6.3 Digital Modulation Modes, Spectral Efficiency, and Weak-Signal Protocols

6.3 Digital Modulation Modes, Spectral Efficiency, and Weak-Signal Protocols

The transition from analog modulation to digital communications represents one of the most profound technological advancements in radio engineering. By encoding intelligence into discrete symbols rather than continuous voltage waveforms, digital systems unlock the mathematical guarantees of information theory, forward error correction, and advanced digital signal processing (DSP). In the amateur radio service, digital modes span a spectrum from high-throughput terrestrial digital voice and microwave networks to ultra-narrowband weak-signal protocols capable of decoding transmissions buried dozens of decibels below the atmospheric noise floor.

---

1. Information Theory Foundations: Shannon, Nyquist, and Baud

The Shannon-Hartley Theorem and Channel Capacity

In 1948, Claude Shannon established the mathematical upper bound for the rate at which error-free information can be transmitted over a communication channel corrupted by Additive White Gaussian Noise (AWGN). The Shannon-Hartley Theorem states:

$$C = B \log_2\left(1 + \frac{S}{N}\right)$$

where:

• $C$ is the channel capacity in bits per second ($\text{bps}$).
• $B$ is the channel bandwidth in Hertz ($\text{Hz}$).
• $S/N$ is the linear signal-to-noise power ratio across the channel ($S/N = 10^{\text{SNR}_{\text{dB}} / 10}$).

Shannon's formulation provides three fundamental engineering deductions:

1. Bandwidth-Power Trade-off: At fixed $S/N$, capacity is proportional to bandwidth. At fixed bandwidth and noise power, it varies as $\log_2(1+S/N)$. If received signal power and white-noise spectral density $N_0$ are fixed instead, $N=N_0B$ grows with bandwidth and $C=B\log_2[1+S/(N_0B)]$ approaches $S/(N_0\ln2)$. At low SNR, increasing signal power can therefore produce an approximately proportional capacity increase.
2. Communication Below the Noise Floor: When signal power is significantly weaker than noise power ($S/N < 1$, corresponding to a negative SNR in decibels), the logarithm can be approximated using the Taylor expansion $\ln(1 + x) \approx x$:

$$C \approx B \left( \frac{S}{N} \right) \log_2(e) \approx 1.4427 \cdot B \left( \frac{S}{N} \right)$$

A negative SNR still gives positive capacity. Reliable communication is theoretically possible at an information rate below this limit with suitable coding and sufficiently long codewords. A low rate or long integration alone does not guarantee error-free reception. Weak-signal modes such as FT8 and WSPR combine coding, synchronization and narrowband detection to work at low SNR.

Baud Rate vs. Bit Rate

A fundamental point of confusion in communications engineering is the distinction between Baud Rate and Bit Rate:

• Baud Rate (Symbol Rate, $S$): The number of transmitted symbols per second. Consecutive symbols may have the same state, so baud does not count only visible transitions.
• Gross Modulation Bit Rate ($R_b$): The rate of bits mapped into symbols, in bits per second. Net user-data rate is lower when some bits or symbol intervals carry FEC, framing or synchronization.

For $M=2^k$ states with $k=\log_2M$ mapped bits per symbol, the gross modulation bit rate is:

$$R_b = S \cdot \log_2(M) = S \cdot k$$

Constellation Encodings:
- Binary Signaling (M = 2,  e.g., BPSK, 2-FSK):  k = log2(2)  = 1 bit/symbol  --> Bit Rate = Baud Rate
- 4-ary Signaling   (M = 4,  e.g., QPSK, 4-FSK):  k = log2(4)  = 2 bits/symbol --> Bit Rate = 2 x Baud Rate
- 8-ary Signaling   (M = 8,  e.g., 8-PSK, 8-FSK):  k = log2(8)  = 3 bits/symbol --> Bit Rate = 3 x Baud Rate
- 16-ary Signaling  (M = 16, e.g., 16-QAM):       k = log2(16) = 4 bits/symbol --> Bit Rate = 4 x Baud Rate

Nyquist Minimum Bandwidth and Pulse Shaping

Harry Nyquist proved that to transmit a symbol rate $S$ over an ideal noiseless low-pass channel without Inter-Symbol Interference (ISI), the minimum theoretical baseband bandwidth is:

$$B_{\text{Nyquist}} = \frac{S}{2}$$

When modulated onto an RF carrier (double-sideband translation), the minimum theoretical RF bandwidth is $B_{\text{RF}} = S$.

