Ferkans — Interactive Telecom Tutor

Beyond Amplitude and Phase

PAM and QAM encode information in the amplitude and/or phase of a carrier. An alternative family encodes information in the frequency of the carrier. Frequency-based modulations offer constant-envelope signals, which are robust to nonlinear amplification — a critical advantage in power-limited systems such as satellite links and early cellular networks.

Definition:
Frequency-Shift Keying (M-FSK)

$M$ -ary Frequency-Shift Keying ( $M$ -FSK) transmits one of $M$ sinusoidal signals during each symbol interval:

$s_m(t) = \sqrt{\frac{2E_s}{T_s}} \cos(2\pi f_m t), \qquad 0 \leq t \leq T_s$

where $f_m = f_c + (2m - 1 - M)\Delta f/2$ for $m = 1, \ldots, M$ .

The minimum tone spacing for orthogonality is $\Delta f = 1/(2T_s)$ (non-coherent detection) or $\Delta f = 1/T_s$ (coherent detection with integer-cycle constraint).

The signal space is $N = M$ dimensional: each tone defines its own basis function. The $M$ signal points lie at the vertices of a regular simplex in $\mathbb{R}^M$ (or on the axes for orthogonal FSK).

The bandwidth grows linearly with $M$ :

$W \approx M \cdot \Delta f = \frac{M}{2T_s}$

so FSK trades spectral efficiency for power efficiency.

,

Definition:
Minimum-Shift Keying (MSK)

Minimum-Shift Keying (MSK) is a special case of binary FSK with the minimum tone spacing that maintains orthogonality while ensuring continuous phase at symbol transitions.

The modulation index is $h = \Delta f \cdot T_b = 0.5$ , where $T_b$ is the bit period and $\Delta f = 1/(4T_b)$ is the frequency deviation.

The MSK signal can be written as

$s(t) = \cos\!\left(2\pi f_c t + \phi(t)\right)$

where the phase $\phi(t)$ varies linearly within each bit interval:

$\phi(t) = \phi(kT_b) + \frac{\pi b_k}{2T_b}(t - kT_b), \qquad kT_b \leq t < (k+1)T_b$

and $b_k \in \{-1, +1\}$ is the $k$ -th data bit.

The phase trajectory is a sequence of $\pm\pi/2$ ramps, creating a continuous-phase signal with:

Constant envelope (ideal for nonlinear amplifiers)
Compact spectrum (main lobe width $= 1.5/T_b$ )
Exact BER identical to BPSK: $P_b = Q(\sqrt{2E_b/N_0})$

,

Theorem: MSK as Continuous-Phase FSK with h = 0.5

MSK is the unique binary CPFSK scheme with modulation index $h = 0.5$ (minimum-shift). This value of $h$ is special because:

The two FSK tones are orthogonal over $[0, T_b]$ .
The phase accumulates exactly $\pm\pi/2$ per bit period, producing a phase trellis where the phase at each bit boundary is a multiple of $\pi/2$ .
The signal can be decomposed into two quadrature half-sinusoid pulse-shaped BPSK streams:

$s(t) = a_I(t)\cos(\pi t / 2T_b)\cos(2\pi f_c t) - a_Q(t)\sin(\pi t / 2T_b)\sin(2\pi f_c t)$

where $a_I(t)$ and $a_Q(t)$ are the even and odd bit streams, each held for $2T_b$ (staggered by $T_b$ ).

The modulation index $h = 0.5$ is the smallest value that ensures the two frequency tones are distinguishable (orthogonal) at the receiver. Any smaller $h$ would merge the tones; any larger $h$ wastes bandwidth. The half-sinusoid pulse shaping in the I/Q decomposition is what gives MSK its compact spectrum.

Proof

Orthogonality condition

Two tones at frequencies $f_c \pm \Delta f/2$ are orthogonal over $[0, T_b]$ when

$\int_0^{T_b} \cos(2\pi(f_c + \Delta f/2)t)\cos(2\pi(f_c - \Delta f/2)t)\,dt = 0$

This requires $\Delta f = k/(2T_b)$ for integer $k$ . The minimum is $k = 1$ , giving $\Delta f = 1/(2T_b)$ and $h = \Delta f \cdot T_b = 0.5$ .

Phase accumulation

The instantaneous frequency is $f_c + b_k/(4T_b)$ , so the phase change per bit is

$\Delta\phi = 2\pi \cdot \frac{b_k}{4T_b} \cdot T_b = \frac{\pi b_k}{2} = \pm\frac{\pi}{2}$

I/Q decomposition

Expanding $s(t) = \cos(2\pi f_c t + \phi(t))$ into I and Q components and identifying the half-sinusoid envelope shape $\cos(\pi t / 2T_b)$ and $\sin(\pi t / 2T_b)$ yields the decomposition. The staggering by $T_b$ between I and Q is the defining property of offset QPSK with sinusoidal pulse shaping. $\blacksquare$

,

Definition:
Gaussian Minimum-Shift Keying (GMSK)

GMSK pre-filters the rectangular frequency pulse of MSK with a Gaussian lowpass filter of bandwidth-time product $BT$ :

$g(t) = \frac{1}{2T_b}\left[Q\!\left(\frac{2\pi B}{\sqrt{\ln 2}}\left(t - \frac{T_b}{2}\right)\right) - Q\!\left(\frac{2\pi B}{\sqrt{\ln 2}}\left(t + \frac{T_b}{2}\right)\right)\right]$

The Gaussian filter smooths the phase transitions, producing a more compact spectrum at the cost of introducing controlled inter-symbol interference (ISI).

