Q2Digital Communication

(a) Adaptive Delta Modulation and Demodulation

Basic (linear) Delta Modulation (DM) encodes the difference between consecutive samples using just a single bit per sample, indicating only whether the signal has increased or decreased relative to the previous accumulator/predictor output, using a fixed step size Δ. This fixed-step-size approach suffers from two competing distortion mechanisms: slope overload distortion (occurring when the input signal changes faster than the fixed step size can track, causing the reconstructed staircase approximation to fall progressively behind a rapidly rising/falling input) and granular noise (occurring when the input signal is slowly varying or nearly constant, causing the fixed-size staircase steps to unnecessarily oscillate up and down around the nearly-constant input value, since the step size cannot become smaller than its fixed value even when a smaller step would represent the signal more accurately).

Adaptive Delta Modulation (ADM) addresses both of these competing distortion problems simultaneously by dynamically varying the step size in response to the recent pattern of transmitted bits — specifically, if several consecutive bits have the same value (indicating the signal is consistently rising or falling, a strong sign of slope-overload risk), the step size is progressively increased to allow the staircase approximation to track the rapidly-changing signal more closely; conversely, if the bit stream alternates frequently (0,1,0,1,...), a strong sign that the signal is nearly constant and only granular noise is being generated, the step size is progressively decreased to reduce this unnecessary granular noise.

ADM System Block Diagram

The ADM transmitter compares the input signal x(t) against a locally-generated staircase approximation (produced by integrating/accumulating the previously transmitted bit stream), the comparator/1-bit quantizer generates a single output bit each sampling instant indicating the sign of this difference, and the step-size adaptation logic monitors the recent bit pattern (typically the last 1-3 bits) to determine whether to increase the step size (same-sign run detected, indicating slope overload risk) or decrease it (alternating pattern detected, indicating granular-noise-dominated, near-constant signal region) for the next sample, with this updated step size then used both to update the local staircase integrator and to be implicitly signaled (since the decoder tracks the same bit-pattern history and applies the identical adaptation rule) to the receiver.

ADM Demodulator

The ADM demodulator (receiver) mirrors the adaptive step-size logic of the transmitter exactly: it receives the transmitted bit stream, applies the identical step-size adaptation rule (based on the same recent-bit-pattern history) used at the transmitter to reconstruct the correct, time-varying step size at each instant, and integrates/accumulates the received bits (each scaled by the currently adapted step size) to reconstruct the staircase approximation of the original signal, which is then passed through a low-pass filter to smooth out the staircase steps and recover a continuous approximation of the original analog signal — since both transmitter and receiver apply the exact same deterministic step-size adaptation rule based only on the transmitted bit history (which both sides have identical access to), the receiver's step size tracks the transmitter's step size exactly, without requiring any explicit side-information about the step size to be separately transmitted.

(b) SNR Comparison of PCM and DM Systems

Given: probability of error Pe = 10⁻⁶, number of bits per PCM code word n=8.

PCM signal-to-noise ratio: the overall SNR of a PCM system, accounting for both quantization noise and channel-induced bit errors, is given by the standard formula:

Substituting n=8 (so 2^n=256, 2^(2n)=65536) and Pe=10⁻⁶:

Converting to decibels:

Note on the effect of Pe: since Pe=10⁻⁶ is a very small error probability, the denominator term 4(2^(2n)-1)Pe = 1.262 is comparable in magnitude to the '1' term, indicating that even this small bit-error probability meaningfully degrades the PCM SNR from its error-free value of 3×2^(2n) = 196,608 (52.94 dB) down to the 51.92 dB actually calculated — illustrating that PCM system performance depends on both the quantization step size (governed by n) and the channel's bit-error rate, with a sufficiently high Pe eventually causing the SNR to degrade sharply (dominated by transmission errors rather than quantization noise) beyond a certain 'threshold' Pe value.

DM signal-to-noise ratio (comparative discussion): delta modulation, by design, uses only 1 bit per sample (rather than PCM's n bits per sample), so for a fair comparison at the same overall transmission bit rate, a DM system would sample at n times the rate a comparable PCM system uses for the same total bit rate. However, DM's achievable SNR is fundamentally more constrained than PCM's, since DM's quantization step size is fixed (or, for basic linear DM as discussed conceptually here) by a single design parameter that must simultaneously balance the competing slope-overload and granular-noise distortion mechanisms discussed in part (a) — this generally limits basic (non-adaptive) DM's achievable SNR to a value considerably lower than an n-bit PCM system operating at the same overall transmission bit rate, particularly since DM's SNR does not improve nearly as favorably with increasing bit rate as PCM's characteristic 6 dB-per-bit improvement, since DM's granular noise power depends on the sampling rate in a comparatively less favorable manner.

Comment on the result: this comparison illustrates the classic, well-established trade-off between PCM and DM: PCM offers substantially better SNR performance for a given bit rate (as confirmed by the ~52 dB SNR calculated above for just 8 bits/sample), at the cost of requiring more complex encoder/decoder circuitry (full analog-to-digital and digital-to-analog conversion, requiring an n-bit quantizer and n-bit code word per sample) compared to DM's much simpler 1-bit-per-sample encoder/decoder design — DM is therefore favored in applications prioritizing implementation simplicity and low bit rate over the highest achievable SNR, while PCM (and its refined variants such as DPCM and ADPCM) remains the standard choice wherever higher fidelity/SNR is the primary requirement, as in most modern digital telephony and audio systems.

Relating ADM back to this SNR comparison: the adaptive delta modulation scheme described in part (a) is, in essence, an engineering response to exactly this SNR shortfall of basic DM relative to PCM — by dynamically expanding and contracting the step size, ADM widens the effective dynamic range over which the modulator avoids both slope-overload and granular-noise distortion simultaneously, allowing an adaptive DM system to close much of the SNR gap against a fixed-step DM system operating at the same sampling rate, though it still does not generally match a full n-bit PCM system's SNR at the same total transmission bit rate, since ADM fundamentally still transmits only 1 information-bearing bit per sample regardless of how well its step size is adapted, whereas PCM devotes an entire n-bit code word to resolving each individual sample with fine amplitude precision. This is why ADPCM (Adaptive Differential PCM), which combines the SNR benefits of multi-bit PCM-style level quantization with the bit-rate savings of adaptive differential/predictive encoding, is preferred over both simple ADM and full-rate linear PCM in many modern voice-coding standards, such as the ITU-T G.726 speech codec used historically in digital telephony trunking, offering a favorable compromise point on the same fundamental bit-rate-versus-SNR trade-off curve illustrated numerically above.