QPSK transmission and detection

Quadrature Phase Shift Keying (QPSK) is a 4-level PSK in which each symbol carries 2 bits, and the carrier takes one of four phases (π/4, 3π/4, 5π/4, 7π/4). For the same bandwidth it carries twice the data rate of BPSK, so it has twice the bandwidth efficiency (2 bps/Hz ideally) with the same bit error rate.

s_i(t) = √(2E_s/T_s) cos(2π f_c t + (2i − 1)π/4),  i = 1..4

It can be written as two BPSK signals in quadrature: an in-phase (I) part on cos and a quadrature (Q) part on sin.

Constellation diagram

              Q
              |
     01  •    |    •  00
              |
   -----------+-----------  I
              |
     11  •    |    •  10
              |

Gray coding is used, so neighbouring points differ by only one bit. Distance between neighbouring points = √(2E_s).

QPSK transmitter

           +-> [I: odd bits ] -> (x) --+
           |   [NRZ, Rb/2   ]    ^     |
bits ->[S/P]                cos ωct    (+)-> QPSK
 (Rb)      |                     v     |
           +-> [Q: even bits] -> (x) --+
               [NRZ, Rb/2   ]  sin ωct

1. The unipolar data at rate R_b is converted to bipolar NRZ (±1).
2. A serial-to-parallel converter splits it into the I stream (odd bits) and Q stream (even bits), each at R_b/2.
3. Each stream may pass a pulse-shaping filter (e.g. raised cosine) to limit bandwidth.
4. The I stream multiplies cos(ω_c t) and the Q stream multiplies sin(ω_c t); the two BPSK signals are added to give QPSK.

QPSK receiver (coherent detection)

          +-> (x) -> [LPF/Integ] -> [Decision] -+
          |    cos ωct                          |
QPSK -> [BPF]                                [P/S]-> bits
          |    sin ωct                          |
          +-> (x) -> [LPF/Integ] -> [Decision] -+
               ^
         [Carrier recovery]

1. The input passes a band-pass filter to remove out-of-band noise.
2. A carrier recovery circuit produces coherent cos and sin references.
3. Two correlators (multiplier + integrator over T_s) recover the I and Q components.
4. Decision circuits compare each output with zero to decide 1 or 0.
5. A parallel-to-serial converter combines the two streams into the original bit stream.

Properties

• Bit error rate: P_b = Q(√(2E_b/N₀)), same as BPSK.
• Phase changes of up to 180° can occur, which causes envelope dips after filtering; OQPSK and π/4-QPSK reduce this.
• Used in satellite links, CDMA (IS-95), WCDMA and LTE.