Continuous-Phase Frequency-Shift Keying (CPFSK)

Continuous-Phase Frequency-Shift Keying (CPFSK)#

Issues of FSK#

In this section, we examine a class of digital modulation techniques in which the signal phase is constrained to remain continuous throughout the transmission process, it means that the phase of the signal does not undergo any abrupt changes or jumps during the modulation process. Instead, it evolves smoothly over time, ensuring that there are no sudden transitions between phase values.

Imposing this requirement on the phase leads to a modulator—either phase-based or frequency-based—that inherently exhibits memory.

The conventional FSK (Frequency-Shift Keying) signal is typically generated by shifting the carrier frequency by an increment $m\Delta f$, where $1 \leq m \leq M$. Here, $M = 2^k$ represents the total number of available frequencies, and each frequency corresponds to one of the possible $k$-bit symbols to be transmitted. In this conventional approach, the frequency shift is achieved by selecting from among $M$ separate oscillators, each tuned to a specific frequency. When a new $k$-bit symbol arrives, the system switches to the oscillator associated with that symbol for the duration of one signaling interval, $T = k/R$ seconds, where $R$ is the symbol rate.

Although straightforward to implement, this abrupt switching from one oscillator output to another at symbol boundaries produces significant spectral side lobes outside the main lobe of the transmitted signal. As a result, conventional FSK systems require a larger bandwidth to accommodate these side lobes.

To mitigate this bandwidth expansion, Continuous-Phase FSK (CPFSK) modifies the carrier frequency in a smooth and continuous manner. Instead of abruptly jumping from one frequency to another, the information-bearing signal continuously modulates a single carrier, ensuring that the phase of the transmitted signal remains continuous. This property leads to a so-called “memory” in the modulator, since the phase at any given time depends on its history.

To develop a mathematical representation of CPFSK, one typically begins by expressing the information signal in the form of a Pulse Amplitude Modulation (PAM) signal. This PAM representation provides a convenient basis for describing how the frequency changes continuously over each symbol interval while preserving phase continuity.

Thus, the motivation of using CPFSK is:

To avoid the large spectral side lobes caused by abrupt frequency changes, CPFSK ensures continuous frequency modulation.
The carrier’s frequency is modulated smoothly, maintaining phase continuity, hence the name continuous-phase FSK.
The memory in CPFSK arises from this phase continuity.
The representation of CPFSK begins with a Pulse Amplitude Modulation (PAM) signal, which serves as the base for defining the frequency-modulated waveform.

CPFSK Waveform#

We define

$ d(t) $ is the modulating baseband signal.
$ s(t) $ is the bandpass modulated signal.
$ v(t) $ is the lowpass equivalent signal of $s(t)$.

PAM Signal Representation#

The CPFSK signal begins with a Pulse Amplitude Modulation (PAM) signal, defined as:

\[ \boxed{ d(t) = \sum_{n} I_n g(t - nT) } \]

$I_n$: Sequence of amplitudes derived by mapping $k$-bit binary blocks into amplitude levels such as $\pm 1, \pm 3, \dots, \pm (M-1)$.
$g(t)$: Rectangular pulse of amplitude $1/2T$ and duration $T$ seconds.
Purpose: This PAM signal $d(t)$ serves as the base for frequency modulation of the carrier.

Lowpass Equivalent Waveform#

The lowpass equivalent waveform $v(t)$ is expressed as:

\[ \boxed{ v(t) = \sqrt{\frac{2\mathcal{E}}{T}} e^{j\left[4\pi T f_d \int_{-\infty}^{t} d(\tau)d\tau + \phi_0\right]} } \]

$\mathcal{E}$: Signal energy.
$f_d$: Peak frequency deviation.
$\phi_0$: Initial carrier phase.
Interpretation: The phase of $v(t)$ depends on the integral of $d(t)$, ensuring phase continuity despite possible amplitude discontinuities in $d(t)$.

Carrier-Modulated Signal#

The carrier-modulated CPFSK signal $s(t)$ is defined as:

\[ \boxed{ s(t) = \sqrt{\frac{2\mathcal{E}}{T}} \cos\left[2\pi f_c t + \phi(t; I) + \phi_0\right] } \]

$f_c$: Carrier frequency.
$\phi(t;I)$: Time-varying phase of the carrier.
Significance: The cosine term shows that the signal is a continuous-phase waveform modulated by the information sequence $\{I_n\}$.

Time-Varying Phase of the Carrier#

The phase $\phi(t;I)$ is defined as:

\[ \phi(t;I) = 4\pi T f_d \int_{-\infty}^{t} \left[\sum_n I_n g(\tau - nT)\right] d\tau \]

This equation illustrates how the phase evolves over time as a result of the PAM input signal $d(t)$.

Phase Calculation Over Symbol Intervals#

Within each signaling interval $ nT \leq t \leq (n+1)T $, the phase can be expressed as:

\[ \phi(t;I) = 2\pi f_d T \sum_{k=-\infty}^{n-1} I_k + 2\pi f_d q(t - nT) I_n \]

Or equivalently:

\[ \phi(t;I) = \theta_n + 2\pi h I_n q(t - nT) \]

Key Parameters

$h$: Modulation index, defined as:

\[ h = 2 f_d T \]

$\theta_n$: Phase memory term that accounts for all previous symbols:

\[ \theta_n = \pi h \sum_{k=-\infty}^{n-1} I_k \]

$q(t)$: Phase transition function:

\[\begin{split} q(t) = \begin{cases} 0 & t < 0 \\ \frac{t}{2T} & 0 \leq t \leq T \\ \frac{1}{2} & t > T \end{cases} \end{split}\]

We can see that:

The phase of CPFSK is determined by both the current input symbol and the cumulative effect of all previous symbols, introducing memory into the modulation process.
The parameter $\theta_n$ embodies this memory, as it is the accumulation of past symbols.
The modulation index $h$ governs the phase variation introduced per symbol.
The function $q(t)$ ensures a smooth, continuous phase transition between symbols.

