Sampling and digital modulation

In short

This topic covers how a continuous-time signal is turned into a stream of bits (sampling, quantizing, coding) and how those bits are then carried on a carrier (ASK, FSK, PSK, QAM and their relatives). It is the spine of every digital link in use in India - the 64 kbps PCM voice channel on BSNL's E1 trunks, GMSK in 2G GSM, QPSK/16-QAM/64-QAM/256-QAM in 4G and 5G NR, QPSK and 8-PSK on DTH satellite links, BPSK on NavIC. In exams it is heavily weighted: GATE EC almost always asks a sampling/aliasing numerical and a BER or bandwidth-efficiency comparison, and university papers ask PCM-vs-DM derivations. In placement rounds (Qualcomm, Samsung R&D, Nokia, Ericsson, Texas Instruments, Analog Devices, Tejas Networks, ISRO, DRDO, BEL) interviewers test whether you can connect a constellation diagram to bandwidth, power and error rate.

Explained

Sampling theorem, aliasing and reconstruction

• Sampling multiplies a signal x(t) by an impulse train of period Ts, producing a spectrum that repeats X(f) at every multiple of fs = 1/Ts, each scaled by fs.
• Nyquist (low-pass) sampling theorem: a signal band-limited to fm Hz is completely recoverable from its samples if fs > 2fm. Nyquist rate = 2fm; Nyquist interval = 1/2fm.
• Aliasing: if fs < 2fm the repeated spectra overlap and a high frequency masquerades as a lower one. A tone at f0 sampled at fs appears at |f0 − k·fs| folded into 0 to fs/2.
• Folding frequency = fs/2. Everything above it folds down; the fold is a mirror about fs/2.
• Anti-aliasing filter: an analog low-pass filter *before* the sampler is mandatory - once aliasing has happened no DSP can undo it. Practical systems sample a little above 2fm to leave a transition band (speech: 3.4 kHz signal sampled at 8 kHz).
• Reconstruction: pass the samples through an ideal LPF of cut-off fs/2; in the time domain this is sinc interpolation, x(t) = Σ x(nTs) sinc[(t − nTs)/Ts]. An ideal brick-wall filter is non-causal, so real DACs use a zero-order hold plus a smoothing filter.
• Types of sampling: ideal (impulse), natural (pulse train of width τ, each pulse follows the signal - spectrum weighted by sinc but each harmonic band undistorted) and flat-top / sample-and-hold (each sample held constant).

Practical sampling: aperture effect, bandpass sampling, oversampling

• Aperture effect: flat-top sampling convolves with a rectangular pulse of width τ, so the spectrum is multiplied by τ·sinc(fτ) - high frequencies are attenuated. Corrected by an equalizer with response ∝ 1/sinc(fτ).
• Sample-and-hold circuit: switch + hold capacitor + buffer; errors are acquisition time, droop, pedestal error and aperture jitter (jitter limits SNR at high input frequencies).
• Bandpass sampling: a signal occupying fL to fH (bandwidth B = fH − fL) needs only fs = 2fH/k, where k = floor(fH/B) - far less than 2fH. Used in IF-sampling SDR receivers and in radar.
• Oversampling (fs >> 2fm) spreads quantization noise over a wider band; the in-band noise falls by 3 dB for every doubling of fs, i.e. 10 log10(OSR) dB, OSR = fs/2W. It also relaxes the anti-aliasing filter.
• Sigma-delta (ΔΣ) ADC: 1-bit quantizer inside a feedback loop pushes quantization noise out of band (noise shaping: ~9 dB, i.e. 1.5 bits, per doubling for first order), then decimation filters it away. Standard in audio codecs and modern SoCs.
• Undersampling on purpose is aliasing used as down-conversion; it also aliases wideband noise, so a pre-select filter is still required.