Rectangular pulses have broad sinc-shaped spectra. Pulse shaping limits this spectrum, while the combined channel and receiver response must meet the Nyquist zero-ISI condition at the sampling instants. A raised-cosine response meets that condition. Commonly, matched root-raised-cosine (RRC) filters at transmitter and receiver combine to give the raised-cosine response; either RRC filter alone is not that response. With symbol rate $S$ and roll-off $\alpha$, the ideal RF bandwidth is:

$$B_{\text{RF}} = S (1 + \alpha)$$

where $\alpha$ is the filter roll-off factor ($0 \le \alpha \le 1$). A filter with $\alpha = 0.35$ limits excess bandwidth to $35\%$ above the Nyquist rate, suppressing out-of-band spectral emissions by tens of decibels.

---

2. Primary Digital Modulation Formats in Amateur Radio

Modulation Formats:
       FSK (Frequency Shift)                 PSK (Phase Shift)                 QAM (Amplitude + Phase)
       f_mark      f_space                     0 deg       180 deg                  +-----+-----+
         |            |                          |            |                     |  *  |  *  |
      ---+------------+---> f                 ---+------------+---> Phase           |  *  |  *  |
                                                                                ----+-----+----+---
                                                                                    |  *  |  *  |
                                                                                    +-----+-----+

1. Frequency Shift Keying (FSK) and Audio FSK (AFSK)

In Frequency Shift Keying, the binary states are represented by shifting the transmitter carrier between two discrete frequencies:

• MARK Frequency ($f_{\text{mark}}$): Represents binary 1 (conventionally the higher frequency).
• SPACE Frequency ($f_{\text{space}}$): Represents binary 0 (conventionally the lower frequency).
• Frequency Shift ($\Delta f$): The difference between mark and space: $\Delta f = |f_{\text{mark}} - f_{\text{space}}|$.

Amateur RTTY (Radioteletype): The classic HF amateur digital mode operates at $45.45\text{ baud}$ using a standard frequency shift of $170\text{ Hz}$ with 5-bit Baudot coding.

Direct FSK vs. Audio FSK (AFSK):

• Direct FSK (FSK): The digital bitstream directly keys the master oscillator frequency (e.g. through a varactor). The RF envelope remains strictly constant, generating a clean spectrum devoid of intermodulation.
• Audio FSK (AFSK): Digital software generates two alternating audio tones (e.g., $2125\text{ Hz}$ Mark and $2295\text{ Hz}$ Space), which are fed into the microphone input of an ordinary SSB transceiver. If the transceiver's audio gain is set too high, speech amplifier clipping and Automatic Level Control (ALC) action generate audio harmonics and severe intermodulation distortion (IMD) products, splattering across adjacent frequencies.

Continuous Phase FSK (CPFSK) and MSK: Switching between two independent oscillators produces abrupt phase discontinuities at bit transitions, generating broad spectral sidelobes. In Continuous Phase FSK (CPFSK), the phase is constrained to transition smoothly without phase jumps. When the frequency shift is set to exactly half the symbol rate ($\Delta f = S / 2$, modulation index $h = 0.5$), it is termed Minimum Shift Keying (MSK)—the theoretical minimum frequency separation for coherent orthogonal detection. Adding a Gaussian pulse-shaping filter yields Gaussian Minimum Shift Keying (GMSK), the format used in D-STAR and GSM.

