Phase Modulation

Phase Modulation#

Phase-Shift Keying (PSK)#

In digital phase modulation, information is conveyed by varying the phase of a sinusoidal carrier signal.

Digital phase modulation is commonly referred to as Phase-Shift Keying (PSK), where each symbol corresponds to a discrete phase shift.

Unlike Pulse Amplitude Modulation (PAM), which encodes information in the signal’s amplitude, PSK modulates the phase while keeping the amplitude constant.

PSK Waveform#

The \( M \) possible phase-modulated signals can be expressed as:

\[ \boxed{ s_m(t) = \Re \left\{ g(t)e^{j \frac{2\pi(m-1)}{M}}e^{j2\pi f_c t} \right\},} \quad m = 1, 2, \ldots, M \]

Expanding the complex exponential:

\[ s_m(t) = g(t) \cos \left( 2\pi f_c t + \frac{2\pi(m-1)}{M} \right) \]

Using the cosine angle sum identity:

\[ s_m(t) = g(t) \cos \left( \frac{2\pi(m-1)}{M} \right) \cos(2\pi f_c t) - g(t) \sin \left( \frac{2\pi(m-1)}{M} \right) \sin(2\pi f_c t) \]

where:

  • \( g(t) \) is the lowpass, real-valued, signal pulse shape, which influences the bandwidth and spectral characteristics.

  • \( \theta_m = \frac{2\pi(m - 1)}{M} \) represents the \( M \) possible discrete phase values that encode information.

  • \( f_c \) is the carrier frequency.

Interpretation of PSK

  • Each symbol \( s_m(t) \) corresponds to a unique phase shift of the carrier.

  • The phase values \( \theta_m \) are spaced equally around the unit circle, ensuring efficient use of the signal space.

Energy of Phase Modulation Waveforms#

One of the fundamental properties of PSK is that all signal waveforms have equal energy. This property ensures constant envelope transmission, which makes PSK signals robust to channel impairments like amplitude fading.

The average symbol energy for phase-modulated signals is given by:

\[ \mathcal{E}_{\text{avg}} = \mathcal{E}_m = \frac{1}{2}\mathcal{E}_g \]

where \( \mathcal{E}_g \) represents the total pulse energy.

The average bit energy is given by:

\[ \boxed{ \mathcal{E}_{\text{b,avg}} = \frac{\mathcal{E}_g}{2 \log_2 M} } \]

where:

  • \( M \) is the number of possible phase states (symbols) in the PSK constellation.

  • \( \log_2 M \) represents the number of bits per symbol.

For convenience, we often use the notations:

\[ \mathcal{E} = \mathcal{E}_{\text{avg}}, \quad \mathcal{E}_b = \mathcal{E}_{\text{b,avg}} \]

which simplify further derivations.

Signal Space Representation of Phase Modulation#

Orthogonality and Basis Functions#

Since phase-modulated signals contain two components in quadrature (cosine and sine terms), they form a two-dimensional signal space. The two fundamental components:

\[ g(t) \cos(2\pi f_c t) \quad \text{and} \quad g(t) \sin(2\pi f_c t) \]

are orthogonal functions, meaning they do not interfere with each other in signal representation.

Using these functions, we define the basis functions:

\[ \boxed{ \phi_1(t) = \sqrt{\frac{2}{\mathcal{E}_g}} g(t) \cos 2\pi f_c t } \]
\[ \boxed{ \phi_2(t) = -\sqrt{\frac{2}{\mathcal{E}_g}} g(t) \sin 2\pi f_c t } \]

where:

  • \( \phi_1(t) \) and \( \phi_2(t) \) are bandpass, unit-energy basis functions, ensuring that they form an orthonormal basis for the signal space.

Signal Representation in Terms of Basis Functions#

Using the defined basis functions, any phase-modulated signal can be expressed as:

\[ s_m(t) = \sqrt{\frac{\mathcal{E}_g}{2}} \cos \left( \frac{2\pi}{M} (m - 1) \right) \phi_1(t) + \sqrt{\frac{\mathcal{E}_g}{2}} \sin \left( \frac{2\pi}{M} (m - 1) \right) \phi_2(t) \]

This equation shows that each PSK symbol is represented as a linear combination of the two orthogonal basis functions, with coefficients that depend on the phase angle.

