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Multistage Multirate Systems

Module by: Douglas L. Jones

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Multistage multirate systems are often more efficient. Suppose one wishes to decimate a signal by an integer factor MM, where MM is a composite integer M= M 1 M 2 … M p =∏i=1p M i M M 1 M 2 … M p i 1 p M i . A decimator can be implemented in a multistage fashion by first decimating by a factor M 1 M 1 , then decimating this signal by a factor M 2 M 2 , etc. (Figure 1)

Figure 1
Multistage decimator
Multistage decimator (imag001.png)
Multistage implementations are of significant practical interest only if they offer significant computational savings. In fact, they often do!

The computational cost of a single-stage interpolator is: NM T 0 tapssec N M T 0 taps sec The computational cost of a multistage interpolator is: N 1 M 1 T 0 + N 2 M 1 M 2 T 0 +…+ N p M T 0 N 1 M 1 T 0 N 2 M 1 M 2 T 0 … N p M T 0 The first term is the most significant, since the rate is highest. Since N i ∝ M i ∝ N i M i for a lowpass filter, it is not immediately clear that a multistage system should require less computation. However, the multistage structure relaxes the requirements on the filters, which reduces their length and makes the overall computation less.

Filter design for Multi-stage Structures

Ostensibly, the first-stage filter must be a lowpass filter with a cutoff at π M 1 M 1 , to prevent aliasing after the downsampler. However, note that aliasing outside the final overall passband |ω|<πM ω M is of no concern, since it will be removed by later stages. We only need prevent aliasing into the band |ω|<πM ω M ; thus we need only specify the passband over the interval |ω|<πM ω M , and the stopband over the intervals ω∈2πk M 1 −πM2πk M 1 +πM ω 2 k M 1 M 2 k M 1 M , for k∈1…M−1 k 1 … M 1 . (Figure 2)

Figure 2
Figure 2 (imag002.png)
Of course, we don't want gain in the transition bands, since this would need to be suppressed later, but otherwise we don't care about the response in those regions. Since the transition bands are so large, the required filter turns out to be quite short. The final stage has no "don't care" regions; however, it is operating at a low rate, so it is relatively unimportant if the final filter turns out to be rather long!

L-infinity Tolerances on the Pass and Stopbands

The overall response is a cascade of multiple filters, so the worst-case overall passband deviation, assuming all the peaks just happen to line up, is 1+ δ p ov =∏i=1p1+ δ p i 1 δ p ov i 1 p 1 δ p i 1− δ p ov =∏i=1p1− δ p i 1 δ p ov i 1 p 1 δ p i So one could choose all filters to have equal specifications and require for each-stage filter. For δ p ov ≪1 ≪ δ p ov 1 , 1+ δ p i + ≤1+ δ p ov p≈1+p-1 δ p ov 1 δ p i + p 1 δ p ov 1 p δ p ov 1− δ p i - ≤1− δ p ov p≈1−p-1 δ p ov 1 δ p i - p 1 δ p ov 1 p δ p ov Alternatively, one can design later stages (at lower computation rates) to compensate for the passband ripple in earlier stages to achieve exceptionally accurate passband response.

δ s δ s remains essentially unchanged, since the worst-case scenario is for the error to alias into the passband and undergo no further suppression in subsequent stages.

Interpolation

Interpolation is the flow-graph reversal of the multi-stage decimator. The first stage has a cutoff at πL L (Figure 3):

Figure 3
Figure 3 (imag003.png)
However, all subsequent stages have large bands without signal energy, due to the earlier stages (Figure 4):
Figure 4
Figure 4 (imag004.png)
The order of the filters is reversed, but otherwise the filters are identical to the decimator filters.

Efficient Narrowband Lowpass Filtering

A very narrow lowpass filter requires a very long FIR filter to achieve reasonable resolution in the frequency response. However, were the input sampled at a lower rate, the cutoff frequency would be correspondingly higher, and the filter could be shorter!

The transition band is also broader, which helps as well. Thus, Figure 5 can be implemented as Figure 6.

Figure 5
Figure 5 (imag005.png)
Figure 6
Figure 6 (imag006.png)
and in practice the inner lowpass filter can be coupled to the decimator or interpolator filters. If the decimator and interpolator are implemented as multistage structures, the overall algorithm can be dramatically more efficient than direct implementation!

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