Theoretical BER vs SNR for binary ASK, FSK, and PSK (with MATLAB Code + Simulator)

Bit Error Rate (BER) Equations

In ASK, noise directly affects the signal amplitude, making it the most vulnerable since the data is carried in amplitude changes. In FSK, data is represented by frequency variations, and because noise typically impacts amplitude more than frequency, FSK is more robust than ASK. In PSK, data is encoded in the signal phase, and BPSK specifically uses 180-degree phase shifts, creating the greatest separation between signal points and therefore achieving the lowest bit error rate (BER) for the same power level. BER formulas for ASK, FSK, and PSK modulation schemes.

ASK

BER = 0.5 × erfc(0.5 × √SNR)

FSK

BER = 0.5 × erfc(√(SNR / 2))

PSK

BER = 0.5 × erfc(√SNR)

erfc / Q-function (Click here)

The Mathematical Foundation

The Bit Error Rate is fundamentally derived from the Conditional Probability of Error, where we integrate the Gaussian Noise PDF over the decision region:

$ P_b = \int_{\mathrm{Threshold}}^{\infty} \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-A)^2}{2\sigma^2}} \, dx = Q\!\left(\frac{A}{\sigma}\right) $

In digital communications, the argument $x$ inside the $Q(x)$ function represents the ratio of the signal's "safety margin" to the noise magnitude. In an AWGN channel, the noise standard deviation is defined as: $\sigma = \sqrt{N_0 / 2}$

BPSK

Distance to boundary is $\sqrt{E_b}$

$x = \frac{\sqrt{E_b}}{\sqrt{N_0/2}}$
$x = \sqrt{\frac{2E_b}{N_0}}$

BFSK

Distance to boundary is $\frac{\sqrt{E_b}}{\sqrt{2}}$

$x = \frac{\sqrt{E_b}/\sqrt{2}}{\sqrt{N_0/2}}$
$x = \sqrt{\frac{E_b}{N_0}}$

BASK

Distance to boundary is $\frac{\sqrt{E_b}}{2}$

$x = \frac{\sqrt{E_b}/2}{\sqrt{N_0/2}}$
$x = \sqrt{\frac{E_b}{2N_0}}$

Why the 0.5 factor?

In 0.5 * erfc(x), the 0.5 represents the 50% probability of transmitting a '0' vs a '1' in a random bitstream.

Eb/N0 vs SNR

While SNR is the power ratio, Eb/N0 is the energy per bit, allowing us to compare schemes with different bandwidths fairly.

Theoretical BER vs SNR for Amplitude Shift Keying (ASK)

The theoretical Bit Error Rate (BER) for binary ASK depends on how binary bits are mapped to signal amplitudes. For typical cases:

If bits are mapped to 1 and -1, the BER is:

BER = Q(√(2 × SNR))

If bits are mapped to 0 and 1, the BER becomes:

BER = Q(√(SNR / 2))

Where:

Q(x) is the Q-function: Q(x) = 0.5 × erfc(x / √2)
SNR: Signal-to-Noise Ratio
N₀: Noise Power Spectral Density

Understanding the Q-Function and BER for ASK

Bit '0' transmits noise only
Bit '1' transmits signal (1 + noise)
Receiver decision threshold is 0.5

BER is given by:

P_b = Q(0.5 / σ), where σ = √(N₀ / 2)

Using SNR = (0.5)² / N₀, we get:

BER = Q(√(SNR / 2))

Theoretical BER vs SNR for Frequency Shift Keying (FSK)

For binary FSK, the theoretical BER is:

BER = Q(√(SNR))

The Q-function is defined as:

Q(x) = 0.5 × erfc(x / √2)

Similarities Between ASK and FSK

Both BERs decrease as SNR increases
Both use the Q-function for analytical BER calculation
FSK generally performs better under noisy conditions

MATLAB Code for Theoretical BER vs SNR

Binary ASK (BASK)

% The code is written by SalimWireless.Com 

clc;
clear all;
close all;

SNRdB = 0:20; 
SNR = 10.^(SNRdB/10); 

BER_th = (1/2) * erfc(0.5 * sqrt(SNR));

semilogy(SNRdB, BER_th, '-rh', 'linewidth', 2.5);
grid on;
title('Theoretical Bit Error Rate vs. SNR for Binary ASK Modulation');
xlabel('SNR (dB)');
ylabel('BER');
legend('Theoretical');
axis([0 20 1e-5 1]);

Binary FSK (BFSK)

% The code is written by SalimWireless.Com 

clc;
clear;
close all;

SNRdB = 0:1:10;              
SNR = 10.^(SNRdB/10);        

BER_th = (1/2) * erfc(sqrt(SNR / 2));

disp('SNR (dB)    Theoretical BER');
disp([SNRdB', BER_th']);

figure;
semilogy(SNRdB, BER_th, '-kh', 'LineWidth', 2);
xlabel('SNR (dB)');
ylabel('Bit Error Rate (BER)');
title('Theoretical BER vs SNR for BFSK');
grid on;

Theoretical BER vs SNR for BPSK -> (Click Here)

BER vs SNR Simulation

Select Modulation:

Enter SNR range (dB) [start:step:end]:

Comparison of ASK, FSK, and PSK Performance

Feature	ASK (OOK)	BFSK	BPSK
Power Efficiency	Low	Medium	High
Bandwidth Efficiency	High	Low	High
Noise Immunity	Poor (Sensitive to Amp)	Good	Excellent
Best Used In	Fiber Optics, RFID	Caller ID, Paging	Deep Space, Satellite

Eb/N0 (Energy per Bit to Noise Power Density):: The normalized SNR measure, also known as the "SNR per bit." It is the most important metric for comparing different modulation schemes regardless of bandwidth. [Read More]
Q-Function:: The Q-function represents the tail probability of the standard normal distribution. In BER terms, it tells us the probability that noise will exceed the decision boundary. Read more
Coherent vs. Non-Coherent Detection:: Coherent detection (used in the formulas above) requires the receiver to be in phase-sync with the transmitter. Non-coherent detection is easier to build but results in a higher BER.

Impact of AWGN on BER vs SNR of FSK and PSK →

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