Key parameter: $BT$ product controls the trade-off between spectral compactness and ISI:

$BT = \infty$ : no filtering, reduces to MSK
$BT = 0.3$ : used in GSM — 99% bandwidth $\approx 1.18/T_b$
$BT = 0.5$ : used in Bluetooth
Smaller $BT$ : more compact spectrum, more ISI, harder to detect

Example: GSM Uses GMSK with BT = 0.3

The GSM cellular standard uses GMSK with $BT = 0.3$ at a bit rate of $R_b = 270.833$ kbps.

(a) What is the 99% power bandwidth of the GSM signal?

(b) What is the channel spacing in GSM, and how does it compare?

(c) Why was GMSK chosen over QPSK for GSM?

Solution

99% bandwidth

For GMSK with $BT = 0.3$ , the 99% power bandwidth is approximately $0.86/T_b = 0.86 \times R_b = 233$ kHz.

Channel spacing comparison

GSM uses 200 kHz channel spacing. The 99% bandwidth of 233 kHz slightly exceeds this, but the rapid spectral roll-off of GMSK ensures adequate adjacent-channel rejection with practical filters.

Why GMSK

In the 1980s when GSM was designed:

Constant envelope was essential because power amplifiers were expensive and nonlinear
Compact spectrum was needed for the 200 kHz channel spacing
Simplicity: GMSK can be generated with a VCO (voltage- controlled oscillator) — no I/Q modulator needed

Modern standards (LTE, 5G NR) use OFDM with QAM because digital signal processing and linear PAs are now affordable, offering much higher spectral efficiency. $\blacksquare$

Multicarrier Modulation as Frequency-Domain Modulation

FSK assigns one symbol to one frequency tone. Orthogonal Frequency Division Multiplexing (OFDM) takes this idea to its logical conclusion: assign one QAM symbol to each of $K$ closely-spaced orthogonal subcarriers simultaneously.

OFDM can be viewed as modulation in the frequency domain: the data vector $\mathbf{X} = [X_0, X_1, \ldots, X_{K-1}]^T$ modulates $K$ subcarriers, and the time-domain signal is obtained via the inverse DFT:

$x[n] = \frac{1}{\sqrt{K}} \sum_{k=0}^{K-1} X_k\, e^{j2\pi kn/K}$

This connection between frequency-domain modulation and multicarrier transmission is developed fully in Chapter 14.

MSK Phase Trellis Animation

Visualise how the MSK phase evolves continuously as data bits are transmitted. Each bit causes a

\pm\pi/2

linear phase ramp, creating the characteristic phase trellis. The continuous-phase property — no abrupt phase jumps — is what gives MSK its compact spectrum.

MSK phase trellis: each bit causes a

\pm\pi/2

phase ramp. The continuous phase produces a compact spectrum.

QPSK Time-Domain Signal

Watch how a sequence of QPSK symbols maps to the transmitted baseband waveform. The animation shows the I and Q components, the constellation trajectory, and the combined signal envelope.

Parameters

Number of symbols8

SNR (dB)20

Why This Matters: From GSM to 5G — The Modulation Evolution

The modulation choices in wireless standards mirror the technology trade-offs of each era:

1G (AMPS): Analog FM — no digital modulation
2G (GSM): GMSK — constant envelope for cheap PAs
2.5G (EDGE): 8-PSK — higher data rate, still constant-ish envelope
3G (UMTS): QPSK with CDMA spreading
4G (LTE): OFDM with up to 64-QAM
5G NR: OFDM with up to 256-QAM (1024-QAM in Rel. 17)
Wi-Fi 7: OFDM with 4096-QAM

Each generation increases modulation order and spectral efficiency as advances in ADCs, DSPs, and PA linearisation permit it.

See full treatment in Principle of OFDM

Quick Check

What is the modulation index $h$ of MSK, and why is it called "minimum-shift"?

$h = 1.0$ ; minimum bandwidth among all FSK

$h = 0.5$ ; minimum $h$ for orthogonal FSK tones

$h = 0.25$ ; minimum shift for continuous phase

$h = 0.5$ ; minimum phase shift per symbol

Correction:

h = 0.5

; minimum

h

for orthogonal FSK tones

MSK uses the minimum modulation index ( $h = 0.5$ ) that maintains orthogonality between the two frequency tones over one bit period. Smaller $h$ would make the tones non-orthogonal and non-separable.

Frequency-Shift Keying (FSK)

A modulation scheme that encodes information in the frequency of the carrier. $M$ -FSK uses $M$ distinct frequencies and has an $M$ -dimensional signal space.

Minimum-Shift Keying (MSK)

Binary continuous-phase FSK with modulation index $h = 0.5$ , the minimum value ensuring orthogonality. Achieves BPSK-equivalent BER with constant envelope and compact spectrum.

Gaussian Minimum-Shift Keying (GMSK)

MSK with a Gaussian pre-filter applied to the frequency pulse. The $BT$ product controls the trade-off between spectral compactness and ISI. Used in GSM ( $BT = 0.3$ ) and Bluetooth ( $BT = 0.5$ ).

Historical Note: The Continuous-Phase Modulation Family

1972-1981

Continuous-phase modulation (CPM) was developed in the 1960s-70s by researchers seeking spectrally efficient constant-envelope modulations. T. Aulin and C.-E. Sundberg (1981) provided the definitive treatment, showing that CPM with appropriate pulse shaping can achieve excellent spectral and power efficiency simultaneously. MSK (de Buda, 1972) and GMSK (Murota and Hirade, 1981) are the most widely deployed CPM variants, having been adopted by GSM, DECT, and Bluetooth.

Frequency and Phase Modulation