Digital Modulation

Tip

Exam Focus

High-yield items: the bit-rate/symbol-rate law $R_b=R_s\log_2 M$ and bits/symbol $k=\log_2 M$; which carrier parameter each scheme varies (ASK amplitude, FSK frequency, PSK phase, QAM both I/Q amplitudes); BPSK and Gray-coded QPSK share $P_b=Q(\sqrt{2E_b/N_0})$ while orthogonal BFSK gives $Q(\sqrt{E_b/N_0})$; BPSK and QPSK constellations and I/Q generation; 16/64-QAM constellations and why higher order needs more SNR; MSK as continuous-phase FSK with $h=0.5$ and GMSK; PAM/PWM/PPM; and spread spectrum with processing gain $G_p=R_c/R_b$ plus FDMA/TDMA/CDMA.

Digital Modulation Fundamentals

Digital modulation maps groups of bits to a finite set of carrier waveforms called symbols. A sinusoidal carrier $c(t)=A_c\cos(2\pi f_c t+\phi)$ may vary in amplitude, frequency, phase, or in both amplitude and phase.

Scheme	Parameter carrying information
ASK	Amplitude
FSK	Frequency
PSK	Phase
QAM	In-phase and quadrature amplitudes (amplitude and phase)

Baseband and Passband

Baseband transmission sends a line-coded pulse waveform in its original low-pass range, with energy near DC (Ethernet over copper, digital buses). Passband transmission maps data onto a carrier centred at nonzero $f_c$ (radio, satellite, modem, microwave). Passband operation permits radiation by practical antennas, channel allocation, frequency translation, and use of bandpass channels that cannot pass DC.

Bit and Symbol Quantities

Quantity	Meaning
$R_b$	Bit rate (bit/s)
$T_b=1/R_b$	Bit duration
$R_s$	Symbol rate (Bd)
$T_s=1/R_s$	Symbol duration
$M$	Number of possible symbols
$k=\log_2 M$	Bits carried by each symbol

Note

Law — Bit-rate / symbol-rate relation

$$R_b=R_s\log_2 M,\qquad k=\log_2 M .$$

Increasing $M$ lowers the symbol rate for a fixed bit rate, but constellation points crowd together at fixed average power and demand higher SNR.

Energy and BER

Caution

Key Formula — Bit and symbol energy

$$E_b=\frac{P_{\text{avg}}}{R_b},\qquad E_s=E_b\log_2 M .$$

The normalised ratio $E_b/N_0$ allows fair power-efficiency comparison across different bit rates and bandwidths. The bit error rate (BER) is the long-run probability of a wrongly decided bit under stated channel and detection assumptions.

Binary Digital Modulation

Binary carrier modulation assigns one of two waveforms during each bit interval $0\le t<T_b$. The three basic schemes vary amplitude, frequency, or phase of a common carrier driven by the same data sequence.

ASK, FSK and PSK carrier waveforms for the common data sequence $1\,0\,1\,1\,0$: ASK gates the carrier on/off, FSK switches between two frequencies, and PSK flips the phase by $180^\circ$ on a $0$ bit.

The corresponding modem architecture shows how mapping, carrier multiplication, coherent recovery, and decision are implemented.

Binary modulator (data $\to$ mapper $\to$ multiply by carrier) and coherent demodulator (multiply by local carrier $\to$ integrate-and-dump $\to$ decision). Coherent PSK also requires carrier synchronization and phase-ambiguity resolution before bit assignment.

Detection	Reference	Benefit	Cost
Coherent	Synchronised local carrier	Best BER	Carrier recovery / ambiguity handling
Noncoherent	No absolute phase	Simpler	Poorer BER / more BW

Carrier recovery for PSK can lock with a rotational ambiguity (for example, $180^\circ$ in BPSK or one of $M$ rotations in $M$-PSK). A known preamble/pilot, differential coding, or an appropriate ambiguity-resolution rule is therefore needed before assigning recovered symbols to bits. Matched filters/correlators maximise output SNR for known waveforms in additive white Gaussian noise (AWGN); the receiver samples at symbol instants and picks the most likely symbol.

