Quantization
Fall 2018
Signals and Noise
Quantizing a sample
Learn It!
Pre-Requisite Knowledge
Goal
Find bit string and max quantization error
Which of the bit strings in the figure represents the value? What is the maximum quantization error possible?
We are to quantize the temperature range -40 ° F to 110 ° F by rounding using the uniform 4-bit quantizer shown to the left. Which bit string will represent an input temperature of 7 ° F? What is the maximum quantization error possible for a temerature between -40 and 110°F?
Part 1
Find the location of 7°F
$Spacing=\frac{110 - -40}{16-1}=10$
To figure out where to put 7°F on our scale, we need to know how far apart those tic marks are. Note that with 16 tic marks for our 16 bits, we have only 15 spaces between them. Keep this "off by one" error in mind!

Lucky for us, this problem has been set up with very convenient, nice-numbered spacing.
We can count up the tic marks. 7 is located between 0 and 10 degrees, but closer to 10.
0101
Since the 0101 point is closest to our actual sample, we round up to 10°F
Part 2
Maximum Quantization Error
Finding the maximum quantization error is pretty easy. The worst case scenario is a point half way between the tic marks where we quantize. So the worst error is half the separation between consecutive levels.
$MaxError=\frac{Spacing}{2}=10/2=5°F$
Applying the formula to our case, we get the result of 5°F.