# FIR filters: direct form / transposed FIR

I am learning about FIR filters and I'm confused. I am trying to find out different types of FIR filters.

1. Is direct form and n-tap FIR filter the same?
2. What does transposed FIR filter do?

A finite impulse response (FIR) digital filter implements the following convolution sum

$$y(n)=\sum_{n=0}^{N-1}h(k)x(n-k)\tag{1}$$

for each output sample $y(n)$, where $x(n)$ is the discrete-time input signal, $h(n)$ is the filter's impulse response, and $N$ is the filter length. The values $h(n)$ are also called filter taps, and $N$ is then referred to as the number of taps. The filter described by Equation (1) is also called an $N$-tap filter.

Direct-form and transposed direct-form are just different implementations, i.e. different ways to compute the sum in (1). In theory they are identical, but when computed with finite precision, there can be differences between the different implementations. The direct-form FIR structure is also called tapped delay line or transversal filter.

The two realizations below are the direct-form structure (transversal filter, tapped delay-line) and the transposed structure (from Oppenheim and Schafer, Discrete-time Signal Processing):

• An important thing to note is also that the filter order is $N-1$, i.e., one less than the number of taps. The different mentioned above typically relates to the word length of the registers (and to some extent the adders). Coefficient quantization affects both structures identically. Another difference is that the transposed direct form has a shorter critical path, i.e., number of cascaded operations. – Oscar Apr 4 '14 at 7:01

Recall the various types of linear phase FIR filters,in which the coefficients h(n) of the transfer function

are assumed to be symmetric or antisymmetric. Since the order of the polynomial in each of these two types can be either odd or even, we have four types of filters with different properties.

• this has nothing to do with the question posted. – Marcus Müller Feb 10 '19 at 15:54