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BER vs SNR for M-ary QAM, M-ary PSK, QPSK, BPSK, ...(MATLAB Code + Simulator)


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M-ary PSK

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M-ary QAM

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Bit Error Rate (BER) & BER Requirements by Application →

Theoretical Error Probability of Digital Modulation

Key Concept: The performance of a modulation scheme is measured by how many errors occur for a given signal strength. We distinguish between Bit Error Rate (BER)—the probability of a single bit being wrong—and Symbol Error Rate (SER)—the probability of a group of bits (a symbol) being wrong.

1. Bit Error Rate (BER) Formulas

BER indicates the end-to-end performance of a data link. For higher-order modulation like QAM, we use approximations based on Gray Coding (where adjacent symbols differ by only one bit).

Binary Phase Shift Keying (BPSK)

The baseline for performance; very robust against noise.

\[ P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \] Quadrature Phase Shift Keying (QPSK)

Carries 2 bits per symbol but maintains the same BER as BPSK.

\[ P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \] M-ary PSK (e.g., 8-PSK) \[ P_b \approx \frac{2}{\log_2(M)} Q\left( \sqrt{\frac{2E_s}{N_0}} \sin\left(\frac{\pi}{M}\right) \right) \] Rectangular M-ary QAM (16, 64, 256-QAM)

As M increases, the distance between points decreases, making them more sensitive to noise.

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

64-QAM: \[ P_b \approx \frac{7}{12} Q\left( \sqrt{\frac{2}{7}\frac{E_b}{N_0}} \right) \]

256-QAM: \[ P_b \approx \frac{15}{32} Q\left( \sqrt{\frac{8}{85}\frac{E_b}{N_0}} \right) \]

2. Symbol Error Rate (SER) Formulas

SER is useful for designing the physical layer and hardware. One symbol error usually results in only one bit error if Gray Coding is used.

BPSK & QPSK

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

QPSK: \[ P_s \approx 2Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \]

General M-QAM Symbol Error Rate

A symbol is "lost" if the noise pushes it outside its decision boundary.

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

64-QAM: \[ P_s \approx \frac{7}{2} Q\left( \sqrt{\frac{2}{7}\frac{E_b}{N_0}} \right) \]

Summary of Parameters

  • \(E_b\): Energy required to send one bit.
  • \(E_s\): Energy required to send one symbol (\(E_s = E_b \cdot \log_2 M\)).
  • \(N_0\): Noise power density.
  • AWGN Channel: These formulas assume a static environment with only Gaussian noise (no fading).
  • Gray Coding: A technique where adjacent constellation points differ by only one bit to minimize BER.

Performance Comparison (in MATLAB)

BER vs. SNR Graph
Download MATLAB BER Code →
Technique Bits/Symbol
BPSK1
QPSK2
8-PSK3
16-QAM4
64-QAM6

Modulation Scheme and Bandwidth Requirement

Modulation | Bits/Symbol | Bandwidth
BPSK 1 Rb
QPSK 2 Rb / 2
8-PSK 3 Rb / 3
16-QAM 4 Rb / 4

Read more about Modulation Scheme & Bandwidth Requirement


QPSK vs. 4-QAM: Why They Should Have Same BER?

In digital communications, QPSK and 4-QAM exhibit identical Bit Error Rate (BER) performance. Although they are categorized differently, they share the same constellation geometry and power efficiency.

Read why they are identical →


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