Amplitude Shift Keying (ASK)

In binary ASK the carrier amplitude takes one of two values. The common on-off keying (OOK) case is

$$s_1(t)=A_c\cos(2\pi f_c t),\qquad s_0(t)=0,\qquad 0\le t<T_b .$$

Generation: multiply unipolar NRZ data by the carrier (product modulator) or switch the carrier on/off. Detection: coherent (multiply by synchronised carrier, integrate, threshold) or noncoherent (envelope/square-law detector plus threshold).

Caution

Key Formula — Rectangular ASK null-to-null bandwidth

$$B_{\text{ASK}}\approx 2R_b .$$

With raised-cosine pulse shaping, binary ASK can instead occupy approximately $B\approx(1+\alpha)R_s=(1+\alpha)R_b$ under the corresponding passband/Nyquist bandwidth convention. A bandwidth answer must state both the pulse shape and the definition used; it must not compare this value directly with a rectangular-pulse null-to-null value. ASK is the simplest scheme but is sensitive to amplitude noise and fading and has a non-constant envelope (linear amplification preferred). Uses: optical intensity links, RFID, remote controls and simple telemetry.

Binary Frequency Shift Keying (BFSK)

BFSK uses two frequencies, the mark $f_1$ and space $f_0$:

$$s_1(t)=A_c\cos(2\pi f_1 t),\qquad s_0(t)=A_c\cos(2\pi f_0 t).$$

Caution

Key Formula — Centre frequency and deviation

$$f_c=\frac{f_1+f_0}{2},\qquad \Delta f=\frac{\left\lvert f_1-f_0\right\rvert}{2}.$$

Orthogonality and Bandwidth

For coherent observation over $T_b$ a common minimum separation is $\left\lvert f_1-f_0\right\rvert=1/(2T_b)$; simple noncoherent orthogonality often uses $1/T_b$. A Carson-like null-to-null estimate for rectangular symbols is

$$B_{\text{BFSK}}\approx 2\Delta f+2R_b .$$

Generation, Detection and Performance

Generate by switching two oscillators, driving a VCO, or using a phase accumulator (continuous phase). Detect coherently with two correlators (choose larger output) or noncoherently with two bandpass filters and envelope detectors. A PLL that follows the two tone states or a frequency discriminator followed by thresholding provides another practical BFSK detector.

Caution

Key Formula — BFSK BER in AWGN

$$P_b^{\text{coh}}=Q\!\left(\sqrt{\tfrac{E_b}{N_0}}\right),\qquad P_b^{\text{noncoh}}=\tfrac12\,\mathrm{e}^{-E_b/2N_0}.$$

BFSK has a constant envelope (efficient nonlinear PAs) and better amplitude-noise tolerance than ASK, but usually wider bandwidth than PSK. Uses: low-speed modems, paging, caller ID, telemetry, low-power radio.

Binary Phase Shift Keying (BPSK)

BPSK uses two antipodal phases separated by $180^\circ$:

$$s_1(t)=A_c\cos(2\pi f_c t),\qquad s_0(t)=-A_c\cos(2\pi f_c t).$$

Map bits to polar levels $a_k\in\{+1,-1\}$ and drive a balanced product modulator, $s(t)=a_k A_c\cos(2\pi f_c t)$.

BPSK constellation: two antipodal points at $0^\circ$ and $180^\circ$ on the in-phase axis ($\pm\sqrt{E_b}$), giving maximum Euclidean distance for a fixed bit energy.

Coherent detection recovers the carrier (Costas loop), correlates over $T_b$, and decides from the sign of the sample:

Caution

Key Formula — Coherent BPSK BER

$$P_b=Q\!\left(\sqrt{\tfrac{2E_b}{N_0}}\right) =\tfrac12\operatorname{erfc}\!\left(\sqrt{\tfrac{E_b}{N_0}}\right).$$

DPSK encodes information in the phase change between symbols, avoiding an absolute phase reference at the cost of error propagation and slightly worse BER.

