Digital Modulation¶

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Baseband transmission sends a pulse/line-coded waveform directly in its original low-pass frequency range, normally with significant energy near DC. No sinusoidal RF carrier is required. Examples are Ethernet over copper, a digital bus and a short wired PCM line. The channel must pass frequencies from near zero through the signal bandwidth, and repeaters recover timing and make symbol decisions.

Passband transmission maps bits to the amplitude, frequency and/or phase of a carrier centered at nonzero \(f_c\). For a simple product translation,

so a baseband spectrum \(X(f)\) is shifted around \(\pm f_c\). Digital ASK, FSK, PSK and QAM use a finite set of such carrier waveforms.

Wireless links require passband operation because efficient antenna size is related to wavelength, practical spectrum is allocated in RF bands, and many users must occupy separate or coded channels. A \(1\,\text{kHz}\) baseband waveform would require an impractically large efficient antenna, whereas translating it onto, for example, a \(100\,\text{MHz}\) carrier permits practical radiation; the information bandwidth does not become \(100\,\text{MHz}\), but is centered there. Passband systems add carrier/timing recovery and frequency/phase errors, while baseband links avoid RF conversion but are limited to low-pass media.

Practice target: 8–9 minutes; draw low-pass versus translated spectra, give the carrier equation, and compare channel, hardware and applications.

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Binary carrier modulation assigns one of two waveforms during each bit interval \(0\leq t<T_b\), where \(R_b=1/T_b\).

ASK/OOK: information changes carrier amplitude. For on-off keying,

A product modulator or RF switch generates it. A coherent receiver correlates with a synchronized carrier and applies a threshold; a noncoherent envelope/square-law detector is simpler. Its envelope varies, so fading, amplitude noise and nonlinear amplification are troublesome.

BFSK: information selects mark and space frequencies:

A VCO or two switched oscillators generates BFSK. Coherent detection uses two matched filters/correlators and chooses the larger output; noncoherent detection compares energies from two bandpass-filter/envelope branches. It has constant envelope and tolerates nonlinear efficient power amplifiers, but normally occupies more bandwidth than PSK.

BPSK: information changes carrier phase by \(180^\circ\):

A polar mapper \(a_k\in\{+1,-1\}\) drives a balanced product modulator. Coherent detection recovers carrier phase, correlates/matched-filters over \(T_b\), and decides from output sign. The two antipodal signal-space points maximize binary Euclidean distance for fixed bit energy.

With rectangular symbols, one commonly quoted null-to-null estimate is \(B_{ASK}\approx2R_b\) and \(B_{BFSK}\approx2\Delta f+2R_b\); numerical values change with pulse shape and bandwidth definition. For coherent orthogonal BFSK, one minimum separation is \(|f_1-f_0|=1/(2T_b)\); noncoherent orthogonality commonly uses \(1/T_b\).

Under ideal coherent detection in AWGN,

These equations assume matched filtering, equal bit-energy normalization and the stated orthogonality; OOK results depend on coherent/noncoherent operation and energy convention. The qualitative BER ordering should not be quoted without those assumptions.

Practice target: 18–20 minutes; draw one shared bit sequence, write all six waveforms, explain three detector paths and qualify the bandwidth/BER comparison.

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Frequency Shift Keying (FSK) represents digital symbols by discrete carrier frequencies while keeping nominal amplitude constant. In binary FSK, bit 1 selects the mark frequency \(f_1\) and bit 0 the space frequency \(f_0\):

Generation uses two switched oscillators, binary control of a VCO, or a phase accumulator/synthesizer. The latter methods can maintain continuous phase. A coherent receiver correlates against synchronized \(f_1\) and \(f_0\) references and selects the greater decision statistic. A noncoherent receiver uses two bandpass filters followed by envelope/energy detectors; a PLL or frequency discriminator is another option.

For observation over \(T_b\), coherent orthogonal BFSK commonly uses \(|f_1-f_0|=1/(2T_b)\), while a simple noncoherent orthogonality condition is often \(1/T_b\). With rectangular binary symbols, an engineering null-to-null estimate is

not a universal occupied-bandwidth formula.

Example: at \(R_b=2\,\text{kbps}\), coherent minimum separation is \(1/(2T_b)=1\,\text{kHz}\). Symmetric tones around \(10\,\text{kHz}\) may be \(9.5\) and \(10.5\,\text{kHz}\), so \(\Delta f=0.5\,\text{kHz}\) and the stated estimate gives \(B\approx5\,\text{kHz}\).

FSK's constant envelope permits efficient nonlinear power amplifiers and gives better amplitude-fading/noise tolerance than ASK, but it usually consumes more bandwidth than BPSK. Applications include low-speed modems, caller ID, paging, telemetry, radio control, RFID and low-power sensor links; M-ary FSK is also used where power efficiency is more important than spectral efficiency.

Practice target: 8–9 minutes; label mark/space frequencies, sketch generator and dual-filter detector, and solve one separation/bandwidth example with assumptions.

