Speech characteristics and 7-bit Hamming code

Characteristics of speech

• Most energy and intelligibility in 300–3400 Hz.
• Non-uniform amplitude pdf: small amplitudes are more likely.
• Strong correlation between adjacent samples, so speech is predictable.
• Non-flat spectrum with formant peaks.
• Voiced sounds are quasi-periodic (pitch 80–350 Hz); unvoiced sounds are noise-like.
• Quasi-stationary over 10–30 ms; about 40–60% silence in conversation.

7-bit Hamming code

The (7,4) Hamming code is a linear block code that adds 3 parity bits to 4 data bits. Minimum distance d_min = 3, so it corrects any single-bit error (or detects two). Parity bits sit at positions that are powers of 2:

Even-parity equations (each parity covers the positions whose binary index has that bit set):

 p1 = d1 ⊕ d2 ⊕ d4   (positions 1,3,5,7)
 p2 = d1 ⊕ d3 ⊕ d4   (positions 2,3,6,7)
 p3 = d2 ⊕ d3 ⊕ d4   (positions 4,5,6,7)

Example – encoding: data 1011 → d1 = 1, d2 = 0, d3 = 1, d4 = 1.

 p1 = 1 ⊕ 0 ⊕ 1 = 0
 p2 = 1 ⊕ 1 ⊕ 1 = 1
 p3 = 0 ⊕ 1 ⊕ 1 = 0
 Codeword (p1 p2 d1 p3 d2 d3 d4) = 0 1 1 0 0 1 1

Example – error correction: suppose bit 5 is flipped; received word = 0 1 1 0 1 1 1.

 s1 = r1 ⊕ r3 ⊕ r5 ⊕ r7 = 0⊕1⊕1⊕1 = 1
 s2 = r2 ⊕ r3 ⊕ r6 ⊕ r7 = 1⊕1⊕1⊕1 = 0
 s3 = r4 ⊕ r5 ⊕ r6 ⊕ r7 = 0⊕1⊕1⊕1 = 1
 Syndrome (s3 s2 s1) = 101 = 5

The syndrome points to position 5, so that bit is inverted back to 0, giving 0110011 and data 1011. A syndrome of 000 means no error.

Code rate = 4/7 ≈ 0.57. Such block codes protect the most important speech-coder bits in mobile systems.