2. Phase Shift Keying (PSK)

In Phase Shift Keying, the carrier frequency remains fixed while its phase is shifted among discrete angular states:

• Binary Phase Shift Keying (BPSK): Two antipodal phases, such as $0^\circ$ and $180^\circ$, convey 1 bit per symbol ($M=2$).
• Quadrature Phase Shift Keying (QPSK): Four constellation phases, such as $45^\circ$, $135^\circ$, $225^\circ$ and $315^\circ$, convey 2 bits per symbol ($M=4$). They use two orthogonal carrier components, I and Q; the four states are not all mutually orthogonal. At the same symbol rate and pulse shaping, QPSK carries twice the gross bit rate of BPSK in the same RF bandwidth. Coherent Gray-coded QPSK and BPSK have the same theoretical bit-error probability at a given $E_b/N_0$ in AWGN.
• Differential PSK (DPSK): Phase shifts represent the change in bit state relative to the preceding symbol rather than absolute phase. This eliminates phase ambiguity at the receiver and avoids the need for complex carrier phase tracking.

The PSK31 Revolution

Developed by British amateur Peter Martinez (G3PLX), PSK31 revolutionized HF amateur keyboard communications by achieving extraordinary spectral efficiency:

1. Symbol Rate: Operates at $31.25\text{ baud}$—a speed matched to human typing cadence ($~50\text{ words per minute}$).
2. Envelope Shaping: In BPSK31, a phase reversal is shaped so that amplitude falls to zero at the reversal and rises smoothly afterward. Where no reversal is required, amplitude can remain at full value. Each symbol does not necessarily produce a separate bell-shaped envelope.
3. Narrow Spectrum: Shaping reduces the broad spectral components caused by abrupt phase changes; it does not eliminate modulation sidebands. The $31.25\text{ baud}$ symbol rate is distinct from occupied bandwidth. As one documented implementation example, MixW describes about $40\text{ Hz}$ width at $-3\text{ dB}$ and $80\text{ Hz}$ at $-40\text{ dB}$. State the measurement level when quoting a bandwidth.
4. Varicode Encoding: Employs a variable-length character code where frequently used letters (such as 'e' and 't') are assigned short codes (e.g. 2 or 3 bits), while rare characters receive longer codes. Consecutive zeros act as character delimiters, maximizing throughput.

3. Quadrature Amplitude Modulation (QAM)

QAM combines both amplitude modulation and phase modulation simultaneously. Constellation diagrams map symbol states on a two-dimensional In-Phase ($I$) and Quadrature ($Q$) plane:

• 16-QAM: Defines 16 constellation points, encoding $k = \log_2(16) = 4\text{ bits/symbol}$.
• 64-QAM: Defines 64 constellation points, encoding $k = \log_2(64) = 6\text{ bits/symbol}$.

Trade-offs: While 16-QAM and 64-QAM deliver high spectral efficiency ($>3-5\text{ bps/Hz}$), they do not possess a constant envelope. They require highly linear Class A or Class AB power amplifiers (backed off well below compression) and demand significantly higher signal-to-noise ratios to distinguish adjacent constellation states. In amateur service, QAM is used on higher-SNR microwave links, digital ATV and mesh networks, and in adaptive HF modems such as PACTOR-4.

4. Multiple Frequency Shift Keying (MFSK)

MFSK utilizes $M$ distinct orthogonal tones (where $M = 4, 8, 16, 32, 64$). Only one tone is active at any instant. By extending the symbol duration across multiple frequencies, MFSK provides exceptional resistance to ionospheric multipath delay spread and selective fading, which severely degrade high-speed single-carrier PSK modes on HF.

---

3. Weak-Signal Protocols: The WSJT-X Architecture

Pioneered by Nobel laureate Joe Taylor (K1JT), the WSJT-X suite transformed weak-signal amateur communications by combining MFSK modulation, precise UTC time synchronisation, and advanced forward error correction.