Quadrature Phase Shift Keying (QPSK)

QPSK uses four phase states and carries two bits/symbol:

$$k=\log_2 4=2,\qquad R_s=\frac{R_b}{2}.$$

Serial bits are grouped into dibits mapped to $I_k,Q_k\in\{+1,-1\}$:

Caution

Key Formula — QPSK signal

$$s_k(t)=A\big[I_k\cos(2\pi f_c t)-Q_k\sin(2\pi f_c t)\big].$$

Gray-coded QPSK constellation at $45^\circ$, $135^\circ$, $225^\circ$, $315^\circ$: adjacent points differ by one bit, so a nearest-neighbour symbol error usually causes only one bit error.

Dibit	$I$	$Q$	Phase
	$+$	$+$	$45^\circ$
01	$-$	$+$	$135^\circ$
11	$-$	$-$	$225^\circ$
10	$+$	$-$	$315^\circ$

QPSK Gray mapping (one common convention).

Under ideal coherent detection and Gray mapping, QPSK has the same bit BER as BPSK at equal $E_b/N_0$, namely $P_b=Q(\sqrt{2E_b/N_0})$; its advantage is twice the bits/symbol, not a $3\,dB$ BER gain. At the receiver, carrier recovery first supplies phase-aligned cosine and negative-sine references; a preamble, pilot or differential rule resolves the fourfold phase ambiguity. The two mixer outputs pass through low-pass matched filters, are sampled at each symbol centre, and are decided by the signs of $I$ and $Q$; the resulting dibits are then converted from parallel to serial.

Offset QPSK (OQPSK) delays the $Q$ stream by half a symbol so $I$ and $Q$ never switch together, limiting the phase jump to $90^\circ$ and easing nonlinear PA operation.

Generation: (a) BPSK maps a bit to $\pm1$ and multiplies by the carrier; (b) QPSK splits serial bits into $I$/$Q$ streams driving two balanced modulators (cosine and $-$sine) whose outputs are summed.

Minimum Shift Keying (MSK)

MSK is continuous-phase binary FSK with modulation index $h=0.5$. Using $h=2\Delta f/R_b$,

$$\Delta f=\frac{R_b}{4},\qquad \left\lvert f_1-f_0\right\rvert=2\Delta f=\frac{R_b}{2},$$

the minimum separation giving orthogonal binary tones while phase accumulates continuously (no jumps at bit boundaries).

MSK phase evolves as a continuous piecewise-linear trellis ($\pm90^\circ$ per bit), whereas QPSK-type modulation makes abrupt phase steps at symbol boundaries.

Properties: carrier phase never jumps; constant envelope allows efficient saturated PAs; lower sidelobes than abrupt FSK; equivalent to OQPSK with half-sinusoidal pulse shaping. GMSK pre-filters data with a Gaussian low-pass filter, narrowing the spectrum at the cost of controlled ISI (set by the time-bandwidth product $BT$); GSM uses GMSK.

Quadrature Amplitude Modulation (QAM)

QAM independently controls the amplitudes of two orthogonal carriers:

Caution

Key Formula — QAM signal and rate

$$s_k(t)=I_k\cos(2\pi f_c t)-Q_k\sin(2\pi f_c t),\qquad k=\log_2 M,\quad R_s=\frac{R_b}{\log_2 M}.$$

Each pair $(I_k,Q_k)$ is one point in the complex constellation.

Square constellations of increasing order: as $M$ rises the points crowd together at fixed average power, so a larger $E_b/N_0$ is needed.

Scheme	Symbols	Bits/symbol	$R_s$ for $R_b$
-QAM	16	4	$R_b/4$
64-QAM	64	6	$R_b/6$
256-QAM	256	8	$R_b/8$

Gray-coded 16-QAM: four levels per axis ($\propto\{-3,-1,+1,+3\}$), one-bit changes between horizontal/vertical neighbours, and rectangular decision boundaries.

