# Connexions

You are here: Home » Content » Digital Signal Processing (Ohio State EE700) » Multi-stage Interpolation and Decimation

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Phil Schniter. E-mail the author

# Multi-stage Interpolation and Decimation

Module by: Phil Schniter. E-mail the author

Summary: This module covers the fundamentals of multistage decimation.

## Multistage Decimation

In the single-stage interpolation structure illustrated in Figure 1, the required impulse response of Hz H z can be very long for large LL.

Consider, for example, the case where L=30 L 30 and the input signal has a bandwidth of ω 0 =0.9πradians ω 0 0.9 radians . If we desire passband ripple δ p =0.002 δ p 0.002 and stopband ripple δ s =0.001 δ s 0.001 , then Kaiser's formula approximates the required FIR filter length to be Nh-10log(δPδS)132.3Δω900 Nh -10 10 δP δS 13 2.3 Δ ω 900 choosing Δω=2π2 ω 0 L Δ ω 2 2 ω 0 L as the width of the first transition band (i.e., ignoring the other transition bands for this rough approximation). Thus, a polyphase implementation of this interpolation task would cost about 900900 multiplies per input sample.

Consider now the two-stage implementation illustrated in Figure 2.

We claim that, when LL is large and ω0ω0 is near Nyquist, the two-stage scheme can accomplish the same interpolation task with less computation.

Let's revisit the interpolation objective of our previous example. Assume that L 1 =2 L 1 2 and L 2 =15 L 2 15 so that L 1 L 2 =L=30 L 1 L 2 L 30 . We then desire a pair FzGz F z G z which results in the same performance as Hz H z . As a means of choosing these filters, we employ a Noble identity to reverse the order of filtering and upsampling (see Figure 3).

It is now clear that the composite filter Gz15Fz G z 15 F z should be designed to meet the same specifications as Hz H z . Thus we adopt the following strategy:

1. Design Gz15 G z 15 to remove unwanted images, keeping in mind that the DTFT Gej15ω G 15 ω is 2π15 2 15 -periodic in ωω.
2. Design Fz F z to remove the remaining images.
The first and second plots in Figure 4 illustrate example DTFTs for the desired signal xn x n and its LL-upsampled version vl v l , respectively. Our objective for interpolation, is to remove all but the shaded spectral image shown in the second plot. The third plot shows that, due to symmetry requirements Gz15 G z 15 will be able to remove only one image in the frequency range 0 2π15 0 2 15 . Due to its periodicity, however, Gz15 G z 15 also removes some of the other undesired images, namely those centered at π15+m2π15 15 m 2 15 for mZ m . Fz F z is then used to remove the remaining undesired images, namely those centered at m2π15 m 2 15 for mZ m such that mm is not a multiple of 1515. Since it is possible that the passband ripples of Fz F z and Gz15 G z 15 could add constructively, we specify δ p =0.001 δ p 0.001 for both Fz F z and Gz G z , half the passband ripple specified for Hz H z . Assuming that the transition bands in Fz F z have gain no greater than one, the stopband ripples will not be amplified and we can set δ s =0.001 δ s 0.001 for both Fz F z and Gz G z , the same specification as for Hz H z .

The computational savings of the multi-stage structure result from the fact that the transition bands in both Fz F z and Gz G z are much wider than the transition bands in Hz H z . From the block diagram, we can infer that the transition band in Gz G z is centered at ω=π2 ω 2 with width π ω 0 =0.1πrad ω 0 0.1 rad . Likewise, the transition bands in Fz F z have width 4π2 ω 0 30=2.230πrad 4 2 ω 0 30 2.2 30 rad . Plugging these specifications into the Kaiser length approximation, we obtain N g 64 N g 64 and N f 88 N f 88 Already we see that it will be much easier, computationally, to design two filters of lengths 6464 and 8888 than it would be to design one 900900-tap filter.

As we now show, the computational savings also carry over to the operation of the two-stage structure. As a point of reference, recall that a polyphase implementation of the one-stage interpolator would require N h 900 N h 900 multiplications per input point. Using a cascade of two single-stage polyphase interpolators to implement the two-stage scheme, we find that the first interpolator would require N g 64 N g 64 per input point xn x n , while the second would require N f 88 N f 88 multiplies per output of Gz G z . Since Gz G z outputs two points per input xn x n , the two-stage structure would require a total of 64+2×88=240 64 2 88 240 multiplies per input. Clearly this is a significant savings over the 900900 multiplies required by the one-stage structure. Note that it was advantageous to choose the first upsampling ratio (L1L1) as small as possible, so that the second stage of interpolation operates at a low rate.

Multi-stage decimation can be formulated in a very similar way. Using the same example quantities as we did for the case of multi-stage interpolation, we have the block diagrams and filter-design methodology illustrated in Figure 5 and Figure 6.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks