$($a$)[7$ marks $]$ The following equation represents an infinite impulse response filter which was discussed

during lectures about recurrent neural networks:

$y[n]=y[n-1]+y[n-2]+{\displaystyle \underset{n=0}{\overset{N}{}}}x[n]+C$

where $x[n]$ is the input sample and $y[n]$ is the output sample at time instance $n$;$C$ is a constant. The

notation $y[n-1]$ indicates an output delayed by one time unit. It can be thought of as having a memory

cell.

$($i$)$ By considering the equation of the filter given above, explain how it achieves the memory using

the following input, $x[n]=[0\mathrm{.}5,0\mathrm{.}4,0\mathrm{.}7,0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}4,0\mathrm{.}8,0\mathrm{.}5],n=0,$dots$7,C=0\mathrm{.}1,N=2,y[-1]=$

$0\mathrm{.}0,y[-2]=0\mathrm{.}0.[$Hint: calculating $y[n]$ up to $n=5$ should be sufficient.$]$

$($ii$)$ In what way does this filter function similarly to an LSTM cell and how are they different? In your

response, you need to explain how LSTM achieves the concept of "memory".

$($b$)[10$ marks $]$ Assume we can model an image captioning task as a translation from image to words,

describe how you will design such a system using transformer model. Your design must mention the

key subsystems $($or algorithms$)$ and the role they play in the overall design. Please limit your response to

no more than a page and half.