A complete I/Q modem then combines Gray mapping, pulse shaping, quadrature modulation, synchronized reception, and nearest-point decisions.

Generic QAM/PSK I/Q modem: Gray mapping and pulse shaping feed the quadrature modulator and linear RF amplifier; after carrier/timing recovery, matched filtering and sampling, the receiver selects the nearest point and Gray-demaps it back to serial bits.

The QAM transmitter groups input bits, applies a Gray constellation mapping, pulse-shapes the resulting $I_k$ and $Q_k$ streams, modulates orthogonal carriers, sums them and uses a sufficiently linear RF amplifier. The receiver performs carrier and symbol-timing recovery, coherent I/Q downconversion, matched filtering and symbol-centre sampling; it then chooses the nearest constellation point, Gray-demaps that point and converts the recovered parallel bits to a serial stream.

For raised-cosine roll-off $\alpha$, a common passband estimate is

$$B\approx(1+\alpha)R_s=(1+\alpha)\frac{R_b}{\log_2 M}.$$

A high-SNR Gray-coded square-QAM BER approximation is

$$P_b\approx\frac{4}{\log_2 M}\Big(1-\frac{1}{\sqrt M}\Big) Q\!\left(\sqrt{\frac{3\log_2 M}{M-1}\,\frac{E_b}{N_0}}\right).$$

At fixed power, larger $M$ shrinks point spacing, so QAM needs higher SNR and more linear amplification. Uses: digital microwave, cable modems, Wi-Fi/cellular OFDM subcarriers, DVB and adaptive modulation.

Overall Comparison

Coherent AWGN BER with equal $E_b$ and matched filtering, sampled from the exact laws $Q(\sqrt{2E_b/N_0})$ (BPSK/Gray QPSK) and $Q(\sqrt{E_b/N_0})$ (orthogonal BFSK). OOK is omitted because a comparison must first state its energy, prior-probability and threshold normalization.

The plotted comparison assumes AWGN, equal energy per information bit and coherent matched-filter receivers; Gray mapping is assumed for QPSK. It shows only the established laws $P_b=Q(\sqrt{2E_b/N_0})$ for BPSK/Gray QPSK and $P_b=Q(\sqrt{E_b/N_0})$ for coherent orthogonal BFSK. OOK is not placed on the same curve ranking because its result depends on how on-symbol versus average bit energy, symbol priors and the decision threshold are normalized.

Feature	OOK/ASK	Orthogonal BFSK	BPSK
Parameter	Amplitude	Frequency	Phase
Envelope	Varies/off	Constant	Constant before filtering
Noncoherent option	Envelope det.	Energy det.	Differential (not plain)
Coherent AWGN BER	State energy/threshold model	$Q(\sqrt{E_b/N_0})$	$Q(\sqrt{2E_b/N_0})$
Bandwidth	Moderate	Widest	Compact
PA needs	Linear	Nonlinear OK	Nonlinear OK if controlled

Binary scheme comparison.

Scheme	Bits/sym	Spectral eff.	Main strength
BPSK		Moderate	Excellent power efficiency
QPSK		Good	BPSK-like BER at half $R_s$
MSK/GMSK		Good, compact	Efficient nonlinear PA
-QAM		High	High throughput, moderate SNR
/256-QAM	/8	Very high	Max throughput on clean channels

$M$-ary scheme comparison.

Power efficiency (small $E_b/N_0$ for a target BER) favours BPSK/QPSK; spectral efficiency (bit/s per Hz) favours higher-order QAM. Improving one usually costs the other, so adaptive systems pick a mode from the channel SNR and amplifier constraints.

Analog Pulse Modulation

Pulse modulation varies a parameter of a periodic pulse train according to an analog message; these schemes stay analog until quantisation and coding (PCM) are added.

Analog pulse modulation of one message: natural PAM follows the input during each aperture, flat-top PAM holds the sampled level, PWM varies width, and PPM varies position about a timing reference.

