Considering that $previous is defined most of the time, it's better to check $line =~ /BEEP/ first. Or just eliminate the check for definedness all together:

my $previous = "";
while (<$handle>) {
    print $previous if /BEEP/;
    $previous = $_;
}
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If "BEEP" is on the first line, it just prints nothing. Which is the same effect as not printing (assuming no assignment to $\ has been done).