These nominal thresholds use a $2500\text{ Hz}$ reference noise bandwidth and approximately $50\%$ decoding probability under the documented test conditions. They are not hard reception limits. Optional a priori (AP) decoding can improve FT8 sensitivity by up to about 4 dB using already known message information, with a slightly greater risk of false decodes.

1. FT8 (Franke-Taylor 8-Tone)

FT8 is the dominant weak-signal mode on amateur HF and 6-metre bands worldwide:

• Modulation: Continuous-phase 8-FSK modulation. Tone spacing is exactly equal to the symbol rate: $\Delta f = 6.25\text{ Hz}$.
• Symbol Rate: $6.25\text{ baud}$ (each symbol lasts $\tau = 1 / 6.25 = 160\text{ ms}$).
• Transmission Framing: FT8 uses repeating $15\text{ s}$ UTC sequences. Nominal transmission starts about $0.5\text{ s}$ after the boundary and lasts $79\times0.160=12.64\text{ s}$, ending near $13.14\text{ s}$. The remaining time permits timing tolerance and processing before the next sequence; it is not a separate $2.36\text{ s}$ interval after a one-second-delayed transmission.
• Occupied Bandwidth: The nominal bandwidth is $8\times6.25=50\text{ Hz}$. The $7\times6.25=43.75\text{ Hz}$ distance between the two outer tone centres is not the complete occupied bandwidth.
• Time Synchronisation: Requires the host computer clock to be synchronised to Universal Coordinated Time (UTC) within $\pm 1.0\text{ second}$ (typically using NTP internet time servers or GPS).
• Message Structure: A 77-bit message plus 14 CRC bits forms 91 bits, encoded by an LDPC (174,91) code into 174 coded bits. These occupy 58 data symbols of 3 bits each. Three 7-symbol Costas arrays provide 21 synchronization symbols, making 79 symbols in total.
• Decoding Capability: The nominal single-message threshold is about $-21\text{ dB}$ in a $2500\text{ Hz}$ noise reference bandwidth. AP-assisted decoding can reach lower SNRs, including around $-24\text{ dB}$ under suitable conditions. A quoted SNR depends on its reference bandwidth; it does not by itself specify whether a listener can hear the signal.

2. FT4

FT4 was designed for rapid contest exchanges. It uses 4-GFSK at about $20.83\text{ baud}$ in $7.5\text{ s}$ T/R sequences, with $5.04\text{ s}$ transmissions and nominal $83.3\text{ Hz}$ bandwidth. It shares the LDPC (174,91) message code with FT8. Its sequence is half as long as FT8's, while its nominal $-17.5\text{ dB}$ threshold in 2500 Hz trades some weak-signal sensitivity for speed.

3. WSPR (Weak Signal Propagation Reporter)

WSPR probes ionospheric propagation paths worldwide using milliwatt-level beacon transmissions:

• 4-FSK modulation with tone spacing of $1.4648\text{ Hz}$.
• Symbol rate of $1.4648\text{ baud}$; transmission spans $110.6\text{ seconds}$ within a 2-minute window.
• Total occupied bandwidth is a minuscule $6\text{ Hz}$.
• Features a heavy convolutional code with constraint length $K=32$, decoding signals down to $-28\text{ dB}$ to $-31\text{ dB}$ SNR in a $2.5\text{ kHz}$ bandwidth. A $1\text{ W}$ WSPR transmitter frequently achieves worldwide reception.

4. JS8Call

Created by Jordan Sherer (KN4CRD), JS8Call adapts the robust 8-FSK physical modulation of FT8 for conversational, keyboard-to-keyboard ragchewing, store-and-forward messaging, and automatic networking, featuring variable speed sub-modes (Slow, Normal, Fast, Turbo).

