Multirate digital signal processingMultirate digital signal processing Prasanta Kumar Ghosh Oct23,...
Transcript of Multirate digital signal processingMultirate digital signal processing Prasanta Kumar Ghosh Oct23,...
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