Quantization, companding and SQNR

• Quantization maps the continuous amplitude to one of L = 2ⁿ levels; the rounding error is the irreversible step in digitization.
• Uniform quantizer: step size Δ = (full-scale range)/L. Mid-rise (no zero level, even L) vs mid-tread (has a zero level - preferred for speech so silence codes to zero).
• Quantization noise is modelled as uniform in (−Δ/2, +Δ/2), independent of the signal, with power Δ²/12 - valid only when Δ is small and the signal is busy.
• Overload (saturation) distortion happens when the signal exceeds full scale; granular (quantization) noise dominates inside the range. Together they define dynamic range.
• Every extra bit doubles the number of levels and gives ~6 dB more SQNR, but adds fs bit/s to the rate - the central design trade-off.
• Non-uniform quantization / companding: speech amplitudes are small most of the time, so a uniform quantizer wastes levels. Compress before quantizing, expand after, so small signals get fine steps. SQNR becomes nearly independent of input level.
• Standards: ITU-T G.711 defines A-law (A = 87.6, 13-segment piecewise approximation) used in India/Europe with the E1 hierarchy, and μ-law (μ = 255, 15-segment) used in North America/Japan. Indian PSTN and TEC Generic Requirements follow A-law, 8 kHz, 8 bits, 64 kbps.
• Equivalent view: companded 8-bit PCM gives roughly the SQNR of 12-13 bit uniform coding for speech.

Waveform coding: PCM, DPCM, DM, ADM and TDM

• PCM chain: anti-aliasing LPF → sample-and-hold → compressor → quantizer → encoder (binary/Gray) → line coder → channel → regenerative repeater → decoder → expander → reconstruction LPF.
• Regenerative repeaters are the big advantage of PCM: they decide and re-transmit clean pulses, so noise does not accumulate as it does in analog links. They do add jitter, which does accumulate.
• Line codes (how bits become pulses on copper): unipolar/polar NRZ, RZ, AMI, HDB3 (used on 2.048 Mbps E1 in India), Manchester (Ethernet 10BASE-T), B8ZS (T1). Needs: no DC, clock recovery, error monitoring.
• DPCM: quantize the *prediction error* e(n) = x(n) − x̂(n) instead of the sample. Because speech and images are correlated, the error has smaller variance - a prediction gain Gp of 6-12 dB, i.e. 1-2 bits saved per sample at the same quality.
• ADPCM (ITU-T G.726) adapts both predictor and step size: toll-quality voice at 32 kbps, half of G.711.
• Delta modulation (DM): 1-bit DPCM - transmit only whether the staircase approximation must step up (+δ) or down (−δ). Very simple hardware; needs a high sampling rate (typically 8-16× Nyquist).
• DM's two errors: slope overload when the staircase cannot keep up (δ·fs < max |dx/dt|) and granular noise when the signal is nearly flat and the staircase hunts about it. A larger δ cures one and worsens the other.
• Adaptive DM (ADM): step size grows after consecutive identical bits and shrinks on alternations - handles both errors; used in CVSD for tactical/military and some Bluetooth voice links.
• TDM: interleaved samples from many sources share one channel. E1 (ITU-T G.732, India): 32 timeslots × 8 bits × 8000 frames/s = 2.048 Mbps, 30 voice channels + TS0 for sync + TS16 for signalling. T1 (1.544 Mbps, 24 channels) is the North American equivalent. Higher orders: E2 8.448, E3 34.368, E4 139.264 Mbps; modern transport uses SDH (STM-1 = 155.52 Mbps).
• Frame synchronization is essential in TDM - a lost frame alignment word scrambles every channel.

Baseband transmission: pulse shaping, ISI and optimum detection

• Inter-symbol interference (ISI) is the smearing of one symbol into its neighbours by a band-limited channel - the dominant impairment in wired and LOS links.
• Nyquist's first criterion: ISI-free transmission of Rs symbols/s needs a minimum baseband bandwidth of Rs/2 Hz (the ideal "brick wall" channel). Equivalently the pulse must be zero at all other sampling instants.
• Raised-cosine filter with roll-off α (0 to 1) is the practical compromise: baseband bandwidth (1+α)Rs/2, excess bandwidth α·Rs/2. Root raised cosine is split between transmitter and receiver so the cascade is raised cosine and the receiver is matched.
• Matched filter / correlation receiver maximizes SNR at the sampling instant; its output SNR = 2E/N0, independent of pulse shape. For equiprobable symbols of equal noise variance the optimum threshold is midway between the two expected levels.
• Eye diagram: overlay of many symbol intervals on a scope. Eye opening = noise margin, width = timing margin, thickness of crossings = jitter, slope at crossing = sensitivity to timing error.
• Equalization undoes residual ISI: linear (zero-forcing, MMSE), decision-feedback (DFE), adaptive (LMS, RLS).
• Scramblers randomize long runs of identical bits so clock recovery and spectral spreading stay healthy.
• Signal space view: any set of M waveforms can be written on an orthonormal basis found by Gram-Schmidt; detection becomes "pick the nearest constellation point", and error probability depends on the minimum Euclidean distance d_min.

