Trade-offs — Bandwidth vs. Power vs. Error Rate | WRL-101

Every lesson in this module handed you a lever, and every lever had a price tag attached. Now we put them on one bench. A radio link has exactly three currencies — bandwidth, power, and the error rate you are willing to tolerate — and the honest statement of the engineering problem is that you can improve any two of them by spending the third. Nobody gets all three for free, and a designer who claims otherwise has hidden the cost somewhere you have not looked yet.

The Communication Triangle

Put the three currencies at the corners of a triangle. You want a high data rate, low transmit power, and a vanishingly small bit error rate. Pick a target for two corners and the third is no longer yours to choose — physics has already set it. Want the same rate at half the power? Then either occupy more bandwidth or accept more errors. Want the same rate in half the bandwidth? Then pay in power, because a denser constellation needs a cleaner channel to survive.

Shannon’s capacity from M6-L1 is the referee, and it becomes much more useful once we write the SNR out in full. Noise power is the noise density N₀ multiplied by the bandwidth you occupy, so SNR = P/(N₀B) — and bandwidth appears twice, in opposite directions:

Capacity in Terms of Power and Bandwidth

C = B \log_2\!\left(1 + \frac{P}{N_0 B}\right) \;\xrightarrow[\;B \to \infty\;]{}\; \frac{P}{N_0 \ln 2}

Widening B multiplies the front factor but shrinks the SNR inside the logarithm, because a wider receiver lets in more noise. The multiplication wins, but with sharply diminishing returns: as B grows without limit, C rises only to P/(N₀ ln 2). Infinite bandwidth does not buy infinite rate at fixed power.

Bandwidth-Limited or Power-Limited?

Before you tune anything, ask which constraint is actually binding, because the two regimes reward opposite moves. A bandwidth-limited link has plenty of signal and too little spectrum; a power-limited link has spectrum to spare and not enough signal. The conventional dividing line is a spectral efficiency of about 2 b/s/Hz — below it you are usually short of power, above it short of hertz.

Regime	Binding constraint	Typical η	What actually helps	Examples
Power-limited	Received signal energy	< 2 b/s/Hz	Lower-order modulation, lower code rate, more bandwidth, bigger antennas	Deep-space probes, satellite uplinks, NB-IoT
Bandwidth-limited	Licensed spectrum, interference	> 2 b/s/Hz	Higher-order modulation, higher code rate, tighter roll-off, MIMO, smaller cells	Urban cellular, WiFi in a crowded flat, microwave backhaul

The regime also tells you which improvement is worth paying for. Add 6 dB of transmit power to a bandwidth-limited link and you gain roughly two extra bits per symbol — useful, but you will run out of constellation soon enough. Add 6 dB to a power-limited link and you may go from “no link at all” to a working one. Same 6 dB, wildly different value.

Trading Power for Bandwidth

The cleanest way to see the power–bandwidth trade is to fix the bit rate and let the bandwidth float. Energy per bit E_b = P/R_b is the fair currency for comparing links that run at different rates, and rewriting Shannon in terms of E_b/N₀ and spectral efficiency η = R_b/B gives the exact price of every choice of η:

The Minimum Energy per Bit

\frac{E_b}{N_0} \;\ge\; \frac{2^{\eta} - 1}{\eta}, \qquad \eta = \frac{R_b}{B}

The absolute minimum E_b/N₀ for reliable communication at any rate. It is a decreasing function of nothing but η: spread the same bits over more hertz and each bit needs less energy. As η → 0 the expression tends to ln 2 = 0.693, which is the −1.59 dB Shannon limit of M6-L1.

Target η	Bandwidth for 1 Mbps	Min E_b/N₀	Min SNR = η · E_b/N₀
→ 0 (infinite B)	—	−1.59 dB	—
0.5 b/s/Hz	2 MHz	−0.82 dB	−3.83 dB
1 b/s/Hz	1 MHz	0.00 dB	0.00 dB
2 b/s/Hz	500 kHz	1.76 dB	4.77 dB
4 b/s/Hz	250 kHz	5.74 dB	11.76 dB
6 b/s/Hz	167 kHz	10.21 dB	17.99 dB
8 b/s/Hz	125 kHz	15.03 dB	24.07 dB

Read the table as a price list. Halving the bandwidth of a 1 Mbps link from 500 kHz to 250 kHz — that is, doubling η from 2 to 4 — costs 5.74 − 1.76 = 3.98 dB of extra energy per bit, and doubling again to 8 b/s/Hz costs a further 9.29 dB. Squeezing spectrum gets progressively more expensive, which is precisely why the top of the modulation ladder is reserved for users standing near the transmitter.