---

4. Error Detection and Forward Error Correction (EDAC)

Noise and fading can cause decision errors. Modern digital links combine error detection, correction and, where the protocol supports it, retransmission:

Information Bits (77 bits) --->+----------------------+        +-----------------------+
                               | CRC Generator        |------->| FEC Encoder           |---> Transmitted
                               | (+14 Parity Check)   |        | (LDPC / Convolutional)|     Codeword
                               +----------------------+        +-----------------------+

1. Error Detection: Cyclic Redundancy Checks (CRC)

A CRC computes check bits using an agreed generator polynomial and frame format. The receiver applies the corresponding check. A failed check detects corruption; a passed check does not prove that every bit is correct because some error patterns are undetected. Detection capability depends on the polynomial, frame length and error pattern, so there is no universal detection percentage for all CRCs. A CRC alone does not reconstruct corrupted data.

ARQ adds retransmission through protocol acknowledgements, sequence numbers and timeouts. Connected AX.25 uses such recovery; AX.25 UI frames, used by APRS, are connectionless and do not themselves guarantee acknowledgement or retransmission. Error detection and retransmission are distinct functions.

2. Forward Error Correction (FEC)

On a one-way link, no return path is available for ARQ; on a two-way link, retransmissions consume airtime and add delay. FEC adds structured redundancy so that a decoder can recover data from errors within the code's capability without requesting another transmission:

• Convolutional Coding and Viterbi Decoding: Data bits pass through an array of shift registers tapped by modulo-2 adders. The Viterbi algorithm performs maximum-likelihood trellis decoding, selecting the most probable transmitted sequence.
• Low-Density Parity-Check (LDPC) Codes: Advanced linear block codes governed by sparse parity-check matrices. When decoded via iterative belief-propagation algorithms, LDPC codes operate within a fraction of a decibel of the theoretical Shannon limit.

3. Bit Interleaving: Mitigating Ionospheric Burst Errors

HF static and fading can produce bursts of adjacent errors, as well as isolated errors. A burst concentrated in one codeword may exceed its correction capability. Interleaving spreads that concentration across several codewords.

Interleaving reorders bits before transmission. Consider three illustrative codewords, each able to correct one bit error:

Original Codeword Stream:  [A1 A2 A3 A4] [B1 B2 B3 B4] [C1 C2 C3 C4]
                              |
Written into Matrix Rows:     | A1  A2  A3  A4 |
                              | B1  B2  B3  B4 |
                              | C1  C2  C3  C4 |
                              |
Read out by Columns:       A1 B1 C1  A2 B2 C2  A3 B3 C3  A4 B4 C4  (Transmitted over RF)

Channel Static Burst Hits:            [XXXXXX] (Destroys B2, C2, A3)

Receiver De-interleaver restores original order:
Reconstructed Block A: [A1 A2 XX A4]  --> One error; within the assumed code capability
Reconstructed Block B: [B1 XX B3 B4]  --> One error; within the assumed code capability
Reconstructed Block C: [C1 XX C3 C4]  --> One error; within the assumed code capability

Here the burst affects B2, C2 and A3, so deinterleaving leaves one error in each codeword and the assumed code can correct all three. A longer burst or different error pattern may leave too many errors in one codeword. Interleaving adds latency and does not guarantee successful decoding.

---

5. Amateur Digital Voice (DV) Systems

Human speech sampled at standard PCM rates ($8000\text{ samples/s} \times 16\text{ bits} = 128\text{ kbps}$) cannot fit inside narrow amateur channels. Digital voice relies on vocoders (voice encoders) that model human vocal tract biology (pitch, formant resonances, and vocal cord excitation) to compress speech down to $2.0\text{ to }3.6\text{ kbps}$:

Vocoder Engines

• AMBE Family: Proprietary vocoders from Digital Voice Systems Inc. DMR uses AMBE+2 with $2450\text{ bps}$ of voice parameters plus $1150\text{ bps}$ of FEC, totaling $3600\text{ bps}$. Classic D-STAR specifies the earlier AMBE2020 format: $2400\text{ bps}$ voice, expanded to $3600\text{ bps}$ with FEC. System Fusion has distinct voice/data and full-rate voice modes; one fixed voice-rate setting does not describe all of them.
• Codec2: A fully open-source, patent-free speech vocoder created by Australian amateur David Rowe (VK5DGR). Supports ultra-low bitrates from $3200\text{ bps}$ down to $700\text{ bps}$, enabling high-quality digital voice over HF SSB transceivers (FreeDV).