Vector Representation and Signal Space#

The vector representation of the phase-modulated signal is:

\[ \vec{s}_m = \left( \sqrt{\frac{\mathcal{E}_g}{2}} \cos \left( \frac{2\pi}{M} (m - 1) \right), \sqrt{\frac{\mathcal{E}_g}{2}} \sin \left( \frac{2\pi}{M} (m - 1) \right) \right), \quad m = 1, 2, \ldots, M \]

This represents the constellation points of PSK as vectors in a two-dimensional Euclidean space.

The dimensionality of the signal space is:

\[ N = 2 \]

which confirms that PSK signals are inherently two-dimensional. This is because they use both in-phase (cosine) and quadrature (sine) components to represent information.

DISCUSSION: Visualize the Signal Space

To visualize the signal space defined by the given orthonormal basis functions, we can think of it as a 2D Euclidean space similar to the Cartesian coordinate plane (\(xy\)-plane).

Signal Space Representation

  • The basis functions \( \phi_1(t) \) and \( \phi_2(t) \) define a two-dimensional orthonormal coordinate system.

  • Any signal \( s(t) \) in this space can be represented as a linear combination of the basis functions: \(s(t) = s_1 \phi_1(t) + s_2 \phi_2(t)\), where \( s_1 \) and \( s_2 \) are the signal coefficients

  • The coefficients \( (s_1, s_2) \) uniquely determine the signal \( s(t) \) within this signal space.

Geometric Interpretation

  • The pair \( (s_1, s_2) \) can be visualized as a point in the 2D plane, just like a Cartesian coordinate system.

  • The two basis functions \( \phi_1(t) \) and \( \phi_2(t) \) form an orthonormal coordinate system (just like unit vectors \( \vec{e}_x \) and \( \vec{e}_y \) in 2D space).

PSK Signal Constellations#

  • Binary Phase-Shift Keying (BPSK): \( M = 2 \)

    • BPSK signals are one-dimensional and correspond to binary PAM signals.

    • These signals are a special case of binary antipodal signaling.

    • The phase shifts are \( 0^\circ \) and \( 180^\circ \).

  • Quadrature Phase-Shift Keying (QPSK): \( M = 4 \)

    • QPSK uses a two-dimensional signal space, with four equally spaced points on a circle in the complex plane.

    • The phase shifts are \( 0^\circ, 90^\circ, 180^\circ, 270^\circ \).

  • 8-PSK: \( M = 8 \)

    • 8-PSK extends the QPSK constellation to eight equally spaced points, increasing the number of bits per symbol.

  • \(M\)-ary PSK (M-PSK): General case where \( M \) different phase shifts are used.

Mapping Bits to Phases#

As with PAM, the \( k \)-bit information sequence (\( M = 2^k \)) is mapped to the \( M \) possible phases in the PSK constellation. The preferred mapping is Gray encoding, which ensures that adjacent points differ by only one bit. This minimizes the bit error rate (BER) by ensuring that noise-induced errors typically result in a single-bit error.

Euclidean Distance Between PSK Signal Points#

The Euclidean distance between two PSK constellation points is a key metric for analyzing the system’s noise performance. The distance between two symbols \( s_m \) and \( s_n \) is given by:

\[ d_{mn} = \sqrt{\|\boldsymbol{\rm s}_m - \boldsymbol{\rm s}_n\|^2} \]

For PSK, this simplifies to:

\[ d_{mn} = \sqrt{\mathcal{E}_g \left[ 1 - \cos \left( \frac{2\pi}{M} (m - n) \right) \right]} \]

where:

  • \( \mathcal{E}_g \) is the energy of the signal pulse \( g(t) \).

  • \( M \) is the number of PSK constellation points (e.g., \( M = 2, 4, 8 \)).

  • \( m \) and \( n \) are the indices of the signal points being compared.

