Sampling & Pulse Modulation Lecture Notes, Summaries of Electrical Engineering

Download Sampling & Pulse Modulation Lecture Notes and more Summaries Electrical Engineering in PDF only on Docsity! Sampling & Pulse Modulation Lecture Notes in “Digital Communications” by: Dr. Mohammed AlMahamdy Electrical Engineering | University of Anbar -17- 1.6.5 Signal to Quantization Noise Ratio It is important to consider the quantization noise in the overall system quality. To calculate SNqR of a uniformly quantized signal, it is suitable to make the following assumptions: (1) Linear quantization (i.e. equal increments between quantization levels). (2) Zero mean signal (i.e. symmetrical PDF around the 0 Volt). (3) Uniform signal PDF (i.e. all signal levels equally likely). Let: 𝐿 be the number of the levels of the quantizer, 𝑙 be the number of bits per a PCM word (𝐿 = 2𝑙), and 𝑉𝑝 be the peak design level of the quantizer. The quantization interval 𝑞 becomes: 𝑞 = 2𝑉𝑝 𝐿 − 1 The PDF of the allowed levels is given by: 𝑝(𝑣) = ∑ 1 𝐿 𝛿 (𝑣 − 𝑞𝑘 2 ) 𝐿 𝑘=−𝐿 𝑘=odd The mean square signal after quantization is: 𝑣2̅̅ ̅ = ∫ 𝑣2𝑝(𝑣)𝑑𝑣 ∞ −∞ = 2 𝐿 [∫ 𝑣2𝛿 (𝑣 − 𝑞 2 ) 𝑑𝑣 ∞ 0 +∫ 𝑣2𝛿 (𝑣 − 3𝑞 2 )𝑑𝑣 ∞ 0 +⋯] = 2 𝐿 ( 𝑞 2 ) 2 [12 + 32 + 52 +⋯+ (𝐿 − 1)2] = 2 𝐿 ( 𝑞 2 ) 2 [ 𝐿(𝐿 − 1)(𝐿 + 1) 6 ] ∴ 𝑣2̅̅ ̅ = 𝑞2 12 (𝐿2 − 1) Lecture Notes in “Digital Communications” Sampling & Pulse Modulation Electrical Engineering | University of Anbar by: Dr. Mohammed AlMahamdy -18- Denoting the quantization error (i.e. the difference between the unquantized and quantized signals) as 휀𝑞, then the PDF of 휀𝑞 is uniform: 𝑝(휀𝑞) = { 1 𝑞 − 𝑞 2 ≤ 휀𝑞 < 𝑞 2 0 elsewhere The mean square quantization error (noise) is: 휀𝑞2̅̅ ̅ = ∫ 휀𝑞 2 𝑝(휀𝑞) 𝑑휀𝑞 𝑞/2 −𝑞/2 = 𝑞2 12 Therefore, the average SNqR will be: SNqR = 𝑣2̅̅ ̅/휀𝑞2̅̅ ̅ = 𝐿2 − 1 Since the peak signal level is 𝑞𝐿 2 Volts then the peak SNqR will be: SNqR = (𝐿𝑞 2⁄ )2 휀𝑞2̅̅ ̅ = 3𝐿2 We may express it in decibels as SNqR = 6.02𝑙 + 𝛼 Where 𝛼 = 4.77 for the peak SNqR, and 𝛼 = 0 for the average SNqR. This equation is called the 6dB rule, and it points out that: an additional 6dB improvement in the SNqR is obtained for each bit added to the PCM word.