Major Digital Voice Protocols

DMR 2-Slot TDMA Architecture:
Repeater Frequency (12.5 kHz Channel):
[ Slot 1 (30 ms) ][ Slot 2 (30 ms) ][ Slot 1 (30 ms) ][ Slot 2 (30 ms) ]
     |                    |               |
  User A               User B          User A
  (Local QSO)        (Worldwide TG)  (Local QSO)

• DMR Tier II: A $60\text{ ms}$ TDMA frame contains two $30\text{ ms}$ slots, supporting two independent logical traffic channels on a $12.5\text{ kHz}$ repeater channel. A mobile sends bursts in its assigned slot, with guard/ramp time. The activated repeater downlink transmits continuously across both slots and the common announcement channel; it does not simply switch its carrier off during the other user's slot.
• D-STAR: True FDMA occupying a narrow $6.25\text{ kHz}$ bandwidth. Features callsign-based routing embedded directly into the packet header.
• Yaesu System Fusion (YSF): Utilizes C4FM (4-level FSK). Features Automatic Mode Selection (AMS), which instantly switches between analog FM and digital C4FM depending on received signal type.

---

6. Worked Calculation: Baud Rate, Bit Rate, and Shannon Capacity

Worked Example 6.3: Shannon Capacity and Digital Voice Throughput

1. A high-speed microwave amateur link utilizes 16-QAM modulation at a symbol rate of $9600\text{ baud}$. Calculate the raw transmission bit rate ($R_b$):

$$k = \log_2(M) = \log_2(16) = 4\text{ bits/symbol}$$ $$R_b = S \cdot k = 9600\text{ baud} \times 4\text{ bits/symbol} = 38400\text{ bps} = 38.4\text{ kbps}$$

1. An HF radio voice channel possesses an analog bandwidth of $B = 3000\text{ Hz}$ and an operating signal-to-noise ratio of $20\text{ dB}$. Calculate the theoretical Shannon channel capacity ($C$):

Convert SNR from decibels to a linear power ratio:

$$\frac{S}{N} = 10^{\text{SNR}_{\text{dB}} / 10} = 10^{20 / 10} = 10^2 = 100$$

Apply the Shannon-Hartley formulation:

$$C = B \log_2\left(1 + \frac{S}{N}\right) = 3000 \times \log_2(1 + 100) = 3000 \times \log_2(101)$$

Using the base change rule $\log_2(x) = \frac{\ln(x)}{\ln(2)}$:

$$\log_2(101) = \frac{\ln(101)}{\ln(2)} \approx \frac{4.61512}{0.69315} \approx 6.6582$$ $$C = 3000 \times 6.6582 \approx 19975\text{ bps} \approx 20.0\text{ kbps}$$

Even over an ordinary $3000\text{ Hz}$ voice channel with a modest $20\text{ dB}$ SNR, Shannon theory demonstrates that modern DSP modems can theoretically achieve almost $20\text{ kbps}$ of throughput—vastly exceeding classical $300\text{ baud}$ HF limits.

---

7. Master Comparison: Amateur Digital Modes

Symbol rate and gross mapped-bit rate differ from net message or speech rate. For example, FreeDV700D uses a $700\text{ bit/s}$ speech codec but an approximately $1900\text{ bit/s}$ modem stream including coding and other overhead; it is not a $700\text{ bit/s}$ uncoded RF stream. FreeDV modes use different codecs and modem waveforms, so the 700D example does not describe every mode.

Use the earlier WSJT table for its documented SNR test conditions. Receiver sensitivity comparisons also require a specified bandwidth, error/decoding criterion, coding and channel model; a single universal minimum-SNR number does not characterize all implementations of a modulation family.