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Part A — BPSK [4 marks]¶

Map binary data to polar symbols \(a_k\in\{+1,-1\}\) and multiply by a carrier in a balanced/product modulator:

Thus the two phases are \(0\) and \(\pi\). A coherent receiver regenerates carrier phase, correlates/matched-filters over \(T_b\), and decides from the sign of the sample.

Its constellation is \((+\sqrt{E_b},0)\) and \((-\sqrt{E_b},0)\) on the I axis. The antipodal spacing \(2\sqrt{E_b}\) gives strong power efficiency. It carries one bit/symbol, so \(R_s=R_b\).

Part B — QPSK [6 marks]¶

QPSK groups serial bits into dibits, hence

A serial-to-parallel converter sends one bit stream to an in-phase mapper \(I_k\) and the other to a quadrature mapper \(Q_k\), each taking \(\pm1\). Product modulators use orthogonal cosine and negative-sine carriers from a \(90^\circ\) splitter, then a summer forms

The four points have equal radius and phases \(45^\circ,135^\circ,225^\circ,315^\circ\) for one common mapping.

This is Gray coding: nearest neighbors differ in one bit, so the most likely nearest-point symbol error usually produces one bit error. A coherent receiver mixes with synchronized cosine and negative-sine references, matched-filters I and Q, samples at symbol centers, decides each sign, Gray-demaps and converts parallel decisions back to serial data.

For raised-cosine pulse shaping of roll-off \(\alpha\), a common passband occupied-bandwidth estimate is

under that convention. Thus QPSK carries the same bit rate at half the BPSK symbol rate. Under ideal coherent AWGN detection and Gray mapping, both have

so QPSK's benefit is spectral efficiency, not an automatic \(3\,\text{dB}\) BER gain. Carrier phase ambiguity and synchronization are practical limitations; differential coding or pilot-assisted recovery can resolve quadrant ambiguity.

Practice target: 16–18 minutes; spend about 6 minutes on BPSK and 10–12 on the I/Q generator, Gray table, four decisions and symbol-rate result.

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Quadrature Amplitude Modulation (QAM) independently modulates two orthogonal carriers. Each symbol selects an in-phase amplitude \(I_k\) and quadrature amplitude \(Q_k\):

The complex point \(I_k+jQ_k\) defines both resultant amplitude and phase. An \(M\)-QAM constellation carries

Square 16-QAM uses four levels on each axis, commonly proportional to \(\{-3,-1,+1,+3\}\), giving \(4\times4=16\) points and 4 bits/symbol. Square 64-QAM uses eight levels per axis, commonly \(\{\pm1,\pm3,\pm5,\pm7\}\), giving \(8\times8=64\) points and 6 bits/symbol. Actual amplitudes are normalized to the required average symbol energy.

Gray mapping makes horizontal or vertical nearest neighbors differ by one bit, reducing bit errors per likely symbol error. At the transmitter, bits are grouped and Gray-mapped, I/Q pulses are shaped, multiplied by cosine/sine carriers and summed. The receiver recovers carrier/timing, coherently downconverts to I and Q, matched-filters, samples, chooses the nearest constellation point and Gray-demaps.

For raised-cosine shaping with roll-off \(\alpha\), one common passband bandwidth estimate is

where the convention and filtering are explicit. For \(R_b=120\,\text{Mb/s}\) and \(\alpha=0.2\), 16-QAM uses \(R_s=30\,\text{MBd}\) and \(B\approx36\,\text{MHz}\); 64-QAM uses \(20\,\text{MBd}\) and \(B\approx24\,\text{MHz}\).

The bandwidth saving has a power/noise cost. If average transmit power is fixed, fitting more points into the same I/Q region reduces minimum Euclidean distance \(d_{min}\). Noise, phase error, gain imbalance or nonlinear distortion then crosses a decision boundary more easily. A common high-SNR Gray-coded square-QAM approximation in AWGN is

under coherent detection and ideal synchronization. The factor involving \(M-1\) shows the increasing \(E_b/N_0\) requirement as order rises. QAM also has a varying envelope/high peak-to-average ratio, so the RF chain must be linear or distortion compresses outer points. Adaptive Wi-Fi, cellular/OFDM, microwave, cable and DVB links therefore select 16/64/256-QAM only when measured channel SNR and linearity support the target error rate.

Practice target: 18–20 minutes; draw both square grids, label bits/symbol and Gray neighbors, trace the I/Q modem and calculate one symbol-rate/bandwidth pair.

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Minimum Shift Keying (MSK) is binary continuous-phase FSK with modulation index

Using the convention \(h=2\Delta f/R_b\),

This is the minimum coherent orthogonal tone separation for binary symbols while phase is accumulated continuously rather than reset at bit boundaries.

The data selects one of the two instantaneous frequencies, but the phase at the end of one bit becomes the starting phase of the next. Consequently there are no abrupt phase jumps. MSK can also be represented as offset QPSK with half-sinusoidal pulse shaping.

