Multirate digital signal

processing

Prasanta Kumar Ghosh

Oct24, 2017

Implementation of Sampling

Rate Conversion

Polyphase filter structure for efficient implementation of

sampling rate converters

M-component polyphase

decomposition Polyphase components

Downsampled and

delayed version

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

Input/output relation of a upsampler

Interchange of filters and downsamplers/upsamplers

Noble identities

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

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.

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

Why compute filter output and

then throw away samples?

Downsampling commutes

with addition

decimation

Polyphase structure for decimation and interpolation filters

With noble identity we get

decimation

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,...

Polyphase structure for decimation and interpolation filters

Commutator model

decimation

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,...

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,...

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.