Quantization Noise

Quantization noise is the error introduced when a continuous amplitude sample is approximated to the nearest discrete quantization level during the PCM encoding process. Since no finite number of quantization levels can exactly represent every possible analog value, this approximation error is unavoidable and fundamentally limits the signal fidelity in any digital communication system. Understanding quantization noise is essential for designing PCM systems with adequate signal quality, and it is a central topic in GATE analog communication questions.

Uniform Quantization and its Noise Characteristics

In uniform quantization, the total amplitude range of the signal is divided into L equal intervals, each of width delta. When a sample falls within an interval, it is assigned the midpoint value of that interval. The error, which is the difference between the true sample value and its assigned quantized value, lies in the range from negative delta divided by 2 to positive delta divided by 2. This error is called the quantization error.

For a large number of quantization levels, the quantization error is modeled as a uniformly distributed random variable over the interval from negative delta divided by 2 to positive delta divided by 2. Under this model, the mean value of the error is zero and its variance, which represents the quantization noise power, equals delta squared divided by 12.

The fundamental problem with uniform quantization is that the noise power is constant (delta squared divided by 12) regardless of the signal amplitude, but the signal power varies with the input level. For large amplitude signals, the signal-to-noise ratio is acceptable. However, for small amplitude signals, the signal power falls while the noise power stays the same, leading to a poor SNR for weak signals. This is particularly problematic for speech signals, where low-level sounds such as whispers carry important information and must not be drowned in quantization noise.

Non-Uniform Quantization

Non-uniform quantization solves the problem of poor SNR for small-amplitude signals by using smaller step sizes for small amplitude values and larger step sizes for large amplitude values. Since most practical signals including speech spend more time at low amplitudes, using finer quantization at those levels improves the average SNR without increasing the number of bits per sample. The result is a nearly constant SQNR across the full dynamic range of the signal.

Non-uniform quantization is implemented in practice through a technique called companding, where the signal is first passed through a compressor (a non-linear amplifier that amplifies weak signals more than strong ones), then uniformly quantized, transmitted, and finally expanded at the receiver using the inverse non-linearity. The name companding comes from the combination of the words compressing and expanding.

Mathematical Expressions

For uniform PCM with a full-load sinusoidal input of amplitude A, the signal power is A squared divided by 2. The quantization noise power is delta squared divided by 12, where delta equals 2A divided by 2 raised to n. Substituting and simplifying gives SQNR equals 3 times 2 raised to 2n divided by 2, which in decibels becomes approximately 6.02n plus 1.76 dB. This derivation is important to understand, not just the final formula.

For a general signal with power Px, the SQNR becomes 3 times 2 raised to 2n times Px divided by A squared. This shows that SQNR depends on the ratio of signal power to maximum amplitude squared, which is essentially the loading factor. When the signal power is much less than A squared (under-loaded quantizer), SQNR drops significantly, which is the root cause of the problem that companding addresses.

Granular Noise and Slope Overload

In delta modulation (a variant of PCM), two specific types of quantization noise appear. Granular noise occurs when the signal changes slowly and the delta modulator over-corrects back and forth around the actual value, producing a staircase that oscillates by one step on either side. Slope overload distortion occurs when the signal changes faster than the delta modulator can track, because the fixed step size is too small to keep up with a steep slope. These two effects set a fundamental trade-off in the step size selection for delta modulation.

Given:
Number of bits per sample n = 8
Signal range: -4V to +4V (A = 4V)
Input: sinusoidal, full-load

Why this formula applies:
Uniform quantization with sinusoidal full-load input.
SQNR = 6.02n + 1.76 dB (standard PCM SQNR formula)

Formula:
delta = 2A / 2^n
Noise power = delta^2 / 12
SQNR (dB) = 6.02n + 1.76

Substitution:
delta = 8 / 256 = 0.03125 V
Noise power = (0.03125)^2 / 12 = 9.765e-4 / 12
            = 8.138e-5 V^2
SQNR = 6.02 x 8 + 1.76 = 49.92 dB

Final Answer:
Step size = 31.25 mV, Noise power = 8.14e-5 V^2,
SQNR = 49.92 dB

Exam Tip: For GATE, the SQNR formula 6.02n + 1.76 dB applies only to uniform quantization with a sinusoidal full-load input. Doubling the number of quantization levels (adding 1 bit) improves SQNR by 6 dB. If the input power drops by half (3 dB below full load), SQNR also drops by 3 dB. Non-uniform quantization maintains near-constant SQNR across the dynamic range using companding.

• Quantization error lies uniformly between negative delta/2 and positive delta/2. Noise power equals delta squared divided by 12.
• SQNR for uniform quantization with sinusoidal full-load input is 6.02n + 1.76 dB. Every bit adds 6 dB.
• Uniform quantization gives poor SNR for small-amplitude signals because noise power is constant while signal power varies.
• Non-uniform quantization places finer steps at low amplitudes and coarser steps at high amplitudes, equalizing SQNR across the dynamic range.
• Non-uniform quantization is implemented via companding: compress signal before quantizer, expand after decoder.

Quick Revision

• Quantization noise power = delta squared / 12, where delta is the step size.
• SQNR = 6.02n + 1.76 dB for uniform quantization, sinusoidal full-load input only.
• Uniform quantization: constant noise power, SQNR poor at low amplitudes.
• Non-uniform quantization: smaller delta at low amplitudes, near-constant SQNR over dynamic range.
• Companding = compress + expand. Implemented using mu-law (North America, Japan) and A-law (Europe, India) standards.
• GATE trap: SQNR formula is not valid for non-sinusoidal or low-power inputs. For a signal loaded at half full-scale, SQNR decreases by 6 dB compared to the full-load value.