Trading Power for Error Rate

Hold the bandwidth and the modulation fixed and the remaining knob is raw power. Bit error rate falls very steeply with SNR — in the Gaussian-noise case it is governed by a Q-function, so near the operating point 1 dB of extra SNR can improve BER by roughly an order of magnitude. That steepness is a double-edged property: a link that meets 10⁻⁶ comfortably will meet 10⁻⁹ with very little more power, but a link 2 dB short of its target is not marginally worse — it is broken.

The other half of the picture is the cost of density, which M5-L3 introduced as roughly 6 dB of SNR per extra 2 bits per symbol. Here is that ladder computed exactly, for uncoded square QAM at a bit error rate of 10⁻⁶ with Gray coding:

Modulation	Bits/symbol	SNR for BER 10⁻⁶	Step in SNR	E_b/N₀ required
QPSK	2	13.5 dB	—	10.5 dB
16-QAM	4	20.4 dB	+6.9 dB	14.4 dB
64-QAM	6	26.6 dB	+6.1 dB	18.8 dB
256-QAM	8	32.5 dB	+6.0 dB	23.5 dB

Notice that the SNR column climbs in ~6 dB steps while the E_b/N₀ column climbs in gentler steps of about 4 dB. Both are correct, and the difference is not a rounding artefact: each rung carries two more bits per symbol, so part of the extra symbol energy is paid back by the extra bits it delivers. This is exactly why you must always state which ratio you are quoting — per symbol or per bit.

Buying Error Rate with Bandwidth: Coding Gain

So far power has been the only currency for buying a lower error rate. Coding gives you a second one. An error-correcting code of rate R_c transmits 1/R_c channel bits for every data bit, so at a fixed data rate it expands the symbol rate — and therefore the occupied bandwidth — by that same factor:

Redundancy Costs Bandwidth

B_{\text{coded}} = \frac{B_{\text{uncoded}}}{R_c}

A rate-1/2 code doubles the bandwidth needed for the same payload rate; a rate-3/4 code adds 33%. In exchange it delivers coding gain: the dB of transmit power you no longer need in order to hit the same BER. Modern LDPC and turbo codes (M6-L1) return several dB — often 5 to 8 dB at a target BER of 10⁻⁶ — for that bandwidth.

This is the trade-off triangle in its purest form: coding converts bandwidth into error-rate performance without touching the power amplifier. And it reframes the whole modulation-and-coding question, because the two levers can cancel each other. Moving from QPSK to 16-QAM doubles the bits per symbol — then spending half of that gain on a stronger code hands back most of the SNR you just demanded. The pairing that matters is never the modulation alone; it is the modulation and the code rate together, which is why real standards ship them as a single indexed choice.

Where the fourth currency hides. The triangle has three named corners and one unnamed one: delay. A retransmission buys reliability with latency, an interleaver buys burst protection with buffering delay, and a long LDPC codeword buys coding gain with decoding delay. For file transfer that is a bargain. For a voice call, a control loop, or a vehicle-to-vehicle warning, delay is the tightest constraint on the list — and then you go back to paying in power and bandwidth.

Adaptive Modulation and Coding

Every trade-off so far has been a design-time decision. But a mobile channel is not a fixed thing: move a phone across a room and its SNR can swing by 20 dB or more. Design for the worst case and every user in good conditions is throttled to cell-edge speeds; design for the best case and everyone else loses the link entirely. Adaptive modulation and coding (AMC) refuses the choice by making the trade-off a run-time decision, re-evaluated continuously.

The loop has three steps. Measure the channel quality; look up the densest modulation-and-coding scheme whose required SNR the channel currently supports, minus a safety margin; transmit with it until the next measurement. Throughput then tracks conditions instead of being frozen at the worst case, and the average spectral efficiency of the cell moves much closer to its peak:

The AMC Selection Rule

m^{\star} = \max\left\{\, m \;:\; \mathrm{SNR}_{\text{req}}(m) \le \widehat{\mathrm{SNR}} - \Delta \,\right\}

Choose the highest-throughput scheme m whose required SNR fits inside the measured SNR less a margin Δ. The margin absorbs measurement error and the channel’s drift between measurement and transmission — which is why AMC works beautifully for a walking pedestrian and struggles at 300 km/h, where the estimate is stale before it is used.

