6.4 Multistage multirate systems

Multistage multirate systems are often more efficient. Suppose one wishes to decimate a signal by an integer factor $M$ , where $M$ is a composite integer $M=M_1{M}_2\dots M_p=\prod_{i=1}^p M_i$ . A decimator can be implemented in a multistage fashion by first decimating by a factor $M_1$ , then decimating this signal by a factor $M_2$ , etc. ( [link] )

Multistage decimator

The computational cost of a single-stage interpolator is: $$\frac{N}{M{T}_0}\frac{\mathrm{taps}}{\sec }$$ The computational cost of a multistage interpolator is: $$\frac{N_1}{M_1{T}_0}+\frac{N_2}{M_1{M}_2{T}_0}+\dots +\frac{N_p}{M{T}_0}$$ The first term is the most significant, since the rate is highest. Since $\propto (N_i, M_i)$ 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 overallcomputation less.

Filter design for multi-stage structures

Ostensibly, the first-stage filter must be a lowpass filterwith a cutoff at $\frac{\pi }{M_1}$ , to prevent aliasing after the downsampler. However, note that aliasing outside the final overall passband $\left|\omega \right|< \frac{\pi }{M}$ is of no concern, since it will be removed by later stages. We only need prevent aliasing into the band $\left|\omega \right|< \frac{\pi }{M}$ ; thus we need only specify the passband over the interval $\left|\omega \right|< \frac{\pi }{M}$ , and the stopband over the intervals $\omega \in \left[\frac{2\pi k}{M_1}-\frac{\pi }{M} , \frac{2\pi k}{M_1}+\frac{\pi }{M}\right]$ , for $k\in \{1, \dots , M-1\}$ . ( [link] )

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 peaksjust happen to line up, is $$1+{\delta }_{p_{\mathrm{ov}}}=\prod_{i=1}^p 1+{\delta }_{p_i}$$ $$1-{\delta }_{p_{\mathrm{ov}}}=\prod_{i=1}^p 1-{\delta }_{p_i}$$ So one could choose all filters to have equal specifications and require for each-stage filter. For $\ll ({\delta }_{p_{\mathrm{ov}}}, 1)$ , $$1+{\delta }_{p_i}^{+}\le \sqrt[p]{1+{\delta }_{p_{\mathrm{ov}}}}\approx 1+p^{(-1)}{\delta }_{p_{\mathrm{ov}}}$$ $$1-{\delta }_{p_i}^{-}\le \sqrt[p]{1-{\delta }_{p_{\mathrm{ov}}}}\approx 1-p^{(-1)}{\delta }_{p_{\mathrm{ov}}}$$ Alternatively, one can design later stages (at lower computation rates) to compensate for the passband ripple inearlier stages to achieve exceptionally accurate passband response.

${\delta }_s$ remains essentially unchanged, since the worst-case scenario is for the error to alias into the passband and undergo nofurther suppression in subsequent stages.

Interpolation

Interpolation is the flow-graph reversal of the multi-stage decimator. The first stage has a cutoff at $\frac{\pi }{L}$ ( [link] ):

Efficient narrowband lowpass filtering

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

The transition band is also broader, which helps as well. Thus, [link] can be implemented as [link] .