In multirate signal processing, we can change the sampling rate from fs to an arbitrary fraction DI fs by combining upsampling

by a factor I and downsampling by a factor D$.$ If I and D are large non$-$prime numbers $($I $=$ I$1$I$2$IM$,$ D $=$ D$1$D$2$DN$),$

then it is possible to implement upsampling and downsampling in M and N steps, respectively. Below is given four different

implementations that all change the sampling rate from fs to $16$ fs$.$ Hm$($z$)$ is a low$-$pass filter with cut$-$off frequency $/$m$.25$

Which of the implementations has the least loss of information for any given input signals, where fs corresponds to the Nyquist rate?

$0\mathrm{.}5$ p$.$

a$)$

b$)$

c$)$

d$)$ e$)$

$4$ H$5($z$)54$ H$5($z$)5$

$4$ H$4($z$)4$ H$5($z$)5$ H$5($z$)5$ H$5($z$)54$ H$4($z$)4$ H$5($z$)5$

$$ H$5($z$)5$ H$5($z$)54$ H$4($z$)4$ H$4($z$)$ It is impossible to decide without more information about the signal.