Scheme	Varied parameter	Note
PAM	Amplitude	Most amplitude-noise sensitive; simplest
PWM/PDM	Width/duration	Limiter-friendly; more bandwidth than PAM
PPM	Time position	Best amplitude-noise immunity; jitter-sensitive

In natural PAM, each pulse top follows the input throughout its sampling aperture; flat-top PAM holds one sampled value and is the form presented to practical PCM quantisation. For a comparable pulse rate and fidelity, PWM generally requires more bandwidth than PAM, while PPM usually imposes the greatest timing precision and often the widest pulse spectrum. PWM is commonly generated by comparing the message with a ramp and recovered by integration or low-pass filtering. A PPM receiver establishes the frame reference, measures pulse displacement, converts that position change to a voltage and then filters the result. PPM rejects amplitude disturbances well after limiting, but timing jitter directly corrupts the recovered message.

Spread Spectrum and Multiple Access

Spread spectrum deliberately occupies far more bandwidth than the information signal using a shared pseudorandom pattern; a receiver knowing the pattern despreads the wanted signal.

DSSS

Multiply each data bit by a much faster bipolar PN chip sequence (chip rate $R_c$). The transmitted bandwidth is set mainly by $R_c$.

Caution

Key Formula — Processing gain

$$G_p=\frac{R_c}{R_b},\qquad G_{p,\text{dB}}=10\log_{10}\frac{R_c}{R_b}.$$

The receiver must first acquire the PN-code phase and then track its drift. Correlation with that aligned replica collapses the wanted signal to data bandwidth while narrowband interference is spread and filtered.

FHSS

A PN generator drives a frequency synthesiser, hopping the carrier among many channels. Slow hopping sends several symbols per hop; fast hopping makes several hops per symbol. FHSS resists narrowband interference and selective fading but needs hop synchronisation and a fast synthesiser.

(a) DSSS: a fast PN sequence spreads the data; the receiver acquires and tracks PN phase before correlation with its aligned replica. (b) FHSS: the carrier hops among frequency channels over time under PN control.

Multiple Access

FDMA gives each user a separate frequency channel with guard bands; users transmit simultaneously in different bands. TDMA gives each user a time slot on a shared frequency. CDMA lets users share the same time and band using distinct spreading codes separated by correlation. In a resolvable multipath channel, a rake receiver aligns correlator fingers to delayed replicas and combines them to recover useful multipath energy.

Feature	FDMA	CDMA
User separation	Frequency channel	Spreading code
Guard resource	Guard bands	Low code cross-corr., power control
Capacity	Hard allocation	Interference-limited (soft)
Sync	Frequency plan	Code timing, power control
Main impairment	Adjacent channel	MAI, near-far problem

Solved Examples

Note

Example — Baseband vs passband transmission (5 marks)

Baseband sends a line-coded waveform in its low-pass range (energy near DC): LAN cable, digital bus. Passband maps data to a carrier, $s(t)=x(t)\cos(2\pi f_c t)$, shifting the spectrum about $\pm f_c$: ASK, FSK, PSK, QAM.

Basis	Baseband	Passband
Spectrum	Near $0$ Hz	Around $f_c$
Operation	Line coding	Carrier mod/demod
Channel	Passes DC	May reject DC
Example	Ethernet, bus	Radio, satellite

Wireless needs passband because efficient antenna size scales with wavelength and spectrum is allocated in RF bands; a 1 kHz baseband signal on a 100 MHz carrier radiates practically without its information bandwidth becoming 100 MHz.
Practice: 8–9 min; draw low-pass vs translated spectra and compare channel, hardware and applications.

[!NOTE]

Example — ASK, BFSK and BPSK (10 marks)

Each bit interval $0\le t<T_b$ carries one of two waveforms. ASK/OOK: $s_1=A_c\cos\omega_c t$, $s_0=0$; product modulator/RF switch; envelope or coherent detection; varying envelope, amplitude-noise sensitive. BFSK: $s_1=A_c\cos\omega_1 t$, $s_0=A_c\cos\omega_0 t$, $\Delta f=\left\lvert f_1-f_0\right\rvert/2$; VCO/two oscillators; two correlators or two filter/envelope branches; constant envelope. BPSK: $s_{1,0}=\pm A_c\cos\omega_c t$; polar mapper $+$ balanced modulator; coherent sign decision; antipodal points maximise distance.

