Q1Digital Communication

Question

Q.1 (a) What is meant by Quantization? Explain the need of quantization of signals and derive an expression for Quantization error. [10]

(b) An audio signal, s(t) = 3cos(2π×500t) is Quantized using 10-bit PCM. Determine the signal-to-quantization noise ratio. [6]

Answer 1

(a) Quantization, Its Need, and Quantization Error

Quantization is the process, within the analog-to-digital conversion chain (specifically within Pulse Code Modulation, PCM), of mapping each continuous-amplitude sample of an analog signal to the nearest value among a finite, discrete set of representation (quantization) levels, so that the sample can subsequently be represented by a finite-length binary code word for digital transmission, storage, or processing.

Need for quantization: an analog signal's amplitude can take on any value within a continuous range (theoretically infinite precision), but a digital communication or storage system can only represent, transmit, and process a finite number of discrete symbols (bits) — quantization is therefore an essential, unavoidable step in converting any real-world analog signal into a digital representation suitable for the many benefits of digital transmission (superior noise immunity through regeneration, ease of multiplexing/encryption/error correction, and compatibility with digital processing and storage systems) — without quantization, the sampled analog values could not be represented by a finite number of bits at all.

Derivation of Quantization Error Expression

Consider a uniform quantizer with L discrete quantization levels spanning the signal's full dynamic range from -Vmax to +Vmax, giving a uniform step size:

Δ = \frac{2 V _{ma x}}{L}

The quantization error e = x - x_q (the difference between the actual continuous sample value x and its quantized representation x_q) is bounded within ±Δ/2 for any input sample (since the quantizer always rounds to the nearest level, at most half a step size away). Assuming this quantization error is uniformly distributed over the interval [-Δ/2, +Δ/2] (a standard, well-justified assumption for a signal with sufficiently fine quantization relative to its dynamic range and reasonably 'busy'/non-constant amplitude variation), the probability density function of the error is f(e) = 1/Δ for -Δ/2 ≤ e ≤ Δ/2, and zero elsewhere.

The mean-square quantization error (quantization noise power) is then found by integrating e² weighted by this uniform PDF:

\overline{e^{2}} = \int_{- Δ/2}^{Δ/2} e^{2} \cdot \frac{1}{Δ} d e = \frac{1}{Δ} [\frac{e ^{3}}{3}]_{- Δ/2}^{Δ/2} = \frac{1}{Δ} \times \frac{2}{3} \times (\frac{Δ}{2})^{3} = \frac{Δ ^{2}}{12}

Result: the mean-square quantization error (average quantization noise power) is Δ²/12, where Δ is the quantizer's step size. This fundamental result shows that quantization noise power decreases with the square of the step size — halving the step size (equivalent to doubling the number of quantization levels, i.e., adding one more bit to the PCM code word) reduces the quantization noise power by a factor of 4 (6 dB), which is the origin of the well-known '6 dB per bit' improvement rule relating PCM signal-to-quantization-noise ratio to the number of bits per sample.

(b) Signal-to-Quantization Noise Ratio for the Given Audio Signal

Given: s(t) = 3cos(2π×500t), a sinusoidal signal with peak amplitude Am=3, quantized using n=10-bit PCM.

For a sinusoidal input signal quantized using n-bit uniform PCM, the standard signal-to-quantization-noise ratio (SQNR) formula is:

S QN R (dB) = 6.02 n + 1.76

This formula is derived from the ratio of the sinusoidal signal's average power (Am²/2, independent of the specific amplitude Am as long as the quantizer's full range is matched to the signal's peak amplitude) to the quantization noise power Δ²/12 (with Δ=2Am/L=2Am/2^n for an n-bit quantizer spanning the full signal range), giving SQNR = 3×2^(2n)/2 in linear terms, which in decibels simplifies to the compact 6.02n+1.76 expression shown above.

Substituting n=10:

S QN R = 6.02 \times 10 + 1.76 = 60.2 + 1.76 = 61.96 dB

Result: the signal-to-quantization-noise ratio for this 10-bit PCM system is approximately 61.96 dB. This result is independent of the specific peak amplitude (3V) and frequency (500 Hz) of the given signal, since the standard SQNR formula for a full-scale sinusoidal input depends only on the number of quantization bits n, confirming that adding more bits to the PCM representation directly and predictably improves the achievable signal-to-quantization-noise ratio by approximately 6 dB per additional bit, a key design trade-off (bit rate/bandwidth vs quantization noise performance) in the design of any PCM-based digital communication system.

