) ¼ 1 ( 2 : 117 )

P

The DC gain of this filter is

X X

DC gain ¼ H ( j v)j v¼ 0 ¼

h ( n ) exp ( j v)j v¼ 0 ¼ h ( n ) ¼ P

( : )

2 118

n n

g ( n )

H 1 ( j ω) H 2 ( j ω) Q

P

f ( n )

FIGURE 2.36 Sampling rate conversion by a factor of Q .

H (ω)

P

g ( n )

Q

f ( n ) P

ω

- BW

BW

FIGURE 2.37 Sampling rate conversion by factor of a Q .