Interpretation

  • The minimum Euclidean distance \( d_{\text{min}} \) corresponds to adjacent points in the constellation, which occurs when \( |m - n| = 1 \).

  • The larger the \( d_{\text{min}} \), the more robust the system is to noise, resulting in fewer errors.

Minimum Distance in PSK#

Minimum Euclidean Distance in PSK#

The minimum distance \( d_{\text{min}} \) between adjacent constellation points (when \( |m - n| = 1 \)) in \( M \)-PSK is given by:

\[ d_{\text{min}} = \sqrt{\mathcal{E}_g \left( 1 - \cos \frac{2\pi}{M} \right)} \]

Using the trigonometric identity \( 1 - \cos x = 2 \sin^2 \frac{x}{2} \), this can be rewritten as:

\[ \boxed{ d_{\text{min}} = \sqrt{2\mathcal{E}_g \sin^2 \frac{\pi}{M}} } \]

This formula highlights how \( d_{\text{min}} \) depends on the pulse energy \( \mathcal{E}_g \) and the number of symbols \( M \). Larger \( M \) results in smaller angular separation between points, reducing \( d_{\text{min}} \).

Relating \( d_{\text{min}} \) to Energy Per Bit#

Using the relationship between \( \mathcal{E}_g \) (pulse energy) and \( \mathcal{E}_b \) (energy per bit), we can express \( d_{\text{min}} \) as:

\[ d_{\text{min}} = 2 \sqrt{\left( \log_2 M \times \sin^2 \frac{\pi}{M} \right) \mathcal{E}_b} \]

where:

  • \( \log_2 M \) accounts for the number of bits per symbol.

  • \( \sin^2 \frac{\pi}{M} \) reflects the spacing between constellation points.

Asymptotic Approximation for Large \( M \)#

Definition of Asymptotic Approximation
An approximation is referred to as asymptotic if it becomes increasingly accurate as a certain parameter approaches a limiting value — in this case, as \( M \to \infty \).

For \( M \)-PSK, the minimum Euclidean distance is:

\[ d_{\text{min}} = \sqrt{2\mathcal{E}_g \sin^2 \frac{\pi}{M}} \]

For large \( M \), we use the approximation \( \sin \frac{\pi}{M} \approx \frac{\pi}{M} \), valid when \( \frac{\pi}{M} \) is small (i.e., \( M \) is large). Substituting this into the expression for \( d_{\text{min}} \), we get:

\[ d_{\text{min}} \approx \sqrt{2\mathcal{E}_g \left(\frac{\pi}{M}\right)^2} = \sqrt{\frac{2\pi^2 \mathcal{E}_g}{M^2}} \]

Expressing \( \mathcal{E}_g \) in terms of \( \mathcal{E}_b \) (energy per bit):

\[ d_{\text{min}} \approx 2 \sqrt{\frac{\pi^2 \log_2 M}{M^2} \mathcal{E}_b} \]

Asymptotic Nature

  • As \( M \to \infty \), the approximation \( \sin \frac{\pi}{M} \approx \frac{\pi}{M} \) becomes increasingly accurate.

  • This means the derived expression for \( d_{\text{min}} \) is an asymptotic representation of the actual minimum distance for large \( M \).

Practical Implication
While the exact formula for \( d_{\text{min}} \) is valid for any \( M \), the asymptotic approximation simplifies analysis for large \( M \), where \( \sin^2 \frac{\pi}{M} \) is computationally similar to \( \left(\frac{\pi}{M}\right)^2 \).

Thus, the approximation is not exact for all \( M \), but it becomes very accurate asymptotically as \( M \) grows.

PSK Variant: \( \pi/4 \)-QPSK#

A popular variant of QPSK, called \( \pi/4 \)-QPSK, introduces an additional phase shift of \( \pi/4 \) in the carrier phase during each symbol interval. This has the following advantages:

  • Improved synchronization: The phase shift facilitates symbol synchronization, making the modulation scheme more robust in practical systems.

  • Constellation structure: The signal points alternate between two QPSK constellations offset by \( \pi/4 \), reducing the likelihood of sudden large phase transitions.