	ASK	BFSK	BPSK
Parameter	Amplitude	Frequency	Phase
Envelope	Varies	Constant	Constant
Bandwidth	$\approx2R_b$	$\approx2\Delta f+2R_b$	Compact
Coherent BER	State OOK model	$Q(\sqrt{E_b/N_0})$	$Q(\sqrt{2E_b/N_0})$

Coherent AWGN: $P_b^{\text{BPSK}}=Q(\sqrt{2E_b/N_0})$, $P_b^{\text{BFSK}}=Q(\sqrt{E_b/N_0})$ (assuming matched filtering and equal energy).
Practice: 18–20 min; one shared bit sequence, all six waveforms, three detectors, and qualified BW/BER comparison.

[!NOTE]

Example — FSK principle and bandwidth (5 marks)

FSK keeps amplitude constant and switches frequency: bit $1\to f_1$ (mark), bit $0\to f_0$ (space), $\Delta f=\left\lvert f_1-f_0\right\rvert/2$. Generate with two oscillators, a VCO, or a synthesiser (continuous phase); detect with two correlators (coherent) or two filter/envelope branches (noncoherent). Null-to-null estimate $B\approx2\Delta f+2R_b$.
Numerical: at $R_b=2\,kbit/s$, coherent minimum separation $1/(2T_b)=1\,kHz$. Tones $9.5$ and 10.5 kHz give $\Delta f=0.5\,kHz$ and $B\approx5\,kHz$. Uses: modems, paging, telemetry, RFID.
Practice: 8–9 min; label mark/space, sketch VCO generator and dual-filter detector, solve one separation example.

[!NOTE]

Example — BPSK and QPSK generation and constellations (4+6 marks)

BPSK [4]: $s_k(t)=a_k A_c\cos\omega_c t$ with $a_k\in\{+1,-1\}$; phases $0$ and $\pi$; two antipodal points $(\pm\sqrt{E_b},0)$; one bit/symbol so $R_s=R_b$; coherent sign decision. QPSK [6]: dibits give $k=\log_2 4=2$, $R_s=R_b/2$. A serial/parallel converter feeds $I_k,Q_k\in\{\pm1\}$ to two balanced modulators ($\cos\omega_c t$, $-\sin\omega_c t$) summed to $s_k=A[I_k\cos\omega_c t-Q_k\sin\omega_c t]$. Four Gray-coded points at $45^\circ,135^\circ,225^\circ,315^\circ$ (00,01,11,10). Ideal Gray QPSK has the same $P_b=Q(\sqrt{2E_b/N_0})$ as BPSK, so its gain is spectral efficiency, not a 3 dB BER improvement.
Practice: 16–18 min; about 6 min BPSK, about 12 min QPSK I/Q generator, Gray table and symbol-rate result.

[!NOTE]

Example — QAM constellations and SNR requirement (10 marks)

QAM modulates two orthogonal carriers: $s_k=I_k\cos\omega_c t-Q_k\sin\omega_c t$, carrying $k=\log_2 M$ bits/symbol, $R_s=R_b/\log_2 M$. Square 16-QAM uses four levels/axis ($\propto\{-3,-1,+1,+3\}$, 4 b/sym); 64-QAM uses eight levels/axis ($\{\pm1,\pm3,\pm5,\pm7\}$, 6 b/sym). Gray mapping makes neighbours differ by one bit.
Numerical: $R_b=120\,Mbit/s$, $\alpha=0.2$. 16-QAM: $R_s=30\,MBd$, $B\approx(1.2)(30)=36\,MHz$. 64-QAM: $R_s=20\,MBd$, $B\approx24\,MHz$. At fixed power the higher order shrinks $d_{\min}$, so it needs more $E_b/N_0$ and linear amplification (high PAPR).
Practice: 18–20 min; draw both grids, label bits/symbol, compute one $R_s$/$B$ pair, explain the SNR penalty.

