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Multirate digital signal
processing
Prasanta Kumar Ghosh
Oct23, 2018
Reconstruct and resample
If has bandwidth less than , then if
Note the infinite summation in the reconstruction (not practical)
Suppose for sampling rate conversion, we need to sample at
with a desired sampling rate of
thus obtained are accurate only if
If , components above should be filtered out before
resampling. if
If , then it is a convolution summation (LTI system)
For
Let
Integer Fraction
Time-varying system for sampling rate conversion
**Inefficient when interpolating function is complicated
It simplifies when where D and I are relatively prime integers
Value of mD
modulo I
Thus, can take only I distinct values
Hence, only I distinct impulse responses are possible and thus, is
periodic
A linear and periodically time-varying
discrete-time system (great simplification!)
Downsampling/Decimation
Only one impulse response for all m
Upsampling/Interpolation
Impulse response is shifted
at an increment of
Downsampling/Decimation
Upsampling/Interpolation
Fractional shifting results in
Other way is to create a new sequence by inserting zero samples
Decimation by a factor D
Is decimation LTI operation?
Decimation by a factor D
Decimation by a factor D
Decimation by a factor D
Evaluate the Z-transform on unit circle with frequency variable
Thus, gets stretched to by down-sampling
If is correctly designed, then aliasing is eliminated and
Decimation by a factor D
Interpolation by a factor I
DTFT:
Interpolation by a factor I
As the frequency component of
are unique in the range
Images beyond that in
should be rejected by low pass
filtering
C = ?
Interpolation by a factor I
is the desired normalization factor
Sampling rate conversion by a rational factor I/D
Sampling rate conversion by a rational factor I/D
Sampling rate conversion by a rational factor I/D
Frequency response of the combined filter
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