# How is the output size of UPFIRDN determined?

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Harry on 26 Oct 2014
Commented: Bradford Watson on 29 Jan 2021
I want to understand resampling better, so I am trying to replicate the functionality of Matlab's built-in upfirdn() function. To clarify, I'm not looking for a fast implementation (at first), I just want to see how to get the exact same result. As such, I understand the basic steps (conceptually) are:
(1) Upsampling by P (zero insertion).
(2) FIR filtering.
(3) Downsampling by Q (throwing away samples).
This is fine. However, what is bugging me is that (as stated by Matlab help) the final output size is:
Ly = ceil( ((Lx-1)*P + Lh)/Q ) where Lx = length(X) and Lh = length(H)
and I don't understand why. With reference to the 3 stages listed above, a naive implementation might give stage-by-stage output sizes:
(1) P*Lx
(2) P*Lx
(3) floor(P*Lx/Q)
Therefore, I really want to know exactly what upfirdn is doing. Can anyone help?
Bradford Watson on 29 Jan 2021
Whenever two sequences are convolved together, the length of the resulting sequence is Lx + Ly - 1. The inner part of this equation is expressing the length of the convolution of the upsampled sequence (Lx-1)*P and the filter length (Lh), though neglecting the -1 term. This is then divided by the decimation, and the ceiling function is taken to get rid of any fraction.