Hi shamrock...
The signed 16 bit decimal equivalent cannot go less that -32768 and no greater than 32767.
The centreline is zero.

Add 32768 to the signed decimal to shift the centreline to 32768, the minimum now sits at zero and the maximum at 65535.

Divide the result by 256 gives a centreline of 128, minimum of zero and maximum of 255.
Try it and find out.

Nothing more sinister, and accurate enough for a further quantise to 4 bit depth for AudioScope.sh...

HTH...

---------- Post updated 01-11-14 at 09:41 AM ---------- Previous update was 31-10-14 at 11:13 PM ----------

Got it into one block realising that the numbers already exist as numbers and not strings...
I am happy with the results now and shaved off another two tenths of a second execution time...

Code:

#!/bin/bash
# 16to8bit_new.sh
> /tmp/sample.raw
> /tmp/leftbinary
> /tmp/signed16bit.txt
dd if=/dev/urandom of=/tmp/sample.raw bs=1 count=176400 > /dev/null 2>&1
od -td2 -An /tmp/sample.raw > /tmp/signed16bit.txt
awk --characters-as-bytes 'BEGIN \
{
	BINMODE=3;
	FS=" ";
}
{
	if ($1=="")
		exit;
	# $1,$3,$5,$7 are the left hand channel.
	# $2,$4,$6,$8 are the right hand channel.
	$1=(int(($1+32768)/256));
	# $2=(int(($2+32768)/256));
	$3=(int(($3+32768)/256));
	# $4=(int(($4+32768)/256));
	$5=(int(($5+32768)/256));
	# $6=(int(($6+32768)/256));
	$7=(int(($7+32768)/256));
	# $8=(int(($8+32768)/256));
	printf("%c%c%c%c",$1,$3,$5,$7) > "/tmp/leftbinary";
	# printf("%c%c%c%c",$2,$4,$6,$8) > "/tmp/rightbinary";
}' < /tmp/signed16bit.txt

EDIT:
Now tested on CygWin and completes the cycle in around 1.5 seconds. Highly acceptable...