UploadFile_6417

482 Chapter 9 SAMPLING RATE CONVERSION
Comments:
1. The sampling theorem interpretation for (9.19) is that the sequence
x(n) was originally sampled at D times higher rate than required;
therefore, downsampling by D simply reduces the effective sampling
rate to the minimum required to prevent aliasing.
2. Equation (9.18) expresses the requirement for zero decimation error
in the sense that no information is lost—i.e., there is no irreversible
aliasing error in the frequency domain.
3. The argument ωy
D
occurs because in our notation ω is expressed in
rad/sample. Thus the frequency of y(m) expressed in terms of the
higher-rate sequence x(n) must be divided by D to account for the
slower rate of y(m).
4. Note that there is a factor 1 D
in (9.19). This factor is required to make
the inverse Fourier transform work out properly and is entirely consistent
with the spectra of the sampled analog signals.
9.2.2 THE IDEAL DECIMATOR
In general, (9.18) will not be exactly true, and the (D ↓ 1) downsampler
would cause irreversible aliasing error. To avoid aliasing, we must first
reduce the bandwidth of x(n) toF x,max = F x /2D or, equivalently, to
ω x,max = π/D. Then we may downsample by D and thus avoid aliasing.
The decimation process is illustrated in Figure 9.6. The input sequence
x(n) is passed through a lowpass filter, characterized by the
impulse response h(n) and a frequency response H D (ω x ), which ideally
satisfies the condition
{
1, |ωx |≤π/D
H D (ω x )=
0, otherwise
(9.20)
Thus, the filter eliminates the spectrum of X(ω x )inthe range π/D <
ω x