PCM Basics

Pulse Code Modulation is the foundation of all modern digital communication systems. In PCM, an analog message signal is converted into a sequence of binary codewords by performing three operations in sequence: sampling, quantization, and encoding. Each binary codeword represents the quantized amplitude of one sample, and the resulting bitstream can be transmitted over digital channels, stored in memory, or processed by digital systems without any loss of the essential information beyond the quantization step.

Three Steps of PCM

The first step is sampling, where the continuous-time message signal is sampled at a rate fs that satisfies the Nyquist criterion: fs must be at least twice the highest frequency component W in the message signal. This produces a PAM sequence of instantaneous sample values. Under-sampling causes aliasing, which permanently distorts the recovered signal, so the sampling rate is a critical design parameter.

The second step is quantization. Since sample values are continuous and can take infinitely many levels, they must be approximated to one of a finite set of discrete levels. The number of quantization levels L equals 2 raised to the power n, where n is the number of bits per sample. The spacing between adjacent levels is called the step size delta, and it equals the total signal range divided by L. Rounding a sample to its nearest quantization level introduces an irreversible error called quantization noise or quantization error.

The third step is encoding, where each quantized level is mapped to its binary codeword of n bits using natural binary coding or Gray coding. Gray coding is preferred in practice because adjacent levels differ by only one bit, so a single bit error at the receiver maps the decoded value to an adjacent level rather than a widely different one, limiting error magnitude.

Mathematical Expressions in PCM

The number of quantization levels is L equals 2 raised to n. The step size for uniform quantization over a signal range from negative A to positive A is delta equals 2A divided by L equals 2A divided by 2 raised to n. The maximum quantization error magnitude is plus or minus delta divided by 2.

The bit rate R of the PCM bitstream equals n times fs bits per second, where fs is the sampling frequency. The minimum transmission bandwidth required equals R divided by 2 equals n times fs divided by 2, using ideal Nyquist signaling. Since fs must be at least 2W, the minimum PCM bandwidth is at least n times W, which is n times larger than the original message bandwidth W. This is the fundamental bandwidth expansion trade-off in PCM.

The signal-to-quantization-noise ratio for a full-load sinusoidal input in a uniform PCM system is given by SQNR in dB approximately equal to 6.02 times n plus 1.76 dB. This is one of the most important GATE formulas in this chapter. Each additional bit in the codeword increases the SQNR by approximately 6 dB, which means quantization noise halves in voltage for every extra bit.

Practical Implications of PCM

PCM is the basis of all digital telephony. Standard telephone PCM uses 8000 samples per second (twice the 4 kHz voice bandwidth) and 8 bits per sample, giving a bit rate of 64 kbps per voice channel. Audio CDs use 44100 samples per second and 16 bits per sample per channel, producing approximately 1.4 Mbps for stereo audio. These numbers follow directly from the PCM formulas and are useful reference points for GATE numerical problems.

PCM is regenerative, meaning repeater stations can fully reconstruct the digital pulses before noise accumulates to dangerous levels. This is the single most important practical advantage of PCM over analog transmission: noise does not accumulate along the transmission path as long as the bit error rate at each repeater is low.

Given:
Message signal bandwidth W = 4 kHz
Number of bits per sample n = 8
Sampling rate fs = 2 x W = 8000 samples/sec
Signal range: -4V to +4V (total 8V)

Why this formula applies:
Uniform PCM with Nyquist sampling. Step size
and SQNR both depend on n and signal range.

Formula:
L = 2^n
delta = 2A / L where A = 4V (half range)
SQNR (dB) = 6.02n + 1.76
Bit rate R = n x fs
Min BW = R / 2

Substitution:
L = 2^8 = 256 levels
delta = 8 / 256 = 0.03125 V
SQNR = 6.02(8) + 1.76 = 49.92 dB
R = 8 x 8000 = 64000 bits/sec = 64 kbps
Min BW = 64000 / 2 = 32 kHz

Final Answer:
L = 256, delta = 31.25 mV, SQNR = 49.92 dB,
Bit rate = 64 kbps, Minimum BW = 32 kHz

Exam Tip: The formula SQNR = 6.02n + 1.76 dB is valid ONLY for uniform quantization with a full-load sinusoidal input. For GATE, memorize that each extra bit adds approximately 6 dB to SQNR. Also note that PCM bandwidth equals n times the message bandwidth, not n times the sampling frequency.

• Sampling converts continuous m(t) to PAM. Nyquist criterion requires fs greater than or equal to 2W to avoid aliasing.
• Quantization rounds each sample to one of L = 2 raised to n levels. Step size delta = 2A / 2 raised to n for bipolar range negative A to A.
• Encoding maps each quantized level to an n-bit binary word using natural binary or Gray code.
• SQNR = 6.02n + 1.76 dB for sinusoidal full-load input. Every extra bit adds 6 dB.
• PCM bit rate = n times fs. Minimum transmission bandwidth = n times W.
• PCM is regenerative: noise does not accumulate along the repeater chain, unlike analog systems.

Quick Revision

• PCM three steps: sampling at fs greater than or equal to 2W, quantization to L = 2 raised to n levels, encoding to n-bit codewords.
• Step size: delta = 2A / 2 raised to n. Max quantization error = plus or minus delta / 2.
• SQNR = 6.02n + 1.76 dB (sinusoidal, full-load, uniform quantization).
• Bit rate R = n times fs. Minimum BW = n times W (Nyquist signaling).
• Voice PCM standard: 8 kHz sampling, 8 bits, 64 kbps per channel.
• GATE trap: SQNR formula is only for uniform quantization with sinusoidal input; for non-uniform or non-sinusoidal signals, the formula is different.
• PCM advantage over analog: regenerative repeaters prevent noise accumulation along transmission path.