Print a column with the name of the output file

Shell Programming and Scripting
1

I have n files and I am using cat to combine them in to one.
Before that simply add the name of the output file to 4th column and then print the output. Is it possible ?

input1

chr    start    end    name    0    +    key

input2

chr    start    end    name    0    +    key

inputn

chr    start    end    name    0    +    key

print a column with the name of the output file

chr    start    end    name_input1    0    +    key
chr    start    end    name_input2    0    +    key
chr    start    end    name_inputn    0    +    key
2 
awk '{$4=sprintf("%s_%s",$4,FILENAME);print}' input*