filtering - 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?

Answer

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):