[!NOTE]

Example — MSK and GMSK (5 marks)

MSK is continuous-phase FSK with $h=0.5$. With $h=2\Delta f/R_b$, $\Delta f=R_b/4$ and $\left\lvert f_1-f_0\right\rvert=R_b/2$ (minimum orthogonal separation). Phase accumulates continuously with no bit-boundary jumps, so sidelobes are low and the spectrum compact; the constant envelope suits saturated PAs. MSK equals OQPSK with half-sinusoidal shaping. GMSK pre-filters with a Gaussian LPF, narrowing the spectrum at the cost of controlled ISI set by $BT$; GSM uses GMSK.
Practice: 8–9 min; derive both frequency relations, sketch continuous phase, state the Gaussian $BT$/ISI trade-off.

[!NOTE]

Example — Comparison of ASK/FSK/PSK/QPSK/MSK/QAM (10 marks)

Compare on a common basis of $E_b/N_0$ (power) and $\eta=R_b/B$ (spectral efficiency).

Scheme	b/sym	Envelope	Note
ASK		Varies	Simplest, noise-sensitive
BFSK		Constant	Widest BW, nonlinear-PA OK
BPSK		Constant	Best binary power efficiency
QPSK		Constant	BPSK BER at half $R_s$
MSK		Constant	Continuous phase, compact
$M$-QAM	$\log_2 M$	Varies	Highest $\eta$, needs SNR

$P_b(\text{BPSK})=P_b(\text{Gray QPSK})=Q(\sqrt{2E_b/N_0})$; coherent orthogonal BFSK $=Q(\sqrt{E_b/N_0})$. Selection: ASK for simplicity, BFSK/MSK for nonlinear PAs, BPSK/QPSK when power is scarce, high-order QAM for throughput on high-SNR linear channels.
Practice: 18–20 min; define the basis, reproduce the table, quote only assumption-backed BER/BW formulas.

[!NOTE]

Example — DSSS, FHSS, FDMA and CDMA (10 marks)

DSSS: multiply each bit by a fast PN chip sequence; processing gain $G_p=R_c/R_b$, $G_{p,\text{dB}}=10\log_{10}(R_c/R_b)$; the correlator despreads the signal and spreads narrowband interference. FHSS: PN-controlled synthesiser hops the carrier (slow $=$ several symbols/hop, fast $=$ several hops/symbol).
Numerical: $R_b=10\,kbit/s$, $R_c=1\,\text{Mchip/s}$ give $N=100$ chips/bit and $G_{p,\text{dB}}=10\log_{10}(100)=20\,dB$ (nominal, not a guaranteed jamming margin).
Access: FDMA separates users by frequency (guard bands); CDMA shares band/time via codes (needs power control for the near-far problem); TDMA uses time slots.
Practice: 18–20 min; draw both spreading paths, compute one $G_p$, distinguish fast/slow hopping, reproduce the FDMA/CDMA table.

[!NOTE]

Key Point — Revision — Digital Modulation

• $R_b=R_s\log_2 M$; QPSK $2$, 16-QAM $4$, 64-QAM $6$ bits/symbol.

• ASK varies amplitude, FSK frequency, PSK phase, QAM both I/Q amplitudes.

• Coherent BPSK and Gray QPSK: $P_b=Q(\sqrt{2E_b/N_0})$; coherent orthogonal BFSK: $Q(\sqrt{E_b/N_0})$.

• BFSK/MSK have constant envelopes (efficient nonlinear PAs); BFSK is wideband, MSK continuous-phase with $h=0.5$.

• Higher-order QAM raises spectral efficiency but shrinks point spacing, needing more SNR and linearity.

• PAM/PWM/PPM are analog pulse schemes; PCM adds quantisation and coding.

• DSSS spreads with a fast PN code ($G_p=R_c/R_b$); FHSS hops the carrier; FDMA/TDMA/CDMA separate users by frequency/time/code.