Modulation Procedure#

In our definition, $s(t)$ represents the physically bandpass transmitted signal, while $v(t)$ is its mathematical lowpass equivalent.

$v(t)$ can be directly obtained from $d(t)$, because $v(t)$ depends explicitly on the integral of $d(t)$.
The passband signal $s(t)$ is derived from $v(t)$.
The process is $d(t) \rightarrow v(t) \rightarrow s(t)$

Start with the PAM signal $d(t)$

The PAM signal $d(t)$ is constructed by mapping $k$-bit blocks into amplitude levels and applying a rectangular pulse $g(t)$.

Use $d(t)$ to Modulate the Phase

The PAM signal $d(t)$ is integrated to determine the time-varying phase $\phi(t;I)$.
This step directly affects the lowpass equivalent signal $v(t)$.
The phase is given by:

\[ \phi(t;I) = 4\pi T f_d \int_{-\infty}^{t} d(\tau)d\tau \]

Construct the Lowpass Equivalent Signal $v(t)$

Once we have $\phi(t;I)$, we directly obtain the lowpass equivalent signal $v(t)$:

\[ v(t) = \sqrt{\frac{2\mathcal{E}}{T}} e^{j\left[ \phi(t;I) + \phi_0 \right]} \]

Obtain the Carrier-Modulated Signal $s(t)$

Finally, $v(t)$ is used to produce the passband signal $s(t)$ by modulating it onto a carrier frequency $f_c$:

\[ s(t) = \sqrt{\frac{2\mathcal{E}}{T}} \cos\left[2\pi f_c t + \phi(t;I) + \phi_0\right] \]

Appendix: The Derivation of $v(t)$ from $s(t)$#

To derive the expression of the equivalent lowpass waveform $ v(t) $ from $ d(t) $, which is an inverse process.

Given:

PAM signal: $$ d(t) = \sum_{n} I_n g(t - nT) $$
$ d(t) $ is used to frequency-modulate a carrier.
The resulting bandpass signal is: $$ s(t) = \sqrt{\frac{2\mathcal{E}}{T}}\cos\left[2\pi f_c t + \varphi(t; I) + \varphi_0\right] $$

where:

\[ \varphi(t; I) = 4\pi T f_d \int_{-\infty}^{t} d(\tau)d\tau \]

Start with the bandpass signal $ s(t) $

Given that we have the bandpass signal $ s(t) $, which is modulated by $ d(t) $:

\[ s(t) = \sqrt{\frac{2\mathcal{E}}{T}}\cos\left[2\pi f_c t + 4\pi T f_d \int_{-\infty}^{t} d(\tau)d\tau + \varphi_0\right] \]

This is a carrier signal with a time-varying phase $\varphi(t; I)$ introduced by $ d(t) $.

Lowpass equivalent signal

The equivalent lowpass representation $ v(t) $ is obtained by:

Multiplying $ s(t) $ by $ e^{-j2\pi f_c t} $ to shift it down to baseband.

\[ v(t) = s(t) \cdot e^{-j2\pi f_c t} \]

Now substitute the expression for $ s(t) $:

\[ v(t) = \left[\sqrt{\frac{2\mathcal{E}}{T}}\cos\left(2\pi f_c t + \varphi(t; I) + \varphi_0\right)\right] \cdot e^{-j2\pi f_c t} \]

Use Euler’s formula:

\[ \cos(x) = \frac{e^{jx} + e^{-jx}}{2} \]

\[ v(t) = \sqrt{\frac{2\mathcal{E}}{T}}\left[\frac{e^{j\left(2\pi f_c t + \varphi(t; I) + \varphi_0\right)} + e^{-j\left(2\pi f_c t + \varphi(t; I) + \varphi_0\right)}}{2}\right] e^{-j2\pi f_c t} \]

Focus on the first exponential term:

\[ = \sqrt{\frac{2\mathcal{E}}{T}}\left[\frac{e^{j\left(\varphi(t; I) + \varphi_0\right)} + e^{-j\left(4\pi f_c t + \varphi(t; I) + \varphi_0\right)}}{2}\right] \]

The second term has high-frequency content (centered at $ 2f_c $) and can be ignored when considering the lowpass equivalent.

So, the lowpass equivalent signal is:

\[ v(t) = \sqrt{\frac{2\mathcal{E}}{T}}e^{j[\varphi(t; I) + \varphi_0]} \]

Derive $\varphi(t; I)$

Recall:

\[ \varphi(t; I) = 4\pi T f_d \int_{-\infty}^{t} d(\tau)d\tau \]

Substituting $ d(t) $:

\[ \varphi(t; I) = 4\pi T f_d \int_{-\infty}^{t} \left[\sum_n I_n g(\tau - nT)\right] d\tau \]

Finally, we have the same equivalent waveform as obtained previously, i.e.:

\[ v(t) = \sqrt{\frac{2\mathcal{E}}{T}}e^{j\left[4\pi T f_d \int_{-\infty}^{t} d(\tau)d\tau + \varphi_0\right]} \]