Digital modulation: ASK, FSK, PSK, QAM

• ASK / OOK: amplitude switched between two values. Simple and cheap (IR remotes, 433 MHz RF modules), but poor noise performance and no constant envelope - unsuitable for power amplifiers driven near saturation.
• BFSK: two carrier frequencies. Constant envelope, so it tolerates non-linear amplifiers; can be detected non-coherently with two filters plus envelope detectors; needs more bandwidth. Sunde's FSK (Δf = Rb) gives orthogonal tones.
• BPSK: phase 0 or π - the most power-efficient binary scheme (antipodal, d_min = 2√Eb). Needs coherent carrier recovery, hence phase ambiguity, solved by differential encoding (DPSK, detected without a carrier reference at ~1 dB penalty).
• QPSK: 2 bits/symbol on two quadrature carriers. Same BER vs Eb/N0 as BPSK but half the bandwidth - the key result of the topic. Gray coding the constellation keeps adjacent symbols one bit apart, so a symbol error usually costs one bit error.
• Envelope control variants: OQPSK (offset QPSK) delays one arm by half a symbol so phase changes never exceed 90°, avoiding zero crossings; π/4-DQPSK (used in older TETRA/DECT-type systems) limits jumps to ±45°, ±135°; MSK is continuous-phase FSK with modulation index h = 0.5 - constant envelope, compact spectrum; GMSK pre-filters with a Gaussian of BT = 0.3 and is the GSM modulation.
• M-ary PSK and QAM: M-PSK keeps constant amplitude but points crowd on a circle, so d_min shrinks fast beyond 8-PSK. M-QAM places points on a rectangular grid, using both amplitude and phase, which is far more distance-efficient for large M - hence 16/64/256-QAM in LTE, 5G NR and microwave backhaul, and up to 1024/4096-QAM on short, high-SNR links.
• The fundamental trade: increasing M raises bandwidth efficiency (log2M bits/symbol) but demands more Eb/N0 for the same BER, and QAM needs a linear amplifier (high PAPR, lower DC efficiency).
• Coherent vs non-coherent: coherent detection needs carrier phase (costly, best performance); non-coherent (envelope, differential) is simpler and robust to phase noise but loses 1-3 dB.
• Carrier and clock recovery: squaring loop and Costas loop for BPSK, fourth-power/decision-directed loops for QPSK, early-late and Gardner detectors for symbol timing.

Comparison, standards and Indian practice

• Power efficiency ranking (best Eb/N0 first, binary): BPSK → coherent FSK ≈ coherent OOK → non-coherent FSK. Bandwidth efficiency ranking: 256-QAM → 64-QAM → 16-QAM → QPSK → BPSK.
• Adaptive modulation and coding (AMC): real 4G/5G links switch between π/2-BPSK, QPSK, 16-, 64-, 256-QAM every millisecond based on reported channel quality - the practical resolution of the power/bandwidth trade.
• Indian deployments: GSM 900/1800 uses GMSK (270.833 kbps in a 200 kHz carrier, BT = 0.3); EDGE added 8-PSK; 3G WCDMA/HSPA uses QPSK and 16-QAM; LTE/5G NR use OFDM-based QPSK to 256-QAM (5G NR adds π/2-BPSK on the uplink for better power efficiency); DTH and DD Free Dish use DVB-S2 QPSK/8-PSK; NavIC/IRNSS uses BPSK for the standard service and BOC for the restricted service; ISRO telemetry/telecommand links use PCM with BPSK/QPSK.
• Regulators and bodies to name in interviews: DoT (licensing), TRAI (tariff and regulation), WPC (spectrum assignment), TEC (Generic Requirements/type approval), BIS for equipment standards; ITU-T G-series for PCM and ITU-R for radio.

