Chapter 3 PCM Noise and Companding Quantization Noise

Chapter 3: PCM Noise and Companding Ø Ø Ø Quantization Noise Signal to Noise Ratio PCM Telephone System Nonuniform Quantization Companding Huseyin Bilgekul Eeng 360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University 1

Quantization Noise Ø The process of quantization can be interpreted as an additive noise process. Signal X Quantized Signal XQ Quantization Noise n. Q • The signal to quantization noise ratio (SNR)Q=S/N is given as: 2

Effects of Noise on PCM Ø Two main effects produce the noise or distortion in the PCM output: – – Quantizing noise that is caused by the M-step quantizer at the PCM transmitter. Bit errors in the recovered PCM signal, caused by channel noise and improper filtering. • If the input analog signal is band limited and sampled fast enough so that the aliasing noise on the recovered signal is negligible, the ratio of the recovered analog peak signal power to the total average noise power is: • The ratio of the average signal power to the average noise power is – – M is the number of quantized levels used in the PCM system. Pe is the probability of bit error in the recovered binary PCM signal at the receiver DAC before it is converted back into an analog signal. 3

Effects of Quantizing Noise • If Pe is negligible, there are no bit errors resulting from channel noise and no ISI, the Peak SNR resulting from only quantizing error is: • The Average SNR due to quantizing errors is: • Above equations can be expresses in decibels as, Where, M = 2 n α = 4. 77 for peak SNR α = 0 for average SNR 4

DESIGN OF A PCM SIGNAL FOR TELEPHONE SYSTEMS • • Assume that an analog audio voice-frequency(VF) telephone signal occupies a band from 300 to 3, 400 Hz. The signal is to be converted to a PCM signal for transmission over a digital telephone system. The minimum sampling frequency is 2 x 3. 4 = 6. 8 ksample/sec. To be able to use of a low-cost low-pass antialiasing filter, the VF signal is oversampled with a sampling frequency of 8 ksamples/sec. This is the standard adopted by the Unites States telephone industry. Assume that each sample values is represented by 8 bits; then the bit rate of the binary PCM signal is 8 • • This 64 -kbit/s signal is called a DS-0 signal (digital signal, type zero). The minimum absolute bandwidth of the binary PCM signal is This B is for a sinx/x type pulse sampling 5

DESIGN OF A PCM SIGNAL FOR TELEPHONE SYSTEMS • If we use a rectangular pulse for sampling the first null bandwidth is given by • We require a bandwidth of 64 k. Hz to transmit this digital voice PCM signal, whereas the bandwidth of the original analog voice signal was, at most, 4 k. Hz. • We observe that the peak signal-to-quantizing noise power ratio is: Note: 1. Coding with parity bits does NOT affect the quantizing noise, 2. However coding with parity bits will improve errors caused by channel or ISI, which will be included in Pe ( assumed to be 0). 6

Nonuniform Quantization Ø Many signals such as speech have a nonuniform distribution. – The amplitude is more likely to be close to zero than to be at higher levels. Ø Nonuniform quantizers have unequally spaced levels – The spacing can be chosen to optimize the SNR for a particular type of signal. Output sample XQ 6 4 Example: Nonuniform 3 bit quantizer 2 -8 -6 -4 -2 2 -2 4 6 8 Input sample X -4 -6 7

Companding • Nonuniform quantizers are difficult to make and expensive. • An alternative is to first pass the speech signal through a nonlinearity before quantizing with a uniform quantizer. • The nonlinearity causes the signal amplitude to be Compressed. – The input to the quantizer will have a more uniform distribution. • At the receiver, the signal is Expanded by an inverse to the nonlinearity. • The process of compressing and expanding is called Companding. 8

-Law Companding Output |x(t)| • Telephones in the U. S. , Canada and Japan use -law companding: – Where = 255 and |x(t)| < 1 Input |x(t)| 9

Non Uniform quantizing • Voice signals are more likely to have amplitudes near zero than at extreme peaks. • For such signals with non-uniform amplitude distribution quantizing noise will be higher for amplitude values near zero. • A technique to increase amplitudes near zero is called Companding. Effect of non linear quantizing can be obtained by first passing the analog signal through a compressor and then through a uniform quantizer. x x’ C(. ) Compressor x’ y Q(. ) Uniform Quantizer 10

Example: -law Companding x[n]=speech /song/ y[n]=C(x[n]) Companded Signal Segment of x[n] Close View of the Signal Segment of y[n] Companded Signal 11

• • • A-law and -law Companding These two are standard companding methods. u-Law is used in North America and Japan A-Law is used elsewhere to compress digital telephone signals 12

SNR of Compander • The output SNR is a function of input signal level for uniform quantizing. • But it is relatively insensitive for input level for a compander 13

SNR Performance of Compander • The output SNR is a function of input signal level for uniform quantizing. • But it is relatively insensitive for input level for a compander. • α = 4. 77 - 20 Log ( V/xrms) for Uniform Quantizer V is the peak signal level and xrms is the rms value • α = 4. 77 - 20 log[Ln(1 + μ)] • α = 4. 77 - 20 log[1 + Ln A] for μ-law companding for A-law companding 14

V. 90 56 -Kbps PCM Computer modem • The V. 90 PC Modem transmits data at 56 kb/s from a PC via an analog signal on a dial-up telephone line. • A μ law compander is used in quantization with a value for μ of 255. • The modem clock is synchronized to the 8 -ksample/ sec clock of the telephone company. • 7 bits of the 8 bit PCM are used to get a data rate of 56 kb/s ( Frequencies below 300 Hz are omitted to get rid of the power line noise in harmonics of 60 Hz). • SNR of the line should be at least 52 d. B to operate on 56 kbps. • If SNR is below 52 d. B the modem will fallback to lower speeds ( 33. 3 kbps, 28. 8 kbps or 24 kbps). 15