Verictor wrote:
>
> Not sure why you can call this "single stage" approach because you
> have to virtually design H1 and H2 before you can implement this
> "single stage" approach.

Agreed. What I'm questioning is once one has the two filters, does one
actually need to bother with two separate filters and two separate
decimators.

> In fact, they are the same in terms of efficiency. Consider the
> conventional approach, i.e.,
>
> --> H1 --> v L1 --> H2 --> v L2 -->
>
> If doing a "noble identity" conversion just using
>
> --> v L1 --> H2
>
> You can get this:
>
> --> H1 --> H2(z^L1) --> v L1 --> vL2 --> (H2(z^L1) means the H2
> upsampling by L1)
>
> Thus, H1(z)*H2(z^L1) is the H filter in your approach. So that you can
> have vL1.L2 too.

Indeed. I completely agree with you up to this point.

> I don't see why it can be "more efficient" than the
> multirate design.

Would you agree that:
* the order of H1(z)*H2(z^L1) is approximately (N1+L1.N2)? (where the
difference is negligible for large L1 and L2).
* the number of operations per input sample of a polyphase decimation
filter is N/L?

From that, it is simple to obtain the number of operations per input
sample of the two approaches (as before):

(N1 / L1) + (N2 / L1.L2) for the two-stage approach
(N1 / L1.L2) + (N2 / L2) for the combined approach

Setting these as an inequality and rearranging, I find that when (N1/N2)
< (L1 - 1)/(L2 - 1), the combined approach will have fewer operations.


> As an exmaple, say, you have a need to achieve
> downsample 12 = 3 * 4 (can't be 2*6, as you specified). You either can
> let H1 corresponds to L1=3 or L1=4 so that L2 is determined
> respectively. That is,
>
> L1=3: --> H1 --> v 3 --> H2 --> v 4 --->
> L1=4: --> H2' --> v 4 ---> H1' --> v 3 -->
>
> Note that H2' doesn't equal to H2 because the decimation filters need
> to be alias-free. These anti-alias feature is what you mean "more
> efficient"?

I'm not entirely sure what you're trying to show with your example.
Could you expand on this a little?


--
Oli