Moving average filter using DF-I and DF-II I understand that the moving average filter is simply the average of a number of points from the input signal to produce the output signal. If x[] is the input and y[] is the output and M is the number of points, then the filter is just $$y[i]=\frac{1}{M}\sum_{j=0}^{M-1}x\left[ i+j \right]$$
This can be done through many ways, but I want to ask about using Direct Form I and Direct Form II. How do I follow the DF-I and DF-II structures to create a moving average filter? Don't I need a two separate multipliers like $$b_{M}$$ and $$a_{N}$$ to use the DF-I and DF-II, which moving average filters don't necessarily have? How about if I want to use DF-I and DF-II to create an autoregressive filter? Isn't it even more incompatible with DF-I and DF-II?
To start, notice that the diagrams are in the $\mathcal{Z}$ domain, multiplying by $z^{-1}$ is the delay of one sample.
DF-I
Writing the equation for $y_n$ of the DF-I
$$y_n = \sum_{k=0}^{M} b_k x_{n-k} - \sum_{l=1}^{N} a_{l} y_{n-l}$$
Comparing with the equation of the moving average
$$y_n = \sum_{k=0}^{M} \frac{1}{M+1} x_{n-k}$$
we see that we could realise the moving average by settings $b_k=1/(M+1)$, and $N$ set to zero.
DF-II
For the DF-II realization you the sequences $a$ and $b$ must have the same size, so what we can do is to set $a_l=0$, and $b_k = 1/(M+1)$.
If you choose to compute a moving sum then $b_k=1$ and you can save the multiplications, the moving average still can be computed by a gain, thus, performing one multiplication per sample instead of $M+1$ multiplications per sample.