Key points

• Nyquist rate - fs must exceed 2fm; equality works only for ideal band-limited signals, never for a sampled sinusoid at exactly 2f.
• Aliasing - irreversible frequency folding about fs/2; only an analog anti-aliasing filter prevents it.
• Bandpass sampling - fs = 2fH/k with k = floor(fH/B) suffices for a band-pass signal.
• Aperture effect - flat-top sampling multiplies the spectrum by sinc(fτ); needs a 1/sinc equalizer.
• Quantization noise power - Δ²/12, independent of signal, set only by step size.
• 6 dB per bit - SQNR rises ~6.02 dB for each added bit; for a full-load sinusoid SQNR = 1.76 + 6.02n dB.
• Companding - A-law (India, A = 87.6) and μ-law (μ = 255) make SQNR almost independent of signal level.
• PCM bit rate - Rb = n·fs; minimum transmission bandwidth = Rb/2.
• E1 - 2.048 Mbps = 32 × 64 kbps; 30 usable voice channels, A-law, HDB3 line code.
• DPCM/ADPCM - encode prediction error; 1-2 bits per sample saved; G.726 gives toll voice at 32 kbps.
• Delta modulation - 1 bit per sample; slope overload vs granular noise set by step size δ; ADM adapts δ.
• Regenerative repeater - noise does not accumulate in digital links, but jitter does.
• Nyquist ISI criterion - Rs symbols/s need at least Rs/2 Hz baseband; raised cosine costs (1+α).
• Matched filter - maximizes output SNR = 2E/N0; the optimum receiver is "nearest constellation point".
• BPSK vs QPSK - identical BER vs Eb/N0; QPSK uses half the bandwidth. The single most asked fact here.
• Constant envelope - FSK, MSK, GMSK survive non-linear PAs; QAM needs linear PAs because of high PAPR.
• Gray coding - adjacent constellation points differ by one bit, so BER ≈ SER/log2M at high SNR.
• Bandwidth efficiency - η = log2M/(1+α) bit/s/Hz for passband M-ary signalling with raised-cosine shaping.

Formulas and examples

Example 1 - aliasing. x(t) = 3cos(2π·1000t) + 2cos(2π·3500t) is sampled at fs = 5 kHz.

• fm = 3500 Hz, so Nyquist rate = 7 kHz > 5 kHz → aliasing occurs.
• 1000 Hz < fs/2 = 2500 Hz, so it survives at 1000 Hz.
• 3500 Hz folds to |3500 − 5000| = 1500 Hz.
• Reconstructed output = 3cos(2π·1000t) + 2cos(2π·1500t) - a tone that was never transmitted. Minimum fs to avoid this = 7 kHz.

Example 2 - PCM design for telephony. Speech band-limited to 3.4 kHz, sampled at 8 kHz, 8 bits/sample, A-law.

• Rb = 8 × 8000 = 64 kbps per channel; minimum transmission bandwidth = 64/2 = 32 kHz.
• L = 2⁸ = 256 levels; for a full-load sinusoid SQNR = 1.76 + 6.02 × 8 = 49.9 dB.
• Companded (μ-law figure for comparison): 6.02 × 8 + 4.77 − 20 log10(5.545) = 48.2 + 4.77 − 14.88 ≈ 38.1 dB, but almost constant over a wide range of input levels - which is what speech needs.
• 30 such channels in an E1: 32 × 64 kbps = 2.048 Mbps.

Example 3 - quantizer sizing. An analog signal spans ±4 V; SQNR of at least 45 dB is required for a sinusoidal test input.

