Quadrature Phase-Shift Keying (QPSK)

QPSK Compared with BPSK

Quadrature Phase Shift Keying (QPSK), also known as 4-PSK, assigns one of four carrier phases to each dibit. BPSK has two phase states and carries one bit per symbol, whereas QPSK carries two bits of information per symbol. At the same symbol rate, the gross bit rate is twice as high. For a fixed bit rate, QPSK needs half the symbol rate, so occupied bandwidth depends on pulse shaping, filtering, roll-off, and implementation. With equal energy per bit, the symbol energy satisfies $E_s = 2E_b$.

Four Phase States

QPSK uses four different phases of the carrier wave, typically separated by 90 degrees. The absolute phase reference and the bit assignment depend on the implementation. Each phase is assigned a unique two-bit sequence, called a dibit.

These four states are best visualized on a Constellation Diagram. At ideal symbol sampling instants, the four points are equally spaced around a circle. Their equal distance from the origin means that the nominal symbol magnitude is constant and the information is carried by phase. Pulse shaping can change the envelope between sampling instants. One common Gray mapping is:

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

Gray Mapping and Bit Errors

In a noisy channel, a small phase error can cause the receiver to mistake one symbol for an adjacent one. To reduce the resulting bit errors, many QPSK systems use Gray coding.

Gray coding assigns dibits to constellation points so that any two adjacent points differ by only one bit. Adjacent points are the most likely pair involved in a small phase error.

Why Gray Coding Matters

Suppose noise causes the receiver to decide on the neighboring symbol '11' instead of the transmitted '01'.

• With Gray Coding: The transmitted '01' becomes '11'. The difference is just one bit (the first bit). The result is a single bit error.
• With another labeling: If the same neighboring point were labelled '10', the transmitted '01' would differ in both bit positions. One symbol error would then produce two bit errors.

With coherent detection in additive white Gaussian noise and the same $E_b/N_0$, Gray mapping makes a nearest-neighbor symbol error usually produce one bit error. For QPSK, the nearest-neighbor approximation gives $P_b \approx P_s/2$. This reduces the Bit Error Rate (BER) compared with a non-Gray labeling at the same bit signal-to-noise ratio.

QPSK Modulator Architecture

A QPSK signal can be generated by combining two BPSK branches whose carriers are 90 degrees apart (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 sinusoid at the carrier frequency ($f_c$). One branch uses it directly ($cos(\omega_c t)$); the other is shifted by 90° ($\pi/2$). The shaped I stream is multiplied by the cosine carrier, and the Q stream is multiplied by the quadrature carrier. The sign of the quadrature branch follows the chosen phase convention.
4. Summation: The mixer outputs are summed into the QPSK signal. Its four nominal phase states depend on the I and Q levels. With the sign convention shown here, the signal is $s(t) = I(t) \cos(2\pi f_c t) - Q(t) \sin(2\pi f_c t)$.

Advantages and Applications

• Two Bits per Symbol: At the same pulse-shaping filter and symbol rate, QPSK carries twice the bit rate of BPSK. At the same bit rate, its lower symbol rate can reduce occupied bandwidth; the exact result is set by filtering and roll-off.
• Predictable Noise Performance: For coherent detection in AWGN with Gray mapping, QPSK has the same bit-error performance as BPSK at the same $E_b/N_0$. Phase noise, nonlinear distortion, and synchronization errors still affect the implementation.
• Applications: QPSK and its variants occur in satellite links, digital video broadcasting (DVB), cable modems, and selected IEEE 802.11 physical-layer options. The exact mapping and coding depend on the applicable standard.