Cross-Check via Direct Power Ratio Computation

It is instructive to verify the 61.96 dB figure directly from first principles rather than relying solely on the compact formula, since this both confirms the arithmetic and clarifies exactly which assumptions the formula embeds. With n=10 bits, the number of quantization levels is L=2^10=1024, and since the quantizer's full range is matched to the signal's peak amplitude Am=3V (so Vmax=Am=3), the step size is Δ=2Vmax/L=6/1024=0.005859 V.

\overline{e^{2}} = \frac{Δ ^{2}}{12} = \frac{( 0.005859 ) ^{2}}{12} \approx 2.861 \times 1 0^{- 6} V^{2}

The average signal power of the sinusoid s(t)=3cos(2π×500t) is Am²/2 = 9/2 = 4.5 W (across a notional 1-ohm reference resistance, as is standard convention in these calculations). Taking the ratio of signal power to quantization noise power directly:

\frac{S}{N _{q}} = \frac{4.5}{2.861 \times 1 0 ^{- 6}} \approx 1.573 \times 1 0^{6}

10 lo g_{10} (1.573 \times 1 0^{6}) \approx 61.97 dB

This matches the 61.96 dB obtained from the compact formula (the small 0.01 dB discrepancy is simply rounding in the intermediate decimal values), confirming both the correctness of the calculation and the internal consistency of the 6.02n+1.76 shortcut formula with the underlying Δ²/12 quantization-noise derivation from part (a).

Physical Interpretation and Design Implications

This result carries direct practical significance for the design of any PCM-based audio or instrumentation system. An SQNR of about 62 dB is broadly comparable to the dynamic range of standard telephone-quality speech coding (where 8-bit companded PCM, using non-uniform mu-law/A-law quantization rather than simple uniform quantization, achieves an effectively similar perceptual SQNR over the voice signal's typical dynamic range), whereas high-fidelity audio applications such as CD-quality digital audio use 16-bit linear PCM specifically because 16 bits deliver an SQNR of about 6.02×16+1.76 ≈ 98 dB, comfortably exceeding the roughly 90-96 dB dynamic range of human hearing under typical listening conditions. This illustrates why the number of bits per sample is chosen based on the specific application's required dynamic range and perceptual/measurement fidelity, rather than an arbitrary or universal value — a telemetry or control-system ADC digitizing a low-dynamic-range sensor signal may need only 8-10 bits, while high-fidelity audio, professional instrumentation, and precision measurement systems require considerably more.

It is also worth noting an important caveat implicit in the derivation: this entire calculation assumes the quantizer's full input range exactly matches the peak amplitude of the input signal (i.e., no headroom or clipping margin, and no signal underutilizing the quantizer's available range). If the actual signal amplitude were smaller than the quantizer's designed full-scale range Vmax (for example, due to conservative gain staging to avoid clipping on louder signal peaks), the effective SQNR achieved would be correspondingly lower than the 61.96 dB calculated here, since the signal would then be quantized using fewer of the available quantization levels than the full L=1024 — this is precisely why proper automatic gain control and headroom management are essential practical considerations in real PCM system design, to ensure the input signal is scaled to use as much of the quantizer's available dynamic range as possible (without clipping) so as to achieve close to the maximum theoretically available SQNR for the chosen bit depth.

Finally, this problem also illustrates the distinction between quantization noise (a deterministic, signal-dependent distortion arising purely from the finite resolution of the analog-to-digital conversion process, present even over an ideal, noiseless transmission channel) and channel/transmission noise (additive noise introduced during transmission of the encoded bits themselves, which can additionally cause bit errors and further degrade the overall received SNR, as explored quantitatively in the following question). The 61.96 dB figure computed here represents only the quantization-noise-limited SQNR of an ideal, error-free PCM link; the overall end-to-end SNR of a practical PCM system additionally depends on the channel's bit-error rate, and can never exceed this quantization-limited value even under perfect, error-free transmission conditions.