• n ≥ (45 − 1.76)/6.02 = 7.18 → choose n = 8 bits, L = 256.
• Δ = 8/256 = 31.25 mV; Nq = Δ²/12 = (0.03125)²/12 = 8.14 × 10⁻⁵ V² (rms error ≈ 9.02 mV).
• Signal power = Vm²/2 = 16/2 = 8 V²; SQNR = 8/(8.14 × 10⁻⁵) = 98,300 ≈ 49.9 dB ✓ (= 1.5 × 256²).
• If fs = 20 kHz, Rb = 8 × 20,000 = 160 kbps.

Example 4 - delta modulation. x(t) = 2 sin(2π·1000t) V, fs = 64 kHz, message bandwidth W = 1 kHz.

• Max slope = 2π × 1000 × 2 = 12,566 V/s.
• Minimum step to avoid slope overload: δ = 12,566/64,000 = 0.196 V (take 0.2 V).
• SNR = 3fs³/(8π²fm²W) = 3(6.4 × 10⁴)³ / (78.96 × 10⁶ × 10³) = 7.864 × 10¹⁴ / 7.896 × 10¹⁰ ≈ 9,960 → ≈ 40 dB.
• Note DM's SNR improves 9 dB per doubling of fs (fs³ law), against PCM's 6 dB per added bit.

Example 5 - digital modulation link budget. Rb = 2 Mbps, 16-QAM, root-raised-cosine α = 0.25, target BER 10⁻⁵.

• Symbol rate Rs = 2 Mbps / log2 16 = 500 ksym/s.
• RF bandwidth B = (1 + 0.25) × 500k = 625 kHz; efficiency η = 2 × 10⁶/6.25 × 10⁵ = 3.2 bit/s/Hz.
• Required Eb/N0 ≈ 13.4 dB (16-QAM); the same 2 Mbps by QPSK would need 1.25 MHz but only 9.6 dB - exactly 3.8 dB of extra power bought 2× the spectrum.
• Check BPSK at 9.6 dB: γ = 10^0.96 = 9.12, Pe = Q(√(2 × 9.12)) = Q(4.27) ≈ 10⁻⁵ ✓. DPSK at the same γ: ½e^−9.12 = 5.5 × 10⁻⁵. Non-coherent FSK: ½e^−4.56 = 5.2 × 10⁻³.

Where it is used

• Telecom networks: BSNL/MTNL and private operators' E1/SDH transport carries A-law 64 kbps PCM; Tejas Networks and Sterlite build the transport gear; GSM/LTE/5G base stations implement GMSK, QPSK and up to 256-QAM, with AMC switching per user per TTI.
• Semiconductor and SoC roles: modem physical-layer design, mapper/demapper, pulse-shaping and timing-recovery blocks at Qualcomm India, Samsung R&D Bengaluru/Noida, MediaTek, Intel, Nokia and Ericsson India - where constellation, BER and PAPR questions are routine.
• Mixed-signal and ADC/DAC design: Texas Instruments and Analog Devices India work on sample-and-hold, sigma-delta and pipeline converters; sampling rate, aperture jitter and SNR-vs-ENOB calculations are the daily vocabulary.
• Space and defence: ISRO uses PCM telemetry with BPSK/QPSK on S- and X-band; NavIC transmits BPSK/BOC signals; DRDO and BEL build radar and secure-communication links using PSK/QAM and digital receivers (IF sampling).
• Broadcast and DTH: DVB-S2 QPSK/8-PSK for satellite TV (Tata Play, DD Free Dish), DVB-T2 with OFDM-carried QAM for terrestrial digital TV, and 48/96 kHz sampled PCM audio in studios (AIR, Prasar Bharati).
• Instrumentation, IoT and medical: data acquisition cards, digital oscilloscopes, smart meters (as specified in BIS/CEA metering requirements), ECG and EEG front ends, and 433/865-867 MHz ASK/FSK modules in the Indian licence-exempt bands authorised by WPC.