LTE link adaptation

LTE is the textbook implementation. The handset measures the downlink reference signals and reports a CQI (Channel Quality Indicator) — a 4-bit index from 1 to 15 — as often as once per 1 ms subframe. The base station’s scheduler combines that report with what it knows about buffer occupancy and fairness, and picks a MCS (Modulation and Coding Scheme) index for the next transmission. The ladder climbs through QPSK, then 16-QAM, then 64-QAM, with 256-QAM added in LTE-Advanced Pro and carried forward into 5G NR.

CQI index	Modulation	Code rate	Efficiency (b/s/Hz)	Where you are
1	QPSK	0.076	0.15	Cell edge, deep fade — nearly all redundancy
6	QPSK	0.59	1.18	Poor but workable coverage
9	16-QAM	0.60	2.41	Mid-cell, the regime boundary
13	64-QAM	0.75	4.52	Good conditions, near the cell centre
15	64-QAM	0.93	5.55	Excellent SNR — almost no coding left

Two things are worth noticing in that table. The spread from CQI 1 to CQI 15 is a factor of about 36 in throughput over the same bandwidth, which is the entire value proposition of AMC. And the code rate and the modulation order rise together as conditions improve, spending the extra SNR on both currencies at once rather than exhausting one first.

HARQ: the safety net under the ladder

AMC deliberately aims high. LTE targets roughly a 10% block error rate on the first transmission, because a scheme that never failed would be leaving throughput unclaimed. What makes that aggression safe is hybrid automatic repeat request (HARQ): a failed block is not discarded but retransmitted with additional parity, and the receiver combines the attempts, so the second try effectively arrives at a lower code rate. AMC is the fast, optimistic guess; HARQ is the cheap correction when the guess was wrong — and the price of that correction is delay, the fourth currency again.

Which Constraint Dominates Decides the Design

There is no universally best point in the triangle, which is why wireless engineering is not a solved problem with a lookup table. What a good design does is identify the binding constraint honestly and then spend the other two currencies freely against it:

Satellite and deep space — power-limited. Bandwidth is comparatively cheap and the received signal is measured in fractions of a picowatt, so these links run low-order modulation with very low code rates, huge antennas, and every dB of coding gain that exists.
Urban cellular — bandwidth- and interference-limited. Spectrum was auctioned for billions and neighbouring cells are the real noise floor, so the money goes to high-order modulation, AMC, MIMO, and smaller cells that reuse the same hertz more often.
IoT sensors — power- and cost-limited. A decade on one coin cell forbids both a big amplifier and an expensive receiver, so these links use narrow channels, robust low-rate modulation, and repetition rather than clever equalisation.
Short-range indoor links — neither, briefly. With gigahertz of unlicensed spectrum and metres of range, both power and bandwidth are abundant, and the binding constraints become interference from the neighbours and the cost of the radio itself.

Key Takeaways

The communication triangle is bandwidth, power, and error rate: improve any two by spending the third. Shannon sets the exchange rate.
Writing SNR as P/(N₀B) shows bandwidth cutting both ways — and capacity saturating at P/(N₀ ln 2) as B grows.
E_b/N₀ ≥ (2^η − 1)/η: squeezing bandwidth costs energy per bit — 3.98 dB to go from 2 to 4 b/s/Hz, 9.29 dB more to reach 8.
Ask first whether a link is power-limited (η < 2 b/s/Hz) or bandwidth-limited (η > 2); the two regimes reward opposite moves.
Coding gain buys error-rate performance with bandwidth, not power — which is why modulation and code rate are always chosen as a pair.
AMC makes the trade-off a run-time decision: LTE reports CQI every 1 ms and climbs QPSK → 16-QAM → 64-QAM → 256-QAM, a ~36× throughput range, with HARQ retransmission as the safety net.

Module 6 complete. That closes Module 6: Bandwidth & Data Rate — Shannon’s ceiling, Nyquist’s symbol rate, spectral efficiency, and the three-way trade you just worked through. Next, Module 7 turns to the hardware that actually launches the signal: antennas.

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