Multirate digital signal
processing
Prasanta Kumar Ghosh
Oct24, 2017
Implementation of Sampling
Rate Conversion
Polyphase filter structure for efficient implementation of
sampling rate converters
Downsampled and
delayed version
M-component polyphase Polyphase components
decomposition
Polyphase filter structure
Polyphase filter structure
Transpose
polyphase
structure
Interchange of filters and downsamplers/upsamplers
Noble identities
Interchange of filters and downsamplers/upsamplers
Noble identities
Input/output relation of a downsampler
Interchange of filters and downsamplers/upsamplers
Noble identities
For the first system
But
Interchange of filters and downsamplers/upsamplers
Noble identities
Interchange of filters and downsamplers/upsamplers
Noble identities
Input/output relation of a upsampler
Interchange of filters and downsamplers/upsamplers
Noble identities
For the first system
For the second system It is possible to interchange
the LTI filtering and
downsampling or upsampling
if we properly modify the
system function of the filter
Sampling rate conversion with cascaded integrator comb filters
Comb filter
Cascaded integrator comb (CIC) filter integrator
Does not require any multiplication or storage for the filter coefficients
HOGENAUER, E.B. 1981. "An Economical Class of Digital Filters for Decimation and Interpolation"
IEEE Trans. on ASSP, Vol. 29(2), pp. 155-162, April.
Sampling rate conversion with cascaded integrator comb filters
To improve the lowpass frequency response required for sampling rate
conversion, we can cascade K CIC filters. As above all integrations can be
done before downsampling and all difference operations after
downsampling
HOGENAUER, E.B. 1981. "An Economical Class of Digital Filters for Decimation and Interpolation"
IEEE Trans. on ASSP, Vol. 29(2), pp. 155-162, April.
Sampling rate conversion with cascaded integrator comb filters
HOGENAUER, E.B. 1981. "An Economical Class of Digital Filters for Decimation and Interpolation"
IEEE Trans. on ASSP, Vol. 29(2), pp. 155-162, April.
Polyphase structure for decimation and interpolation filters
decimation
Why compute filter output and
then throw away samples?
Downsampling commutes
with addition
Polyphase structure for decimation and interpolation filters
decimation
With noble identity we get
Polyphase structure for decimation and interpolation filters
decimation
x0,x3,x6,...
x0,x1,x2,x3,x4,x5,x6,x7,...
x1,x4,x7,...
x2,x5,x8,...
Only needed samples are computed and all multiplication
and additions are performed at lower sampling rate
Polyphase structure for decimation and interpolation filters
decimation
Commutator model
Polyphase structure for decimation and interpolation filters
interpolation
Polyphase structure for decimation and interpolation filters
interpolation
Polyphase structure for decimation and interpolation filters
interpolation
u1,u2,u3,... u1,0,0,u2,0,0,u3,...
u1,v1,w1,u2,v2,w2,...
v1,v2,v3,... v1,0,0,v2,0,0,v3,...
w1,w2,w3,...
w1,0,0,w2,0,0,w3,...
Polyphase structure for decimation and interpolation filters
interpolation
Commutator model
Structures for rational sampling rate conversion I/D
Polyphase interpolation by a downsampler. But no need for
computing all I interpolated values as only one in D outputs
are keptc
Polyphase subfilter index
Sampling rate conversion for bandpass signals is
achieved by finding an equivalent lowpass
signal, in general.
Sampling rate conversion by an arbitrary factor
What if I/D = 1023/511 ? Or the exact factor is not known when the
rate converter is designed ? Or the actual rate may not be rational
fraction of the input rate?
Polyphase interpolator
Consider polyphase interpolator with I subfilters. It generates
samples with spacing . If this spacing is too small
1. such that the signal values changes by less than
quantization step, then the value at can be
approximated by nearest-neighbor (zero-order hold interpolation)
2. two point linear interpolation can be performed
RAMSTAD, T. A. 1984. "Digital Methods for Conversion Between Arbitrary Sampling
Frequencies," IEEE Trans. Acoustics, Speech, and Signal Processing, Vol. ASSP-32, pp. 577-
591, June.