Interview questions

• Q: State the sampling theorem and explain why sampling exactly at 2fm is unsafe in practice? A: A signal band-limited to fm is recoverable if fs > 2fm. At exactly 2fm the spectral images just touch, so an ideal brick-wall reconstruction filter would be needed; and a sinusoid at fm can be sampled at its zero crossings and vanish.
• Q: Why does aliasing need an analog filter and not a digital one? A: Aliasing happens during sampling - the folded components become indistinguishable from genuine low-frequency content, so no post-sampling processing can separate them.
• Q: A signal occupies 20 to 24 MHz. What is the minimum sampling rate? A: B = 4 MHz, k = floor(24/4) = 6, so fs = 2 × 24/6 = 8 MHz - bandpass sampling, far below 48 MHz.
• Q: Why is quantization noise power Δ²/12 and when does the model fail? A: The error is assumed uniform over (−Δ/2, Δ/2), whose variance is Δ²/12. It fails for small or highly correlated signals, where the error becomes signal-dependent and audible as distortion rather than noise.
• Q: Why does telephony use companding instead of more bits? A: Speech spends most of its time at low amplitude; A-law gives fine steps there, achieving roughly 12-13-bit uniform quality within 8 bits, so the channel stays at 64 kbps.
• Q: What is slope overload and how do you remove it without worsening granular noise? A: The DM staircase cannot track a fast signal when δfs < max slope. Raising δ fixes it but worsens granular noise; the correct answer is adaptive DM, which varies δ with the local slope.
• Q: QPSK carries twice the bits of BPSK. Why is its BER the same? A: QPSK is two independent BPSK streams on orthogonal (I and Q) carriers; each uses half the symbol energy but carries one bit, so Eb/N0 per bit is unchanged - only bandwidth halves.
• Q: Why is 64-QAM used for a user near the cell centre and QPSK at the edge? A: 64-QAM has smaller minimum distance and needs ~8 dB more Eb/N0; adaptive modulation picks the densest constellation the measured SNR can support.
• Q: Why did GSM choose GMSK rather than QPSK? A: GMSK has a constant envelope, so the handset's power amplifier can run in saturation for battery efficiency, and the Gaussian pre-filter keeps adjacent-channel leakage within the 200 kHz grid.
• Q: What does an eye diagram tell you that a BER figure does not? A: It shows *why* errors occur - vertical opening gives noise margin, horizontal width gives timing margin, and the thickness of the crossings exposes jitter and ISI.
• Q: What is the function of a matched filter and what SNR does it give? A: It correlates the received waveform with the known symbol shape to maximize SNR at the decision instant; the maximum is 2E/N0, independent of pulse shape.
• Q: Why does raised-cosine filtering with α = 0.35 cost bandwidth, and why accept it? A: The ideal Nyquist filter is unrealizable and its sinc pulse decays too slowly (severe sensitivity to timing error); α buys a realizable filter and timing robustness for 35% extra bandwidth.
• Q: Differentiate DPSK from DQPSK and state the price of differential detection. A: Both encode information in phase *changes*, binary and quaternary respectively; detection without a carrier reference avoids phase ambiguity but costs roughly 1-3 dB in Eb/N0 and correlates errors in pairs.

Common mistakes

• Treating fs = 2fm as sufficient, or forgetting that the Nyquist rate depends on the *highest* frequency present, not the one of interest.
• Computing an alias without folding about fs/2, and reporting a frequency above fs/2.
• Confusing SQNR formulas: 1.76 + 6.02n applies to a full-load sinusoid; 6.02n applies to a signal uniformly distributed over full scale. Quote the assumption.
• Mixing up minimum PCM bandwidth (Rb/2) with bit rate (n·fs), or quoting bandwidth in bits per second.
• Claiming QPSK is "better" than BPSK in error performance - it is better in bandwidth, identical in BER versus Eb/N0.
• Comparing modulation schemes at the same Es/N0 (per symbol) when the fair comparison for different M is at the same Eb/N0 (per bit).
• Applying the Δ²/12 quantization-noise formula to delta modulation; DM's granular error spans ±δ, giving δ²/3 before the in-band fraction is taken.
• Forgetting the 1/sinc aperture equalizer after flat-top sampling, or ignoring the reconstruction filter after the DAC in a lab setup.
• Omitting Gray coding on a QAM constellation in a design answer, then wondering why BER is several times SER/log2M.

Related topics

• Signals and systems · Fourier, Laplace and Z-transforms · Information theory and coding · Noise in communication systems · OFDM and MIMO · Spread spectrum and CDMA