PCM and ADPCM¶

Exam-ready answer

Pulse Code Modulation (PCM) represents each analog sample by an \(n\)-bit binary word. The complete system must preserve sampling, amplitude coding, bit timing and frame/channel position.

Part A — PCM transmitter [5 marks]¶

1. Anti-aliasing LPF: limits message bandwidth to \(f_m<f_s/2\), preventing spectral replicas from overlapping after sampling.
2. Sampler/sample-and-hold: takes uniform samples at \(f_s\geq2f_m\) and holds each value constant during quantization. For standard speech, \(f_s=8\,\text{kHz}\) and \(T_s=125\,\mu\text{s}\).
3. Compressor, if used: A-law or \(\mu\)-law compresses speech dynamic range to improve low-level quantization performance.
4. Quantizer: rounds the sample to one of \(L=2^n\) levels. For ideal span \(V_{FS}\), \(\Delta=V_{FS}/2^n\) and nearest-level error is approximately within \(\pm\Delta/2\).
5. Encoder and serializer/line coder: assigns the selected level an \(n\)-bit word, serializes words, inserts framing/signaling as required, and produces a clock-recoverable physical waveform.

The uncoded single-channel payload rate is

Part B — PCM receiver [5 marks]¶

1. Input filter/equalizer and 3R regenerator: limits noise/ISI, recovers clock, and re-amplifies, reshapes and retimes each line symbol by threshold decisions.
2. Line decoder, de-framer and deserializer: removes the physical line code, finds frame/slot boundaries and reconstructs each \(n\)-bit sample word. Bit timing identifies bits; frame timing identifies channel positions.
3. PCM decoder: maps each valid codeword back to its quantized amplitude. An expander applies the inverse A/\(\mu\) law when companding was used.
4. DAC/hold: converts each word to an analog level and holds it for one sample period, producing a staircase/pulse waveform.
5. Reconstruction LPF: passes the wanted baseband and suppresses sampling images/switching components before the destination.

The recovered waveform differs from the source by quantization error, filtering distortion and any uncorrected bit errors. For example, 8 bits/sample at \(8\,\text{kHz}\) gives \(64\,\text{kbps}\) before framing overhead. A significant-bit error can cause a much larger sample error than an LSB error, so regeneration and synchronization are essential.

Practice target: 16–18 minutes; divide time equally between the two five-mark chains and label clock, frame, companding and reconstruction paths.

Exam-ready answer

Pulse Code Modulation (PCM) is a digital waveform-coding method in which every uniformly timed analog sample is quantized and represented by a finite binary word. It has three defining operations: sampling discretizes time, quantization discretizes amplitude, and encoding assigns an \(n\)-bit code to one of \(2^n\) levels.

The transmitter uses anti-alias LPF \(\rightarrow\) S/H \(\rightarrow\) optional compressor \(\rightarrow\) quantizer \(\rightarrow\) encoder/line coder. Sampling must satisfy \(f_s\geq2f_m\) with practical guard band; the S/H keeps the input fixed during conversion. The receiver performs 3R regeneration and timing/frame recovery, decodes the word, applies optional expansion, converts it through a DAC/hold, and removes sampling images with a reconstruction LPF. The payload rate is

Advantages: threshold-based 3R repeaters prevent progressive analog noise accumulation; quality is reproducible over long links while decisions remain correct; PCM naturally supports TDM, digital switching and framing; binary data is easy to encrypt, store, process and protect with error-control coding; voice, data and signaling can share digital infrastructure; and IC implementation is stable and repeatable.

PCM is not cost-free. Finite levels create quantization noise, accurate bit/word/frame synchronization is required, and its bit stream commonly needs much more bandwidth than the original analog message. For standard telephone speech, \(n=8\) and \(f_s=8\,\text{kHz}\) give \(64\,\text{kbps}\) before framing. A-law/\(\mu\)-law companding improves weak-speech SQNR but does not eliminate quantization error. The complete five-mark answer should therefore state both the digital-regeneration benefit and the bandwidth/quantization trade-off.

Practice target: 8–9 minutes; define all three conversion operations, trace both directions, list five advantages and close with two limitations.

Exam-ready answer

Given sampling frequency \(f_s=8\,\text{kHz}\) and word length \(n=8\) bits/sample, the uncoded single-channel PCM payload rate is

For ideal two-level baseband signaling with zero intersymbol interference, Nyquist's pulse-transmission limit is \(R_b=2B\) bit/s. Therefore

This is a theoretical lower bound, not the bandwidth of every PCM line. Actual occupied bandwidth depends on line code, pulse shape, raised-cosine roll-off, equalization and the definition of bandwidth. Manchester coding needs more spectrum; framing and signaling raise the transmitted bit rate above the \(64\,\text{kbps}\) payload.

For an ideal uniform \(n\)-bit quantizer driven by a full-scale sinusoid, the quantization signal-to-noise ratio is

The assumptions matter: the error is modeled as uniformly distributed with power \(\Delta^2/12\), there is no clipping, jitter, thermal noise or nonlinearity, and the sine uses full scale. A signal backed off by \(A\,\text{dB}\) has approximately \(A\,\text{dB}\) lower uniform-quantizer SQNR. Telephone A/\(\mu\)-law companding deliberately changes SQNR versus input level, so \(49.92\,\text{dB}\) is not a universal measured speech-channel value.

Practice target: 7–8 minutes; show units on all three boxed results and state the signaling and full-scale-sine assumptions.

