multirate digital signal processing · 2017-10-31 · multirate digital signal processing prasanta...
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![Page 1: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/1.jpg)
Multirate digital signal
processing
Prasanta Kumar Ghosh
Oct24, 2017
![Page 2: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/2.jpg)
Implementation of Sampling
Rate Conversion
![Page 3: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/3.jpg)
Polyphase filter structure for efficient implementation of
sampling rate converters
M-component polyphase
decomposition Polyphase components
Downsampled and
delayed version
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Polyphase filter structure
![Page 5: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/5.jpg)
Polyphase filter structure
Transpose
polyphase
structure
![Page 6: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/6.jpg)
Interchange of filters and downsamplers/upsamplers
Noble identities
![Page 7: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/7.jpg)
Interchange of filters and downsamplers/upsamplers
Noble identities
Input/output relation of a downsampler
![Page 8: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/8.jpg)
Interchange of filters and downsamplers/upsamplers
Noble identities
For the first system
But
![Page 9: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/9.jpg)
Interchange of filters and downsamplers/upsamplers
Noble identities
![Page 10: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/10.jpg)
Input/output relation of a upsampler
Interchange of filters and downsamplers/upsamplers
Noble identities
![Page 11: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/11.jpg)
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
![Page 12: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/12.jpg)
Sampling rate conversion with cascaded integrator comb filters
integrator
Comb filter
Cascaded integrator comb (CIC) filter
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.
![Page 13: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/13.jpg)
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.
![Page 14: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/14.jpg)
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.
![Page 15: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/15.jpg)
Polyphase structure for decimation and interpolation filters
Why compute filter output and
then throw away samples?
Downsampling commutes
with addition
decimation
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Polyphase structure for decimation and interpolation filters
With noble identity we get
decimation
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Polyphase structure for decimation and interpolation filters
Only needed samples are computed and all multiplication
and additions are performed at lower sampling rate
decimation
x0,x1,x2,x3,x4,x5,x6,x7,... x0,x3,x6,...
x1,x4,x7,...
x2,x5,x8,...
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Polyphase structure for decimation and interpolation filters
Commutator model
decimation
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Polyphase structure for decimation and interpolation filters
interpolation
![Page 20: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/20.jpg)
Polyphase structure for decimation and interpolation filters
interpolation
![Page 21: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/21.jpg)
Polyphase structure for decimation and interpolation filters
interpolation u1,u2,u3,...
v1,v2,v3,...
w1,w2,w3,...
u1,0,0,u2,0,0,u3,...
v1,0,0,v2,0,0,v3,...
w1,0,0,w2,0,0,w3,...
u1,v1,w1,u2,v2,w2,...
![Page 22: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/22.jpg)
Polyphase structure for decimation and interpolation filters
interpolation
Commutator model
![Page 23: Multirate digital signal processing · 2017-10-31 · Multirate digital signal processing Prasanta Kumar Ghosh Oct24, 2017 . Implementation of Sampling Rate Conversion. Polyphase](https://reader030.vdocuments.net/reader030/viewer/2022040909/5e81971225104743f73c4731/html5/thumbnails/23.jpg)
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
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Sampling rate conversion for bandpass signals is
achieved by finding an equivalent lowpass signal, in general.
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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.