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488 Chapter 9 SAMPLING RATE CONVERSION
|X(ω x )| |V(ω x )|
ω x
−π 0 π
0
− 3π − 2π − π
I I I
FIGURE 9.10
Spectra of x(n) and v(m) where V (ω y)=X(ω yI)
π
I
2π
I
3π
I
ω y = ω x
I
The spectra X(ω x ) and V (ω y ) are illustrated in Figure 9.10. We observe
that the sampling rate increase, obtained by the addition of I − 1 zero
samples between successive values of x(n), results in a signal whose spectrum
V (ω y )isanI-fold periodic repetition of the input signal spectrum
X(ω x ).
9.3.2 THE IDEAL INTERPOLATOR
Since only the frequency components of x(n) inthe range 0 ≤ ω y ≤
π/I are unique, the images of X(ω) above ω y = π/I should be rejected
by passing the sequence v(m) through a lowpass filter with a frequency
response H I (ω y ) that ideally has the characteristic
H I (ω y )=
{
C, 0 ≤|ωy |≤π/I
0, otherwise
(9.30)
where C is a scale factor required to properly normalize the output
sequence y(m). Consequently, the output spectrum is
{
CX(ωy I), 0 ≤|ω y |≤π/I
Y (ω y )=
0, otherwise
(9.31)
The scale factor C is selected so that the output y(m) = x(m/I) for
m =0, ±I,±2I,.... For mathematical convenience, we select the point
m =0.Thus,
y(0) = 1 ∫ π
Y (ω y )dω y = C ∫ π/I
X(ω y I)dω y (9.32)
2π −π 2π −π/I
Since ω y = ω x /I, (9.32) can be expressed as
y(0) = C I
1
2π
∫ π
therefore, C = I is the desired normalization factor.
−π
X(ω x )dω x = C x(0) (9.33)
I
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