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MATLAB Code for QPSK Modulation and Demodulation


 

Quadrature Phase Shift Keying (QPSK) is a digital modulation scheme that conveys two bits per symbol by changing the phase of the carrier signal. Each pair of bits is mapped to one of four possible phase shifts: 0°, 90°, 180°, or 270°

00  ===> 0 degree phase shift of carrier signal

01  ===> 90 degree

11  ===> 180 degree

10  ===> 270 degree

 

MATLAB Script


 

Result

data
1
0
2
2
0
2
1
.
.
.

modData
-1.00000000000000 + 1.22464679914735e-16i
-1.83697019872103e-16 - 1.00000000000000i
1.00000000000000 + 0.00000000000000i
6.12323399573677e-17 + 1.00000000000000i
6.12323399573677e-17 + 1.00000000000000i
-1.83697019872103e-16 - 1.00000000000000i
1.00000000000000 + 0.00000000000000i
-1.00000000000000 + 1.22464679914735e-16i
6.12323399573677e-17 + 1.00000000000000i
-1.83697019872103e-16 - 1.00000000000000i
.
.
.


Fig 1: Constellation Diagram of Transmitted QPSK (or 4 PSK) Signal




Fig 2: Constellation Diagram of Received QPSK (or 4 PSK) Signal through Noisy Channel


Further Reading

  1. MATLAB Code for Constellation Diagram of QPSK
  2. Constellation Diagram of BPSK

QPSK (Passband) Signal Generation 

Choose QPSK as the modulation type in the simulation to visualize the QPSK signal.

Try the Online Simulator for QPSK Signal Generator (Click here)


Spectral Efficiency in QPSK

M = 4 Modulation

In Quadrature Phase Shift Keying (QPSK), the system transmits two bits per symbol. This allows for a significantly higher Spectral Efficiency (η) compared to binary modulation, doubling the data rate within the same frequency allocation.

Fundamental Measurement

η = Rb / B

For QPSK (M = 4), each symbol carries 2 bits. Therefore:

Rb = 2 × Rs

Where Rb is Bit Rate and Rs is Symbol Rate

Efficiency Benchmarks (η)

Nyquist Limit (B = Rs) 2.0 bps/Hz
Null-to-Null (B = 2Rs) 1.0 bps/Hz

The QPSK Calculation Logic

The measurement of efficiency in a QPSK system is driven by three core technical factors:

  1. 1
    Bit-to-Symbol Mapping: Because QPSK utilizes four distinct phases, it maps bit pairs (00, 01, 10, 11) to a single symbol. This means the Symbol Rate (Rs) is exactly half of the Bit Rate (Rb).
  2. 2
    Bandwidth Efficiency: The Required Bandwidth (B) of a signal is physically determined by the Symbol Rate. Since QPSK has a lower symbol rate than BPSK for the same bit rate, it consumes 50% less bandwidth to move the same amount of data.
  3. 3
The Sweet Spot Result: Under identical pulse-shaping conditions, QPSK provides an Efficiency (η) that is exactly double that of BPSK, reaching a theoretical max of 2 bits/s/Hz.

Why Engineers Choose QPSK

While 16-QAM or 64-QAM offer even higher efficiency, they are extremely fragile in noisy environments. QPSK is the industry's robust middle-ground, used extensively in Satellite Links (DVB-S) and LTE Control Channels where both speed and reliability are critical.

Read more: about Spectral Efficiency for various Modulation Schemes

QPSK Performance Analysis: Bit and Symbol Error Rates

Quadrature Phase Shift Keying (QPSK) is widely used due to its ability to carry twice the data rate of BPSK within the same bandwidth while maintaining the same bit error performance.

1. QPSK Bit Error Rate (BER)

For Gray-coded QPSK, the probability of bit error is identical to BPSK, as the quadrature components are independent:

\[ P_{b, QPSK} = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \]

2. QPSK Symbol Error Rate (SER)

A QPSK symbol consists of two bits. An error occurs if either (or both) bits are detected incorrectly. The approximation for SER is:

\[ P_{s, QPSK} \approx 2Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \]

Note: The exact expression is \( P_s = 1 - [1 - P_b]^2 \).

Comparative BER Performance

How QPSK compares to other modulation formats in terms of \(P_b\):

BPSK (Same as QPSK)

\[ P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \]

8-PSK (Higher Order)

\[ P_b \approx \frac{2}{3} Q\left( \sqrt{\frac{2E_s}{N_0}} \sin\left(\frac{\pi}{8}\right) \right) \]

16-QAM (Gray Coded)

\[ P_b \approx \frac{3}{4} Q\left( \sqrt{\frac{4}{5}\frac{E_b}{N_0}} \right) \]

64-QAM / 256-QAM

\[ P_{b, 64} \approx \frac{7}{12} Q\left( \sqrt{\frac{2}{7}\frac{E_b}{N_0}} \right), \quad P_{b, 256} \approx \frac{15}{32} Q\left( \sqrt{\frac{8}{85}\frac{E_b}{N_0}} \right) \]

Comparative Symbol Error Rate (SER)

Performance benchmarks for symbol detection:

8-PSK Approximation

\[ P_s \approx 2Q\left( \sqrt{\frac{2E_s}{N_0}} \sin\left(\frac{\pi}{8}\right) \right) \]

16-QAM Approximation

\[ P_s \approx 3Q\left( \sqrt{\frac{4}{5}\frac{E_b}{N_0}} \right) \]

Higher Order QAM (64 & 256)

\[ P_{s, 64} \approx \frac{7}{2} Q\left( \sqrt{\frac{2}{7}\frac{E_b}{N_0}} \right), \quad P_{s, 256} \approx \frac{15}{4} Q\left( \sqrt{\frac{8}{85}\frac{E_b}{N_0}} \right) \]
Technical Deep Dive: Q-function & SNR Calculation
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