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


Bit Error Rate (BER) & SNR Guide

Analyze communication system performance with our interactive simulators and MATLAB tools.

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What is Bit Error Rate (BER)?

The BER indicates how many corrupted bits are received compared to the total number of bits sent. It is the primary figure of merit for any digital communication link.

BER = (number of bits received in error) / (total number of transmitted bits)

What is Signal-to-Noise Ratio (SNR)?

SNR is the ratio of signal power to noise power, typically expressed in decibels (dB) to handle large variations in signal strength.

SNR (dB) = 10 * log10(Signal Power / Noise Power)

Example: An SNR of 3 dB means signal power is 2x stronger than noise.


M-ary PSK: (M = modulation order)
BER ≈ (1 / log₂M) × erfc( √(Es/N0) × sin(π/M) )
M-ary QAM (Square):
BER ≈ [ 2(1 - 1/√M) / log₂M ] × erfc( √( 1.5 × Es/N0 / (M - 1) ) )
Q-function / erfc (click here)

Interactive BER Calculator

Calculate theoretical BER for PSK and QAM systems instantly.

M-ary PSK

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M-ary QAM

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Performance Comparison

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


BER vs Eb/N0 Simulation



*Changes update the plot in real-time.

Mathematical Background

The Bit Error Rate (BER) is the probability that a bit is misidentified due to noise. We plot this against $E_b/N_0$ (Energy per bit to noise power spectral density ratio).

1. Energy Conversion

For M-ary modulations, each symbol carries $k = \log_2(M)$ bits. The Symbol Energy ($E_s$) relates to Bit Energy ($E_b$) as:

Es/N0 = (Eb/N0) × log2(M)

2. The Complementary Error Function

Errors in Gaussian noise (AWGN) are calculated using the erfc(x) function. It is related to the Q-function by: Q(x) = ½ erfc(x/√2).

3. Modulation Formulas

M-ary PSK:
BER ≈ (1 / log₂M) × erfc( √(Es/N0) × sin(π/M) )
M-ary QAM (Square):
BER ≈ [ 2(1 - 1/√M) / log₂M ] × erfc( √( 1.5 × Es/N0 / (M - 1) ) )
Binary ASK (On-Off Keying):
BER = ½ erfc( √(Eb/4N0) )
Binary FSK (Coherent):
BER = ½ erfc( √(Eb/2N0) )
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📚 Also Read About

Frequently Asked Questions

What is a good BER for wireless communication?

For voice transmission, a BER of 10^-3 is often acceptable. For high-speed data services, a BER of 10^-6 or lower is typically required.

How does SNR affect BER?

As SNR increases, the signal becomes clearer relative to the noise, resulting in a significantly lower Bit Error Rate.

BER vs SNR Main Page →
Modulation Techniques Main Page >
Online Signal Processing Simulations Main Page >

Advanced M-ary Modulation Simulator (with Q-function)

Explore how Modulation Order (M) and SNR affect the "Distance to Error."

Bits/Symbol (k): 1
Min Distance (d): 2.0
Symbol Error Rate: 0.000
Constellation (Signal + Noise)
Q-Function Tail (Error Probability)

Understanding the Gap

In digital communication, we send symbols. Noise pushes these symbols away from their original spot. If the noise is strong enough to push a symbol across the Decision Boundary, an error occurs.

x = √(2 * Eb/N0)
Move the slider to see how increasing SNR pushes the symbols further apart relative to the noise cloud.

Step-by-Step Comprehension

1. Modulation Order (M)

As M increases (e.g., from BPSK to 64-QAM), we transmit more data per symbol. However, to keep the average power constant, the symbols must be squeezed closer together.

2. The Normalized Distance (x)

The Q(x) function looks at the ratio of Distance / Noise. Even if the noise stays the same, increasing M makes the distance smaller, forcing x to decrease and the error rate to explode.

Intermediate Summary Table

Modulation Energy Efficiency Complexity Why it errors?
BPSK High Low Phase must flip 180°
8-PSK Medium Medium Phase only needs to shift >22.5°
16-QAM Low High Amplitude OR Phase noise can cause error

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