Exam-ready answer

The three coders differ mainly in what is quantized. PCM sends an absolute sample; DPCM sends prediction error; ADPCM also changes predictor and/or quantizer parameters as signal statistics change.

PCM: sample \(x[n]\) is directly quantized and encoded. It needs no predictor and a conventional telephone example uses 8 bits/sample at \(8\,\text{kHz}\):

It is robust and transparent for general waveforms, but it ignores sample-to-sample correlation.

DPCM: the encoder predicts \(\hat{x}[n]\) from previous locally reconstructed samples and quantizes only

The encoder uses reconstructed, not original, past samples so its predictor state matches the decoder. Speech samples are correlated, so a good predictor makes \(\operatorname{var}(e)\) much smaller than \(\operatorname{var}(x)\). The smaller dynamic range can be represented by fewer bits for comparable distortion. Poor prediction, channel errors or rapidly changing inputs reduce this gain.

ADPCM: it retains the prediction-error loop but adapts quantizer step size, predictor coefficients or both from recent reconstructed data.

The sequence is predict \(\rightarrow\) subtract \(\rightarrow\) quantize error \(\rightarrow\) transmit compact code \(\rightarrow\) reconstruct \(\rightarrow\) update identical encoder/decoder states. Rapid or large errors increase step size to avoid overload; quiet intervals reduce it to lower granular quantization noise. A common ITU-T G.726 mode uses 4 bits/sample at \(8\,\text{kHz}\):

half the payload rate of 8-bit PCM for useful telephony speech quality.

At equal line coding/pulse shaping, lower \(R_b\) permits proportionally lower channel bandwidth. The saving is statistical, not lossless: ADPCM exploits correlation and accepts adaptive quantization/prediction distortion, more state complexity and possible error propagation. It performs less predictably on uncorrelated or high-fidelity arbitrary audio than on speech.

Practice target: 18–20 minutes; draw both feedback coders, write the prediction equations, calculate 64/32 kbps and reproduce the comparison table.

Exam-ready answer

Delta modulation (DM) is a one-bit predictive coder. At each high-rate sample it compares the input \(x[n]\) with a local staircase estimate \(x_q[n-1]\) and sends only the sign of the error. A 1 raises the estimate by fixed step \(\Delta\); a 0 lowers it:

The transmitter comparator produces the one-bit sequence and a feedback accumulator/integrator constructs the same staircase used for the next comparison. The receiver applies the bits to an identical accumulator and then low-pass filters the staircase. DM uses one bit/sample but normally samples far above the Nyquist rate, so its bit rate is \(R_b=f_s\) rather than necessarily being small.

Slope-overload distortion: the staircase can change by only \(\Delta\) in one period \(T_s\), so its maximum tracking slope is

For \(x(t)=A_m\sin(2\pi f_m t)\), the maximum source slope is \(2\pi f_mA_m\). Avoiding sustained overload requires approximately

If \(\Delta\) or \(f_s\) is too small, long runs of identical bits command the steepest staircase but it still falls behind the waveform.

Granular noise: when the input is nearly constant or changes slowly and \(\Delta\) is too large, the estimate repeatedly jumps above and below it. Alternating up/down bits create a grain-like fluctuation. Decreasing \(\Delta\) reduces this error but worsens slope overload, giving the fixed-step trade-off.

Numerical check: for \(A_m=1\,\text{V}\), \(f_m=1\,\text{kHz}\) and \(f_s=32\,\text{kHz}\),

A fixed \(0.10\,\text{V}\) step can supply only \(3200\,\text{V/s}\) against the required \(6283\,\text{V/s}\) and will overload near the steepest crossing. A \(0.25\,\text{V}\) step can track that slope but may chatter badly on flat portions.

Adaptive delta modulation (ADM) resolves much of the conflict by varying \(\Delta[n]\). Several identical consecutive bits indicate sustained slope, so the controller increases the step; alternating bits indicate a slowly varying region, so it decreases the step. Both transmitter and receiver apply the same update rule to remain synchronized. ADM reduces, but cannot completely eliminate, overload and granular error; channel bit errors can also perturb the adaptive state.

Practice target: 18–20 minutes; draw both feedback accumulators, derive the slope condition with units, sketch both errors and solve one minimum-step example.

Exam-ready answer

PCM and ADPCM are waveform coders, but PCM quantizes absolute sample amplitude whereas ADPCM quantizes a smaller, adaptively scaled prediction error.

ADPCM saves rate because adjacent speech samples are correlated. If prediction is good, \(\operatorname{var}(e)\ll\operatorname{var}(x)\), so fewer quantizer bits can describe the error with useful speech quality. At \(8\,\text{kHz}\) the common 4-bit example halves payload rate from 64 to \(32\,\text{kbps}\); under ideal two-level Nyquist signaling its lower-bound bandwidth correspondingly changes from 32 to \(16\,\text{kHz}\). Those bandwidth values exclude framing and depend on signaling assumptions.

PCM is simpler, more robust to state disturbance and more transparent for arbitrary signals. ADPCM is more bandwidth-efficient for correlated speech but is state-dependent, can propagate channel errors and may perform poorly when the signal changes unpredictably. The lower rate is therefore achieved by exploiting source redundancy, not by carrying the same absolute samples losslessly.

Practice target: 8–9 minutes; reproduce the seven comparison rows, the prediction equation and the 64-to-32 kbps calculation.