Chapter 5 PROGRAMMING EXERCISE
Mu-Law Encoding in C++
Mu-law encoding is a non-linear companding method that can be used to reduce the bit
depth of a digital audio signal in a way that preserves the dynamic range of samples at low
amplitudes. The word “companding” is used because this method works by compressing
the bit depth of an audio signal and then expanding it again after the signal has been
transmitted. Mu-law encoding is non-linear because it uses more bits to quantize low
amplitude samples as opposed to high amplitude ones. This method works well for
telephone communication because it reduces the signal-to-noise ratio in the area where it
matters most – in low amplitude values, which are common in human speech and are
particularly subject to noise distortion. The equation for non-linear compression by mu-
law encoding is
ln 1 x
m( x) sign( x)
ln(1 )
where 1 x 1 , and sign(x) is 1 if x is negative and 1 otherwise. μ is 255 when samples
are being quantized to 8 bits. (In some sources, you will see this equation using log2 rather
than the natural log. The definitions are equivalent.)
Decompression operates by the inverse equation:
1 x 1
d ( x) sign( x)
Here is a scenario in which mu-law encoding could be used. A 16-bit digital audio signal –
for example, a telephone signal – is to be transmitted across a network. It is reduced to 8-
bits by mu-law encoding and then transmitted. At the receiving end, it is decoded back to a
16-bit signal. Because the amount of error for low amplitude samples is less than it would
have been in a linear encoding, the effect is that a dynamic range of 72 dB is achieved with
mu-law encoding (as opposed to 48 dB that you would expect from an 8-bit audio signal).
It’s possible to implement mu-law encoding by applying the equations above directly.
However, the same effect can be achieved with a bit manipulation method that resembles
the way floating point numbers are represented. Here’s a sketch of the algorithm:
Digital Sound and Music PROGRAMMING EXERCISE
Mu-Law Encoding in C++ Page 1
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
Step 1. Assuming the 16-bit sample value is in sign-and-magnitude format, save the
sign bit.
Step 2. Clip the magnitude to 32,635. (The purpose of the clipping is to avoid
overflow when the bias is added in the next step. Note that the values for 16-bit
samples range from −2n-1 to 2n-1−1. The largest magnitude is 32,767.)
Step 3. Add a bias of 132 to the magnitude. (The purpose of adding the bias is to
ensure that a 1-bit will appear somewhere in the exponent region, as determined in
Step 4 below.)
Step 4. Let the exponent region be defined as the eight bits to the right of the sign
bit. Let p be the position of the leftmost 1-bit in the exponent region, counting
positions from 0 to 7 from right to left.
Step 5. Let the mantissa region be defined as the 4 bits to the right of position p.
Step 6. Encode eight bits by making the leftmost bit the sign bit (saved in Step 1),
the next three bits the binary encoding of p, and the last four bits the mantissa
region identified in Step 5.
Algorithm 1 mu-law encoding
Let’s try this on an example. The original sample is 0010 1110 1001 1011. (We insert a
blank space after every fourth bit for readability.)
Step 1. Sign bit is 0.
Step 2. No clipping necessary.
Step 3. 0010 1110 1001 1011 + 1000 0100 = 0010 1111 0001 1111 (base 2)
Step 4. Exponent region underlined and in boldface.
0010 1111 0001 1111
Leftmost 1-bit in exponent region is in position 6. That’s 110 base 2.
Step 5. Mantissa region is underlined and in boldface.
0010 1111 0001 1111
Step 6. Encode eight bits as
0110 0111
Thus, 0110 0111 is the 8-bit μ-law encoding of 0010 1110 1001 1011. (In fact, this value
would probably be put in one's or two's complement format after encoding, but that step is
not important to your understanding of the concept of μ-law encoding.)
This procedure effectively encodes the samples in a logarithmic manner.
Neighboring quantization values at high amplitudes correspond to larger differences in
amplitudes, as compared to neighboring values at low amplitudes. This lowers the signal-
to-quantization noise ratio at lower amplitudes, where the noise is otherwise most
pronounced.
Decoding reverses this process. The steps are given in Algorithm 2.
Step 1. Save the sign bit (the most significant bit of the eight bits given).
Step 2. Save the exponent region (the three bits to the right of the sign bit) as a 3-
bit field with value e.
Digital Sound and Music PROGRAMMING EXERCISE
Mu-Law Encoding in C++ Page 2
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
Step 3. Save the mantissa region (the remaining four bits) as a four-bit field called
m, which we view as a binary field.
Step 4.
a =3+e
b = (1000 0100 left shifted by e bits) – 1000 0100
decoded sample = (m left shifted by a bits) + b
Step 5. Affix the sign bit as the leftmost bit.
Algorithm 2 mu-law decoding
For our example, the steps work as follows:
Step 1. Sign bit is 0.
Step 2. Exponent region is 1102 = 610.
Step 3. Mantissa region is 0111.
Step 4. a=3+6=9
b = (1000 0100 left shifted 6 bits) – 1000 0100 = 10 0000 0111 1100
decoded sample = (0111 left shifted by 9 bits) + b =
0 1110 0000 0000 + 10 0000 0111 1100 = 10 1110 0111 1100
Step 5. With sign bit affixed, decoded sample value is 0010 1110 0111 1100
The decoded sample is (0010 1110 0111 11002 = 11,90010), while the original sample
value was (00101110100110112 = 11,93110 ). If you try this algorithm on a sample of
smaller magnitude, you'll see that the percent error is much smaller compared to using
truncation to reduce the bit depth. Using the algorithm, a sample value of 19310 is decoded
as 19610, with an error of 1.5%. Using truncation to achieve eight bits, 19310
(00000000110000012) is encoded as 0, for a 100% error. Mu-law encoding distributes the
error in a way that matches human hearing better than linear quantization methods. It
increases the SQNR in low amplitude signals, which is where the noise is most easily
perceived.
Your assignment is to implement mu-law encoding by both methods: applying the
equations, and then using bit manipulations. Compare the results.
Digital Sound and Music PROGRAMMING EXERCISE
Mu-Law Encoding in C++ Page 3
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