Why useful: continuous phase avoids discontinuity-generated high-frequency sidelobes and produces a compact spectrum; constant envelope allows limiting and efficient saturated/nonlinear power amplifiers without converting amplitude distortion into large symbol errors; and the minimum frequency spacing balances power and spectral efficiency. These properties suit mobile and power-limited radios, though coherent detection and phase-state tracking are more complex than simple noncoherent FSK.

Gaussian MSK (GMSK) first passes the binary data through a Gaussian low-pass filter and then applies MSK. The Gaussian filter smooths frequency transitions further and suppresses sidelobes, but pulse spreading introduces controlled intersymbol interference. The time-bandwidth product \(BT\) sets the trade-off: smaller \(BT\) gives a narrower spectrum but more ISI/detection difficulty. GSM uses standardized GMSK because its continuous phase and constant envelope work well with efficient handset power amplifiers. Exact occupied bandwidth depends on \(BT\), pulse truncation and the measurement convention, so it should not be stated from \(h=0.5\) alone.

Practice target: 8–9 minutes; derive the two frequency relations, sketch continuous phase, and state the Gaussian-filter bandwidth/ISI trade-off.

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A fair comparison states bit/symbol energy, pulse shaping, receiver and channel. Let \(E_b/N_0\) measure power efficiency and \(\eta=R_b/B\) bit/s/Hz measure spectral efficiency under a declared bandwidth convention.

For coherent AWGN with matched filtering,

whereas coherent orthogonal BFSK gives

These exact expressions do not apply unchanged to fading, noncoherent reception, coding or unequal energy normalization. ASK's result likewise depends on whether zero and one have equal prior probability and how average bit energy is defined.

With raised-cosine roll-off \(\alpha\), linear QPSK/QAM commonly use

Thus QPSK doubles bit/symbol without reducing ideal coherent Gray-coded BER, while 16/64-QAM raise spectral efficiency to 4/6 bits per symbol but reduce normalized point spacing. Rectangular BFSK is often estimated by \(B\approx2\Delta f+2R_b\); MSK/GMSK spectra require their actual continuous-phase/Gaussian pulse and \(BT\), so a single universal multiple of \(R_b\) is unsafe.

Selection: ASK wins simplicity; BFSK and MSK tolerate efficient nonlinear PAs; BPSK/QPSK are strong when power is scarce; high-order QAM wins throughput on high-SNR linear channels. Practical adaptive systems lower modulation order as SNR falls rather than treating one scheme as universally best.

Practice target: 18–20 minutes; define the common comparison basis, reproduce the six-row table, write only assumption-backed BER/bandwidth formulas and conclude by channel constraint.

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Spread spectrum deliberately occupies much more bandwidth than the uncoded information signal using a shared pseudorandom pattern. The receiver knowing that pattern despreads the wanted signal.

Direct-Sequence Spread Spectrum (DSSS): multiply each data bit by a much faster bipolar PN chip sequence. If chip rate is \(R_c\) and data rate \(R_b\), there are \(N=R_c/R_b\) chips/bit and approximate processing gain

The transmitted spectrum is governed mainly by \(R_c\). A synchronized receiver correlates with the same code: the desired signal collapses back to data bandwidth, while uncorrelated narrowband interference is spread before filtering. Code acquisition/tracking and low cross-correlation are essential.

Frequency-Hopping Spread Spectrum (FHSS): a PN generator controls a frequency synthesizer, moving the modulated carrier among many channels. The receiver follows the same hop order and timing to dehop. Slow hopping sends several data symbols per hop; fast hopping makes several hops during one symbol. A narrowband interferer blocks only hops landing in its channel, but FHSS needs a fast synthesizer, hop synchronization and enough hopping spectrum.

Worked DSSS example: \(R_b=10\,\text{kbps}\) and \(R_c=1\,\text{Mcchip/s}\) give \(N=100\) chips/bit and

This is nominal processing gain, not a guaranteed 20 dB end-to-end jamming margin; implementation loss, code correlation and receiver limits also matter.

FDMA and CDMA are multiple-access methods, whereas DSSS/FHSS describe spreading. CDMA often uses DSSS codes, but the terms are not identical.

In FDMA, increasing users consumes more separate channels and guard bands. In CDMA, a strong nearby transmitter can overwhelm a weak distant user's correlator; tight uplink power control mitigates this near-far problem. CDMA also requires code acquisition/tracking and suitable code families, but can exploit multipath with Rake combining in systems designed for it. DSSS is useful against narrowband interference and for code access; FHSS is useful against narrowband jamming and frequency-selective fading. Neither provides secrecy by itself, because PN sequences and protocols may be discoverable.

Practice target: 18–20 minutes; draw both spreading paths, calculate one processing gain, distinguish fast/slow hopping, and reproduce the FDMA/CDMA resource table.