A Leap in Efficiency from BPSK

Quadrature Phase Shift Keying (QPSK), also known as 4-PSK, is a natural and powerful evolution of BPSK. While BPSK is robust, its limitation is that it can only encode one bit per symbol. QPSK overcomes this by using four distinct phase states, allowing it to encode two bits of information per symbol. This instantly doubles the data transmission rate without requiring any additional bandwidth.

The Core Principle: Four Phases

The fundamental idea of QPSK is to use four different phases of the carrier wave, typically separated by 90 degrees. Each of these phases is assigned a unique two-bit sequence, called a dibit.

These four states are best visualized on a Constellation Diagram. The four points are equally spaced around a circle, signifying that the signal's amplitude is constant, and only its phase changes. A common mapping is:

• The dibit '01' corresponds to a phase shift of $45^{\circ}$
• The dibit '00' corresponds to a phase shift of $135^{\circ}$
• The dibit '10' corresponds to a phase shift of $225^{\circ}$ or $-135^{\circ}$
• The dibit '11' corresponds to a phase shift of $315^{\circ}$ or $-45^{\circ}$

An Intelligent Optimization: Gray Coding

In a noisy channel, a small phase error can cause the receiver to mistake one symbol for an adjacent one. To minimize the impact of such errors, QPSK systems almost always use Gray coding.

The principle of Gray coding is simple yet brilliant: assign dibits to the constellation points in such a way that any two adjacent points differ by only one bit.

Why Gray Coding Matters

Imagine that due to noise, the receiver mistakes the symbol for '01' with its neighbor '11'.

• With Gray Coding: The transmitted '01' becomes '11'. The difference is just one bit (the first bit). The result is a single bit error.
• Without Gray Coding (e.g., if '11' was '10'): The transmitted '01' might become '10'. The difference is two bits (both bits are flipped). The result is two bit errors from a single symbol error.

By using Gray coding, we significantly reduce the Bit Error Rate (BER) for the same signal-to-noise ratio, making the transmission much more robust.

QPSK Modulator Architecture

A QPSK signal is generated by combining two BPSK signals on carriers that are 90 degrees out of phase (in quadrature). The block diagram below illustrates this process.

1. Serial-to-Parallel Conversion: The incoming serial bitstream is grouped into dibits. One bit from each pair (the I-bit, or 'in-phase' bit) is sent down one path, and the other bit (the Q-bit, or 'quadrature' bit) is sent down a second path.
2. Pulse Shaping: The I and Q streams, which are simple voltage levels, are passed through shaping filters (e.g., Root-Raised-Cosine filters). This limits the signal's spectrum and minimizes inter-symbol interference.
3. Quadrature Mixing: A carrier oscillator generates a sine wave at the carrier frequency ($f_c$). This signal is split. One path uses it directly ($cos(\omega_c t)$), while the other path is phase-shifted by 90° ($\pi/2$), creating a sine wave ($sin(\omega_c t)$). The shaped I-stream is multiplied (mixed) with the cosine carrier, and the Q-stream is multiplied with the sine carrier.
4. Summation: The outputs from the two mixers are summed together to create the final QPSK signal, which has four possible phase states depending on the values of the I and Q bits. The resulting signal can be described as: $s(t) = I(t) \cos(2\pi f_c t) - Q(t) \sin(2\pi f_c t)$.

Advantages and Applications

• Doubled Spectral Efficiency: Compared to BPSK, QPSK can transmit twice the data in the same bandwidth, or the same amount of data in half the bandwidth. This makes it a highly efficient modulation scheme.
• Good Robustness: While not as robust as BPSK, the 90-degree separation between states provides good resistance to noise, making it a reliable choice for many applications.
• Widespread Use: QPSK and its variants are cornerstones of modern digital communications, used in satellite communications, cable modems, digital video broadcasting (DVB), and some Wi-Fi standards.

Interactive QPSK – Constellation and Time Waveform

QPSK modulated signal

Constellation diagram