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476 Chapter 9 SAMPLING RATE CONVERSION
(
Resampled analog signal: x a m T )
= ∑ 2
k
x a (kT) sin [ π ( m T 2 − kT) /T ]
π ( m T 2 − kT) /T
= ∑ k
x a (kT) sin [ π ( m
2 − k)]
π ( m
2 − k) (9.3)
resulting in high-rate discrete signal: y(m) △ =x a
(
m T 2
)
(9.4)
In this formulation of ideal interpolation, the discrete signal was converted
to the analog signal and then back to the discrete signal at twice the rate.
In the subsequent sections we will study how to avoid this roundabout
approach and perform sampling rate conversion completely in the digital
domain.
The process of sampling rate conversion in the digital domain can
be viewed as a linear filtering operation, as illustrated in Figure 9.2a.
The input signal x(n) ischaracterized by the sampling rate F x =1/T x ,
and the output signal y(m) ischaracterized by the sampling rate F y =
1/T y , where T x and T y are the corresponding sampling intervals. In our
treatment, the ratio F y /F x is constrained to be rational
F y
F x
= I D
(9.5)
where D and I are relatively prime integers. We shall show that the
linear filter is characterized by a time-variant impulse response, denoted
x(n)
Rate F = 1 x T y
Linear Filter
h(n, m)
(a)
y(m)
F y = 1 T y
y(m)
y(m + 1)
τ i
x(n)
FIGURE 9.2
x(n + 2) x(n + 3) x(n + 4) x(n + 5)
x(n + 1)
y(m + 2)
y(m + 5) y(m + 6)
y(m + 3) y(m + 4)
(b)
Sampling rate conversion viewed as a linear filtering process
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