Quadrature Phase Shift Keying (QPSK)
What is QPSK?
QPSK is a digital modulation scheme that conveys data by changing the phase of a carrier wave. Unlike BPSK, which uses two phases, QPSK uses four distinct phases separated by \(\pi/2\) (90°). This allows each symbol to represent two bits, effectively doubling the data rate within the same bandwidth.
\(s(t) = A \cos(2\pi f_c t + \theta_n)\) where \(\theta_n \in \{0, \pi/2, \pi, 3\pi/2\}\)
- '00': \(A \cos(2\pi f_c t)\)
- '01': \(A \cos(2\pi f_c t + 90^\circ)\)
- '11': \(A \cos(2\pi f_c t + 180^\circ)\)
- '10': \(A \cos(2\pi f_c t + 270^\circ)\)
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QPSK vs. 4-PSK vs. BPSK
| Modulation | Phases | Bits/Symbol | Usage |
|---|---|---|---|
| BPSK | 2 (0, 180°) | 1 | High noise environments, deep space |
| QPSK (4-PSK) | 4 (90° apart) | 2 | Satellite, 4G LTE, Wi-Fi |
| 16-QAM | 16 (Mixed) | 4 | High-speed cable internet |
Note: Technically, 4-PSK is the general term, while QPSK is the practical implementation using In-phase (I) and Quadrature (Q) carriers.
Modulation Performance (BER vs SNR)
Compare the Bit Error Rate of different schemes against the QPSK baseline.
Error Probability Formulas (AWGN)
BPSK / QPSK (Gray Coded):
\[ P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \]16-QAM (Approximation):
\[ P_b \approx \frac{3}{4} Q\left(\sqrt{\frac{4}{5}\frac{E_b}{N_0}}\right) \]64-QAM (Approximation):
\[ P_b \approx \frac{7}{12} Q\left(\sqrt{\frac{2}{7}\frac{E_b}{N_0}}\right) \]Constellation Mapping
The constellation points are represented in the complex plane as {1, j, -1, -j}.
01 (j)
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11 *--+--* 00 (1)
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*
10 (-j)
If the input bitstream is 110111, the symbols will be {-1, j, -1}.